# On the trace theorem to Volterra-type equations with local or non-local
derivatives

Jae-Hwan Choi  Department of Mathematical Sciences, Korea Advanced Institute
of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic
of Korea<EMAIL_ADDRESS>,  Jin Bong Lee  Research Institute of
Mathematics, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul 08826,
Republic of Korea<EMAIL_ADDRESS>,  Jinsol Seo  School of Mathematics,
Korea Institute for Advanced Study, 85 Hoegiro Dongdaemun-gu, Seoul 02455,
Republic of Korea<EMAIL_ADDRESS>and  Kwan Woo  Department of
Mathematical Sciences, Ulsan National Institute of Science and Technology,
Unist-gil 50, Eonyang-eup, Ulju-gun, Ulsan 44919, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

This paper considers traces at the initial time for solutions of evolution
equations with local or non-local derivatives in vector-valued $L_{p}$ spaces
with $A_{p}$ weights. To achieve this, we begin by introducing a generalized
real interpolation method. Within the framework of generalized interpolation
theory, we make use of stochastic process theory and two-weight Hardy’s
inequality to derive our trace and extension theorems. Our results encompass
findings applicable to time-fractional equations with broad temporal weight
functions.

###### Key words and phrases:

Volterra integral equations, Time fractional equations, Trace theorem, Initial
value problem, Generalized real interpolation, Weighted function spaces

###### 2020 Mathematics Subject Classification:

46E35, 45D05, 26A33, 46B70, 60G51

###### Contents

  1. 1 Introduction and Main results
    1. 1.1 Main results
  2. 2 Preliminaries
    1. 2.1 Functions of type $\mathcal{I}(a,b)$ and $\mathcal{I}_{o}(a,b)$ and their properties
    2. 2.2 Volterra-type equations associated with Bernstein functions
  3. 3 Generalized real interpolation
  4. 4 Proof of Main Results
    1. 4.1 Proof of Theorems 1.1 and 1.5: local derivative case
      1. 4.1.1 Proof of Theorem 1.1
      2. 4.1.2 Proof of Theorem 1.5
    2. 4.2 Proof of Theorems 1.2 and 1.6: non-local derivative case
      1. 4.2.1 Proof of Theorem 1.2
      2. 4.2.2 Proof of Theorem 1.6
  5. 5 Trace theorems for finite time interval
  6. A Examples of generalized real interpolation spaces
  7. B Extenstions from $(0,T)$ to $\mathbb{R}_{+}$

##  1\. Introduction and Main results

In this article, we are interested in the trace and extension theorem for the
following evolution equations in vector-valued $L_{p}$ spaces with $A_{p}$
weights:

$\displaystyle\partial_{t}u(t)=f(t),\quad u(0)=u_{0},$ (1.1)
$\displaystyle\partial_{t}^{\kappa}u(t)=f(t),\quad u(0)=u_{0}.$ (1.2)

Here, $\partial_{t}^{\kappa}$ stands for a (generalized) time-fractional
derivative with a kernel $\kappa$ given by

$\partial_{t}^{\kappa}u(t):=\partial_{t}\left(\int_{0}^{t}\kappa(t-s)(u(s)-u(0))~{}\mathrm{d}s\right).$

Note that $\partial_{t}^{\kappa}$ becomes the Caputo fractional derivative
$\partial_{t}^{\alpha}$ for $\kappa(t)=t^{-\alpha}/\Gamma(1-\alpha)$ with
$\alpha\in(0,1)$, where $\Gamma$ is the Gamma function.

Our trace theorems also hold for the following Volterra-type equations:

$\int_{0}^{t}(u(s)-u_{0})\mathrm{d}s=\int_{0}^{t}\varkappa(t-s)f(s)\mathrm{d}s.$
(1.3)

Under our assumptions on kernel $\kappa$, the evolution equation (1.2) can be
rewritten as the Volterra-type equation (1.3), where $\varkappa$ is a kernel
that is uniquely determined by $\kappa$. We discuss the equivalence at the end
of Section 2.2.

In the studies of trace theorems, interpolation theory is mainly used in the
literature. That is, one expects to obtain the following inequality:

$\displaystyle\|u_{0}\|_{(X_{0},X_{1})_{\theta,p}}\lesssim\|u\|_{L_{p}(\mathbb{R}_{+},t^{\gamma_{1}}\,\mathrm{d}t;X_{0})}+\|f\|_{L_{q}(\mathbb{R}_{+},t^{\gamma_{2}}\,\mathrm{d}t;X_{1})}.$
(1.4)

Note that $(X_{0},X_{1})_{\theta,p}$ is an interpolation space whose norm is
given for a functional $K$ by

$\displaystyle\|u_{0}\|_{(X_{0},X_{1})_{\theta,p}}:=\left(\int_{0}^{\infty}t^{-\theta
p}K\left(t,u_{0};X_{0},X_{1}\right)^{p}\,\frac{\mathrm{d}t}{t}\right)^{1/p},\quad\theta\in(0,1),\,\,p\in[1,\infty).$
(1.5)

Precise information for $K$ and other interpolation methods are given in
Section 3. The idea of trace inequalities in the literature is that one can
express the functional $K$ as certain quantities of $\|u\|_{X_{0}}$ and
$\|f\|_{X_{1}}$, and the remained term $t^{-\theta p}$ contributes to a time
weight in the right-hand side of (1.4). For detailed argument, we recommend
[1, 14, 25] and references therein.

