#  Truncated smooth function spaces

Oscar Domínguez  O. Domínguez, Departamento de Análisis Matemático y
Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de
Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain<EMAIL_ADDRESS>and
Sergey Tikhonov  S. Tikhonov, Centre de Recerca Matemàtica
Campus de Bellaterra, Edifici C 08193 Bellaterra (Barcelona), Spain; ICREA,
Pg. Lluív Companys 23, 08010 Barcelona, Spain, and Universitat Autònoma de
Barcelona<EMAIL_ADDRESS>
###### Abstract.

We introduce truncated Besov and Triebel–Lizorkin function spaces and
investigate their main properties: embeddings, interpolation, duality,
lifting, traces. These new scales allow us to improve several known results in
functional analysis and PDE’s.

###### Key words and phrases:

Truncated function spaces, embeddings, interpolation, duality, lifting
property, Fourier and wavelet characterizations

###### 2010 Mathematics Subject Classification:

Primary 46E35, 42B35; Secondary 26A15, 46E30, 46B70, 65T60

This research was partially supported by French National Research Agency
(ANR-10-LABX-0070), (ANR-11-IDEX-0007) (O.D.) and by PID2020-114948GB-I00,
2017 SGR 358, AP 14870758, the CERCA Programme of the Generalitat de
Catalunya, and Severo Ochoa and María de Maeztu Program for Centers and Units
of Excellence in R$\&$D (CEX2020-001084-M) (S.T.).

###### Contents

  1. 1 Introduction
  2. 2 A quick review on classical function spaces: Besov, Triebel–Lizorkin and Lipschitz
  3. 3 Truncated Besov and Triebel–Lizorkin spaces: definitions and basic properties
  4. 4 Characterizations by interpolation
  5. 5 Characterizations of truncated Besov spaces in terms of approximation
  6. 6 Characterizations by differences
  7. 7 Characterization via wavelets
  8. 8 Lifting property
  9. 9 Duality
  10. 10 Embeddings between truncated Besov spaces
  11. 11 Embeddings between truncated Besov and Triebel–Lizorkin spaces with fixed integrability parameter
  12. 12 Embeddings between truncated Triebel–Lizorkin spaces
  13. 13 Franke–Jawerth embeddings for truncated spaces
  14. 14 Embeddings in the space of locally integrable functions
  15. 15 Embeddings in the space of continuous functions
  16. 16 Characterizations and embeddings for general monotone functions
  17. 17 Some applications of truncated function spaces
  18. A Fourier-analytical description of Lipschitz spaces
  19. B Some auxiliary inequalities
  20. C List of symbols

##  1\. Introduction

Smooth function spaces are basic notion in analysis. The main examples are
Besov and Triebel–Lizorkin spaces, since those scales contain various
distinguished spaces of distributions such as standard Sobolev,
Gagliardo–Slobodeckiĭ, fractional Sobolev, Bessel potential spaces,
Hölder–Zygmund spaces, spaces of BMO-type, Hardy spaces, and Lebesgue spaces.
Another important scale is Lipschitz spaces, in particular containing the
space of bounded variation. All mentioned scales have shown to be crucial in
functional and harmonic analysis and PDE’s, among other fields. Key properties
of these spaces, such as duality, interpolation, embeddings,
characterizations, traces, and lifting, have been extensively developed since
the middle of the last century. See e.g. the monographs [BS88, BL76, BIN78,
N75, Pe76, ST87, Ste70, Tri78, Tri83, Tri92, Tri01].

We also stress that in recent times a big deal of attention has been devoted
to function spaces with generalized smoothness. Indeed, in many questions
arising in different areas it has become apparent that the scope of classical
smoothness is rather limited and thus the finer tuning given by generalized
smoothness becomes very useful. Here we mention pointwise multipliers
[LSYY22], elliptic PDE’s and geometric measure theory [MMPT22], probability
theory and stochastic processes [FL06, KSV22], sharp embedding theorems
[Ha07], fractal analysis and a related spectral theory [Tri97], capacity
theory [LWXY21], regularity properties of Euler solutions [OT00, CCW11],
regularity of flows for the transport and continuity equation [CDeL08, Le18,
BN20], or optimal recovery [HKNV22]. The prototype of function spaces with
generalized smoothness is the logarithmic Besov spaces
$B^{s,b}_{p,q}(\mathbb{R}^{d})$ and Triebel–Lizorkin spaces
$F^{s,b}_{p,q}(\mathbb{R}^{d})$, which can be defined through Fourier-
analytical decompositions as follows:

(1.1)
$\|f\|_{B^{s,b}_{p,q}(\mathbb{R}^{d})}=\bigg{(}\sum_{j=0}^{\infty}2^{jsq}(1+j)^{bq}\|(\varphi_{j}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{\frac{1}{q}}$

and

(1.2)
$\|f\|_{F^{s,b}_{p,q}(\mathbb{R}^{d})}=\bigg{\|}\bigg{(}\sum_{j=0}^{\infty}2^{jsq}(1+j)^{bq}|(\varphi_{j}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}.$

In the special case $b=0$ one recovers the classical spaces
$B^{s}_{p,q}(\mathbb{R}^{d})$ and $F^{s}_{p,q}(\mathbb{R}^{d})$, respectively.
Interestingly, the most unexpected effects can be discovered dealing with zero
smoothness, that is, $s=0.$ In particular, the space
$B^{0,b}_{p,q}(\mathbb{R}^{d})$ does not coincide with its counterpart
${\bf{B}}^{0,b}_{p,q}(\mathbb{R}^{d})$ defined by differences. In fact, the
lack of adequate Littlewood–Paley decompositions makes the analysis of
$\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$ rather tricky; we refer the reader to
[DT18] for a detailed study of $\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$,
including some discussions on applications of these spaces to different areas
of analysis (see [DT18, Section 1.2]).

The main goal of this work is to introduce the truncated scale of smooth
function spaces. Let us illustrate this idea by presenting truncated Besov and
Triebel–Lizorkin spaces:

(1.3)
$\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}=\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\right)^{1/r},$
(1.4)
$\|f\|_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}=\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{r}\right)^{1/r}.$

Roughly speaking, the additional summability parameter $r$ enables to refine
classical norms via truncations with respect to dyadic decomposition.

There are several aspects of the matter worth pointing out, which show that
truncated spaces are extremely important in the theory of function spaces,
PDE’s, interpolation theory, potential theory and applications. Next we
highlight some of them.

1\. A unified approach to classical spaces and applications to PDE’s. As one
may expect, choosing appropriately the smoothness and summability parameters
$b$ and $r$, respectively, in $T^{b}_{r}X$ we recover the prototypical space
$X$. For example,
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})=B^{s,b}_{p,q}(\mathbb{R}^{d})$ for $q=r$
and, in particular,
$T^{0}_{q}B^{s}_{p,q}(\mathbb{R}^{d})=B^{s}_{p,q}(\mathbb{R}^{d})$. More than
that, specifying parameters in the definitions of truncated Besov and Triebel-
Lizorkin spaces, one can recover in a unifying fashion all classical scales of
smooth function spaces: Besov $B^{s,b}_{p,q}(\mathbb{R}^{d})$, Lipschitz
$\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})$, Triebel–Lizorkin
$F^{s,b}_{p,q}(\mathbb{R}^{d})$, Besov spaces with zero smoothness
${{B}}^{0,b}_{p,q}(\mathbb{R}^{d})$ and
${\textbf{B}}^{0,b}_{p,q}(\mathbb{R}^{d})$. In particular, we obtain

$T^{b+1/q}_{q}F^{0}_{p,2}(\mathbb{R}^{d})={\textbf{B}}^{0,b}_{p,q}(\mathbb{R}^{d}),\qquad
b>-1/q,$

and

$T^{b+1/q}_{q}F^{s}_{p,2}(\mathbb{R}^{d})=\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d}),\qquad
b<-1/q;$

cf. Proposition 3.8.

Another motivation to introduce truncated smooth function spaces comes from
applications to PDE’s. In his fundamental paper [Vi99], Vishik was able to
extend the classical Yudovich theory for uniqueness of non-Lipschitz $2$D
Euler flows via Besov-type $B_{\Gamma}$ spaces defined in terms of growth
properties of partial sums involving $L_{\infty}$-norms of dyadic frequencies:

$\|f\|_{B_{\Gamma}(\mathbb{R}^{d})}=\sup_{j\geq
0}\,\frac{1}{\Gamma(j)}\,\sum_{\nu=0}^{j}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{\infty}(\mathbb{R}^{d})},$

where $\Gamma$ is a positive and increasing function such that
$\sum_{j=0}^{\infty}\frac{1}{\Gamma(j)}=\infty$. Thus the main result of
[Vi99, Theorem 7.1] guarantees uniqueness of the Euler flow for related
vorticities that are bounded in $B_{\Gamma}(\mathbb{R}^{d})$, with
$\Gamma(j)=j$. In particular, for this special choice of $\Gamma$, the
embeddings

$bmo(\mathbb{R}^{d})\hookrightarrow
B^{0}_{\infty,\infty}(\mathbb{R}^{d})\hookrightarrow
B_{\Gamma}(\mathbb{R}^{d}),$

leads to uniqueness of Euler flows for $bmo(\mathbb{R}^{d})$ vorticities,
extending thus the classical $L_{\infty}$-based theory due to Yudovich [Yu63,
Yu95].

Observe that $B_{\Gamma}(\mathbb{R}^{d}),\,\Gamma(j)=j^{\alpha},\,\alpha\leq
1,$ is a special case of truncated Besov spaces, namely,

$B_{\Gamma}(\mathbb{R}^{d})=T^{-\alpha}_{\infty}B^{0}_{\infty,1}(\mathbb{R}^{d}),$

see Proposition 3.5. Accordingly, truncated constructions (1.3) and (1.4)
arise naturally in the analysis of PDE’s.

2\. Embeddings. Truncated smooth function spaces allow us to sharpen classical
embeddings. To clarify this, we mention the well-known embedding between Besov
and Triebel–Lizorkin (in particular, Sobolev) spaces

$B^{s}_{p,\min\\{p,q\\}}(\mathbb{R}^{d})\hookrightarrow
F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
B^{s}_{p,\max\\{p,q\\}}(\mathbb{R}^{d}),$

which is not sharp anymore in the scale of truncated spaces. Namely, we have
(cf. Corollary 3.11)

$B^{s}_{p,\min\\{p,q\\}}(\mathbb{R}^{d})\hookrightarrow
T_{\min\\{p,q\\}}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T_{\max\\{p,q\\}}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
B^{s}_{p,\max\\{p,q\\}}(\mathbb{R}^{d})$

and any of these embeddings are sharp. Here,
$T_{r}F^{s}_{p,q}(\mathbb{R}^{d})=T^{0}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$.

As already mentioned above, the scales of
$\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$ and
$\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})$ spaces are not contained in the
scales of Besov and Triebel–Lizorkin spaces (cf. (1.1)-(1.2)). The known
embeddings between them involve log-shifts of smoothness:

(1.5)
$B^{0,b+1/\min\\{2,p,q\\}}_{p,q}(\mathbb{R}^{d})\hookrightarrow\textbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})\hookrightarrow
B^{0,b+1/\max\\{2,p,q\\}}_{p,q}(\mathbb{R}^{d})$

and

(1.6)
$B^{s,b+1/\min\\{2,p,q\\}}_{p,q}(\mathbb{R}^{d})\hookrightarrow\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})\hookrightarrow
B^{s,b+1/\max\\{2,p,q\\}}_{p,q}(\mathbb{R}^{d}).$

Furthermore, these embeddings are optimal within the scale of classical Besov
spaces; cf. [DHT20] and [DT18]. As a byproduct of general embedding theorems
between truncated spaces, we are now able to show that both (1.5) and (1.6)
can be considerably sharpened by dealing with the refined scale of truncated
spaces. Namely, we establish (cf. Corollaries 11.9 and 11.10)

$T^{b+1/q}_{q}B^{0}_{p,\min\\{2,p\\}}(\mathbb{R}^{d})\hookrightarrow\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T^{b+1/q}_{q}B^{0}_{p,\max\\{2,p\\}}(\mathbb{R}^{d}),$

and

$T^{b+1/q}_{q}B^{s}_{p,\min\\{2,p\\}}(\mathbb{R}^{d})\hookrightarrow\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T^{b+1/q}_{q}B^{s}_{p,\max\\{2,p\\}}(\mathbb{R}^{d}).$

Furthermore, they are optimal.

3\. Interpolation. One of the main properties of Besov spaces is that they are
closed under real interpolation. A more complete result states that, for
$A,\widetilde{A}\in\\{B,F\\}$,

(1.7)
$(A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),\widetilde{A}^{s_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\theta,r}=B^{s}_{p,r}(\mathbb{R}^{d}),$

where $s=(1-\theta)s_{0}+\theta s_{1}$ and $s_{0}\neq s_{1}$. It is a natural
question to understand whenever this property holds for truncated spaces. We
positively answer this question (cf. Theorem 4.17):

$(T^{b_{0}}_{r_{0}}A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}\widetilde{A}^{s_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\theta,r}=B^{s,b}_{p,r}(\mathbb{R}^{d}),\qquad
s_{0}\neq s_{1}.$

Moreover, in the borderline case $s_{0}=s_{1}=s$, we obtain that (cf. Theorems
4.7 and 4.14)

$(T^{b_{0}}_{r_{0}}A^{s}_{p,q}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}A^{s}_{p,q}(\mathbb{R}^{d}))_{\theta,r}=T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d}),\qquad
b_{0}\neq b_{1}.$

Another intrinsic question can be stated as follows. Can one obtain the
truncated spaces as a result of interpolation between the classical ones? An
answer is again positive and rather surprising since modern interpolation
methods are naturally appearing: _limiting interpolation_. In particular, we
have for $s_{0}>s$ and $b>0$

(1.8)
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})=(B^{s}_{p,q}(\mathbb{R}^{d}),A^{s_{0}}_{p,q_{1}}(\mathbb{R}^{d}))_{(0,b-1/r),r},$

and for $s_{0}<s$ and $b<0$

(1.9)
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})=(A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r};$

see Theorem 4.2. A similar result holds for truncated Triebel–Lizorkin spaces
(cf. Theorem 4.8). As special cases, we can characterize zero-smoothness Besov
and Lipschitz spaces via limiting interpolation:

$\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})=(L_{p}(\mathbb{R}^{d}),A^{s_{0}}_{p,q_{1}}(\mathbb{R}^{d}))_{(0,b),q},\qquad
b>-1/q,$
$\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})=(A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),H^{s}_{p}(\mathbb{R}^{d}))_{(1,b),q},\qquad
b<-1/q.$

The formulae (1.8)-(1.9) will play a key role in our arguments, since they put
truncated function spaces into the powerful framework provided by limiting
interpolation. In particular, this enables to transfer many properties of
classical spaces to truncated spaces.

4\. Lifting properties.  Consider the lifting operator

$I_{\sigma}=(\text{id}-\Delta)^{\frac{\sigma}{2}},\qquad\sigma\in\mathbb{R}.$

In particular, if $\sigma<0$ then $I_{\sigma}$ is the classical _Bessel
potential_. A crucial property of Besov and Triebel–Lizorkin spaces is their
stability under liftings. More precisely, for $A\in\\{B,F\\}$, $I_{\sigma}$
acts as an isomorphism from $A^{s,b}_{p,q}(\mathbb{R}^{d})$ onto
$A^{s-\sigma,b}_{p,q}(\mathbb{R}^{d})$ and

$\|I_{\sigma}f\|_{A^{s-\sigma,b}_{p,q}(\mathbb{R}^{d})}\asymp\|f\|_{A^{s,b}_{p,q}(\mathbb{R}^{d})}.$

On the other hand, the mapping properties of liftings on Besov spaces of zero
smoothness and Lipschitz spaces are much more delicate and not completely
understood. For instance, a partial answer asserts that

$I_{\sigma}:\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})\to
B^{-\sigma,b+1/\max\\{2,p,q\\}}_{p,q}(\mathbb{R}^{d})$

and

$I_{\sigma}:B^{\sigma,b+1/\min\\{2,p,q\\}}_{p,q}(\mathbb{R}^{d})\to\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d}).$

Furthermore, the shifts given by $\frac{1}{\max\\{2,p,q\\}}$ and
$\frac{1}{\min\\{2,p,q\\}}$ are optimal. In particular, this result tells us
that classical Besov spaces are not sharp enough to characterize lifting
properties of $\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$. This drawback can be
now overcome with the help of truncated spaces.

In Section 8.2, we show that $I_{\sigma}$ acts as an isomorphism from
$\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$ onto
$T^{b+1/q}_{q}F^{-\sigma}_{p,2}(\mathbb{R}^{d})$ and

$\|I_{\sigma}f\|_{T^{b+1/q}_{q}F^{-\sigma}_{p,2}(\mathbb{R}^{d})}\asymp\|f\|_{\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})}.$

Recall that
$\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})=T^{b+1/q}_{q}F^{0}_{p,2}(\mathbb{R}^{d})$.
Accordingly, lifting properties of $\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$
should be understood within the more general setting provided by truncated
spaces. A similar statement also holds true for
$\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})$.

5\. Duality. In Section 9, we obtain that the scale of truncated spaces is
closed under duality, extending then the well-known duality assertions related
to classical Besov and Triebel–Lizorkin spaces. Specifically (cf. Theorem 9.1
for precise statement) we have

$(T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d}))^{\prime}=T^{-b}_{r^{\prime}}A^{-s}_{p^{\prime},q^{\prime}}(\mathbb{R}^{d}),\qquad
A\in\\{B,F\\}.$

In particular, this yields a complete answer to the open question of
characterizing the dual spaces of $\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$ and
$\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})$:

$(\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d}))^{\prime}=T^{-b-1/q}_{q^{\prime}}F^{0}_{p^{\prime},2}(\mathbb{R}^{d}),\qquad(T^{-b-1/q}_{q^{\prime}}F^{0}_{p^{\prime},2}(\mathbb{R}^{d}))^{\prime}=\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$

and

$(\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d}))^{\prime}=T^{-b-1/q}_{q^{\prime}}F^{-s}_{p^{\prime},2}(\mathbb{R}^{d}),\qquad(T^{-b-1/q}_{q^{\prime}}F^{-s}_{p^{\prime},2}(\mathbb{R}^{d}))^{\prime}=\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d}).$

6\. Applications. We would like to mention here at least three new
applications of truncated spaces in different areas.

(i) The trace operator can be defined (at least formally) by

$(\text{Tr }f)(x)=(\text{Tr }_{\\{x_{d}=0\\}}f)(x)=f(x^{\prime},0),\qquad
x=(x^{\prime},x_{d}),\quad x^{\prime}\in\mathbb{R}^{d-1},\qquad d\geq 2.$

The well-known trace theorem for Besov spaces states that

$\text{Tr }:B^{s}_{p,q}(\mathbb{R}^{d})\longrightarrow
B^{s-1/p}_{p,q}(\mathbb{R}^{d-1})$

for $p\geq 1,\,s>1/p$, and $0<q\leq\infty$. Furthermore, Tr admits a right
inverse Ex, which is a bounded linear operator from
$B^{s-1/p}_{p,q}(\mathbb{R}^{d-1})$ into $B^{s}_{p,q}(\mathbb{R}^{d})$. See
e.g. [Tri83, Section 2.7.2].

In the limiting case $p=1$ and/or $s=1/p$, we mention the classical results by
Gagliardo [Ga57]:

(1.10) $\text{Tr}:\text{BV}(\mathbb{R}^{d+1}_{+})\to L_{1}(\mathbb{R}^{d})$

and by Peetre [Pe75], Agmon–Hörmander [AH76], and Frazier–Jawerth [FJ85]:

(1.11) $\text{Tr }:B^{1/p}_{p,1}(\mathbb{R}^{d})\longrightarrow
L_{p}(\mathbb{R}^{d-1})$

and there exist bounded extension operators. Importantly, there does not exist
a bounded linear extension related to (1.10) or (1.11), which yields serious
obstructions; cf. [Pe79] (see also [Tri83] for further historical comments).