From the perspective of the classical approaches, to obtain (1.4) in terms of
general temporal weights, it is natural to consider a generalization of
$(X_{0},X_{1})_{\theta,p}$-interpolation in the sense that we put
$\phi(t^{-1})$ rather than just $t^{-\theta}$. One of the novelties of this
paper is that we suggest a function class, $\mathcal{I}_{o}(a,b)$, introduced
in [12] for such $\phi$ as well as construct general interpolation space
$(X_{0},X_{1})_{\phi,p}$, whose norm is given by

$\displaystyle\|u_{0}\|_{(X_{0},X_{1})_{\phi,p}}:=\left(\int_{0}^{\infty}\phi\left(t^{-1}\right)^{p}K\left(t,u_{0};X_{0},X_{1}\right)^{p}\,\frac{\mathrm{d}t}{t}\right)^{1/p},\quad\phi\in\mathcal{I}_{o}(0,1).$
(1.6)

The definition of $\mathcal{I}_{o}(a,b)$ and its properties are given in the
first part of Section 2, and we explain the interpolation in Section 3. For
short introduction of $\mathcal{I}_{o}(a,b)$, we note that
$\phi\in\mathcal{I}_{o}(a,b)$ if and only if there exists $\varepsilon>0$ such
that

$\lambda^{a+\varepsilon}\lesssim\frac{\phi(\lambda
t)}{\phi(t)}\lesssim\lambda^{b-\varepsilon},\quad\forall t>0,\,\,\lambda\geq
1.$

The advantage of the general interpolation is that we can handle not only
trace inequalities for time local equations (1.1) but also time non-local
equations (1.2), including time-fractional derivatives for broader weight
classes than those in the literature. To the best of our knowledge, our result
is new even for the local derivative case, in the sense that trace and
extension theorems hold with Muckenhoupt’s $A_{p}$-weight classes.

Before introducing the main results of this paper, we give a brief overview of
the literature related to initial value problems in (weighted) Sobolev spaces
with or without non-local derivatives. The initial value problem of parabolic
equations in $L_{p}$-Sobolev spaces with local time derivatives has been
studied for a long time, and trace (and extension) theorem with or without
power-type temporal weights can be found in, for example, Weidemaier [26],
Prüss [22], Lindemulder and Veraar [17] and references therein. In the case of
the parabolic equations with non-local derivatives, Zacher [27, 28] obtained
an unweighted $L_{p}$-theory for the Volterra-type equations, including the
time-fractional heat equations. In particular, the kernel $\varkappa$ of (1.3)
considered in [27, 28] belongs to a certain class, and its Laplace transform
$\mathcal{L}[\varkappa]$ satisfies some growth conditions near zero and
infinity, in which case the solution space for (1.3) is given as the vector-
valued Bessel potential spaces $H_{p}^{\alpha}$ with $\alpha>1/p$. See [21]
for a detailed description of kernel $\kappa$ of $\partial_{t}^{\kappa}$ and
solution spaces used in [27, 28]. In Meyries and Schnaubelt [18], Meyries and
Veraar [19], Agresti, Lindemulder, and Veraar [1], the initial trace results
in [27, 28] were extended to the case of vector-valued Triebel-Lizorkin spaces
containing power-type temporal weight:

$\text{$F_{p,q}^{\alpha}(\mathbb{R}_{+},w_{\gamma};X_{0})\cap
L_{p}(\mathbb{R}_{+},w_{\gamma};X_{1})$, $w_{\gamma}(t)=|t|^{\gamma}$},$

where $\gamma\in(-1,p-1)$ and $\alpha>(1+\gamma)/p$. While the aforementioned
results are basically based on the semigroup approach, Dong and Kim [7], Dong
and Liu [8], and Kim and Woo [14] use different approaches to solve initial
value problems for time-fractional parabolic equations and derive initial
trace estimates for solutions. Particularly, the authors in [14] construct new
solution spaces suitable for discussing the initial value problem of time-
fractional evolution equations and investigate the initial behavior of the
solution using an appropriate representation via standard mollification. The
trace estimates in [14] are partially consistent with Theorem 1.2 in this
paper. (See Remark 1.7.) For a certain type of evolution equation with local
derivatives, Choi, Kim, and Lee [5] prove an extension theorem with general
temporal weights by solving an initial value problem. We emphasize that our
main results allow one to consider $A_{p}$ weights in time variable and
generalized time-fractional derivatives, which contain power-type temporal
weights and standard time-fractional derivatives, respectively.

###  1.1. Main results

The main results of this paper are trace and extension theorems that identify
the optimal function space for the initial conditions of equations in Sobolev
spaces with $A_{p}$ weights. Notably, the initial data spaces are closely
related to the temporal weights and the properties of local or non-local time
derivatives.

We introduce trace theorems with local and non-local derivatives,
respectively. Let $(X_{0},X_{1})$ be an interpolation couple and $X_{0}+X_{1}$
be a Banach space with the norm

$\|a\|_{X_{0}+X_{1}}:=\inf\\{\|a_{0}\|_{X_{0}}+\|a_{1}\|_{X_{1}}:a=a_{0}+a_{1},\,a_{0}\in
X_{0},\,a_{1}\in X_{1}\\}.$

######  Theorem 1.1 (Trace theorem with local derivative; half line).

Let $p\in(1,\infty)$ and $w\in A_{p}$. Suppose that $u\in
L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})$, $f\in
L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})$, $u_{0}\in X_{0}+X_{1}$, and

$u(t)=u_{0}+\int_{0}^{t}f(s)\mathrm{d}s$ (1.7)

for $t\in\mathbb{R}_{+}$. Then $u_{0}\in(X_{0},X_{1})_{W^{1/p},p}$ and

$\displaystyle\|u_{0}\|_{(X_{0},X_{1})_{W^{1/p},p}}\lesssim_{p,[w]_{A_{p}}}\|u\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})}+\|f\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})},$
(1.8)

where $W(t):=\int_{0}^{t}w(s)~{}\mathrm{d}s$.

For (1.2), assume that $\kappa:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a right-
continuous decreasing function with $\kappa\in\mathcal{I}_{o}(-1,0)$. Note
that $\mathcal{I}_{o}(-1,0)$ contains the kernel
$t^{-\alpha}/\Gamma(1-\alpha)$ generating the Caputo fractional derivative of
order $\alpha\in(0,1)$. Let us denote

$\kappa^{\ast}(t):=\kappa^{-1}\big{(}1/t\big{)}.$ (1.9)

Here, $\kappa^{-1}$ is a generalized inverse of $\kappa$ defined in (2.19).
Then the following theorem holds:

######  Theorem 1.2 (Trace theorem with non-local derivative; half line).

Let $p\in(1,\infty)$, $w\in A_{p}$ and $\kappa\in\mathcal{I}_{o}(-1,0)$.
Suppose that $W\circ\kappa^{*}\in\mathcal{I}_{o}(0,p)$, $u\in
L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})$, $f\in
L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})$, $u_{0}\in X_{0}+X_{1}$, and

$\int_{0}^{t}\kappa(t-s)\left(u(s)-u_{0}\right)\,\mathrm{d}s=\int_{0}^{t}f(s)\,\mathrm{d}s,$
(1.10)

for $t\in\mathbb{R}_{+}$. Then
$u_{0}\in(X_{0},X_{1})_{(W\circ\kappa^{*})^{1/p},p}$ and

$\displaystyle\|u_{0}\|_{(X_{0},X_{1})_{(W\circ\kappa^{*})^{1/p},p}}\lesssim_{p,[w]_{A_{p}},\kappa}\|u\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})}+\|f\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})}.$
(1.11)

######  Remark 1.3.

If $\kappa$ is a strictly decreasing continuous function, then the assumption
$\kappa\in\mathcal{I}_{o}(0,1)$ and
$W\circ\kappa^{\ast}\in\mathcal{I}_{o}(0,p)$ can be replaced by the following:
there exist constants $0<a_{1}<a_{2}$ and $0<b_{1}<b_{2}<p$ such that for any
$t>0$ and $\lambda\geq 1$,

$\lambda^{a_{1}}\lesssim\left(\frac{\kappa(t)}{\kappa(\lambda
t)}\right)^{a_{2}}\lesssim\frac{W(\lambda
t)}{W(t)}\lesssim\left(\frac{\kappa(t)}{\kappa(\lambda
t)}\right)^{b_{2}}\lesssim\lambda^{b_{1}}.$

For this, see Lemma 2.3-$(vi)$.

For the Volterra-type equations (1.3), it is shown in Section 2.2 that $u$ and
$f$ of (1.3) satisfy (1.10). Thus (1.11) also holds for $u$ and $f$ of (1.3).
The same is true for Theorem 1.6.

######  Remark 1.4.

In Theorems 1.1 and 1.2, we assume that $u$ (and $f$) is defined on
$\mathbb{R}_{+}$, but if $X_{0}$ is continuously embedded into $X_{1}$, i.e.,
$X_{0}\subset X_{1}$, we also obtain trace estimates of $u$ defined on a
finite time interval $(0,T)$. See Section 5 for this.

Another novelty is that we construct $u$ and $f$ satisfying (1.10) by means of
probability theory, which is motivated by spectral theory. Regarding the case
of (1.1), it is natural to consider its fundamental solution as one-parameter
semigroups. Then together with generalized interpolation theory given in
Section 3, we have the following result:

######  Theorem 1.5 (Extension theorem with local derivative).

Let $p\in(1,\infty)$, $w\in A_{p}$, and $a\in(X_{0},X_{1})_{W^{1/p},p}$. Then
there exist $u\in L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})$ and $f\in
L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})$ such that $u(0)=a$ and

$u(t)=a+\int_{0}^{t}f(s)\,\mathrm{d}s,$

for $t\in\mathbb{R}_{+}$. Furthermore,

$\|u\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})}+\|f\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})}\lesssim_{p,[w]_{A_{p}}}\|a\|_{(X_{0},X_{1})_{W^{1/p},p}}.$