Motivated by the above discussion, very recently Malý, Shanmugalingam and
Snipes [MSS18] considered the Besov space with zero smoothness111The
corresponding norm appears frequently in the literature under the name of
_Dini-type condition_. $\mathbf{B}^{0}_{1,1}=\mathbf{B}^{0,0}_{1,1}$, which is
a slightly smaller space than $L_{1}$. It turns out that in this case a
bounded linear extension operator does exist:

(1.12) $\text{Ex}:\mathbf{B}^{0}_{1,1}(\partial\Omega)\to\text{BV}(\Omega)$

such that $\text{Tr}\circ\text{Ex}=\text{id}$. Here $\Omega$ is a smooth
bounded domain in $\mathbb{R}^{d}$.

Working with truncated spaces allows us to give a full solution to the
trace/extension problem in the limiting case $s=1/p$ (cf. (1.10) and (1.11)).
In particular, we can substantially improve the above mentioned result by
Malý, Shanmugalingam and Snipes as follows (cf. Theorem 17.1)

$\text{Tr
}:T^{b+1/q}_{q}B^{1/p}_{p,1}(\mathbb{R}^{d})\longrightarrow\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d-1})$

and there exists a linear extension operator Ex which is continuous from
$\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d-1})$ into
$T^{b+1/q}_{q}B^{1/p}_{p,1}(\mathbb{R}^{d})$ with $\text{Tr }\circ\text{Ex
}=\text{id}.$ Specializing this result with $b=0$ and $p=q=1$, we obtain

$\text{Tr
}:B^{1,1}_{1,1}(\mathbb{R}^{d})\longrightarrow\mathbf{B}^{0}_{1,1}(\mathbb{R}^{d-1})$

(cf. (1.10)-(1.11)) and there exists a bounded linear extension operator

$\text{Ex}:\mathbf{B}^{0}_{1,1}(\mathbb{R}^{d-1})\to
B^{1,1}_{1,1}(\mathbb{R}^{d})$

with $\text{Tr }\circ\text{Ex }=\text{id}.$ Note that
$B^{1,1}_{1,1}(\mathbb{R}^{d})\subsetneq\text{BV}(\mathbb{R}^{d})$ (see
(1.12)).

(ii) An important embedding for elliptic and parabolic equations is the
prominent result by Trudinger [Tru67]:

$W^{d/p}_{p}(\mathbb{T}^{d})\hookrightarrow L_{\infty}(\log
L)_{-1/p^{\prime}}(\mathbb{T}^{d}),\qquad p\in(1,\infty).$

Later on this result was sligtly improved with respect to both domain and
target spaces:

$B^{d/p}_{p,q}(\mathbb{T}^{d})\hookrightarrow L_{\infty,q}(\log
L)_{-1}(\mathbb{T}^{d}).$

In its turn, this embedding can be naturally considered in the more general
setting of Besov spaces of logarithmic smoothness. Namely, if $b<-1/q$ then

(1.13) $B^{d/p,b+1/\min\\{1,q\\}}_{p,q}(\mathbb{T}^{d})\hookrightarrow
L_{\infty,q}(\log L)_{b}(\mathbb{T}^{d}).$

The last embedding has been the object of intensive research since 1979 (see
the paper [DRS79]) until now (see [CaMou04b, CaFar06, CaLeo06, Mar08, MNP08]).
It is known that (1.13) is optimal within the scale of Besov spaces of
logarithmic smoothness. However, we will show that one can sharpen (1.13) with
the help of the truncated Besov spaces $T^{b}_{r}B^{s}_{p,q}(\mathbb{T}^{d})$.
Namely, we show (cf. Theorem 17.10)

$T^{b+1/q}_{q}B^{d/p}_{p,1}(\mathbb{T}^{d})\hookrightarrow L_{\infty,q}(\log
L)_{b}(\mathbb{T}^{d})$

and this embedding is optimal (cf. Remark 17.11).

The counterpart of (1.13) with $b>-1/q$ is formulated in terms of the zero-
smoothness Besov space $\mathbf{B}_{\infty,q}^{0,b}(\mathbb{R}^{d})$. More
precisely, if $b>-1/q$ then

(1.14)
$B^{d/p,b+1/\min\\{1,q\\}}_{p,q}(\mathbb{R}^{d})\hookrightarrow\mathbf{B}_{\infty,q}^{0,b}(\mathbb{R}^{d}).$

This embedding is a crux of matter in function spaces. It served as a basis
for the theory of continuity envelopes developed in detail by Haroske–Triebel
[Tri01, Ha07] and it has been further investigated in [MNP09, MNS11, MNS14]
and the references within. Similarly as (1.13), the embedding (1.14) is
optimal within the scale of classical Besov spaces, but we can now prove that
it admits the following improvements within the scale of truncated spaces (cf.
Theorem 17.13)

$T^{b+1/q}_{q}B^{d/p}_{p,1}(\mathbb{R}^{d})\hookrightarrow\mathbf{B}^{0,b}_{\infty,q}(\mathbb{R}^{d})$

and this is optimal.

(iii) The celebrated Bourgain–Brezis–Mironescu formula [BBM01] states that

$\lim_{s\to
1-}(1-s)^{1/p}\|f\|_{\dot{W}^{s,p}(\mathbb{R}^{d})}=c_{d,p}\,\|\nabla
f\|_{L_{p}(\mathbb{R}^{d})}.$

In some sense it fixes the well-known defect of the fractional Sobolev
(Gagliardo–Slobodeckiĭ) seminorms

$\|f\|_{\dot{W}^{s,p}(\mathbb{R}^{d})}=\
\bigg{(}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{|f(x)-f(y)|^{p}}{|x-y|^{d+sp}}\,dx\,dy\bigg{)}^{1/p},\quad
s\in(0,1),$

that they do not converge to the standard Sobolev seminorm $\|\nabla
f\|_{L_{p}(\mathbb{R}^{d})}$ as $s\to 1-$. The explicit value of the constant
$c_{d,p}$ is known, depending only on $d$ and $p$. This result has been widely
considered recently with respect to different perspectives: interpolation
theory [Mil05], higher-order Besov spaces [KMX05], Triebel–Lizorkin spaces
[BSY21], anisotropic Sobolev spaces [Lu14], etc.

We will show that the family of truncated norms on Besov and Triebel–Lizorkin
spaces satisfies the Bourgain–Brezis–Mironescu phenomenon, in the sense that
under a certain normalization, one can attain the classical norms
$\|\cdot\|_{A^{s}_{p,q}(\mathbb{R}^{d})},\,A\in\\{B,F\\}$, via limits.
Specifically, one has (cf. Theorem 17.20)

$\lim_{b\to
0-}(1-2^{br})^{1/r}\,\|f\|^{*}_{T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d})}=\|f\|_{A^{s}_{p,q}(\mathbb{R}^{d})}.$

To close this introduction, we would like to note that truncated Besov and
Triebel–Lizorkin spaces introduced in this paper can be understood within the
abstract setting provided by interpolation theory. In the forthcoming work
[DT22], we propose a novel interpolation method, _the truncated interpolation_
, which contains as distinguished examples truncated Besov and
Triebel–Lizorkin spaces. It is worth mentioning that this methodology goes
further beyond the setting of smooth function spaces and it enables us to
define many other truncated scales $T^{b}_{r}X$ with the special choice of a
space $X$.

_Notation._ Given two non-negative quantities $A$ and $B$, we use $A\lesssim
B$ for the estimate $A\leq CB$, where $C$ is a positive constant which is
independent of all essential parameters in $A$ and $B$. If $A\lesssim
B\lesssim A$, then we write $A\asymp B$.

Let $X$ and $Y$ be quasi-Banach spaces. We write $X\hookrightarrow Y$ if the
identity operator from $X$ into $Y$ is continuous, that is,
$\|f\|_{Y}\lesssim\|f\|_{X}$ for $f\in X$. By $X=Y$ we mean that
$X\hookrightarrow Y\hookrightarrow X$, i.e., $\|f\|_{X}\asymp\|f\|_{Y}$. We
use $X^{\prime}$ to denote the dual space of $X$ and $p^{\prime}$ is the dual
exponent of $p\in(0,\infty)$, i.e., $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ if
$p\geq 1$ and $p^{\prime}=\infty$ if $p\leq 1$.

As usual, $\mathbb{R}^{d}$ denotes the $d$-dimensional Euclidean space,
$\mathbb{R}=\mathbb{R}^{1}$, $\mathbb{C}$ is the set of all complex numbers,
$\mathbb{T}^{d}$ is the $d$-dimensional torus, $\mathbb{T}=\mathbb{T}^{1}$,
$\mathbb{N}$ is the collection of all natural numbers,
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$, $\mathbb{Z}^{d}$ is the
$d$-dimensional integer lattice, $\mathbb{Z}=\mathbb{Z}^{1}$. For some
$a\in\mathbb{R}$, we let $a_{+}=\max\\{a,0\\}.$

Given a quasi-Banach space $X$ and $p\in(0,\infty)$, the Bochner space
$L_{p}(\mathbb{R}^{d};X)$ is formed by all strongly measurable functions
$f:\mathbb{R}^{d}\to X$ such that

$\|f\|_{L_{p}(\mathbb{R}^{d};X)}=\bigg{(}\int_{\mathbb{R}^{d}}\|f(\xi)\|_{X}^{p}\,d\xi\bigg{)}^{1/p}<\infty.$

In the special case $X=\mathbb{R}$ or $X=\mathbb{C}$, we simply write
$L_{p}(\mathbb{R}^{d})$, the classical Lebesgue space.

Let $s\in\mathbb{R},q\in(0,\infty]$ and let $X$ be a quasi-Banach space. Then
$\ell^{s}_{q}(X)$ is the sequence space formed by all those $x=(x_{j})\subset
X$ such that

$\|x\|_{\ell^{s}_{q}(X)}=\bigg{(}\sum_{j=0}^{\infty}[2^{js}\|x_{j}\|_{X}]^{q}\bigg{)}^{1/q}<\infty$

(with the usual modification if $q=\infty$). In the special case
$X=\mathbb{C}$ (or $X=\mathbb{R}$), we simply write $\ell^{s}_{q}$. In
addition, if $s=0$ then we recover $\ell_{q}$.

##  2\. A quick review on classical function spaces: Besov, Triebel–Lizorkin
and Lipschitz

We collect here basic definitions and standard notations in the theory of
function spaces. By $\mathcal{S}(\mathbb{R}^{d})$ we denote the Schwartz space
of all complex-valued, infinitely differentiable and rapidly decreasing
functions on $\mathbb{R}^{d}$ and by $\mathcal{S}^{\prime}(\mathbb{R}^{d})$
the dual space of all tempered distributions in $\mathbb{R}^{d}$. If
$\varphi\in\mathcal{S}(\mathbb{R}^{d})$ then

(2.1)
$\widehat{\varphi}(\xi)=\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}\varphi(x)\,dx,\qquad\xi\in\mathbb{R}^{d},$

denotes the Fourier transform of $\varphi$. The symbol $\varphi^{\vee}$ stands
for the inverse Fourier transform and is given by the right-hand side of (2.1)
with $i$ in place of $-i$. Both the Fourier transform and the inverse Fourier
transform are extended to $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ in the usual
way. Let $(\varphi_{j})_{j=0}^{\infty}$ be the usual dyadic resolution of
unity in $\mathbb{R}^{d}$.

Let $p,q\in(0,\infty]$ and $s,b\in\mathbb{R}$. Then
$B^{s,b}_{p,q}(\mathbb{R}^{d})$ is the _Besov space_ formed by all
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ such that

(2.2)
$\|f\|_{B^{s,b}_{p,q}(\mathbb{R}^{d})}=\left(\sum_{j=0}^{\infty}2^{jsq}(1+j)^{bq}\|(\varphi_{j}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\right)^{\frac{1}{q}}<\infty$

(with the usual convention if $q=\infty$). The _Triebel–Lizorkin space_
$F^{s,b}_{p,q}(\mathbb{R}^{d}),\,p<\infty,$ is defined as the collection of
all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ such that

(2.3)
$\|f\|_{F^{s,b}_{p,q}(\mathbb{R}^{d})}=\bigg{\|}\bigg{(}\sum_{j=0}^{\infty}2^{jsq}(1+j)^{bq}|(\varphi_{j}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}<\infty$

(with the usual modification made if $q=\infty$). Standard references for the
theory of these spaces are [KL87], [Mou01] and [FL06]; see also [DT18] for a
recent account. In particular, in the absence of logarithmic smoothness (i.e.,
$b=0$) one recovers classical Besov spaces, $B^{s}_{p,q}(\mathbb{R}^{d})$, and
Triebel–Lizorkin spaces, $F^{s}_{p,q}(\mathbb{R}^{d})$, studied systematically
in the monographs [N75, Pe76, Tri83, Tri01].

For $s\in\mathbb{R}$ and $p\in(1,\infty)$, the _(fractional) Sobolev space_
$H^{s}_{p}(\mathbb{R}^{d})$ (also known as _Bessel-potential space_) is formed
by all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ such that

$\|f\|_{H^{s}_{p}(\mathbb{R}^{d})}=\|((1+|\xi|^{2})^{s/2}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}<\infty.$

Recall that $H^{s}_{p}(\mathbb{R}^{d})=F^{s}_{p,2}(\mathbb{R}^{d})$ (with
equivalence of norms) and, in particular,
$W^{k}_{p}(\mathbb{R}^{d})=F^{k}_{p,2}(\mathbb{R}^{d}),\,k\in\mathbb{N}_{0}$,
the classical _Sobolev space_ formed by all $f\in L_{p}(\mathbb{R}^{d})$ whose
weak derivatives $D^{l}f,\,|l|\leq k$, belong to $L_{p}(\mathbb{R}^{d})$ and
equipped with the norm

$\|f\|_{W^{k}_{p}(\mathbb{R}^{d})}:=\sum_{|l|\leq
k}\||D^{l}f|\|_{L_{p}(\mathbb{R}^{d})}.$

It is well known that Besov–Triebel–Lizorkin spaces can be equivalently
introduced in terms of differences, see e.g. [Tri83, 2.5.12], [KL87] and
[HaMou04]. For $k\in\mathbb{N}$ and $f\in L_{p}(\mathbb{R}^{d}),\,1\leq
p\leq\infty$, the _$k$ -th order modulus of smoothness_ $\omega_{k}(f,t)_{p}$
is defined by

(2.4) $\omega_{k}(f,t)_{p}=\sup_{|h|\leq
t}\|\Delta^{k}_{h}f\|_{L_{p}(\mathbb{R}^{d})},\qquad t>0,$

where $\Delta^{k}_{h}$ is the _$k$ -th difference with step
$h\in\mathbb{R}^{d}$_ given by

$(\Delta^{1}_{h}f)(x)=(\Delta_{h}f)(x)=f(x+h)-f(x),\qquad x\in\mathbb{R}^{d},$

and $\Delta_{h}^{k+1}=\Delta_{h}^{k}\Delta_{h}$. It is clear that

(2.5) $\Delta^{k}_{h}f(x)=\sum_{j=0}^{k}(-1)^{j}{k\choose j}f(x+(k-j)h).$

Let $1\leq p\leq\infty,0<q\leq\infty,0\leq s<k$ and $b\in\mathbb{R}$. Then
$\mathbf{B}^{s,b}_{p,q}(\mathbb{R}^{d})$ is the set of all $f\in
L_{p}(\mathbb{R}^{d})$ having finite quasi-norm

(2.6)
$\|f\|_{\mathbf{B}^{s,b}_{p,q}(\mathbb{R}^{d})}=\|f\|_{L_{p}(\mathbb{R}^{d})}+\bigg{(}\int_{0}^{1}(t^{-s}(1-\log
t)^{b}\omega_{k}(f,t)_{p})^{q}\frac{dt}{t}\bigg{)}^{1/q}$

(with the usual modification if $q=\infty$). It is well known that

(2.7)
$\|f\|_{\mathbf{B}^{s,b}_{p,q}(\mathbb{R}^{d})}\asymp\|f\|_{L_{p}(\mathbb{R}^{d})}+\bigg{(}\int_{|h|<1}|h|^{-sq-d}(1-\log|h|)^{bq}\|\Delta^{k}_{h}f\|_{L_{p}(\mathbb{R}^{d})}^{q}dh\bigg{)}^{1/q}$

and different values of $k$ with $k>s$ in the right-hand sides of (2.6) and
(2.7) give equivalent quasi-norms on $\mathbf{B}^{s,b}_{p,q}(\mathbb{R}^{d})$.
We also mention that these spaces admit extensions to the case $p\in(0,1)$.
For the special choice $b=0$, we deal with classical spaces
$\mathbf{B}^{s}_{p,q}(\mathbb{R}^{d})$.

The limiting parameter $s=0$ in $\mathbf{B}^{s,b}_{p,q}(\mathbb{R}^{d})$
deserves special attention. In this case, we may assume $b\geq-1/q\,(b>0\text{
if }q=\infty)$. The latter condition on $b$ is not restrictive since
$\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})=L_{p}(\mathbb{R}^{d})$ for $b<-1/q$.
In particular, the elements in $\mathbf{B}^{0}_{p,1}(\mathbb{R}^{d})$ are
defined in terms of the validity of the classical _$L_{p}$ -Dini condition_

(2.8) $\int_{0}^{1}\omega_{k}(f,t)_{p}\frac{dt}{t}<\infty.$

This condition is widely used in functional analysis, probability theory,
harmonic analysis and PDE’s. The logarithmic refinement of (2.8) provided by
(2.6) (with $s=0$) is also important in applications. In this regard we only
refer to the survey contained in [DT18, Section 1.2].

As already mentioned above, a classical result in the theory of function
spaces asserts that

(2.9)
$B^{s,b}_{p,q}(\mathbb{R}^{d})=\mathbf{B}^{s,b}_{p,q}(\mathbb{R}^{d}),\qquad
s\in(0,k),\qquad p\in[1,\infty].$

However, the limiting case $s=0$ in (2.9) fails to be true. Similarly, putting
$s=k$ in (2.6) we do not recover $B^{s,b}_{p,q}(\mathbb{R}^{d})$. In fact, the
expression given in the right-hand side of (2.6) with $s=k$ corresponds to a
special case of Lipschitz space. More precisely, for $1\leq
p\leq\infty,0<q\leq\infty,s>0$ and $b<-1/q\,(b\leq 0\text{ if }q=\infty)$, the
space $\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})$ is formed by all those $f\in
L_{p}(\mathbb{R}^{d})$ such that

$\int_{0}^{1}t^{-sq}(1-\log t)^{bq}\omega_{s}(f,t)_{p}^{q}\frac{dt}{t}<\infty$

(with the usual change if $q=\infty$). Here $\omega_{s}(f,t)_{p}$ denotes the
classical (if $s\in\mathbb{N}$) or fractional modulus of smoothness, which can
be naturally introduced after replacing the classical differences
$\Delta^{k}_{h}$ (cf. (2.5)) in (2.4) by

$\Delta^{s}_{h}f(x)=\sum_{j=0}^{\infty}(-1)^{j}{s\choose j}f(x+(s-j)h)$

with ${s\choose j}=\frac{s(s-1)\ldots(s-j+1)}{j!}$ and ${s\choose 0}=1$, more
precisely,

$\omega_{s}(f,t)_{p}=\sup_{|h|\leq
t}\|\Delta^{s}_{h}f\|_{L_{p}(\mathbb{R}^{d})}.$

The space $\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})$ is endowed with the quasi-
norm

$\|f\|_{\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})}=\|f\|_{L_{p}(\mathbb{R}^{d})}+\bigg{(}\int_{0}^{1}t^{-sq}(1-\log
t)^{bq}\omega_{s}(f,t)_{p}^{q}\frac{dt}{t}\bigg{)}^{1/q}.$

The assumption $b<-1/q$ is natural, otherwise the space
$\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})$ becomes trivial (in the sense that it
is formed only by the zero element). These spaces play a key role in embedding
theorems and PDE’s and they were intensively investigated in [EH99, EH00,
Ha07] (and the references quoted there) if $s=1$ and in [DHT20] for general
$s>0$. In particular, a bounded function $f$ belongs to
$\text{Lip}^{1,b}_{\infty,\infty}(\mathbb{R}^{d})$ if and only if

$|f(x+h)-f(h)|\lesssim|h|(1-\log|h|)^{-b}\qquad\text{for all}\qquad|h|<1.$

In the special case $b=0$ one recovers the classical Lipschitz condition. On
the other hand,

(2.10)
$\text{Lip}^{s,0}_{p,\infty}(\mathbb{R}^{d})=H^{s}_{p}(\mathbb{R}^{d}).$

As already mentioned above

$\textbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})\neq
B^{0,b}_{p,q}(\mathbb{R}^{d})\qquad\text{and}\qquad\text{Lip}^{k,b}_{p,q}(\mathbb{R}^{d})\neq
B^{k,b}_{p,q}(\mathbb{R}^{d})$

(cf. (2.9)). However, it was recently shown in [DHT20, Theorem 4.1] and [DT18,
Section 9.1] that it is still possible to establish relations between these
three scales of function spaces. Namely, let $p\in(1,\infty)$ and
$q\in(0,\infty]$ then

(2.11)
$B^{0,b+1/\min\\{2,p,q\\}}_{p,q}(\mathbb{R}^{d})\hookrightarrow\textbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})\hookrightarrow
B^{0,b+1/\max\\{2,p,q\\}}_{p,q}(\mathbb{R}^{d}),\qquad\text{if}\qquad b>-1/q,$

and, if $s>0$ then

(2.12)
$B^{s,b+1/\min\\{2,p,q\\}}_{p,q}(\mathbb{R}^{d})\hookrightarrow\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})\hookrightarrow
B^{s,b+1/\max\\{2,p,q\\}}_{p,q}(\mathbb{R}^{d}),\qquad\text{if}\qquad b<-1/q.$

Furthermore, these embeddings are optimal within the class of Besov spaces. We
mention that versions of (2.11) and (2.12) in the limiting cases $p=1,\infty$
are also available in [DHT20, DT18].