Even if we have a non-local kernel $\kappa$ as in (1.2), we can accomplish a
similar result. It is because the kernel $\kappa$ corresponds to a Bernstein
function $\phi$, and $\phi$ yields a subordinator $S=(S_{t})_{t\geq 0}$ whose
Laplace exponent is $\phi$. Then one can establish a solution to (1.2) by
making use of the subordinator $S$. This procedure is given in the second part
of Section 2, and particularly in Proposition 2.12. Finally, we have the
following theorem:

######  Theorem 1.6 (Extension theorem with non-local derivative).

Let $p\in(1,\infty)$ and $w\in A_{p}$. Suppose that
$\kappa\in\mathcal{I}_{o}(-1,0)$, $W\circ\kappa^{*}\in\mathcal{I}_{o}(0,p)$,
and $a\in(X_{0},X_{1})_{(W\circ\kappa^{*})^{1/p},p}$. Then there exist $u\in
L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})$ and $f\in
L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})$ such that

$\int_{0}^{t}\kappa\left(t-s\right)\left(u(s)-a\right)\,\mathrm{d}s=\int_{0}^{t}f(s)\,\mathrm{d}s$
(1.12)

for $t\in\mathbb{R}_{+}$ and $u(0)=a$. Furthermore,

$\|u\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})}+\|f\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})}\lesssim_{p,[w]_{A_{p}},\kappa,W\circ\kappa^{*}}\|a\|_{(X_{0},X_{1})_{(W\circ\kappa^{*})^{1/p},p}}$
(1.13)

Note that (1.13) in Theorem 1.6 depends on $W\circ\kappa^{*}$ while (1.11) in
Theorem 1.2 does not. The difference arises because equivalence of
$\|a\|_{(X_{0},X_{1})_{(W\circ\kappa^{*})^{1/p},p}^{K}}$ and
$\|a\|_{(X_{0},X_{1})_{(W\circ\kappa^{*})^{1/p},p}^{J}}$ depends on
$W\circ\kappa^{*}$, and we use $K$ and $J$ methods to prove Theorems 1.2 and
1.6, respectively. In other words, the trace and extension theorems with non-
local derivatives essentially depend on $p$, $[w]_{A_{p}}$ and $\kappa$.

######  Remark 1.7.

In this paper, while the results on the local derivative (Theorem 1.1,
Corollary 5.1 and Theorem 1.5) hold for any $A_{p}$ weights, the results on
the non-local derivatives (Theorems 1.2, 5.3 and 1.6) hold for $w\in A_{p}$
with $W\circ\kappa^{*}\in\mathcal{I}_{o}(0,p)$. In particular, if we consider
the Caputo fractional derivative $\partial_{t}^{\alpha}$ in (1.2), then
Theorems 1.2, 5.3 and 1.6 hold for $w(t)=|t|^{\gamma}$ with $-1<\gamma<p-1$
and $\gamma+1<p\alpha$, which is a special case of main results in [1] and
[14].

Indeed, let $\alpha\in(0,1)$, $q\in(1,\infty)$, and $w(t)=|t|^{\gamma}$ where
$\gamma\in(-1,q-1)$ and $1+\gamma<q\alpha$. Note that $w\in
A_{q}(\mathbb{R})$. If we take
$\kappa(t)=\big{(}\Gamma(1-\alpha)\big{)}^{-1}t^{-\alpha}$ in the definition
of $I^{\kappa}$ (see (2.12)), for sufficiently smooth function $u=u(t,x)$
defined on $[0,\infty)\times\Omega$,

$I^{\kappa}u(t,x)=\int_{0}^{t}\kappa(t-s)\big{(}u(s,x)-u(0,x)\big{)}\,\mathrm{d}s=I^{1-\alpha}\big{(}u(\cdot,x)-u(0,x)\big{)}(t),$

where $I^{1-\alpha}$ stands for the classical fractional integral, i.e., the
Riemann-Liouville fractional integral. Also let $X_{0}=H_{p}^{2}$ and
$X_{1}=L_{p}$, $1<p<\infty$, where $H_{p}^{2}$ is a usual Sobolev space for
the second order PDEs. By the choice of $\kappa$, it is clear that
$\kappa\in\mathcal{I}_{o}(-1,0)$ and

$\kappa^{\ast}(t)=\big{(}\Gamma(1-\alpha)\big{)}^{1/\alpha}t^{1/\alpha}.$

Moreover, since $W(t)\simeq t^{1+\gamma}$, we also have
$(W\circ\kappa^{\ast})(\lambda)\simeq\lambda^{(1+\gamma)/\alpha}\in\mathcal{I}_{o}(0,q)$
by the choice of $\alpha$, $q$, and $\gamma$. Then by Theorem 1.2,

$u(0,\cdot)\in(X_{0},X_{1})_{(W\circ\kappa^{\ast})^{1/q},q}=(H_{p}^{2},L_{p})_{(W\circ\kappa^{\ast})^{1/q},q}$

where $(W\circ\kappa^{\ast})^{1/q}(\lambda)=\lambda^{(1+\gamma)/q\alpha}$. On
the other hand, by Proposition A.3 (ii), we have that
$(H_{p}^{2},L_{p})_{(W\circ\kappa^{\ast})^{1/q},q}=B_{p,q}^{(W\circ\kappa^{\ast})^{1/q}(2,0)}$.
That is, we have

$\displaystyle\|f\|_{B_{p,q}^{(W\circ\kappa^{\ast})^{1/q}(0,2)}}\,$
$\displaystyle=\|S_{0}f\|_{p}+\left(\sum_{j\geq
1}2^{2qj}\big{(}W\circ\kappa^{\ast}\big{)}(2^{-2j})\|\Delta_{j}f\|_{L_{p}}^{q}\right)^{\frac{1}{q}}$
$\displaystyle\simeq\|S_{0}f\|_{p}+\left(\sum_{j\geq
1}2^{2(1-\frac{1+\gamma}{q\alpha})qj}\|\Delta_{j}f\|_{L_{p}}^{q}\right)^{\frac{1}{q}}=\|f\|_{B_{p,q}^{2-\frac{2(\gamma+1)}{q\alpha}}}.$

Thus, $u(0,\cdot)\in B_{p,q}^{2-\frac{2(\gamma+1)}{q\alpha}}$ and this
coincides with [14, Theorem 3.11] for $k=0$ in there. The extension part can
be dealt with similar way.

We also remark that if we take $X_{0}$ and $X_{1}$ as weighted Sobolev spaces,
the generalized interpolation space $(X_{0},X_{1})$ becomes a weighted Besov
space with variable smoothness, which is characterized by means of difference
operators. For more detail, see [6].

## Notations

Throughout this paper, we will use the following standard notations.

$\bullet$ $\mathbb{R}$, $\mathbb{Q}$, and $\mathbb{Z}$ are the set of all real
numbers, rational numbers, and integers, respectively.

$\bullet$ $\mathbb{R}_{+}=(0,\infty)$, and
$\mathbb{Q}_{+}=\mathbb{Q}\cap\mathbb{R}_{+}$.

$\bullet$ $a\wedge b=\min(a,b)$ and $a\vee b=\max(a,b)$ for
$a,b\in\mathbb{R}$.

$\bullet$ For a set $\\{a,b,\ldots\\}$ and $X,Y\in\mathbb{R}$, we say
$X\lesssim_{a,b,\ldots}Y$, if $X\leq NY$ holds for some $N=N(a,b,\ldots)>0$.
Sometimes we omit $a,b,\ldots$ and just say $X\lesssim Y$ if the dependency on
$a,b,\ldots$ is clear from the context.

$\bullet$ For $p\in[1,\infty]$, $p^{\prime}:=\frac{p}{p-1}$ is the Hölder
conjugate ($\frac{1}{0}:=\infty$ and $\frac{\infty}{\infty}:=1$).

$\bullet$ $\mathbbm{1}_{A}$ is a characteristic function on a set $A$.

$\bullet$ For a positive function $f$ defined on $\mathbb{R}_{+}$, we use the
standard big-$O$ notation $O\big{(}f(\lambda)\big{)}$ and little-$o$ notation
$o\big{(}f(\lambda)\big{)}$.

##  2\. Preliminaries

###  2.1. Functions of type $\mathcal{I}(a,b)$ and $\mathcal{I}_{o}(a,b)$ and
their properties

We introduce a class $\mathcal{I}(a,b)$ and $\mathcal{I}_{o}(a,b)$. In
particular, functions in $\mathcal{I}_{o}(0,1)$ are essential in a generalized
interpolation introduced in Section 3.

######  Definition 2.1.

For a function $\phi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$, we define

$s_{\phi}(\lambda)=\sup_{t>0}\frac{\phi(\lambda t)}{\phi(t)}\,,$

so that $s_{\phi}:\mathbb{R}_{+}\rightarrow(0,\infty]$. Observe that
$s_{\phi}$ is submultiplicative, i.e., $s_{\phi}(\lambda\tau)\leq
s_{\phi}(\lambda)s_{\phi}(\tau)$ for $\lambda,\tau>0$. Together with the
definition of $s_{\phi}(\lambda)$ we introduce the following class of
functions. For $a,b\in\mathbb{R}$,

$\displaystyle\mathcal{I}(a,b):=\\{\phi:\text{$s_{\phi}(\lambda)=O(\lambda^{a})$
as $\lambda\to 0$, and $s_{\phi}(\lambda)=O(\lambda^{b})$ as
$\lambda\to\infty$}\\}.$ (2.1)

We also define $\mathcal{I}_{o}(a,b)$, where $o$ denotes that we replace
$O$-notation with $o$-notation in (2.1). Clearly,
$\mathcal{I}_{o}(a,b)\subsetneq\mathcal{I}(a,b)$.