##  3\. Truncated Besov and Triebel–Lizorkin spaces: definitions and basic
properties

The main objects that we study in this paper are new spaces of smooth
functions, which are obtained via truncation methods of the classical quasi-
norms (2.2) and (2.3) with $b=0$. In particular, these spaces contain as
distinguished examples the Besov spaces $B^{s,b}_{p,q}(\mathbb{R}^{d})$, the
Triebel–Lizorkin spaces $F^{s,b}_{p,q}(\mathbb{R}^{d})$, the Besov spaces of
smoothness near zero $\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$ and the
Lipschitz spaces $\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})$.

######  Definition 3.1.

  1. (i)

Let $p,q,r\in(0,\infty]$ and $s,b\in\mathbb{R}$. The truncated Besov space
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ is formed by all
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ for which

$\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}=\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\right)^{1/r}$

is finite (with the usual convention if $q=\infty$ and/or $r=\infty$).

  2. (ii)

Let $p\in(0,\infty),q,r\in(0,\infty]$ and $s,b\in\mathbb{R}$. The truncated
Triebel–Lizorkin space $T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ is formed by all
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ for which

$\|f\|_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}=\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{r}\right)^{1/r}$

is finite (with the usual convention if $q=\infty$ and/or $r=\infty$).

  3. (iii)

Let $p\in(0,\infty),q,r\in(0,\infty]$ and $s,b\in\mathbb{R}$. The inner
truncated Triebel–Lizorkin space
$\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ is formed by all
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ for which

$\|f\|_{\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}=\bigg{\|}\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}$

is finite (with the usual convention if $q=\infty$ and/or $r=\infty$).

######  Notation 3.2.

_For $A\in\\{B,F\\}$, we let
$T_{r}A^{s}_{p,q}(\mathbb{R}^{d}):=T^{0}_{r}A^{s}_{p,q}(\mathbb{R}^{d})$ and
$\mathfrak{T}_{r}F^{s}_{p,q}(\mathbb{R}^{d}):=\mathfrak{T}^{0}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$._

######  Remark 3.3.

  1. (i)

Let $A\in\\{B,F\\}$. It is plain to see that
$T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d})$ and
$\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ are quasi-Banach spaces
(Banach space if $p,q,r\geq 1$).

  2. (ii)

One can also introduce the counterparts of the spaces
$T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d})$ and
$\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ related to general smooth
resolutions of unity in the sense of [Tri83, Section 2.3.1, page 45]. However
the definition of these spaces do not depend (up to equivalence of quasi-
norms) on the choice of the resolution of unity. The proof of this fact is a
simple consequence of multiplier assertions (cf. [Tri83, Section 2.3.2, page
46] for further details).

  3. (iii)

The periodic counterparts
$T^{b}_{r}A^{s}_{p,q}(\mathbb{T}^{d}),\,A\in\\{B,F\\},$ and
$\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{T}^{d})$ can be introduced in the
same fashion.

######  Proposition 3.4.

Let $A\in\\{B,F\\},p,q,r\in(0,\infty]\,(p<\infty\text{ if }A=F)$ and
$s,b\in\mathbb{R}$. Then

(3.1)
$\|f\|_{T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d})}\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{\|}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}(\varphi_{\nu}\widehat{f})^{\vee}\bigg{\|}^{r}_{A^{s}_{p,q}(\mathbb{R}^{d})}\bigg{)}^{1/r}.$

###### Proof.

Set $\varphi_{-1}\equiv 0$ and
$\widetilde{\varphi}_{l}:=\varphi_{l-1}+\varphi_{l}+\varphi_{l+1}$ for
$l\in\mathbb{N}_{0}$,. By basic Fourier multiplier assertions (cf. [Tri83,
Theorem 1.5.2]),

$\displaystyle\bigg{\|}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}(\varphi_{\nu}\widehat{f})^{\vee}\bigg{\|}_{B^{s}_{p,q}(\mathbb{R}^{d})}^{q}$
$\displaystyle=\sum_{l=0}^{\infty}2^{lsq}\bigg{\|}\bigg{(}\varphi_{l}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}\varphi_{\nu}\widehat{f}\bigg{)}^{\vee}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{q}$
$\displaystyle=\sum_{l=2^{j}-2}^{2^{j+1}-1}2^{lsq}\bigg{\|}\bigg{(}\varphi_{l}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}\varphi_{\nu}\widehat{f}\bigg{)}^{\vee}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{q}$
$\displaystyle\lesssim\sum_{l=2^{j}-2}^{2^{j+1}-1}2^{lsq}\|(\varphi_{l}\widetilde{\varphi}_{l}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}$
$\displaystyle=\sum_{l=2^{j}-2}^{2^{j+1}-1}2^{lsq}\|(\varphi_{l}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}.$

On the other hand, for $j\geq 2$,

$\displaystyle\sum_{l=2^{j}-2}^{2^{j+1}-1}2^{lsq}\bigg{\|}\bigg{(}\varphi_{l}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}\varphi_{\nu}\widehat{f}\bigg{)}^{\vee}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{q}$
$\displaystyle\geq\sum_{l=2^{j}}^{2^{j+1}-3}2^{lsq}\bigg{\|}\bigg{(}\varphi_{l}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}\varphi_{\nu}\widehat{f}\bigg{)}^{\vee}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{q}$
$\displaystyle=\sum_{l=2^{j}}^{2^{j+1}-3}2^{lsq}\|(\varphi_{l}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}.$

This gives (3.1) with $A=B$. The corresponding assertion with $A=F$ follows
similar ideas as above but now relying on vector-valued multiplier theorems,
as can be found e.g. in [Tri83, Theorem 1.6.3] ∎

We provide some elementary equivalent quasi-norms on truncated function
spaces.

######  Proposition 3.5.

Let $p,q,r\in(0,\infty],s\in\mathbb{R}$ and $b\in\mathbb{R}\backslash\\{0\\}$.

  1. (i)

Assume $b>0$. Then

$\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|f\|^{*}_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}:=\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{\infty}2^{\nu
sq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\right)^{1/r},$

and if additionally $p<\infty$ then

$\|f\|_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|f\|^{*}_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}:=\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{\infty}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{r}\right)^{1/r},$
$\|f\|_{\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|f\|^{*}_{\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}:=\bigg{\|}\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{\infty}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}.$

  2. (ii)

Assume $b<0$. Then

$\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|f\|^{*}_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}:=\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=0}^{2^{j}}2^{\nu
sq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\right)^{1/r},$

and if additionally $p<\infty$ then

$\|f\|_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|f\|^{*}_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}:=\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{\|}\bigg{(}\sum_{\nu=0}^{2^{j}}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{r}\right)^{1/r},$
$\|f\|_{\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|f\|^{*}_{\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}:=\bigg{\|}\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=0}^{2^{j}}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}.$

######  Remark 3.6.

We stress that the definitions of
$\|\cdot\|_{T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d})}^{*}$ for $A\in\\{B,F\\}$ and
$\|\cdot\|_{\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}$ depend on the
sign of $b$.

###### Proof of Proposition 3.5.

We will only prove

$\|f\|_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|f\|^{*}_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}.$

The corresponding assertions for $T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ and
$\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ follow similar ideas.

Obviously
$\|f\|_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}\leq\|f\|^{*}_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}$.
To show the converse estimate, we can argue as follows. Let $b>0$, by triangle
inequality and Hardy’s inequality (B.4),

$\displaystyle\|f\|^{*}_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle=\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{\|}\sum_{\nu=2^{j}-1}^{\infty}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{\|}_{L_{p/q}(\mathbb{R}^{d})}^{r/q}\right)^{1/r}$
$\displaystyle\leq\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{l=j}^{\infty}\bigg{\|}\sum_{\nu=2^{l}-1}^{2^{l+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{\|}^{\min\\{1,p/q\\}}_{L_{p/q}(\mathbb{R}^{d})}\bigg{)}^{r/(q\min\\{1,p/q\\})}\right)^{1/r}$
$\displaystyle\asymp\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}^{r}_{L_{p}(\mathbb{R}^{d})}\right)^{1/r}=\|f\|_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}.$

The case $b<0$ can be obtained similarly but now invoking (B.3). ∎

######  Remark 3.7.

The case $b=0$ in $T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d}),\,A\in\\{B,F\\},$ and
$\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ needs special care. This
assertion can be justified by the following two facts. Firstly, Proposition
3.5 provides two natural candidates for quasi-norms on
$T^{0}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$, namely

(3.2) $\left(\sum_{j=0}^{\infty}\bigg{(}\sum_{\nu=2^{j}-1}^{\infty}2^{\nu
sq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\right)^{1/r}$

and

(3.3) $\left(\sum_{j=0}^{\infty}\bigg{(}\sum_{\nu=0}^{2^{j}}2^{\nu
sq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\right)^{1/r}.$

However (3.3) is not a meaningful condition since it is only satisfied by zero
function:

(3.4) $\sum_{j=0}^{\infty}\bigg{(}\sum_{\nu=0}^{2^{j}}2^{\nu
sq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}<\infty\iff
f\equiv 0.$

Indeed, the trivial estimate

$\sum_{j=0}^{\infty}\bigg{(}\sum_{\nu=0}^{2^{j}}2^{\nu
sq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\geq\sum_{j=0}^{\infty}\|(\varphi_{0}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}$

holds. On the other hand, condition (3.2) does make sense and then
$T^{*}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ is formed by all
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ such that

(3.5)
$\|f\|_{T^{*}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}:=\left(\sum_{j=0}^{\infty}\bigg{(}\sum_{\nu=2^{j}-1}^{\infty}2^{\nu
sq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\right)^{1/r}<\infty.$

In particular, in the special case $r=q$ we have

(3.6)
$\|f\|_{T^{*}_{q}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|f\|_{B^{s,0,1/q}_{p,q}(\mathbb{R}^{d})}:=\left(\sum_{j=0}^{\infty}[2^{js}(1+\log(1+j))^{1/q}\|(\varphi_{j}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\right)^{1/q},$

a _Besov space of iterated logarithmic smoothness_ (cf. [DT18, p. 32]).
Analogously, the space $T^{*}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ is defined by

(3.7)
$\|f\|_{T^{*}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}:=\left(\sum_{j=0}^{\infty}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{\infty}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{r}\right)^{1/r}<\infty.$

Clearly

(3.8)
$T^{*}_{\infty}A^{s}_{p,q}(\mathbb{R}^{d})=A^{s}_{p,q}(\mathbb{R}^{d}),\qquad
A\in\\{B,F\\}.$

Secondly

$T^{*}_{r}A^{s}_{p,q}(\mathbb{R}^{d})\neq
T_{r}A^{s}_{p,q}(\mathbb{R}^{d}),\qquad A\in\\{B,F\\}.$

Indeed, to fix some ideas consider first the special case $r=q$ and $A=B$, by
above considerations (cf. (3.6)) we have
$T^{*}_{q}B^{s}_{p,q}(\mathbb{R}^{d})=B^{s,0,1/q}_{p,q}(\mathbb{R}^{d})$, but
on the other hand it is clear that
$T_{q}B^{s}_{p,q}(\mathbb{R}^{d})=B^{s}_{p,q}(\mathbb{R}^{d})$ (we will see in
Proposition 3.8 below further relationships between truncated and classical
function spaces). Note that $B^{s,0,1/q}_{p,q}(\mathbb{R}^{d})\subsetneq
B^{s}_{p,q}(\mathbb{R}^{d})$. In fact, this particular case illustrates what
happens in the general setting. More precisely, it is clear that
$T^{*}_{r}A^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T_{r}A^{s}_{p,q}(\mathbb{R}^{d})$. However, this embedding is strict. To see
this one may consider lacunary Fourier series of the form

(3.9) $f(x)\sim\sum_{j=0}^{\infty}a_{j}e^{i2^{j}x_{1}}\psi(x),\qquad
x=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d},$

where $\psi\in\mathcal{S}(\mathbb{R}^{d})\backslash\\{0\\}$ with $\text{supp
}\psi\subset\mathbb{T}^{d}$ and $(a_{j})_{j\in\mathbb{N}_{0}}$ is a scalar-
valued sequence. It is plain to check that

(3.10)
$(\varphi_{j}\widehat{f})^{\vee}(x)=a_{j}e^{i(2^{j}-2)x_{1}}\psi(x),\qquad
j\geq 0.$

In particular, if $a_{j}=2^{-js}j^{-1/q}(1+\log(1+j))^{-\varepsilon}$ where
$\max\\{1/r,1/q\\}<\varepsilon<1/r+1/q$, then

$\displaystyle\|f\|_{T^{*}_{r}A^{s}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle\asymp\left(\sum_{j=0}^{\infty}\bigg{(}\sum_{\nu=2^{j}}^{\infty}(1+\log\nu)^{-\varepsilon
q}\frac{1}{\nu}\bigg{)}^{r/q}\right)^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}j^{-\varepsilon
r+r/q}\bigg{)}^{1/r}=\infty$

but

$\|f\|_{T_{r}A^{s}_{p,q}(\mathbb{R}^{d})}\asymp\left(\sum_{j=0}^{\infty}j^{-\varepsilon
r}\right)^{1/r}<\infty.$

Hence $f\in T_{r}A^{s}_{p,q}(\mathbb{R}^{d})\backslash
T^{*}_{r}A^{s}_{p,q}(\mathbb{R}^{d})$.

###  3.1. Relationships between truncated and classical spaces

The following result provides a table of coincidences between the scale of
spaces
$T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d}),\,A\in\\{B,F\\},\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$,
introduced in Definition 3.1 and the classical spaces considered in Section 2.

######  Proposition 3.8.

  1. (i)

Let $p,q\in(0,\infty]$ and $s,b\in\mathbb{R}$. Then

$T^{b}_{q}B^{s}_{p,q}(\mathbb{R}^{d})=B^{s,b}_{p,q}(\mathbb{R}^{d}).$

  2. (ii)

Let $p\in(0,\infty),q\in(0,\infty]$ and $s,b\in\mathbb{R}$. Then

$\mathfrak{T}^{b}_{q}F^{s}_{p,q}(\mathbb{R}^{d})=F^{s,b}_{p,q}(\mathbb{R}^{d}).$

  3. (iii)

Let $p\in(1,\infty),q\in(0,\infty]$ and $b>-1/q$. Then

$T^{b+1/q}_{q}F^{0}_{p,2}(\mathbb{R}^{d})=\emph{{B}}^{0,b}_{p,q}(\mathbb{R}^{d}).$

  4. (iv)

Let $p\in(1,\infty),q\in(0,\infty],s>0$ and $b<-1/q$. Then

$T^{b+1/q}_{q}F^{s}_{p,2}(\mathbb{R}^{d})=\emph{Lip}^{s,b}_{p,q}(\mathbb{R}^{d}).$

###### Proof.

(i): We have

$\|f\|_{T^{b}_{q}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\left(\sum_{j=0}^{\infty}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}(1+\nu)^{bq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\right)^{1/q}=\|f\|_{B^{s,b}_{p,q}(\mathbb{R}^{d})}.$

The item (ii) can be obtained in a similar fashion as (i).

(iii): It was shown in [CDT16, Theorem 4.3] that the space
$\textbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$ can be equivalently defined through
the Fourier transform, namely,

(3.11)
$\|f\|_{\textbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})}\asymp\left(\sum_{j=0}^{\infty}(1+j)^{bq}\bigg{\|}\bigg{(}\sum_{\nu=j}^{\infty}|(\varphi_{\nu}\widehat{f})^{\vee}|^{2}\bigg{)}^{1/2}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{q}\right)^{1/q}.$

Therefore the proof will be completed if we show that the right-hand side of
the previous expression is equivalent to
$\|f\|_{T^{b+1/q}_{q}F^{0}_{p,2}(\mathbb{R}^{d})}$. To proceed with, we first
note that, by (3.11) and monotonicity properties,

(3.12)
$\|f\|_{\textbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})}\asymp\left(\sum_{j=0}^{\infty}2^{j(b+1/q)q}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{\infty}|(\varphi_{\nu}\widehat{f})^{\vee}|^{2}\bigg{)}^{1/2}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{q}\right)^{1/q}$

and thus
$\|f\|_{T^{b+1/q}_{q}F^{0}_{p,2}(\mathbb{R}^{d})}\lesssim\|f\|_{\textbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})}$.
To prove the converse estimate, according to (3.12), the fact that
$\ell_{1}\hookrightarrow\ell_{2}$ and the triangle inequality, we obtain

$\displaystyle\|f\|_{\textbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle\asymp\left(\sum_{j=0}^{\infty}2^{j(b+1/q)q}\bigg{\|}\bigg{(}\sum_{l=j}^{\infty}\sum_{\nu=2^{l}-1}^{2^{l+1}-2}|(\varphi_{\nu}\widehat{f})^{\vee}|^{2}\bigg{)}^{1/2}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{q}\right)^{1/q}$
(3.13)
$\displaystyle\leq\left(\sum_{j=0}^{\infty}2^{j(b+1/q)q}\bigg{(}\sum_{l=j}^{\infty}\bigg{\|}\bigg{(}\sum_{\nu=2^{l}-1}^{2^{l+1}-2}|(\varphi_{\nu}\widehat{f})^{\vee}|^{2}\bigg{)}^{1/2}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}\bigg{)}^{q}\right)^{1/q}.$

If $q\leq 1$ then, using (3.13), the embedding
$\ell_{q}\hookrightarrow\ell_{1}$ and changing the order of summation, we get

$\displaystyle\|f\|_{\textbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\left(\sum_{j=0}^{\infty}2^{j(b+1/q)q}\sum_{l=j}^{\infty}\bigg{\|}\bigg{(}\sum_{\nu=2^{l}-1}^{2^{l+1}-2}|(\varphi_{\nu}\widehat{f})^{\vee}|^{2}\bigg{)}^{1/2}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{q}\right)^{1/q}$
$\displaystyle\asymp\bigg{(}\sum_{l=0}^{\infty}2^{l(b+1/q)q}\bigg{\|}\bigg{(}\sum_{\nu=2^{l}-1}^{2^{l+1}-2}|(\varphi_{\nu}\widehat{f})^{\vee}|^{2}\bigg{)}^{1/2}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{1/q}=\|f\|_{T^{b+1/q}_{q}F^{0}_{p,2}(\mathbb{R}^{d})}.$

On the other hand, if $q>1$ we can apply Hardy’s inequality in (3.13) (since
$b+1/q>0$) resulting in

$\|f\|_{\textbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})}\lesssim\left(\sum_{j=0}^{\infty}2^{j(b+1/q)q}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}|(\varphi_{\nu}\widehat{f})^{\vee}|^{2}\bigg{)}^{1/2}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{q}\right)^{1/q}=\|f\|_{T^{b+1/q}_{q}F^{0}_{p,2}(\mathbb{R}^{d})}.$

(iv): See Theorem A.1 in Appendix A.