######  Remark 2.2.

Due to $1=s_{\phi}(1)\leq s_{\phi}(\lambda)\,s_{\phi}(\lambda^{-1})$, we
obtain that $\mathcal{I}(a,b)=\emptyset$ for $a>b$, and
$\mathcal{I}_{o}(a,b)=\emptyset$ for $a\geq b$. In addition, for $a<b$,
$\phi\in\mathcal{I}_{o}(a,b)$ if and only if there exists a bounded function
$K_{\phi}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ such that

$s_{\phi}(\lambda)\leq\left(\lambda^{a}+\lambda^{b}\right)K_{\phi}(\lambda)\quad\textrm{and}\quad\lim_{\lambda\rightarrow
0}K_{\phi}(\lambda)=\lim_{\lambda\rightarrow\infty}K_{\phi}(\lambda)=0.$

See, for example, [12, Proposition 1.1].

The following lemma provides useful information about the classes
$\mathcal{I}(a,b)$ and $\mathcal{I}_{o}(a,b)$:

######  Lemma 2.3.

Let $a,b\in\mathbb{R}$ and $\phi\in\mathcal{I}(a,b)$.

  1. (i)

For $\alpha\in\mathbb{R}$,
$t^{\alpha}\phi(t)\in\mathcal{I}(a+\alpha,b+\alpha)$.

  2. (ii)

For $\alpha\geq 0$, $\phi(t^{\alpha}),\phi(t)^{\alpha}\in\mathcal{I}(\alpha
a,\alpha b)$.

  3. (iii)

For $\alpha\leq 0$, $\phi(t^{\alpha}),\phi(t)^{\alpha}\in\mathcal{I}(\alpha
b,\alpha a)$.

  4. (iv)

Let $a,b>0$ and $\phi$ be strictly increasing, continuous, and

$\lim_{\lambda\uparrow\infty}\phi(\lambda)=\infty.$

Then its inverse $\phi^{-1}$ is of class
$\mathcal{I}(\frac{1}{b},\frac{1}{a})$.

  5. (v)

For $a<c$, $b>d$, $\mathcal{I}(c,d)\subset\mathcal{I}_{o}(a,b)$ and

$\displaystyle\bigcup_{c>a,\,d<b}\mathcal{I}(c,d)=\mathcal{I}_{o}(a,b).$ (2.2)

  6. (vi)

$\phi\in\mathcal{I}_{o}(a,b)$ if and only if there exists $\varepsilon>0$ such
that

$\displaystyle\lambda^{a+\varepsilon}\lesssim\frac{\phi(\lambda
t)}{\phi(t)}\lesssim\lambda^{b-\varepsilon}\quad\forall\,t\geq 0,\,\lambda\geq
1.$ (2.3)

  7. (vii)

If $p\geq 0$ and $\phi\in\mathcal{I}_{o}(0,p)$, then

$\displaystyle\int_{0}^{\infty}\left(1\wedge\frac{x}{t}\right)^{p}\phi(t)\frac{\mathrm{d}t}{t}\lesssim\phi(x).$
(2.4)

  8. (viii)

If $p\geq 0$ and $\phi\in\mathcal{I}_{o}(-p,0)$, then

$\displaystyle\int_{0}^{\infty}\left(1\wedge\frac{t}{x}\right)^{p}\phi(t)\frac{\mathrm{d}t}{t}\lesssim\phi(x),$

  9. (ix)

For $\phi\in\mathcal{I}_{o}(a,b)$, the assertions $(i)$ – $(iii)$ also hold
for $\mathcal{I}_{o}(a,b)$ instead of $\mathcal{I}(a,b)$.

###### Proof.

$(i)$–$(iii)$ and $(ix)$ follows from direct computations.

For $(iv)$, first, we observe that there exists a constant $C>0$ such that

$\phi(\lambda t)\leq C\lambda^{a}\phi(t)$

for $t\in\mathbb{R}_{+}$ and $\lambda\in(0,1]$. On the other hand, for
$x\in\mathbb{R}_{+}$, by the assumption for $\phi$, there exists
$t\in\mathbb{R}_{+}$ such that $C\phi(t)=x$. Then, again by the assumption for
$\phi$, we have $\lambda\phi^{-1}(x/C)\leq\phi^{-1}(\lambda^{a}x)$, and hence

$\frac{\phi^{-1}(x/(C\lambda^{a}))}{\phi^{-1}(x)}\leq\frac{1}{\lambda}$

for all $x\in\mathbb{R}_{+}$ and $\lambda\in(0,1]$. This certainly implies
that $s_{\phi^{-1}}(\lambda)=O(\lambda^{1/a})$ as $\lambda\to\infty$.
Similarly, we also have $s_{\phi^{-1}}(\lambda)=O(\lambda^{1/b})$ as
$\lambda\to 0$.