∎

We continue with some elementary relations for
$T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d})$ for $A\in\\{B,F\\}$ and
$\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$. In particular, we show the
important property that embeddings between classical spaces
$A^{s}_{p,q}(\mathbb{R}^{d})$ are preserved under the action of truncation
$T^{b}_{r}$. To be more precise, we establish the following result.

######  Proposition 3.9.

Let $A,\widetilde{A}\in\\{B,F\\}$. Let
$p_{i}\in(0,\infty]\,(p_{0}\in(0,\infty)\text{ if }A=F\text{ and
}p_{1}\in(0,\infty)\text{ if }\widetilde{A}=F),q_{i}\in(0,\infty]$ and
$s_{i}\in\mathbb{R},$ where $i=0,1$. Assume that

$A^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d})\hookrightarrow\widetilde{A}^{s_{1}}_{p_{1},q_{1}}(\mathbb{R}^{d}).$

Then, for $r\in(0,\infty]$ and $b\in\mathbb{R}$, we have

$T^{b}_{r}A^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d})\hookrightarrow
T^{b}_{r}\widetilde{A}^{s_{1}}_{p_{1},q_{1}}(\mathbb{R}^{d}).$

###### Proof.

Apply (3.1). ∎

######  Proposition 3.10.

  1. (i)

Let $A\in\\{B,F\\}$. Let $p\in(0,\infty]\,(p\in(0,\infty)\text{ if
}A=F),q,r\in(0,\infty],s\in\mathbb{R}$ and $b>0$. Then

(3.14) $T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
A^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T^{-b}_{r}A^{s}_{p,q}(\mathbb{R}^{d})$

and

(3.15) $\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow\mathfrak{T}^{-b}_{r}F^{s}_{p,q}(\mathbb{R}^{d}).$

  2. (ii)

Let $A,\widetilde{A}\in\\{B,F\\}$. Let $p\in(0,\infty]\,(p\in(0,\infty)\text{
if }A=F\text{ or
}\widetilde{A}=F),q_{0},q_{1},r\in(0,\infty],s,b\in\mathbb{R}$ and
$\varepsilon>0$. Then

(3.16)
$T^{b}_{r}A^{s+\varepsilon}_{p,q_{0}}(\mathbb{R}^{d})\hookrightarrow\widetilde{A}^{s}_{p,q_{1}}(\mathbb{R}^{d})\hookrightarrow
T^{b}_{r}A^{s-\varepsilon}_{p,q_{0}}(\mathbb{R}^{d}),$ (3.17)
$\mathfrak{T}^{b}_{r}F^{s+\varepsilon}_{p,q_{0}}(\mathbb{R}^{d})\hookrightarrow
A^{s}_{p,q_{1}}(\mathbb{R}^{d})\hookrightarrow\mathfrak{T}^{b}_{r}F^{s-\varepsilon}_{p,q_{0}}(\mathbb{R}^{d}).$

  3. (iii)

Let $p\in(0,\infty),q,r\in(0,\infty]$ and $s,b\in\mathbb{R}$. Then

(3.18) $T^{b}_{r}B^{s}_{p,\min\\{p,q\\}}(\mathbb{R}^{d})\hookrightarrow
T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T^{b}_{r}B^{s}_{p,\max\\{p,q\\}}(\mathbb{R}^{d}).$

In particular

$T^{b}_{r}F^{s}_{p,p}(\mathbb{R}^{d})=T^{b}_{r}B^{s}_{p,p}(\mathbb{R}^{d}).$

  4. (iv)

Let $p\in(0,\infty),q,r\in(0,\infty]$ and $s,b\in\mathbb{R}$. Then

(3.19)
$T^{b}_{\min\\{p,r\\}}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T^{b}_{\max\\{p,r\\}}F^{s,b}_{p,q}(\mathbb{R}^{d}).$

In particular

$T^{b}_{p}F^{s}_{p,q}(\mathbb{R}^{d})=\mathfrak{T}^{b}_{p}F^{s}_{p,q}(\mathbb{R}^{d}).$

  5. (v)

Let $p\in(0,\infty)$ and $s,b\in\mathbb{R}$. Then

$T^{b}_{p}B^{s}_{p,p}(\mathbb{R}^{d})=T^{b}_{p}F^{s}_{p,p}(\mathbb{R}^{d})=\mathfrak{T}^{b}_{p}F^{s}_{p,p}(\mathbb{R}^{d}).$

Before we give the proof of this proposition, we observe that the new scale of
spaces $T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d})$ enables to sharpen classical
embeddings. For instance, let us illustrate this phenomenon with the well-
known embeddings

(3.20) $B^{s}_{p,\min\\{p,q\\}}(\mathbb{R}^{d})\hookrightarrow
F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
B^{s}_{p,\max\\{p,q\\}}(\mathbb{R}^{d}),$

where $p\in(0,\infty),q\in(0,\infty]$ and $s\in\mathbb{R}$, and their special
case (taking $q=2$), for $1<p<\infty$ and $s\in\mathbb{R}$,

(3.21) $B^{s}_{p,\min\\{p,2\\}}(\mathbb{R}^{d})\hookrightarrow
H^{s}_{p}(\mathbb{R}^{d})\hookrightarrow
B^{s}_{p,\max\\{p,2\\}}(\mathbb{R}^{d}).$

######  Corollary 3.11.

  1. (i)

Let $s\in\mathbb{R},p\in(0,\infty)$ and $q\in(0,\infty]$. Then

$\displaystyle B^{s}_{p,\min\\{p,q\\}}(\mathbb{R}^{d})$
$\displaystyle\hookrightarrow T_{\min\\{p,q\\}}F^{s}_{p,q}(\mathbb{R}^{d})$
(3.22) $\displaystyle\hookrightarrow
F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T_{\max\\{p,q\\}}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
B^{s}_{p,\max\\{p,q\\}}(\mathbb{R}^{d}).$

  2. (ii)

Let $s\in\mathbb{R}$ and $p\in(1,\infty)$. Then

$\displaystyle B^{s}_{p,\min\\{p,2\\}}(\mathbb{R}^{d})$
$\displaystyle\hookrightarrow T_{\min\\{p,2\\}}F^{s}_{p,2}(\mathbb{R}^{d})$
$\displaystyle\hookrightarrow H^{s}_{p}(\mathbb{R}^{d})\hookrightarrow
T_{\max\\{p,2\\}}F^{s}_{p,2}(\mathbb{R}^{d})\hookrightarrow
B^{s}_{p,\max\\{p,2\\}}(\mathbb{R}^{d}).$

###### Proof.

(i): Since
$F^{s}_{p,q}(\mathbb{R}^{d})=\mathfrak{T}_{q}F^{s}_{p,q}(\mathbb{R}^{d})$ and
$B^{s}_{p,q}(\mathbb{R}^{d})=T_{q}B^{s}_{p,q}(\mathbb{R}^{d})$ (cf.
Proposition 3.8), it follows from (3.18)-(3.19) that

$F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T_{\max\\{p,q\\}}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T_{\max\\{p,q\\}}B^{s}_{p,\max\\{p,q\\}}(\mathbb{R}^{d})=B^{s}_{p,\max\\{p,q\\}}(\mathbb{R}^{d})$

and

$F^{s}_{p,q}(\mathbb{R}^{d})\hookleftarrow
T_{\min\\{p,q\\}}F^{s}_{p,q}(\mathbb{R}^{d})\hookleftarrow
T_{\min\\{p,q\\}}B^{s}_{p,\min\\{p,q\\}}(\mathbb{R}^{d})=B^{s}_{p,\min\\{p,q\\}}(\mathbb{R}^{d}).$

Specializing (i) with $q=2$ and $p\in(1,\infty)$ one gets (ii). ∎

######  Remark 3.12.

Embeddings given in Corollary 3.11 are non trivial improvements of the
classical ones (3.20) and (3.21). For instance, assume $p>q$ then by (3.22),

$F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T_{p}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow B^{s}_{p,p}(\mathbb{R}^{d}).$

Furthermore these embeddings are strict. Indeed, consider the lacunary Fourier
series $f$ defined by (3.9). Since (3.10) holds, we have

$\|f\|_{F^{s}_{p,q}(\mathbb{R}^{d})}\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jsq}|a_{j}|^{q}\bigg{)}^{1/q},$
$\|f\|_{T_{p}F^{s}_{p,q}(\mathbb{R}^{d})}\asymp\bigg{(}\sum_{k=0}^{\infty}\bigg{(}\sum_{j=2^{k}-1}^{2^{k+1}-2}2^{jsq}|a_{j}|^{q}\bigg{)}^{p/q}\bigg{)}^{1/p},$

and

$\|f\|_{B^{s}_{p,p}(\mathbb{R}^{d})}\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jsp}|a_{j}|^{p}\bigg{)}^{1/p}.$

Taking $a_{j}=2^{-js}(j+1)^{-1/q}(1+\log(1+j))^{-\varepsilon}$ where
$1/p<\varepsilon<1/q$, we have

$\|f\|_{F^{s}_{p,q}(\mathbb{R}^{d})}^{q}\asymp\sum_{j=1}^{\infty}(1+\log
j)^{-\varepsilon q}\frac{1}{j}=\infty$

and

$\|f\|_{T_{p}F^{s}_{p,q}(\mathbb{R}^{d})}^{p}\asymp\sum_{k=1}^{\infty}k^{-\varepsilon
p}<\infty.$

On the other hand, if $a_{j}=2^{-js}(1+j)^{-\varepsilon}$ with
$1/p<\varepsilon<1/q$ then

$\|f\|_{B^{s}_{p,p}(\mathbb{R}^{d})}^{p}\asymp\sum_{j=1}^{\infty}j^{-\varepsilon
p}<\infty$

and

$\|f\|_{T_{p}F^{s}_{p,q}(\mathbb{R}^{d})}^{p}\asymp\sum_{k=0}^{\infty}\bigg{(}\sum_{j=2^{k}}^{2^{k+1}}j^{-\varepsilon
q}\bigg{)}^{p/q}\asymp\sum_{k=0}^{\infty}2^{-kp(\varepsilon-1/q)}=\infty.$

The above argument also shows that the embeddings

$B^{s}_{p,p}(\mathbb{R}^{d})\subsetneqq
T_{p}F^{s}_{p,q}(\mathbb{R}^{d})\subsetneqq F^{s}_{p,q}(\mathbb{R}^{d}),\qquad
p<q$

are strict.

###### Proof of Proposition 3.10.

(i): We start by proving the right-hand side embedding with $A\in\\{B,F\\}$.
It is clear that

$\|f\|_{T^{-b}_{r}A^{s}_{p,q}(\mathbb{R}^{d})}\leq\bigg{(}\sum_{j=0}^{\infty}2^{-jbr}\bigg{)}^{1/r}\|f\|_{A^{s}_{p,q}(\mathbb{R}^{d})}$

where the last sum is convergent since $b>0$. We also have

$\bigg{(}\sum_{j=0}^{\infty}2^{-jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}(x)|^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}\lesssim\bigg{(}\sum_{\nu=0}^{\infty}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}(x)|^{q}\bigg{)}^{1/q}$

for $x\in\mathbb{R}^{d}$ and thus
$\|f\|_{\mathfrak{T}^{-b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}\lesssim\|f\|_{F^{s}_{p,q}(\mathbb{R}^{d})}.$

Concerning the left-hand side embedding of (3.14) with $A=B$, we have

$\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}\leq
2^{-jbq}\,\|f\|^{q}_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})},\qquad j\geq 0,$

for $f\in T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$. Summing up the previous
estimates, we obtain

$\displaystyle\|f\|^{q}_{B^{s}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle=\sum_{j=0}^{\infty}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{q}$
$\displaystyle\leq\|f\|^{q}_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}\sum_{j=0}^{\infty}2^{-jbq}\asymp\|f\|^{q}_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}.$

Similarly, to deal with $A=F$ in (3.14), we note that

(3.23) $\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}\leq
2^{-jb}\,\|f\|_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})},\qquad j\geq 0,$

for $f\in T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$. Furthermore, we claim that

(3.24)
$\|f\|_{F^{s}_{p,q}(\mathbb{R}^{d})}\leq\bigg{(}\sum_{j=0}^{\infty}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{\min\\{p,q\\}}\bigg{)}^{1/\min\\{p,q\\}}.$

Assuming momentarily that (3.24) is true, then the embedding
$T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
F^{s}_{p,q}(\mathbb{R}^{d})$ follows directly from (3.23).

To prove (3.24), we shall consider two possible cases. Firstly, if $p\geq q$
we can estimate $\|f\|_{F^{s}_{p,q}(\mathbb{R}^{d})}$ using the Minkowski’s
inequality, specifically,

$\displaystyle\|f\|_{F^{s}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle=\bigg{\|}\bigg{(}\sum_{j=0}^{\infty}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}$
$\displaystyle\leq\bigg{(}\sum_{j=0}^{\infty}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{1/q}.$

Secondly, if $p<q$ then we make use of the fact that
$\ell_{p}\hookrightarrow\ell_{q}$ and apply Fubini’s theorem in order to get

$\|f\|_{F^{s}_{p,q}(\mathbb{R}^{d})}\leq\bigg{(}\sum_{j=0}^{\infty}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{p}\bigg{)}^{1/p}.$

Hence (3.24) holds.

The proof of the left-hand side embedding in (3.15), i.e.,
$\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
F^{s}_{p,q}(\mathbb{R}^{d})$ follows also from (3.24) since (cf. (3.23))

$\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}\leq
2^{-jb}\,\|f\|_{\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})},\qquad j\geq
0,$

for $f\in\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$.

(ii): Concerning the left-hand side embedding in (3.16), in light of (3.20)
and Proposition 3.9, it is enough to prove that

$T^{b}_{r}B^{s+\varepsilon}_{p,\infty}(\mathbb{R}^{d})\hookrightarrow
B^{s}_{p,\xi}(\mathbb{R}^{d})$

for some $\xi>0$ sufficiently small. Applying Hölder’s inequality, we get

$\displaystyle\|f\|_{B^{s}_{p,\xi}(\mathbb{R}^{d})}$
$\displaystyle=\bigg{(}\sum_{j=0}^{\infty}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}(2^{\nu
s}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})})^{\xi}\bigg{)}^{1/\xi}$
$\displaystyle\lesssim\bigg{(}\sum_{j=0}^{\infty}2^{-2^{j}\varepsilon\xi}\sup_{\nu=2^{j}-1,\ldots,2^{j+1}-2}\,\\{2^{\nu(s+\varepsilon)}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}\\}^{\xi}\bigg{)}^{1/\xi}$
$\displaystyle\lesssim\bigg{(}\sum_{j=0}^{\infty}2^{-2^{j}\varepsilon\xi}2^{-jb\xi}\bigg{)}^{1/\xi}\,\|f\|_{T^{b}_{\infty}B^{s+\varepsilon}_{p,\infty}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\|f\|_{T^{b}_{r}B^{s+\varepsilon}_{p,\infty}(\mathbb{R}^{d})}.$

On the other hand, in a similar fashion as above, the right-hand side
embedding in (3.16) would follow from the special case

$B^{s}_{p,\infty}(\mathbb{R}^{d})\hookrightarrow
T^{b}_{r}B^{s-\varepsilon}_{p,\xi}(\mathbb{R}^{d}),\qquad\xi>0.$

To prove the latter embedding, one can reason as follows:

$\displaystyle\|f\|_{T^{b}_{r}B^{s-\varepsilon}_{p,\xi}(\mathbb{R}^{d})}$
$\displaystyle=\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu(s-\varepsilon)\xi}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{\xi}\bigg{)}^{r/\xi}\right)^{1/r}$
$\displaystyle\lesssim\bigg{(}\sum_{j=0}^{\infty}2^{-2^{j}\varepsilon
r}2^{jbr}\bigg{)}^{1/r}\,\|f\|_{B^{s}_{p,\infty}(\mathbb{R}^{d})}\asymp\|f\|_{B^{s}_{p,\infty}(\mathbb{R}^{d})}.$

Next we turn our attention to the left-hand side embedding in (3.17). Note
that it is enough to show that

(3.25)
$\mathfrak{T}^{b}_{r}F^{s+\varepsilon}_{p,q_{0}}(\mathbb{R}^{d})\hookrightarrow
B^{s}_{p,\xi}(\mathbb{R}^{d})$

for some $\xi>0$ sufficiently small; cf. (3.20). Let $j\in\mathbb{N}_{0}$ and
$\nu\in\\{2^{j}-1,\ldots,2^{j+1}-2\\}$. Since

$\displaystyle\|2^{\nu
s}(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}$
$\displaystyle\lesssim
2^{-2^{j}\varepsilon}\,\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu(s+\varepsilon)q_{0}}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q_{0}}\bigg{)}^{1/q_{0}}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}$
$\displaystyle\lesssim
2^{-2^{j}\varepsilon}\,2^{-jb}\,\|f\|_{\mathfrak{T}^{b}_{r}F^{s+\varepsilon}_{p,q_{0}}(\mathbb{R}^{d})},$

we have

$\displaystyle\|f\|_{B^{s}_{p,\xi}(\mathbb{R}^{d})}$
$\displaystyle=\bigg{(}\sum_{j=0}^{\infty}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}\|2^{\nu
s}(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}^{\xi}\bigg{)}^{1/\xi}$
$\displaystyle\lesssim\bigg{(}\sum_{j=0}^{\infty}2^{-2^{j}\varepsilon\xi}\,2^{j(1-b\xi)}\bigg{)}^{1/\xi}\,\|f\|_{\mathfrak{T}^{b}_{r}F^{s+\varepsilon}_{p,q_{0}}(\mathbb{R}^{d})}$
$\displaystyle\asymp\|f\|_{\mathfrak{T}^{b}_{r}F^{s+\varepsilon}_{p,q_{0}}(\mathbb{R}^{d})}.$

This completes the proof of (3.25). Furthermore, similar ideas as above can be
applied to show the right-hand side embedding in (3.17). This is left to the
interested reader.

(iii): It follows immediately from Proposition 3.9 and (3.20).

(iv): We only provide the proof of the left-hand side embedding of (3.19). If
$p\geq r$ then we can apply Minkowski’s inequality so that

$\displaystyle\|f\|_{\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle=\bigg{\|}\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}$
$\displaystyle\leq\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{r}\bigg{)}^{1/r}=\|f\|_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}.$

On the other hand, if $r\geq p$ then

$\displaystyle\|f\|_{\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle\leq\bigg{\|}\bigg{(}\sum_{j=0}^{\infty}2^{jbp}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{p/q}\bigg{)}^{1/p}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}$
$\displaystyle=\bigg{(}\sum_{j=0}^{\infty}2^{jbp}\bigg{\|}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|(\varphi_{\nu}\widehat{f})^{\vee}|^{q}\bigg{)}^{1/q}\bigg{\|}_{L_{p}(\mathbb{R}^{d})}^{p}\bigg{)}^{1/p}=\|f\|_{T^{b}_{p}F^{s}_{p,q}(\mathbb{R}^{d})}.$

Item (v) is an immediate consequences of Fubini’s theorem.

∎

######  Remark 3.13.

A detailed study of embedding theorems between
$T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d})$-spaces will be postponed until Sections
10–13. In particular, the optimality of (3.18) will be addressed in Corollary
11.3 below, namely,

$T^{b}_{r}B^{s}_{p,u}(\mathbb{R}^{d})\hookrightarrow
T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})\iff u\leq\min\\{p,q\\}$

and

$T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})\hookrightarrow
T^{b}_{r}B^{s}_{p,u}(\mathbb{R}^{d})\iff u\geq\max\\{p,q\\}.$

##  4\. Characterizations by interpolation

###  4.1. Limiting interpolation. General facts.

The goal of this section is to show that the spaces
$T^{b}_{r}A^{s}_{p,q}(\mathbb{R}^{d}),\\\ A\in\\{B,F\\}$, and
$\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ can be generated as
interpolation spaces of the classical spaces $B^{s}_{p,q}(\mathbb{R}^{d})$ and
$F^{s}_{p,q}(\mathbb{R}^{d})$ (cf. Theorems 4.2, 4.8 and 4.20 below). As a
consequence, we shall conclude not only that these new scales are closed under
interpolation (cf. Theorems 4.7 and 4.14 below), but also new interpolation
formulae involving $\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$ and
$\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})$ (cf. Corollary 4.19). To get all
these results we shall apply the machinery provided by limiting interpolation.
Next we briefly recall the construction of this method.

Let $(A_{0},A_{1})$ be a couple of quasi-Banach spaces. Given $t>0$ and $f\in
A_{0}+A_{1}$, the _Peetre’s $K$-functional_ is defined by

(4.1)
$K(t,f):=K(t,f;A_{0},A_{1}):=\inf\\{\|f_{0}\|_{A_{0}}+t\|f_{1}\|_{A_{1}}:f=f_{0}+f_{1},\,f_{j}\in
A_{j}\\}.$

For $\theta\in(0,1),\,q\in(0,\infty]$ and $b\in\mathbb{R}$, the interpolation
space $(A_{0},A_{1})_{\theta,q,b}$ is formed by all those $f\in A_{0}+A_{1}$
such that

(4.2)
$\|f\|_{(A_{0},A_{1})_{\theta,q,b}}:=\bigg{(}\int_{0}^{\infty}(t^{-\theta}(1+|\log
t|)^{b}K(t,f))^{q}\frac{dt}{t}\bigg{)}^{1/q}<\infty$

(with the usual modification if $q=\infty$). See [Gu78], [DRS79] and [GOT05].
In the special case $b=0$ one recovers the classical interpolation space
$(A_{0},A_{1})_{\theta,q}$, i.e.,

(4.3)
$\|f\|_{(A_{0},A_{1})_{\theta,q}}:=\bigg{(}\int_{0}^{\infty}(t^{-\theta}K(t,f))^{q}\frac{dt}{t}\bigg{)}^{1/q}$

cf. [BL76], [BS88] and [Tri78]. Since
$K(t,f;A_{0},A_{1})=tK(t^{-1},f;A_{1},A_{0})$, one has

(4.4) $(A_{0},A_{1})_{\theta,q,b}=(A_{1},A_{0})_{1-\theta,q,b}$

with equality of quasi-norms.

The introduction of limiting interpolation spaces requires some non-trivial
modifications of the quasi-norms (4.2) and (4.3). Indeed, note that setting
$\theta=0$ or $\theta=1$ we only obtain in general the trivial space
$\\{0\\}$. To overcome this issue we make the elementary observation that,
working with ordered couples (i.e., $A_{1}\hookrightarrow A_{0}$),

(4.5)
$\|f\|_{(A_{0},A_{1})_{\theta,q}}\asymp\bigg{(}\int_{0}^{1}(t^{-\theta}K(t,f))^{q}\frac{dt}{t}\bigg{)}^{1/q},\qquad\theta\in(0,1).$

This follows from the fact $K(t,f)\asymp\|f\|_{A_{0}}$ for $t>1$. Accordingly,
one may consider the integral given in the right-hand side of (4.5) to
introduce the space $(A_{0},A_{1})_{\theta,q}$. This together with the
additional refinement given by logarithmic weights allow us to define limiting
interpolation spaces. More precisely, given an ordered couple of quasi-Banach
spaces $(A_{0},A_{1})$, for $\theta\in\\{0,1\\},q\in(0,\infty]$ and
$b\in\mathbb{R}$, the _limiting interpolation space_
$(A_{0},A_{1})_{(\theta,b),q}$ is the set of all those $f\in A_{0}$ for which

(4.6)
$\|f\|_{(A_{0},A_{1})_{(\theta,b),q}}=\bigg{(}\int_{0}^{1}(t^{-\theta}(1-\log
t)^{b}K(t,f))^{q}\frac{dt}{t}\bigg{)}^{1/q}$

is finite (with the usual modification if $q=\infty$). See [JM91, GOT05,
CD15a, ALM19] and the references given there. It is not hard to check that if
$b<-1/q\,(b\leq 0\text{ if }q=\infty)$ then $(A_{0},A_{1})_{(0,b),q}=A_{0}$
and if $b\geq-1/q\,(b>0\text{ if }q=\infty)$ then
$(A_{0},A_{1})_{(1,b),q}=\\{0\\}$. Accordingly, we may assume without loss of
generality that $b\geq-1/q\,(b>0\text{ if }q=\infty)$ if $\theta=0$ and
$b<-1/q\,(b\leq 0\text{ if }q=\infty)$ if $\theta=1$. Under these assumptions,
the $((\theta,b),q)$-method produces intermediate spaces between $A_{0}$ and
$A_{1}$, that is,

(4.7) $A_{1}\hookrightarrow(A_{0},A_{1})_{(\theta,b),q}\hookrightarrow A_{0},$

and it satisfies the interpolation property for bounded linear operators.
Furthermore, the left-hand side embedding in (4.7) is dense whenever
$q<\infty$. We remark that it is possible to introduce limiting interpolation
spaces for general couples of quasi-Banach spaces (not necessarily ordered)
with the help of adequate slowly varying functions, however for the purposes
of this paper is enough to restrict ourselves to the ordered case.

The following relations between classical and limiting interpolation will be
useful; cf. [EOP02, Corollaries 7.8 and 7.11] and [DT18, Lemma 2.2].

######  Lemma 4.1.

Let $\theta\in(0,1),\,q,r,r_{0},r_{1}\in(0,\infty]$ and $\xi\in\mathbb{R}$.

  1. (i)

If $b\geq-1/r$ then

$(A_{0},(A_{0},A_{1})_{\theta,q,\xi})_{(0,b),r}=(A_{0},A_{1})_{(0,b),r}.$

  2. (ii)

If $b<-1/r$ then

$((A_{0},A_{1})_{\theta,q,\xi},A_{1})_{(1,b),r}=(A_{0},A_{1})_{(1,b),r}.$

  3. (iii)

If $0<b_{0}+1/r_{0}<b_{1}+1/r_{1}$ then

$((A_{0},A_{1})_{(0,b_{0}),r_{0}},(A_{0},A_{1})_{(0,b_{1}),r_{1}})_{\theta,q}=(A_{0},A_{1})_{(0,\gamma),q}$

where $\gamma=(1-\theta)(b_{0}+1/r_{0})+\theta(b_{1}+1/r_{1})-1/q$.

  4. (iv)

If $b_{1}+1/r_{1}<b_{0}+1/r_{0}<0$ then

$((A_{0},A_{1})_{(1,b_{0}),r_{0}},(A_{0},A_{1})_{(1,b_{1}),r_{1}})_{\theta,q}=(A_{0},A_{1})_{(1,\gamma),q}$

where $\gamma=(1-\theta)(b_{0}+1/r_{0})+\theta(b_{1}+1/r_{1})-1/q$.

  5. (v)

If $b>-1/r$ then

$((A_{0},A_{1})_{(0,b),r},A_{1})_{\theta,q}=(A_{0},A_{1})_{\theta,q,(1-\theta)(b+1/r)}.$

  6. (vi)

If $b<-1/r$ then

$(A_{0},(A_{0},A_{1})_{(1,b),r})_{\theta,q}=(A_{0},A_{1})_{\theta,q,\theta(b+1/r)}.$

###  4.2. $T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ spaces via interpolation

######  Theorem 4.2 (Characterization of
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ via limiting interpolation).

Let $A\in\\{B,F\\}$. Let $p,q,q_{0},r\in(0,\infty]\,(p<\infty\text{ if
}A=F),s,s_{0}\in\mathbb{R},s\neq s_{0}$, and $b\in\mathbb{R},\,b\neq 0$.

  1. (i)

If $s_{0}>s$ and $b>0$ then

(4.8)
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})=(B^{s}_{p,q}(\mathbb{R}^{d}),A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}))_{(0,b-1/r),r}.$

  2. (ii)

If $s_{0}<s$ and $b<0$ then

(4.9)
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})=(A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r}.$

######  Remark 4.3.

Note that the assumption $s_{0}>s$ in (i) implies that
$A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d})\hookrightarrow
B^{s}_{p,q}(\mathbb{R}^{d})$ (cf. [Tri83, Section 2.3.2, page 47]). A similar
comment applies to (ii).

######  Remark 4.4.

The case $b=0$ in Theorem 4.2 is more delicate since the outcome is not the
expected spaces $T_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ but
$T^{*}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ (cf. (3.5)). This will be treated in
detail in Remark 4.5 below.

###### Proof of Theorem 4.2.

The proof relies on the retraction method (see [Tri78, Section 1.2.4] for
details on this method) and limiting interpolation techniques. Assume first
$p\in[1,\infty]$ and $A=B$. In this case, it is well known that
$B^{s}_{p,q}(\mathbb{R}^{d})$ is a retract of
$\ell^{s}_{q}(L_{p}(\mathbb{R}^{d}))$ and $B^{s_{0}}_{p,q}(\mathbb{R}^{d})$ is
a retract of $\ell^{s_{0}}_{q}(L_{p}(\mathbb{R}^{d}))$ with the corresponding
co-retraction operator being

(4.10) $\mathfrak{J}f:=((\varphi_{j}\widehat{f})^{\vee})_{j=0}^{\infty}.$

Furthermore, elementary computations lead to

(4.11)
$K(t,x;\ell^{s}_{q}(L_{p}(\mathbb{R}^{d})),\ell^{s_{0}}_{q}(L_{p}(\mathbb{R}^{d})))\asymp\bigg{(}\sum_{\nu=0}^{\infty}[\min(2^{\nu
s},t2^{\nu s_{0}})\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{1/q}$

for $t>0$ and
$x=(x_{\nu})_{\nu=0}^{\infty}\in\ell^{s}_{q}(L_{p}(\mathbb{R}^{d}))+\ell^{s_{0}}_{q}(L_{p}(\mathbb{R}^{d}))$.

(i): Assume $s_{0}>s$. According to (4.11), for every $j\geq 0$,

$\displaystyle
K(2^{-j(s_{0}-s)},x;\ell^{s}_{q}(L_{p}(\mathbb{R}^{d})),\ell^{s_{0}}_{q}(L_{p}(\mathbb{R}^{d})))$
(4.12) $\displaystyle\asymp 2^{-j(s_{0}-s)}\bigg{(}\sum_{\nu=0}^{j}[2^{\nu
s_{0}}\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{1/q}+\bigg{(}\sum_{\nu=j+1}^{\infty}[2^{\nu
s}\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{1/q}.$

Therefore,

$\displaystyle\|x\|_{(\ell^{s}_{q}(L_{p}(\mathbb{R}^{d})),\ell^{s_{0}}_{q}(L_{p}(\mathbb{R}^{d})))_{(0,b-1/r),r}}$
$\displaystyle\asymp\left(\sum_{j=0}^{\infty}[(1+j)^{b-1/r}K(2^{-j(s_{0}-s)},x;\ell^{s}_{q}(L_{p}(\mathbb{R}^{d})),\ell^{s_{0}}_{q}(L_{p}(\mathbb{R}^{d})))]^{r}\right)^{1/r}$
(4.13)
$\displaystyle\asymp\left(\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}2^{-j(s_{0}-s)r}\bigg{(}\sum_{\nu=0}^{j}[2^{\nu
s_{0}}\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}$
$\displaystyle+\left(\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=j}^{\infty}[2^{\nu
s}\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}$
$\displaystyle=:I+II.$

Furthermore,

(4.14) $I\lesssim II.$

Indeed, it follows from Hardy’s inequality (B.3) that

$I\lesssim\left(\sum_{j=0}^{\infty}\left[(1+j)^{b-1/r}2^{js}\|x_{j}\|_{L_{p}(\mathbb{R}^{d})}\right]^{r}\right)^{1/r}\leq
II.$

Combining (4.13) and (4.14), we obtain that

(4.15)
$\|x\|_{(\ell^{s}_{q}(L_{p}(\mathbb{R}^{d})),\ell^{s_{0}}_{q}(L_{p}(\mathbb{R}^{d})))_{(0,b-1/r),r}}\asymp\left(\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=j}^{\infty}[2^{\nu
s}\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}.$

In virtue of the retraction method and (4.15) we get

$\displaystyle\|f\|_{(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q}(\mathbb{R}^{d}))_{(0,b-1/r),r}}$
$\displaystyle\asymp\|\mathfrak{J}f\|_{(\ell^{s}_{q}(L_{p}(\mathbb{R}^{d})),\ell^{s_{0}}_{q}(L_{p}(\mathbb{R}^{d})))_{(0,b-1/r),r}}$
(4.16)
$\displaystyle\asymp\left(\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=j}^{\infty}[2^{\nu
s}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}.$

Assume further $b>0$. We claim

(4.17)
$\left(\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=j}^{\infty}[2^{\nu
s}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}\asymp\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}.$

Assuming momentarily the validity of the previous assertion, we conclude by
(4.16) that (i) holds for $A=B,\,p\in[1,\infty]$ and $q_{0}=q\in(0,\infty]$.
Next we show (4.17). By monotonicity properties and Hardy’s inequality (note
that $b>0$; cf. (B.4))

$\displaystyle\left(\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=j}^{\infty}[2^{\nu
s}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}$
$\displaystyle\asymp$
$\displaystyle\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}}^{\infty}[2^{\nu
s}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}$
(4.18)
$\displaystyle\asymp\left(\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}[2^{\nu
s}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}=\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}.$

The non-diagonal case $q_{0}\neq q$ can reduced to the previous case via the
reiteration formula given in Lemma 4.1(i). Indeed, let $s_{1}$ be such that
$s_{0}\in(s,s_{1})$. By the well-known interpolation properties of Besov
spaces (cf. [BL76, Theorem 6.4.5], [Tri78, 2.4.1] and [Tri83, 2.4.2])

(4.19)
$B^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d})=(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{1}}_{p,q}(\mathbb{R}^{d}))_{\theta,q_{0}}$

where $\theta\in(0,1)$ is given by $s_{0}=(1-\theta)s+\theta s_{1}$. Applying
now Lemma 4.1(i) we have

$\displaystyle(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}))_{(0,b-1/r),r}$
$\displaystyle=(B^{s}_{p,q}(\mathbb{R}^{d}),(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{1}}_{p,q}(\mathbb{R}^{d}))_{\theta,q_{0}})_{(0,b-1/r),r}$
$\displaystyle=(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{1}}_{p,q}(\mathbb{R}^{d}))_{(0,b-1/r),r}=T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d}).$

This completes the proof of (i) in the case $A=B$ and $p\in[1,\infty]$.
Furthermore, the case $p\in(0,1)$ can be done in a similar fashion but now
replacing $L_{p}(\mathbb{R}^{d})$ by the Hardy space $h_{p}(\mathbb{R}^{d})$
so that the operator $\mathfrak{J}$ (cf. (4.10)) is well defined and the
retraction method can be also applied.

The formula (i) with $A=F$ follows immediately from the case $A=B$ and the
well-known embeddings

(4.20) $B^{s_{0}}_{p,\min\\{p,q_{0}\\}}(\mathbb{R}^{d})\hookrightarrow
F^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d})\hookrightarrow
B^{s_{0}}_{p,\max\\{p,q_{0}\\}}(\mathbb{R}^{d})$

(see, e.g., [Tri83, (9), page 47]).

(ii): Assume $s_{0}<s$ and $b<0$. According to (4.11), for every $j\geq 0$,

$\displaystyle
K(2^{-j(s-s_{0})},x;\ell^{s_{0}}_{q}(L_{p}(\mathbb{R}^{d})),\ell^{s}_{q}(L_{p}(\mathbb{R}^{d})))$
$\displaystyle\asymp 2^{-j(s-s_{0})}\bigg{(}\sum_{\nu=0}^{j}[2^{\nu
s}\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{1/q}+\bigg{(}\sum_{\nu=j+1}^{\infty}[2^{\nu
s_{0}}\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{1/q}.$

Thus, by Hardy’s inequality (B.4), we have

$\displaystyle\|x\|_{(\ell^{s_{0}}_{q}(L_{p}(\mathbb{R}^{d})),\ell^{s}_{q}(L_{p}(\mathbb{R}^{d})))_{(1,b-1/r),r}}$
$\displaystyle\asymp\left(\sum_{j=0}^{\infty}[2^{j(s-s_{0})}(1+j)^{b-1/r}K(2^{-j(s-s_{0})},x;\ell^{s_{0}}_{q}(L_{p}(\mathbb{R}^{d})),\ell^{s}_{q}(L_{p}(\mathbb{R}^{d})))]^{r}\right)^{1/r}$
$\displaystyle\asymp\left(\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=0}^{j}[2^{\nu
s}\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}$
$\displaystyle+\left(\sum_{j=0}^{\infty}2^{j(s-s_{0})r}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=j}^{\infty}[2^{\nu
s_{0}}\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}$
$\displaystyle\asymp\left(\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=0}^{j}[2^{\nu
s}\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}.$

Invoking now the retraction method we establish that

$\|f\|_{(B^{s_{0}}_{p,q}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r}}\asymp\left(\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=0}^{j}[2^{\nu
s}\|(\varphi_{\nu}\widehat{f})^{\vee}\|_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{r/q}\right)^{1/r}$

and thus a similar argument as in (4.18) now with $b<0$ yields that

$\|f\|_{(B^{s_{0}}_{p,q}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r}}\asymp\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}$

or equivalently,

(4.21)
$(B^{s_{0}}_{p,q}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r}=T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d}).$

This covers (ii) with $A=B,p\in[1,\infty]$ and $q_{0}=q\in(0,\infty]$.