$(v)$ Clearly, $\mathcal{I}(c,d)\subset\mathcal{I}_{o}(a,b)$ for $a<c$ and
$b>d$, and thus

$\bigcup_{a<c,b>d}\mathcal{I}(c,d)\subset\mathcal{I}_{o}(a,b).$

Therefore, it suffices to show that “$\supset$” holds instead of “$=$” in
(2.2). Let $\phi\in\mathcal{I}_{o}(a,b)$ and take a constant $N>0$ such that
$s_{\phi}(\lambda)\leq\frac{1}{2}\lambda^{a}$ if $\lambda\leq 2^{-N}$, and
$s_{\phi}(\lambda)\leq\frac{1}{2}\lambda^{b}$ if $\lambda\geq 2^{N}$. Then for
$\lambda\leq 2^{-N}$, there is $k\in\mathbb{N}$ such that
$2^{-(k+1)N}<\lambda\leq 2^{-kN}$, which yields

$\displaystyle
s_{\phi}\left(\lambda\right)\leq\left(s_{\phi}\left(2^{-N}\right)\right)^{k-1}s_{\phi}\left(2^{N\left(k-1\right)}\lambda\right)\leq
2^{-k}\lambda^{a}\leq 2\lambda^{a+\frac{1}{N}}$ (2.5)

by the submultiplicativity. Similarly, for $\lambda\geq 2^{N}$ there is
$l\in\mathbb{N}$ such that $2^{lN}\leq\lambda<2^{(l+1)N}$, which yields

$\displaystyle
s_{\phi}\left(\lambda\right)\leq\left(s_{\phi}\left(2^{N}\right)\right)^{l-1}s_{\phi}\left(2^{-N\left(l-1\right)}\lambda\right)\leq
2^{-l}\lambda^{b}\leq 2\lambda^{b-\frac{1}{N}}.$ (2.6)

By (2.5) and (2.6), $\phi\in\mathcal{I}(a+\frac{1}{N},b-\frac{1}{N})$. This
proves (2.2).

$(vi)$ This is a direct result of $(v)$ and its proof with Remark 2.2.

$(vii)$ By $(v)$ and $(vi)$, there exists $\varepsilon>0$ such that
$\phi\in\mathcal{I}(\varepsilon,p-\varepsilon)$ and

$\frac{\phi(t)}{\phi(x)}\lesssim\left(\frac{t}{x}\right)^{\varepsilon}\mathbbm{1}_{t\leq
x}+\left(\frac{t}{x}\right)^{p-\varepsilon}\mathbbm{1}_{t>x}.$

Therefore,

$\displaystyle\int_{0}^{\infty}\left(1\wedge\frac{x}{t}\right)^{p}\frac{\phi(t)}{\phi(x)}\frac{\mathrm{d}t}{t}$
$\displaystyle=\int_{0}^{x}\cdots\,\,+\int_{x}^{\infty}\cdots$ (2.7)
$\displaystyle\lesssim\int_{0}^{x}\left(\frac{t}{x}\right)^{\varepsilon}\frac{\mathrm{d}t}{t}+\int_{x}^{\infty}\left(\frac{x}{t}\right)^{p}\left(\frac{t}{x}\right)^{p-\varepsilon}\frac{\mathrm{d}t}{t}\lesssim
1.$

$(viii)$ Since $\phi\in\mathcal{I}_{o}(-p,0)$, by $(i)$,
$t^{p}\phi(t)\in\mathcal{I}_{o}(0,p)$. Using $(vii)$,

$\displaystyle\int_{0}^{\infty}\left(1\wedge\frac{t}{x}\right)^{p}\phi(t)\,\frac{\mathrm{d}t}{t}=x^{-p}\int_{0}^{\infty}\left(1\wedge\frac{x}{t}\right)^{p}t^{p}\phi(t)\,\frac{\mathrm{d}t}{t}\lesssim\phi(x).$

The lemma is proved. ∎

When a kernel $\kappa$ of $\partial_{t}^{\kappa}$ is of class
$\mathcal{I}_{o}(-1,0)$, we have the following proposition:

######  Proposition 2.4.

Let $\kappa$ be a positive function defined on $\mathbb{R}_{+}$.

  1. (i)

If $\kappa\in\mathcal{I}_{o}(-1,0)$, then for $\lambda\in\mathbb{R}_{+}$,

$\displaystyle\mathcal{L}[\kappa](\lambda):=\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}\kappa(t)\,\mathrm{d}t\simeq\frac{\kappa(\lambda^{-1})}{\lambda},$ (2.8)

where the equivalence ($\simeq$) depends only on $K_{\phi}$ in Remark 2.2.
Moreover, $\mathcal{L}[\kappa]\in\mathcal{I}_{o}(-1,0)$ (by Lemma 2.3-$(ix)$).

  2. (ii)

If $\kappa$ is a decreasing function on $\mathbb{R}_{+}$, and
$\mathcal{L}[\kappa]\in\mathcal{I}_{o}(-1,0)$, then (2.8) holds, and the
equivalence in (2.8) depends only on $K_{\mathcal{L}[\kappa]}$. Moreover,
$\kappa\in\mathcal{I}_{o}(-1,0)$ (by Lemma 2.3-$(ix)$).