Assume $q_{0}\in(0,\infty],\,q_{0}\neq q$. Given $s_{0}<s$ we take $s_{1}$
with $s_{1}<s_{0}$ and define $\theta\in(0,1)$ such that
$s_{0}=(1-\theta)s_{1}+\theta s$. Since
$B^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d})=(B^{s_{1}}_{p,q}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))_{\theta,q_{0}}$
(cf. (4.19)) we can apply (4.21) and Lemma 4.1(ii) to get

$\displaystyle T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$
$\displaystyle=(B^{s_{1}}_{p,q}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r}$
$\displaystyle=((B^{s_{1}}_{p,q}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))_{\theta,q_{0}},B^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r}$
$\displaystyle=(B^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r},$

that is, (ii) holds with $A=B$ and $p\in[1,\infty]$. Furthermore, the
modification of this method obtained by replacing $L_{p}(\mathbb{R}^{d})$ by
$h_{p}(\mathbb{R}^{d})$ enables us to cover the full range $p\in(0,\infty]$.
Finally, the case $A=F$ follows from the $B$-case via the embeddings (4.20). ∎

######  Remark 4.5.

Let us consider the limiting case $b=0$ in (4.8). Namely (cf. (4.16)) we have

(4.22)
$(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q}(\mathbb{R}^{d}))_{(0,-1/r),r}=T^{*}_{r}B^{s}_{p,q}(\mathbb{R}^{d}),$

see (3.5). In the special case $r=q$, we have (cf. (3.6))

$(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q}(\mathbb{R}^{d}))_{(0,-1/q),q}=B^{s,0,1/q}_{p,q}(\mathbb{R}^{d}).$

######  Remark 4.6.

The limiting cases $p=1,\infty$ in Theorem 4.2 deserve special attention.
Unlike in the case $p\in(1,\infty)$, the Sobolev spaces
$W^{k}_{1}(\mathbb{R}^{d})$ and $W^{k}_{\infty}(\mathbb{R}^{d})$,
$k\in\mathbb{N}_{0}$, do not fit into the scale of Triebel–Lizorkin spaces
$F^{s}_{p,q}(\mathbb{R}^{d})$. However, the interpolation formulae (4.8) and
(4.9) still are valid with $W^{k}_{p}(\mathbb{R}^{d}),\,p=1,\infty$. More
precisely, let $p=1,\infty$ and $q,r\in(0,\infty]$, then

(4.23)
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})=(B^{s}_{p,q}(\mathbb{R}^{d}),W^{k}_{p}(\mathbb{R}^{d}))_{(0,b-1/r),r},\qquad
s<k,\qquad b>0$

and

(4.24)
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})=(W^{k}_{p}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r},\qquad
s>k,\qquad b<0.$

To prove (4.23), we make use of the well-known embeddings (cf. [Tri83, Section
2.5.7])

$B^{k}_{p,1}(\mathbb{R}^{d})\hookrightarrow
W^{k}_{p}(\mathbb{R}^{d})\hookrightarrow B^{k}_{p,\infty}(\mathbb{R}^{d}).$

By (4.8), we derive

$\displaystyle T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$
$\displaystyle=(B^{s}_{p,q}(\mathbb{R}^{d}),B^{k}_{p,1}(\mathbb{R}^{d}))_{(0,b-1/r),r}\hookrightarrow(B^{s}_{p,q}(\mathbb{R}^{d}),W^{k}_{p}(\mathbb{R}^{d}))_{(0,b-1/r),r}$
$\displaystyle\hookrightarrow(B^{s}_{p,q}(\mathbb{R}^{d}),B^{k}_{p,\infty}(\mathbb{R}^{d}))_{(0,b-1/r),r}=T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d}).$

The proof of (4.24) can be obtained similarly but now invoking (4.9).

###  4.3. $T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ spaces are closed under
interpolation.

Our next result proves that real interpolation between spaces
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ with $s,p$ and $q$ fixed produces a
space of the same type.

######  Theorem 4.7.

Let $0<p,q,r,r_{0},r_{1}\leq\infty,-\infty<s,b_{0},b_{1}<\infty$ and
$0<\theta<1$. Assume further $b_{0}\neq b_{1}$ and $b_{0}b_{1}>0$ and let
$b=(1-\theta)b_{0}+\theta b_{1}$. Then

$(T^{b_{0}}_{r_{0}}B^{s}_{p,q}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}B^{s}_{p,q}(\mathbb{R}^{d}))_{\theta,r}=T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d}).$

###### Proof.

The assumption $b_{0}b_{1}>0$ means that both $b_{0}$ and $b_{1}$ are either
positive or negative. Assume first that $0<b_{0}<b_{1}$. Let $s_{0}>s$.
According to Theorem 4.2(i),

$T^{b_{i}}_{r_{i}}B^{s}_{p,q}(\mathbb{R}^{d})=(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q}(\mathbb{R}^{d}))_{(0,b_{i}-1/r_{i}),r_{i}},\qquad
i=0,1.$

Let $\gamma:=b-1/r$. In light of Lemma 4.1(iii), one has

$\displaystyle(T^{b_{0}}_{r_{0}}B^{s}_{p,q}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}B^{s}_{p,q}(\mathbb{R}^{d}))_{\theta,r}$
$\displaystyle=$
$\displaystyle((B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q}(\mathbb{R}^{d}))_{(0,b_{0}-1/r_{0}),r_{0}},(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q}(\mathbb{R}^{d}))_{(0,b_{1}-1/r_{1}),r_{1}})_{\theta,r}$
$\displaystyle=(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q}(\mathbb{R}^{d}))_{(0,\gamma),r}=T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d}),$

where the last step follows again from Theorem 4.2(i).

The case $b_{0}>b_{1}>0$ can be reduced to the previous case. More precisely,
basic properties of the real interpolation method (cf. (4.4)) yield that

$(T^{b_{0}}_{r_{0}}B^{s}_{p,q}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}B^{s}_{p,q}(\mathbb{R}^{d}))_{\theta,r}=(T^{b_{1}}_{r_{1}}B^{s}_{p,q}(\mathbb{R}^{d}),T^{b_{0}}_{r_{0}}B^{s}_{p,q}(\mathbb{R}^{d}))_{1-\theta,r}=T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d}).$

This completes the proof under $b_{0},b_{1}>0$. The situation $b_{0},b_{1}<0$
can be settled following similar ideas but now employing Theorem 4.2(ii) and
Lemma 4.1(iv).

∎

In order to show that $T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ spaces are closed
under interpolation, we will need the results from the next section.

###  4.4. $T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ spaces via interpolation

The goal of this section is to show that the scale
$\\{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})\\}$ can be generated as limiting
interpolation involving the classical Triebel–Lizorkin spaces
$F^{s}_{p,q}(\mathbb{R}^{d})$. To be more precise, we establish the following

######  Theorem 4.8 (Characterization of
$T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ via limiting interpolation).

Let $A\in\\{B,F\\}$. Let
$p\in(1,\infty),q,q_{0},r\in(0,\infty],s,s_{0}\in\mathbb{R},s\neq s_{0}$, and
$b\in\mathbb{R}\backslash\\{0\\}$.

  1. (i)

If $s_{0}>s$ and $b>0$ then

(4.25)
$T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})=(F^{s}_{p,q}(\mathbb{R}^{d}),A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}))_{(0,b-1/r),r}.$

  2. (ii)

If $s_{0}<s$ and $b<0$ then

(4.26)
$T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})=(A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),F^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r}.$

The proof of Theorem 4.8 will rely on two interpolation lemmas, which are of
independent interest.

######  Lemma 4.9.

Let $(A_{0},A_{1})$ be a quasi-Banach pair and let $0<p<\infty$. Then

(4.27)
$K(t,f;L_{p}(\mathbb{R}^{d};A_{0}),L_{p}(\mathbb{R}^{d};A_{1}))\asymp\bigg{(}\int_{\mathbb{R}^{d}}K(t,f(x);A_{0},A_{1})^{p}\,dx\bigg{)}^{1/p}$

for $t>0$ and $f\in L_{p}(\mathbb{R}^{d};A_{0})+L_{p}(\mathbb{R}^{d};A_{1})$.

###### Proof.

This result was already stated without proof in [Per84, p. 218]. For the sake
of completeness, next we provide details of the proof.

Without loss of generality, we may assume that
$f(x)=\sum_{j=1}^{N}a_{j}\chi_{\Omega_{j}}$ where $a_{j}\in A_{0}\cap A_{1}$,
the $\Omega_{j}$’s are measurable sets in $\mathbb{R}^{d}$ of finite measure
with $\Omega_{i}\cap\Omega_{j}=\emptyset$ if $i\neq j$ and $\chi_{\Omega_{j}}$
are the related characteristic functions. This assumption allows us to avoid
unnecessary complications related to measurability. Then

$\displaystyle
K(t,f;L_{p}(\mathbb{R}^{d};A_{0}),L_{p}(\mathbb{R}^{d};A_{1}))^{p}$
$\displaystyle\asymp\inf_{f=f_{0}+f_{1}}(\|f_{0}\|_{L_{p}(\mathbb{R}^{d};A_{0})}^{p}+t^{p}\|f_{1}\|^{p}_{L_{p}(\mathbb{R}^{d};A_{1})})$
$\displaystyle=\inf_{f=f_{0}+f_{1}}\int_{\mathbb{R}^{d}}(\|f_{0}(x)\|_{A_{0}}^{p}+t^{p}\|f_{1}(x)\|_{A_{1}}^{p})\,dx.$

Assume $f=f_{0}+f_{1}$ with $f_{i}\in L_{p}(\mathbb{R}^{d};A_{i}),\,i=0,1$. In
particular, $f(x)=f_{0}(x)+f_{1}(x)$ which implies

$\displaystyle K(t,f(x);A_{0},A_{1})^{p}$
$\displaystyle\asymp\inf_{f(x)=a_{0}+a_{1}}(\|a_{0}\|_{A_{0}}^{p}+t^{p}\|a_{1}\|_{A_{1}}^{p})$
$\displaystyle\leq\|f_{0}(x)\|_{A_{0}}^{p}+t^{p}\|f_{1}(x)\|_{A_{1}}^{p}$

and integrating this inequality

$\int_{\mathbb{R}^{d}}K(t,f(x);A_{0},A_{1})^{p}\,dx\lesssim\int_{\mathbb{R}^{d}}(\|f_{0}(x)\|_{A_{0}}^{p}+t^{p}\|f_{1}(x)\|_{A_{1}}^{p})\,dx.$

Taking now the infimum over all possible decompositions $f=f_{0}+f_{1}$, we
arrive at

$\int_{\mathbb{R}^{d}}K(t,f(x);A_{0},A_{1})^{p}\,dx\lesssim
K(t,f;L_{p}(\mathbb{R}^{d};A_{0}),L_{p}(\mathbb{R}^{d};A_{1}))^{p}.$

Conversely, given $x\in\mathbb{R}^{d}$, consider any decomposition
$f(x)=f_{0}(x)+f_{1}(x)$ with $f_{i}(x)\in A_{i},\,i=0,1$, such that

$\|f_{0}(x)\|_{A_{0}}^{p}+t^{p}\|f_{1}(x)\|_{A_{1}}^{p}\lesssim
K(t,f(x);A_{0},A_{1})^{p}.$

Integrating over all $x\in\mathbb{R}^{d}$, we derive

$\int_{\mathbb{R}^{d}}(\|f_{0}(x)\|_{A_{0}}^{p}+t^{p}\|f_{1}(x)\|_{A_{1}}^{p})\,dx\lesssim\int_{\mathbb{R}^{d}}K(t,f(x);A_{0},A_{1})^{p}\,dx.$

In particular,

$K(t,f;L_{p}(\mathbb{R}^{d};A_{0}),L_{p}(\mathbb{R}^{d};A_{1}))^{p}\lesssim\int_{\mathbb{R}^{d}}K(t,f(x);A_{0},A_{1})^{p}\,dx.$

∎

######  Lemma 4.10.

Let $0<p<\infty$ and $0<q,r\leq\infty$.

  1. (i)

Assume $-\infty<s<s_{0}<\infty$ and $b>0$. Then

$\|f\|_{(L_{p}(\mathbb{R}^{d};\ell^{s}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q}))_{(0,b-1/r),r}}\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=2^{j}}^{2^{j+1}-1}[2^{\nu
s}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}$

(with the corresponding modifications if $q=\infty$ and/or $r=\infty$).

  2. (ii)

Assume $-\infty<s_{0}<s<\infty$ and $b<0$. Then

$\|f\|_{(L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s}_{q}))_{(1,b-1/r),r}}\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=2^{j}}^{2^{j+1}-1}[2^{\nu
s}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}$

(with the corresponding modifications if $q=\infty$ and/or $r=\infty$).

###### Proof.

(i): Specializing (4.27) with $(A_{0},A_{1})=(\ell^{s}_{q},\ell^{s_{0}}_{q})$
and using (cf. (4.11))

(4.28)
$K(t,x;\ell^{s}_{q},\ell^{s_{0}}_{q})\asymp\bigg{(}\sum_{\nu=0}^{\infty}[\min(2^{\nu
s},t2^{\nu s_{0}})|x_{\nu}|]^{q}\bigg{)}^{1/q},$

we derive

(4.29)
$K(t,f;L_{p}(\mathbb{R}^{d};\ell^{s}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q}))\asymp\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=0}^{\infty}[\min(2^{\nu
s},t2^{\nu s_{0}})|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{1/p}.$

Inserting this formula into the definition of limiting interpolation space
(4.6),

$\displaystyle\|f\|_{(L_{p}(\mathbb{R}^{d};\ell^{s}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q}))_{(0,b-1/r),r}}$
$\displaystyle\asymp$
$\displaystyle\bigg{(}\sum_{j=0}^{\infty}[(1+j)^{b-1/r}K(2^{-j(s_{0}-s)},f;L_{p}(\mathbb{R}^{d};\ell^{s}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q})]^{r}\bigg{)}^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=0}^{\infty}[2^{\nu
s}\min(1,2^{(\nu-j)(s_{0}-s)})|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}2^{j(s-s_{0})r}(1+j)^{(b-1/r)r}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=0}^{j}[2^{\nu
s_{0}}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}$
$\displaystyle+\bigg{(}\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=j}^{\infty}[2^{\nu
s}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}$
$\displaystyle=:I+II.$

Next we prove

(4.30) $I\lesssim II,$

which immediately leads to

(4.31)
$\|f\|_{(L_{p}(\mathbb{R}^{d};\ell^{s}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q}))_{(0,b-1/r),r}}\asymp
II.$

To proceed with (4.30), we consider two cases. Firstly, we assume $q\geq p$.
Then, for each $x\in\mathbb{R}^{d}$ and $j\in\mathbb{N}_{0}$,

$\bigg{(}\sum_{\nu=0}^{j}[2^{\nu
s_{0}}|f_{\nu}(x)|]^{q}\bigg{)}^{1/q}\leq\bigg{(}\sum_{\nu=0}^{j}[2^{\nu
s_{0}}|f_{\nu}(x)|]^{p}\bigg{)}^{1/p}$

which yields, by Hardy’s inequality (B.3) (note that $s_{0}>s$),

$\displaystyle I$
$\displaystyle\leq\bigg{(}\sum_{j=0}^{\infty}2^{j(s-s_{0})r}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=0}^{j}2^{\nu
s_{0}p}\|f_{\nu}\|_{L_{p}(\mathbb{R}^{d})}^{p}\bigg{)}^{r/p}\bigg{)}^{1/r}$
$\displaystyle\lesssim\bigg{(}\sum_{j=0}^{\infty}2^{jsr}(1+j)^{(b-1/r)r}\|f_{j}\|_{L_{p}(\mathbb{R}^{d})}^{r}\bigg{)}^{1/r}\leq
II.$

Secondly, if $q<p$ we can apply Minkowski’s inequality to get

$\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=0}^{j}[2^{\nu
s_{0}}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{1/p}\leq\bigg{(}\sum_{\nu=0}^{j}2^{\nu
s_{0}q}\|f_{\nu}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{1/q}.$

Accordingly, by Hardy’s inequality (B.3),

$\displaystyle I$
$\displaystyle\leq\bigg{(}\sum_{j=0}^{\infty}2^{j(s-s_{0})r}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=0}^{j}2^{\nu
s_{0}q}\|f_{\nu}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}$
$\displaystyle\lesssim\bigg{(}\sum_{j=0}^{\infty}2^{jsr}(1+j)^{(b-1/r)r}\|f_{j}\|^{r}_{L_{p}(\mathbb{R}^{d})}\bigg{)}^{1/r}\leq
II.$

This completes the proof of (4.30).

Furthermore, we make the following claim

(4.32)
$II\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=2^{j}}^{2^{j+1}-1}[2^{\nu
s}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}.$

Indeed, by basic monotonicity properties,

$II\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=2^{j}}^{\infty}[2^{\nu
s}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}$

and thus the estimate $\gtrsim$ in (4.32) is clear. Conversely, one can
rewrite the previous estimate as

$II\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{l=j}^{\infty}F_{l}(x)^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}$

where

$F_{l}(x):=\bigg{(}\sum_{\nu=2^{l}}^{2^{l+1}-1}[2^{\nu
s}|f_{\nu}(x)|]^{q}\bigg{)}^{1/q},\qquad l\in\mathbb{N}_{0},\qquad
x\in\mathbb{R}^{d}.$

If $q\geq p$ then the embedding $\ell_{p}\hookrightarrow\ell_{q}$ and Hardy’s
inequality (B.4) (note that $b>0$) imply

$\displaystyle II$
$\displaystyle\lesssim\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{l=j}^{\infty}\|F_{l}\|_{L_{p}(\mathbb{R}^{d})}^{p}\bigg{)}^{r/p}\bigg{)}^{1/r}$
$\displaystyle\lesssim\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\|F_{l}\|_{L_{p}(\mathbb{R}^{d})}^{r}\bigg{)}^{1/r}$

which gives (4.32). On the other hand, if $q<p$ we can apply Minkowski’s
inequality together with Hardy’s inequality (B.4) to estimate

$\displaystyle II$
$\displaystyle\lesssim\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{l=j}^{\infty}\|F_{l}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}$
$\displaystyle\lesssim\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\|F_{j}\|_{L_{p}(\mathbb{R}^{d})}^{r}\bigg{)}^{1/r}.$

This finishes the proof of (4.32).

Putting together (4.31) and (4.32),

$\|f\|_{(L_{p}(\mathbb{R}^{d};\ell^{s}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q}))_{(0,b-1/r),r}}\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=2^{j}}^{2^{j+1}-1}[2^{\nu
s}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}.$

(ii): It follows from (4.29) (after replacing the roles played by $s$ and
$s_{0}$) that

$\displaystyle\|f\|_{(L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s}_{q}))_{(1,b-1/r),r}}$
$\displaystyle\asymp$
$\displaystyle\bigg{(}\sum_{j=0}^{\infty}[2^{j(s-s_{0})}(1+j)^{b-1/r}K(2^{-j(s-s_{0})},f;L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s}_{q})]^{r}\bigg{)}^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}2^{j(s-s_{0})r}(1+j)^{(b-1/r)r}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=0}^{\infty}[2^{\nu
s_{0}}\min(1,2^{(\nu-j)(s-s_{0})})|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}(1+j)^{(b-1/r)r}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=0}^{j}[2^{\nu
s}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}$
$\displaystyle+\bigg{(}\sum_{j=0}^{\infty}2^{j(s-s_{0})r}(1+j)^{(b-1/r)r}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=j}^{\infty}[2^{\nu
s_{0}}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}$
$\displaystyle=:\mathcal{I}+\mathcal{II}.$

Furthermore, we claim

$\mathcal{II}\lesssim\mathcal{I}$

and

$\mathcal{I}\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=2^{j}}^{2^{j+1}-1}[2^{\nu
s}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}.$

We omit the proofs of these estimates since they are obvious modifications of
those in (4.30) and (4.32), respectively. Hence

$\|f\|_{(L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s}_{q}))_{(1,b-1/r),r}}\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\int_{\mathbb{R}^{d}}\bigg{(}\sum_{\nu=2^{j}}^{2^{j+1}-1}[2^{\nu
s}|f_{\nu}(x)|]^{q}\bigg{)}^{p/q}\,dx\bigg{)}^{r/p}\bigg{)}^{1/r}.$

∎

We are now ready to give the

###### Proof of Theorem 4.8.

(i): We start by proving (4.25) with $A=F$ and $q_{0}=q$, i.e.

(4.33)
$T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})=(F^{s}_{p,q}(\mathbb{R}^{d}),F^{s_{0}}_{p,q}(\mathbb{R}^{d}))_{(0,b-1/r),r}.$

Since $F^{s}_{p,q}(\mathbb{R}^{d})$ is a retract of
$L_{p}(\mathbb{R}^{d};\ell^{s}_{q})$, the interpolation space
$(F^{s}_{p,q}(\mathbb{R}^{d}),F^{s_{0}}_{p,q}(\mathbb{R}^{d}))_{(0,b-1/r),r}$
can be identified, via the retraction method, with
$(L_{p}(\mathbb{R}^{d};\ell^{s}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q}))_{(0,b-1/r),r}$.
This space was computed in Lemma 4.10(i). Accordingly, the desired formula
(4.33) follows.

The formula (4.25) for $A=B$ and $0<q_{0}\leq\infty$ follows from the previous
case (i.e., (4.33)) and the well-known interpolation formula (cf. [Tri83,
Theorem 2.4.2, p. 64])

(4.34)
$(F^{s}_{p,q}(\mathbb{R}^{d}),F^{s_{0}}_{p,q}(\mathbb{R}^{d}))_{\theta,q_{0}}=B^{(1-\theta)s+\theta
s_{0}}_{p,q_{0}}(\mathbb{R}^{d})$

for $\theta\in(0,1)$. More precisely, given $s_{0}>s$ we choose $s_{1}>s_{0}$
and $\theta\in(0,1)$ such that $s_{0}=(1-\theta)s+\theta s_{1}$. Then, by
(4.34), Lemma 4.1(i), and (4.33),

$\displaystyle(F^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}))_{(0,b-1/r),r}$
$\displaystyle=(F^{s}_{p,q}(\mathbb{R}^{d}),(F^{s}_{p,q}(\mathbb{R}^{d}),F^{s_{1}}_{p,q}(\mathbb{R}^{d}))_{\theta,q_{0}})_{(0,b-1/r),r}$
$\displaystyle=(F^{s}_{p,q}(\mathbb{R}^{d}),F^{s_{1}}_{p,q}(\mathbb{R}^{d}))_{(0,b-1/r),r}=T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d}).$

It remains to show (4.25) with $A=F$ and $0<q_{0}\leq\infty$ with $q_{0}\neq
q$. This is a simple consequence of the embeddings (4.20) and the
corresponding assertion with $A=B$.

(ii): The proof of (4.26) follows similar ideas as those given in (4.25).
Accordingly, we will only provide a sketch of the argument. We first
concentrate on the case $A=F$ with $q_{0}=q$ and we claim that

(4.35)
$T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})=(F^{s_{0}}_{p,q}(\mathbb{R}^{d}),F^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r}.$

Indeed, this is an immediate consequence of Lemma 4.10(ii) and the retraction
method. Once formula (4.35) has been established, its extension from
$F^{s_{0}}_{p,q}(\mathbb{R}^{d})$ to any $A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d})$
with $s_{0}<s$ and $0<q_{0}\leq\infty$ can be obtained via Lemma 4.1(ii) and
(4.20).

∎

######  Remark 4.11.

Similarly as in Remark 4.5, the proof of Theorem 4.8 can be applied for the
limiting case $b=0$. Namely, if
$p\in(1,\infty),q\in(0,\infty],r\in(0,\infty),-\infty<s<s_{0}<\infty$, then
(cf. (3.7))

(4.36)
$(F^{s}_{p,q}(\mathbb{R}^{d}),F^{s_{0}}_{p,q}(\mathbb{R}^{d}))_{(0,-1/r),r}=T^{*}_{r}F^{s}_{p,q}(\mathbb{R}^{d}).$

Recall that $T_{r}F^{s}_{p,q}(\mathbb{R}^{d})\neq
T^{*}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$; see Remark 3.7.

Next we write down two important special cases of Theorem 4.8. Namely

(4.37)
$T^{b+1/q}_{q}F^{0}_{p,2}(\mathbb{R}^{d})=\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})\qquad\text{and}\qquad
T^{b+1/q}_{q}F^{s}_{p,2}(\mathbb{R}^{d})=\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d}),$

cf. Proposition 3.8(iii) and (iv) for assumptions on the involved parameters.
Accordingly, the following result is an immediate application of Theorem 4.8.

######  Corollary 4.12 ($\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$ and
$\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})$ via limiting interpolation).

Let $A\in\\{B,F\\}$.

  1. (i)

Let $1<p<\infty,0<q,q_{1}\leq\infty,s>0$ and $b>-1/q$. Then

(4.38)
$\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})=(L_{p}(\mathbb{R}^{d}),A^{s}_{p,q_{1}}(\mathbb{R}^{d}))_{(0,b),q}.$

  2. (ii)

Let $1<p<\infty,0<q,q_{0}\leq\infty,s>0,s_{0}<s$ and $b<-1/q$. Then

(4.39)
$\emph{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})=(A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),H^{s}_{p}(\mathbb{R}^{d}))_{(1,b),q}.$

######  Remark 4.13.

The formula (4.38) was already obtained in [CDT16, Theorem 3.4] using the
well-known fact

$K(t^{k},f;L_{p}(\mathbb{R}^{d}),W^{k}_{p}(\mathbb{R}^{d}))\asymp
t^{k}\|f\|_{L_{p}(\mathbb{R}^{d})}+\omega_{k}(f,t)_{p},\qquad t\in(0,1),$

and reiteration formulae for limiting interpolation (cf. Lemma 4.1).
Furthermore this method also works with the limiting values $p=1,\infty$ in
(4.38).

On the other hand, (4.39) with
$A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d})=L_{p}(\mathbb{R}^{d})$ (i.e., $s_{0}=0$
and $q_{0}=2$) reads

(4.40)
$\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})=(L_{p}(\mathbb{R}^{d}),H^{s}_{p}(\mathbb{R}^{d}))_{(1,b),q};$

see [DHT20, (2.13)] for an alternative approach to (4.40) based on
$K$-functionals and (fractional) moduli of smoothness.

$(T^{b_{0}}_{r_{0}}A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}\tilde{A}^{s_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\theta,r}=B^{s,b}_{p,r}(\mathbb{R}^{d}),\qquad
s_{0}\neq s_{1},$

$(T^{b_{0}}_{r_{0}}F^{s}_{p,q}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}F^{s}_{p,q}(\mathbb{R}^{d}))_{\theta,r}=T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d}),\qquad
b_{0}\neq b_{1}$

###  4.5. $T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ spaces are closed under
interpolation.

As an application of Theorem 4.8, we establish the closedness of the scale of
spaces $T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ under classical real
interpolation. More precisely, we establish the following

######  Theorem 4.14.

Let $1<p<\infty,0<q,r,r_{0},r_{1}\leq\infty,-\infty<s,b_{0},b_{1}<\infty$ and
$0<\theta<1$. Assume further $b_{0}\neq b_{1}$ and $b_{0}b_{1}>0$ and let
$b=(1-\theta)b_{0}+\theta b_{1}$. Then

$(T^{b_{0}}_{r_{0}}F^{s}_{p,q}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}F^{s}_{p,q}(\mathbb{R}^{d}))_{\theta,r}=T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d}).$

###### Proof.

Assume e.g. that $b_{0},b_{1}<0$. It is not restrictive to assume further
$b_{0}<b_{1}$. Let $s_{0}<s$. By Theorem 4.8(ii),

$T^{b_{j}}_{r_{j}}F^{s}_{p,q}(\mathbb{R}^{d})=(F^{s_{0}}_{p,q}(\mathbb{R}^{d}),F^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b_{j}-1/r_{j}),r_{j}},\qquad
j=0,1.$

These formulae and Lemma 4.1(iv) imply

$\displaystyle(T^{b_{0}}_{r_{0}}F^{s}_{p,q}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}F^{s}_{p,q}(\mathbb{R}^{d}))_{\theta,r}$
$\displaystyle=$
$\displaystyle((F^{s_{0}}_{p,q}(\mathbb{R}^{d}),F^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b_{0}-1/r_{0}),r_{0}},(F^{s_{0}}_{p,q}(\mathbb{R}^{d}),F^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b_{1}-1/r_{1}),r_{1}})_{\theta,r}$
$\displaystyle=(F^{s_{0}}_{p,q}(\mathbb{R}^{d}),F^{s}_{p,q}(\mathbb{R}^{d}))_{(1,b-1/r),r}=T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$

where the last step follows from Theorem 4.8(ii).

The case $b_{0},b_{1}>0$ can be treated analogously, but now applying Theorem
4.8(i). ∎

######  Remark 4.15.

Theorems 4.8 and 4.14 can be extended to deal with $0<p<\infty$. We refer to
Remark 7.14 below.

######  Remark 4.16.

Theorems 4.2 and 4.7 as well as Theorems 4.8 and 4.14 show some striking
differences between classical interpolation (cf. (4.2) and (4.3)) and limiting
interpolation (cf. (4.6)). Let $A,\tilde{A}\in\\{B,F\\}$. The following holds

$(A^{s}_{p,q}(\mathbb{R}^{d}),\tilde{A}^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}))_{\theta,r,b}=B^{(1-\theta)s+\theta
s_{0},b}_{p,r}(\mathbb{R}^{d})$

provided that $s,s_{0},b\in\mathbb{R},\,s\neq
s_{0},\theta\in(0,1),0<p<\infty,0<q,q_{0},r\leq\infty$; cf. [Tri83, Section
2.4.2] if $b=0$ and [CF88] if $b\in\mathbb{R}$. In particular, this formula
tells us that any interpolation pair formed by Besov or Triebel–Lizorkin
spaces with fixed integrability parameter $p$ produces as an outcome the Besov
space $B^{(1-\theta)s+\theta s_{0},b}_{p,r}(\mathbb{R}^{d})$. Note that
classical interpolation does not depend on the secondary parameters $q$ and
$q_{0}$. Importantly, these assertions may fail to be true dealing with
limiting interpolation, as can be illustrated by

$\displaystyle T^{b+1/r}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$
$\displaystyle=(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}))_{(0,b),r}$
$\displaystyle\neq(F^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}))_{(0,b),r}=T^{b+1/r}_{r}F^{s}_{p,q}(\mathbb{R}^{d}),$

where $q\neq p$, and

$\displaystyle T^{b+1/r}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$
$\displaystyle=(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}))_{(0,b),r}$
$\displaystyle\neq(B^{s}_{p,\tilde{q}}(\mathbb{R}^{d}),B^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}))_{(0,b),r}=T^{b+1/r}_{r}B^{s}_{p,\tilde{q}}(\mathbb{R}^{d}),$

where $q\neq\tilde{q}$.

###  4.6. Interpolation in the case of non-fixed smoothness

Our next result extends the classical embedding

$(A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),\tilde{A}^{s_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\theta,r}=B^{s}_{p,r}(\mathbb{R}^{d}),\qquad
A,\tilde{A}\in\\{B,F\\},$

and, on the other hand, it complements Theorems 4.7 and 4.14 to pairs of
spaces
$(T^{b}_{r}A^{s_{0}}_{p,q}(\mathbb{R}^{d}),T^{b}_{r}A^{s_{1}}_{p,q}(\mathbb{R}^{d}))$
with $s_{0}\neq s_{1}$. In sharp contrast with the case $s_{0}=s_{1}$ studied
in these theorems, the resulting interpolation space with $s_{0}\neq s_{1}$ is
always a classical Besov space. In particular, we are now able to compute the
interpolation space relative to
$(\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d}),\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d}))$,
which seems to be unknown in the literature; see Corollary 4.19 below.

######  Theorem 4.17.

Let $A,\tilde{A}\in\\{B,F\\}$. Assume
$0<p,q_{0},q_{1},r,r_{0},r_{1}\leq\infty\,(p<\infty\text{ when }A=F\text{ or
}\tilde{A}=F),b_{0},b_{1}\in\mathbb{R}\backslash\\{0\\},s_{0},s_{1}\in\mathbb{R},s_{0}\neq
s_{1},0<\theta<1$. Let

$s=(1-\theta)s_{0}+\theta s_{1}\qquad\text{and}\qquad b=(1-\theta)b_{0}+\theta
b_{1}.$

Then

(4.41)
$(T^{b_{0}}_{r_{0}}A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}\tilde{A}^{s_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\theta,r}=B^{s,b}_{p,r}(\mathbb{R}^{d}).$

In particular, if $\frac{b_{0}}{b_{0}-b_{1}}\in(0,1)$ then

$(T^{b_{0}}_{r_{0}}A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}\tilde{A}^{s_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\frac{b_{0}}{b_{0}-b_{1}},r}=B^{s}_{p,r}(\mathbb{R}^{d}).$

######  Remark 4.18.

The outcome in (4.41) does not depend on $q_{0}$ and $q_{1}$. This is in
contrast with Theorems 4.7 and 4.14.

###### Proof of Theorem 4.17.

We will make use of embeddings between truncated and classical function spaces
given in Corollaries 10.5 and 11.7 below. More precisely, we have

$B^{s_{i},b_{i}}_{p,\min\\{q_{i},r_{i}\\}}(\mathbb{R}^{d})\hookrightarrow
T^{b_{i}}_{r_{i}}A^{s_{i}}_{p,q_{i}}(\mathbb{R}^{d})\hookrightarrow
B^{s_{i},b_{i}}_{p,\max\\{q_{i},r_{i}\\}}(\mathbb{R}^{d}),\qquad i=0,1.$

Therefore

$\displaystyle(T^{b_{0}}_{r_{0}}A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}\tilde{A}^{s_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\theta,r}$
$\displaystyle\hookrightarrow$
$\displaystyle(B^{s_{0},b_{0}}_{p,\max\\{q_{0},r_{0}\\}}(\mathbb{R}^{d}),B^{s_{1},b_{1}}_{p,\max\\{q_{1},r_{1}\\}}(\mathbb{R}^{d}))_{\theta,r}=B^{s,b}_{p,r}(\mathbb{R}^{d})$

where we have applied well-known interpolation properties of Besov spaces of
logarithmic smoothness in the last step (cf. [CF88, Theorem 5.3] and [DT18,
Lemma 2.3]). Analogously, one can prove

$B^{s,b}_{p,r}(\mathbb{R}^{d})\hookrightarrow(T^{b_{0}}_{r_{0}}A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d}),T^{b_{1}}_{r_{1}}\tilde{A}^{s_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\theta,r}.$

∎

######  Corollary 4.19 (Interpolation involving
$\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$ and
$\text{Lip}^{s,b}_{p,q}(\mathbb{R}^{d})$).

Let $0<\theta<1,1<p<\infty,s>0,0<q,q_{0},q_{1}\leq\infty,b_{0}>-1/q_{0}$ and
$b_{1}<-1/q_{1}$.

  1. (i)

(4.42)
$(\mathbf{B}^{0,b_{0}}_{p,q_{0}}(\mathbb{R}^{d}),\emph{Lip}^{s,b_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\theta,q}=B^{\theta
s,(1-\theta)(b_{0}+1/q_{0})+\theta(b_{1}+1/q_{1})}_{p,q}(\mathbb{R}^{d}).$

In particular, if
$\theta=\frac{b_{0}+\frac{1}{q_{0}}}{b_{0}+\frac{1}{q_{0}}-(b_{1}+\frac{1}{q_{1}})}$
then

$(\mathbf{B}^{0,b_{0}}_{p,q_{0}}(\mathbb{R}^{d}),\emph{Lip}^{s,b_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\theta,q}=B^{\theta
s}_{p,q}(\mathbb{R}^{d}).$

  2. (ii)

$(L_{p}(\mathbb{R}^{d}),\emph{Lip}^{s,b_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\theta,q}=B^{\theta
s,\theta(b_{1}+1/q_{1})}_{p,q}(\mathbb{R}^{d}).$

  3. (iii)

$(\mathbf{B}^{0,b_{0}}_{p,q_{0}}(\mathbb{R}^{d}),H^{s}_{p}(\mathbb{R}^{d}))_{\theta,q}=B^{\theta
s,(1-\theta)(b_{0}+1/q_{0})}_{p,q}(\mathbb{R}^{d}).$

###### Proof.

(i): According to (4.37), we have

$T^{b_{0}+1/r_{0}}_{r_{0}}F^{0}_{p,2}(\mathbb{R}^{d})=\mathbf{B}^{0,b_{0}}_{p,r_{0}}(\mathbb{R}^{d})\qquad\text{and}\qquad
T^{b_{1}+1/r_{1}}_{r_{1}}F^{s}_{p,2}(\mathbb{R}^{d})=\text{Lip}^{s,b_{1}}_{p,r_{1}}(\mathbb{R}^{d}).$

Then (4.42) follows from Theorem 4.17.

(ii): Combining (4.39) (see also (4.40)) and Lemma 4.1(vi), we get

$\displaystyle(L_{p}(\mathbb{R}^{d}),\text{Lip}^{s,b_{1}}_{p,q_{1}}(\mathbb{R}^{d}))_{\theta,q}$
$\displaystyle=(L_{p}(\mathbb{R}^{d}),(L_{p}(\mathbb{R}^{d}),H^{s}_{p}(\mathbb{R}^{d}))_{(1,b_{1}),q_{1}})_{\theta,q}$
$\displaystyle=(L_{p}(\mathbb{R}^{d}),H^{s}_{p}(\mathbb{R}^{d}))_{\theta,q,\theta(b_{1}+1/q_{1})}=B^{\theta
s,\theta(b_{1}+1/q_{1})}_{p,q}(\mathbb{R}^{d})$

where the last step makes use of well-known interpolation properties of
classical Besov and Triebel–Lizorkin spaces (see, e.g., [CF88, Theorem 5.3]
and [DT18, (2.37)]).

(iii): By (4.38) (with $A=F$ and $q_{1}=2$) and Lemma 4.1(v),

$\displaystyle(\mathbf{B}^{0,b_{0}}_{p,q_{0}}(\mathbb{R}^{d}),H^{s}_{p}(\mathbb{R}^{d}))_{\theta,q}$
$\displaystyle=((L_{p}(\mathbb{R}^{d}),H^{s}_{p}(\mathbb{R}^{d}))_{(0,b_{0}),q_{0}},H^{s}_{p}(\mathbb{R}^{d}))_{\theta,q}$
$\displaystyle=(L_{p}(\mathbb{R}^{d}),H^{s}_{p}(\mathbb{R}^{d}))_{\theta,q,(1-\theta)(b_{0}+1/q_{0})}=B^{\theta
s,(1-\theta)(b_{0}+1/q_{0})}_{p,q}(\mathbb{R}^{d}).$

∎

###  4.7. $\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ spaces via
interpolation

Recall that the operator $\mathfrak{J}$ is defined in (4.10). Since
$\mathfrak{J}$ is a retract from $B^{s}_{p,q}(\mathbb{R}^{d})$ to
$\ell^{s}_{q}(L_{p}(\mathbb{R}^{d}))$ and from $F^{s}_{p,q}(\mathbb{R}^{d})$
to $L_{p}(\mathbb{R}^{d};\ell^{s}_{q})$, Theorem 4.2 (see, e.g., (4.8) with
$A=B$) can be interpreted as

$f\in
T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})\iff\mathfrak{J}(f)\in(\ell^{s}_{q}(L_{p}(\mathbb{R}^{d})),\ell^{s_{0}}_{q_{0}}(L_{p}(\mathbb{R}^{d})))_{(0,b-1/r),r}$

and

$\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|\mathfrak{J}(f)\|_{(\ell^{s}_{q}(L_{p}(\mathbb{R}^{d})),\ell^{s_{0}}_{q_{0}}(L_{p}(\mathbb{R}^{d})))_{(0,b-1/r),r}}.$

In the same fashion, Theorem 4.8 (specifically, (4.25) with $A=F$) can be
rewritten as

$f\in
T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})\iff\mathfrak{J}(f)\in(L_{p}(\mathbb{R}^{d};\ell^{s}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q_{0}}))_{(0,b-1/r),r}$

and

$\|f\|_{T^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|\mathfrak{J}(f)\|_{(L_{p}(\mathbb{R}^{d};\ell^{s}_{q}),L_{p}(\mathbb{R}^{d};\ell^{s_{0}}_{q_{0}}))_{(0,b-1/r),r}}.$

Our next result shows that $\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$
can be generated in terms of
$L_{p}(\mathbb{R}^{d};(\ell^{s}_{q},\ell^{s_{0}}_{q_{0}})_{(0,b-1/r),r})$,
i.e., after changing the roles of $L^{p}(\mathbb{R}^{d})$ and
$((0,b-1/r),r)$-interpolation in the construction of truncated $F$-spaces
given above.

######  Theorem 4.20 (Characterization of
$\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})$ via limiting interpolation).

Let $p\in(0,\infty),q,q_{0},r\in(0,\infty],s,s_{0}\in\mathbb{R},s\neq s_{0}$,
and $b\in\mathbb{R}\backslash\\{0\\}$.

  1. (i)

If $s_{0}>s$ and $b>0$ then

$f\in\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})\quad\text{if and only
if}\quad\mathfrak{J}(f)\in
L^{p}(\mathbb{R}^{d};(\ell^{s}_{q},\ell^{s_{0}}_{q_{0}})_{(0,b-1/r),r}).$

Furthermore

$\|f\|_{\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|\mathfrak{J}(f)\|_{L^{p}(\mathbb{R}^{d};(\ell^{s}_{q},\ell^{s_{0}}_{q_{0}})_{(0,b-1/r),r})}.$

  2. (ii)

If $s_{0}<s$ and $b<0$ then

$f\in\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})\quad\text{if and only
if}\quad\mathfrak{J}(f)\in
L^{p}(\mathbb{R}^{d};(\ell^{s_{0}}_{q_{0}},\ell^{s}_{q})_{(1,b-1/r),r}).$

Furthermore

$\|f\|_{\mathfrak{T}^{b}_{r}F^{s}_{p,q}(\mathbb{R}^{d})}\asymp\|\mathfrak{J}(f)\|_{L^{p}(\mathbb{R}^{d};(\ell^{s_{0}}_{q_{0}},\ell^{s}_{q})_{(1,b-1/r),r})}.$

###### Proof.

The proof follows similar ideas as the proof of Theorem 4.2. More precisely,
it was already shown in (4.15) and (4.17) (modulo replacing
$\|x_{\nu}\|_{L_{p}(\mathbb{R}^{d})}$ by $|x_{\nu}|$) that

(4.43)
$\|x\|_{(\ell^{s}_{q},\ell^{s_{0}}_{q})_{(0,b-1/r),r}}\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}|x_{\nu}|^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}$

provided that $s_{0}>s$ and $b>0$. This proves (i) with $q_{0}=q$. The general
case $q_{0}\in(0,\infty]$ follows from reiteration formulas. Indeed, we make
use of the well-known fact that (cf. [BL76, Theorem 5.6.1] and [Tri78,
1.18.2])

$\ell^{s_{0}}_{q_{0}}=(\ell^{s}_{q},\ell^{s_{1}}_{q})_{\theta,q_{0}},$

where $s<s_{0}<s_{1}$ and $\theta\in(0,1)$ such that $s_{0}=(1-\theta)s+\theta
s_{1}$.

∎

##  5\. Characterizations of truncated Besov spaces in terms of approximation

The goal of this section is to establish characterizations of the spaces
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ via approximation procedures. We
restrict our attention to approximation procedures in $L_{p}(\mathbb{R}^{d})$
with $p\in[1,\infty]$ (and thus smoothness parameter $s>0$). However, many of
the results presented in this section admit analogues in
$L_{p}(\mathbb{R}^{d})$ with $p>0$ (and so $s>d\max\\{\frac{1}{p}-1,0\\}$).

Let us recall two well-known characterization of Besov spaces (see, e.g.,
[Pi81], [Tri83, Section 2.5.3]). Namely, if
$p\in[1,\infty],q\in(0,\infty],s>0$ and $b\in\mathbb{R}$ then

(5.1)
$\|f\|_{B^{s,b}_{p,q}(\mathbb{R}^{d})}\asymp\bigg{(}\sum_{j=0}^{\infty}[2^{js}(1+j)^{b}E_{2^{j}-1}(f)_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{1/q}$

and if, additionally, $\alpha>0$ and $r\in(0,\infty]$ then

(5.2)
$\|f\|_{B^{s+\alpha,b}_{p,r}(\mathbb{R}^{d})}\asymp\bigg{(}\sum_{j=0}^{\infty}[2^{j\alpha}(1+j)^{b}E_{2^{j}-1}(f)_{B^{s}_{p,q}(\mathbb{R}^{d})}]^{r}\bigg{)}^{1/r}.$

Here, for $X=L_{p}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d})$, we use the
notation

$E_{j}(f)_{X}:=\inf_{g}\|f-g\|_{X},\qquad j\geq 1,$

where the infimum is taken over all entire functions $g$ of exponential type
$j$ and $E_{0}(f)_{X}:=\|f\|_{X}$. The limiting case $s=0$ in (5.1)
characterizes $\mathbf{B}^{0,b}_{p,q}(\mathbb{R}^{d})$ (cf. [DT18, Section
10.4]). Our next result deals with the case $\alpha=0$ in (5.2) by showing the
connection with the spaces $T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$.

Let $\mu_{j}:=2^{2^{j}}$ for $j\geq 1$ and $\mu_{0}=0$.

######  Theorem 5.1.

Let $p\in[1,\infty],q,r\in(0,\infty]$ and $s>0$.

  1. (i)

If $b>0$ then

(5.3)
$\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\bigg{(}\sum_{j=0}^{\infty}[2^{jb}E_{\mu_{j}}(f)_{B^{s}_{p,q}(\mathbb{R}^{d})}]^{r}\bigg{)}^{1/r}.$

  2. (ii)

If $b\in\mathbb{R}\backslash\\{0\\}$ then

(5.4)
$\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}(2^{\nu
s}E_{2^{\nu}-1}(f)_{L_{p}(\mathbb{R}^{d})})^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}.$

######  Remark 5.2.

Under the assumptions of Theorem 5.1(ii), letting $q=r$ in (5.4), we have

$\|f\|_{T^{b}_{q}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\bigg{(}\sum_{\nu=0}^{\infty}(2^{\nu
s}(1+\nu)^{b}E_{2^{\nu}-1}(f)_{L_{p}(\mathbb{R}^{d})})^{q}\bigg{)}^{1/q}\asymp\|f\|_{B^{s,b}_{p,q}(\mathbb{R}^{d})},$

where the last estimate follows from (5.1). Compare with Proposition 3.8.

###### Proof of Theorem 5.1.

(i): Let $u>0$. In virtue of Theorem 4.2(i), we have

(5.5)
$T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})=(B^{s}_{p,q}(\mathbb{R}^{d}),B^{s+1/u}_{p,u}(\mathbb{R}^{d}))_{(0,b-1/r),r}.$

Making use of the known estimate (see [Nil82, Proposition 6.3])

$K(2^{j/u},f;B^{s+1/u}_{p,u}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))\asymp\bigg{(}\sum_{\nu=0}^{j}2^{\nu}E_{2^{\nu}-1}(f)_{B^{s}_{p,q}(\mathbb{R}^{d})}^{u}\bigg{)}^{1/u},\qquad
j\in\mathbb{N}_{0},$

we can rewrite (5.5) as follows

$\displaystyle\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}[(1+j)^{b-1/r}K(2^{-j/u},f;B^{s}_{p,q}(\mathbb{R}^{d}),B^{s+1/u}_{p,u}(\mathbb{R}^{d}))]^{r}\bigg{)}^{1/r}$
$\displaystyle\asymp\left(\sum_{j=0}^{\infty}2^{-jr/u}(1+j)^{(b-1/r)r}\bigg{(}\sum_{\nu=0}^{j}2^{\nu}E_{2^{\nu}-1}(f)_{B^{s}_{p,q}(\mathbb{R}^{d})}^{u}\bigg{)}^{r/u}\right)^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}[(1+j)^{b-1/r}E_{2^{j}-1}(f)_{B^{s}_{p,q}(\mathbb{R}^{d})}]^{r}\bigg{)}^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}[2^{jb}E_{\mu_{j}}(f)_{B^{s}_{p,q}(\mathbb{R}^{d})}]^{r}\bigg{)}^{1/r},$

where we have applied Hardy’s inequality (B.3) in the penultimate step. The
proof of (5.3) is complete.

(ii): The equivalence (5.4) with $b>0$ follows from (5.3) and the fact that

$E_{j-1}(f)_{B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\bigg{(}\sum_{\nu=j}^{\infty}(\nu^{s}E_{\nu-1}(f)_{L_{p}(\mathbb{R}^{d})})^{q}\frac{1}{\nu}\bigg{)}^{1/q},\qquad
j\in\mathbb{N}_{0}$

(cf. [Nil82, Proposition 6.3]). To be more precise, by Hardy’s inequality
(B.4) (note that $b>0$) and basic monotonicity properties, we have

$\displaystyle\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}[2^{jb}E_{\mu_{j}}(f)_{B^{s}_{p,q}(\mathbb{R}^{d})}]^{r}\bigg{)}^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=\mu_{j}}^{\infty}(\nu^{s}E_{\nu-1}(f)_{L_{p}(\mathbb{R}^{d})})^{q}\frac{1}{\nu}\bigg{)}^{r/q}\bigg{)}^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=\mu_{j}}^{\mu_{j+1}-1}(\nu^{s}E_{\nu-1}(f)_{L_{p}(\mathbb{R}^{d})})^{q}\frac{1}{\nu}\bigg{)}^{r/q}\bigg{)}^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}E_{2^{\nu}-1}(f)_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}.$

Next we deal with (5.4) under $b<0$. Since (cf. [Nil82, p. 322])

$K(2^{js},f;L_{p}(\mathbb{R}^{d}),B^{s}_{p,q}(\mathbb{R}^{d}))\asymp\bigg{(}\sum_{\nu=0}^{j}2^{\nu
sq}E_{2^{\nu}-1}(f)_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{1/q},\qquad
j\in\mathbb{N}_{0},$

it follows from Theorem 4.2(ii) (with
$A^{s_{0}}_{p,q_{0}}(\mathbb{R}^{d})=F^{0}_{p,2}(\mathbb{R}^{d})=L_{p}(\mathbb{R}^{d})$
if $p\in(1,\infty)$) and (4.24) (with $k=0$ if $p=1,\infty$) that

$\displaystyle\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}[(1+j)^{b-1/r}K(2^{js},f;B^{s}_{p,q}(\mathbb{R}^{d}),L_{p}(\mathbb{R}^{d}))]^{r}\bigg{)}^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}(1+j)^{br}\bigg{(}\sum_{\nu=0}^{j}2^{\nu
sq}E_{2^{\nu}-1}(f)_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\frac{1}{1+j}\bigg{)}^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=0}^{2^{j}}2^{\nu
sq}E_{2^{\nu}-1}(f)_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}$
$\displaystyle\asymp\bigg{(}\sum_{j=0}^{\infty}2^{jbr}\bigg{(}\sum_{\nu=2^{j}-1}^{2^{j+1}-2}2^{\nu
sq}E_{2^{\nu}-1}(f)_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}$

where we have applied Hardy’s inequality (B.3) (taking into account that
$b<0$) in the last estimate. ∎

######  Remark 5.3.

The above proof also deals with the limiting case $b=0$ in (5.3) and (5.4).
More precisely, if $s>0,1\leq p\leq\infty,0<q\leq\infty,$ and $0<r<\infty$
then (cf. (3.5))

$\|f\|_{T^{*}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\bigg{(}\sum_{j=0}^{\infty}[E_{\mu_{j}}(f)_{B^{s}_{p,q}(\mathbb{R}^{d})}]^{r}\bigg{)}^{1/r}$

and

(5.6)
$\|f\|_{T^{*}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\bigg{\\{}\sum_{j=0}^{\infty}\bigg{[}\sum_{\nu=2^{j}-1}^{\infty}(2^{\nu
s}E_{2^{\nu}-1}(f)_{L_{p}(\mathbb{R}^{d})})^{q}\bigg{]}^{r/q}\bigg{\\}}^{1/r}.$

Furthermore, if $b=0$ and $r=\infty$ then we have the trivial estimate (cf.
(3.8))

$\|f\|_{B^{s}_{p,q}(\mathbb{R}^{d})}=\sup_{j\in\mathbb{N}_{0}}E_{\mu_{j}}(f)_{B^{s}_{p,q}(\mathbb{R}^{d})}=\|f\|_{T^{*}_{\infty}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\sup_{j\in\mathbb{N}_{0}}\bigg{(}\sum_{\nu=2^{j}-1}^{\infty}[2^{\nu
s}E_{2^{\nu}-1}(f)_{L_{p}(\mathbb{R}^{d})}]^{q}\bigg{)}^{1/q}.$

For $p\in[1,\infty]$ we define the class

$\mathfrak{U}_{p}:=\\{a=(a_{k})_{k\in\mathbb{N}_{0}}:a_{k}\in
L_{p}(\mathbb{R}^{d}),\quad\text{supp
}\widehat{a}_{k}\subset\\{x\in\mathbb{R}^{d}:|x|\leq 2^{k}\\}\\},$

which is formed by _functions of exponential type_ in $L_{p}(\mathbb{R}^{d})$.
Let $0<q\leq\infty$ and $s>0$. A classical representation theorem in Besov
spaces asserts (see, e.g., [Tri83, Section 2.5.3]) that $f\in
B^{s}_{p,q}(\mathbb{R}^{d})$ if and only if $f$ can be decomposed as

(5.7)
$f=\sum_{k=0}^{\infty}f_{k}\quad\text{with}\quad(f_{k})_{k\in\mathbb{N}_{0}}\in\mathfrak{U}_{p}$

(convergence being in $L_{p}(\mathbb{R}^{d})$) such that

(5.8) $\sum_{k=0}^{\infty}2^{ksq}\|f_{k}\|_{L_{p}(\mathbb{R}^{d})}^{q}<\infty$

(with the usual modification if $q=\infty$). Furthermore

(5.9)
$\|f\|_{B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\inf\left(\sum_{k=0}^{\infty}2^{ksq}\|f_{k}\|_{L_{p}(\mathbb{R}^{d})}^{q}\right)^{\frac{1}{q}},$

where the infimum runs over all possible decompositions (5.7) satisfying
(5.8).

######  Theorem 5.4.

Let $p\in[1,\infty],q,r\in(0,\infty],s>0$ and
$b\in\mathbb{R}\backslash\\{0\\}$. Then $f\in
T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ if and only if $f$ admits decomposition
(5.7) such that

(5.10)
$\sum_{k=0}^{\infty}2^{kbr}\bigg{(}\sum_{\nu=2^{k}-1}^{2^{k+1}-2}(2^{\nu
s}\|f_{\nu}\|_{L_{p}(\mathbb{R}^{d})})^{q}\bigg{)}^{\frac{r}{q}}<\infty$

(with the usual modification if $q=\infty$ and/or $r=\infty$). Furthermore

(5.11)
$\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\inf\left(\sum_{k=0}^{\infty}2^{kbr}\bigg{(}\sum_{\nu=2^{k}-1}^{2^{k+1}-2}(2^{\nu
s}\|f_{\nu}\|_{L_{p}(\mathbb{R}^{d})})^{q}\bigg{)}^{\frac{r}{q}}\right)^{\frac{1}{r}},$

where the infimum runs over all possible decompositions (5.7) satisfying
(5.10).

###### Proof.

Assume $f\in T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$. Let
$(g_{k})_{k\in\mathbb{N}_{0}}\in\mathfrak{U}_{p}$ be such that

$\|f-g_{k}\|_{L_{p}(\mathbb{R}^{d})}\lesssim
E_{2^{k}}(f)_{L_{p}(\mathbb{R}^{d})},\qquad k\in\mathbb{N}_{0}.$

Consequently $f=\lim_{k\to\infty}g_{k}$. Define

$f_{0}=g_{0}\quad\text{and}\quad f_{k}=g_{k}-g_{k-1},\quad k\geq 1.$

Clearly $f=\sum_{k=0}^{\infty}f_{k}$ with
$(f_{k})_{k\in\mathbb{N}_{0}}\in\mathfrak{U}_{p}$. Furthermore

$\|f_{0}\|_{L_{p}(\mathbb{R}^{d})}\lesssim\|f\|_{L_{p}(\mathbb{R}^{d})}\quad\text{and}\quad\|f_{k}\|_{L_{p}(\mathbb{R}^{d})}\lesssim
E_{2^{k-1}}(f)_{L_{p}(\mathbb{R}^{d})},\quad k\geq 1.$

Therefore, by (5.4),

$\displaystyle\sum_{k=0}^{\infty}2^{kbr}\bigg{(}\sum_{\nu=2^{k}-1}^{2^{k+1}-2}(2^{\nu
s}\|f_{\nu}\|_{L_{p}(\mathbb{R}^{d})})^{q}\bigg{)}^{\frac{r}{q}}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}2^{kbr}\bigg{(}\sum_{\nu=2^{k}-1}^{2^{k+1}-2}(2^{\nu
s}E_{2^{\nu}-1}(f)_{L_{p}(\mathbb{R}^{d})})^{q}\bigg{)}^{\frac{r}{q}}$
$\displaystyle\asymp\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}<\infty.$

Conversely, assume $f$ can be decomposed as (5.7) with (5.10). Given
$k\in\mathbb{N}_{0}$, we have

$E_{2^{k}}(f)_{L_{p}(\mathbb{R}^{d})}\lesssim\sum_{\nu=k}^{\infty}\|f_{\nu}\|_{L_{p}(\mathbb{R}^{d})}.$

Inserting this estimate into (5.4) and applying Hardy’s inequality (B.4), we
get

$\displaystyle\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\bigg{(}\sum_{k=0}^{\infty}2^{kbr}\bigg{(}\sum_{\nu=2^{k}-1}^{2^{k+1}-2}2^{\nu
sq}\bigg{(}\sum_{l=\nu}^{\infty}\|f_{l}\|_{L_{p}(\mathbb{R}^{d})}\bigg{)}^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}$
$\displaystyle\lesssim\bigg{(}\sum_{k=0}^{\infty}2^{kbr}\bigg{(}\sum_{\nu=2^{k}}^{2^{k+1}-1}2^{\nu
sq}\|f_{\nu}\|_{L_{p}(\mathbb{R}^{d})}^{q}\bigg{)}^{r/q}\bigg{)}^{1/r}.$

Taking now the infimum over all possible decompositions (5.7) with (5.10), we
achieve the desired result. ∎

######  Remark 5.5.

(i) Assume $b>0$. Under the assumptions of Theorem 5.4, the condition (5.10)
implies automatically that the series $\sum_{k=0}^{\infty}f_{k}$ converges
absolutely in $L_{p}(\mathbb{R}^{d})$. For instance, if $q,r\geq 1$ then we
can apply Hölder’s inequality twice so that

$\displaystyle\sum_{k=0}^{\infty}\|f_{k}\|_{L_{p}(\mathbb{R}^{d})}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{\nu=2^{k}-1}^{2^{k+1}-2}\|f_{\nu}\|_{L_{p}(\mathbb{R}^{d})}\lesssim\sum_{k=0}^{\infty}\bigg{(}\sum_{\nu=2^{k}-1}^{2^{k+1}-2}(2^{\nu
s}\|f_{\nu}\|_{L_{p}(\mathbb{R}^{d})})^{q}\bigg{)}^{\frac{1}{q}}$
$\displaystyle\lesssim\bigg{(}\sum_{k=0}^{\infty}2^{kbr}\bigg{(}\sum_{\nu=2^{k}-1}^{2^{k+1}-2}(2^{\nu
s}\|f_{\nu}\|_{L_{p}(\mathbb{R}^{d})})^{q}\bigg{)}^{\frac{r}{q}}\bigg{)}^{\frac{1}{r}}<\infty.$

The proof is even easier if $q<1$ or $r<1$.

(ii) Theorem 5.4 can be complemented by the following decomposition theorem in
terms of classical Besov spaces. Namely, let $p\in[1,\infty],q,r\in(0,\infty]$
and $s,b>0$, the function $f\in T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})$ if and
only if $f$ can be decomposed as

(5.12) $f=\sum_{k=0}^{\infty}f_{k}$

(convergence in $B^{s}_{p,q}(\mathbb{R}^{d})$) such that $\text{supp
}\widehat{f}_{k}\subset\big{\\{}x\in\mathbb{R}^{d}:|x|\leq\mu_{k}\big{\\}}$
and

(5.13)
$\sum_{k=0}^{\infty}2^{kbr}\|f_{k}\|_{B^{s}_{p,q}(\mathbb{R}^{d})}^{r}<\infty$

(with the usual modification if $r=\infty$). Furthermore

(5.14)
$\|f\|_{T^{b}_{r}B^{s}_{p,q}(\mathbb{R}^{d})}\asymp\inf\left(\sum_{k=0}^{\infty}2^{kbr}\|f_{k}\|_{B^{s}_{p,q}(\mathbb{R}^{d})}^{r}\right)^{\frac{1}{r}},$