###### Proof.

$(i)$ First note that $\mathcal{L}[\kappa](\lambda)$ is well-defined on
$\mathbb{R}_{+}$ due to $\kappa\in\mathcal{I}_{o}(0,1)$. Then for
$\lambda\in\mathbb{R}_{+}$,

$\mathcal{L}[\kappa](\lambda)=\int_{0}^{\lambda^{-1}}\mathrm{e}^{-\lambda
t}\kappa(t)\mathrm{d}t+\int_{\lambda^{-1}}^{\infty}\mathrm{e}^{-\lambda
t}\kappa(t)\mathrm{d}t=:I_{0}(\lambda)+I_{1}(\lambda).$

Since $\kappa\in\mathcal{I}_{o}(-1,0)$, by Lemma 2.3-$(vi)$, there exists
$\varepsilon>0$ such that

$(s^{-1}t)^{-1+\varepsilon}\kappa(s)\lesssim\kappa(t)\lesssim(s^{-1}t)^{-\varepsilon}\kappa(s)\quad\forall\,\,0<s<t<\infty.$

Therefore, we have

$\kappa(\lambda^{-1})\int_{0}^{\lambda^{-1}}\mathrm{e}^{-\lambda t}(\lambda
t)^{-1+\varepsilon}\mathrm{d}t\lesssim
I_{0}(\lambda)\lesssim\kappa(\lambda^{-1})\int_{0}^{\lambda^{-1}}\mathrm{e}^{-\lambda
t}(\lambda t)^{-\varepsilon}\mathrm{d}t,$

and

$\kappa(\lambda^{-1})\int_{\lambda^{-1}}^{\infty}\mathrm{e}^{-\lambda
t}(\lambda t)^{-1+\varepsilon}\mathrm{d}t\lesssim
I_{1}(\lambda)\lesssim\kappa(\lambda^{-1})\int_{\lambda^{-1}}^{\infty}\mathrm{e}^{-\lambda
t}(\lambda t)^{-\varepsilon}\mathrm{d}t.$

This implies

$\mathcal{L}[\kappa](\lambda)=I_{0}(\lambda)+I_{1}(\lambda)\simeq\frac{\kappa(\lambda^{-1})}{\lambda}.$

$(ii)$ We assume that $\mathcal{L}[\kappa]\in\mathcal{I}_{o}(-1,0)$. Observe
that

$\mathcal{L}[\kappa](\lambda^{-1})=\lambda\int_{0}^{\infty}\mathrm{e}^{-z}\kappa(\lambda
z)\mathrm{d}z.$

Since $\kappa$ is a decreasing function, we see that

$\displaystyle\mathcal{L}[\kappa](\lambda^{-1})\geq\lambda\int_{0}^{1}\mathrm{e}^{-z}\kappa(\lambda
z)\mathrm{d}z\geq\mathrm{e}^{-1}\lambda\kappa(\lambda).$ (2.9)

On the other hand, by (2.9), for any $\delta\in\mathbb{R}_{+}$,

$\lambda\int_{0}^{\delta}\mathrm{e}^{-z}\kappa(\lambda
z)\mathrm{d}z\leq\mathrm{e}\int_{0}^{\delta}\frac{\mathrm{e}^{-z}}{z}\mathcal{L}[\kappa](\lambda^{-1}z^{-1})\,\mathrm{d}z.$

For $0<\delta\leq 1$, by Lemma 2.3-$(vi)$ with the fact that
$\mathcal{L}[\kappa]\in\mathcal{I}_{o}(-1,0)$, we have

$\displaystyle\int_{0}^{\delta}\frac{\mathrm{e}^{-z}}{z}\mathcal{L}[\kappa](\lambda^{-1}z^{-1})\mathrm{d}z$
$\displaystyle=\mathcal{L}[\kappa](\lambda^{-1})\int_{0}^{\delta}\frac{\mathrm{e}^{-z}}{z}\frac{\mathcal{L}[\kappa](\lambda^{-1}z^{-1})}{\mathcal{L}[\kappa](\lambda^{-1})}\,\mathrm{d}z$
$\displaystyle\lesssim\mathcal{L}[\kappa](\lambda^{-1})\int_{0}^{\delta}\mathrm{e}^{-z}z^{\varepsilon-1}\,\mathrm{d}z,$

for some $\varepsilon>0$. Hence

$\lambda\int_{0}^{\delta}\mathrm{e}^{-z}\kappa(\lambda
z)\mathrm{d}z\leq\frac{1}{2}\mathcal{L}[\kappa](\lambda^{-1}),$

for sufficiently small $\delta=\delta(K_{\mathcal{L}[\kappa]})\in(0,1)$. Then
we have

$\displaystyle\mathcal{L}[\kappa](\lambda^{-1})$
$\displaystyle=\lambda\int_{0}^{\delta}\mathrm{e}^{-z}\kappa(\lambda
z)\,\mathrm{d}z+\lambda\int_{\delta}^{\infty}\mathrm{e}^{-z}\kappa(\lambda
z)\,\mathrm{d}z$
$\displaystyle\leq\frac{1}{2}\mathcal{L}[\kappa](\lambda^{-1})+\lambda\int_{\delta}^{\infty}\mathrm{e}^{-z}\kappa(\lambda
z)\,\mathrm{d}z.$

Since $\kappa$ is decreasing,

$\mathcal{L}[\kappa](\lambda^{-1})\leq
2\lambda\int_{\delta}^{\infty}\mathrm{e}^{-z}\kappa(\lambda
z)\,\mathrm{d}z\leq 2\lambda\kappa(\delta\lambda),$

and then,

$\mathcal{L}[\kappa](\lambda^{-1})\leq\mathcal{L}[\kappa](\delta\lambda^{-1})\leq
2\delta^{-1}\lambda\kappa(\lambda)$

for all $\lambda>0$, where the first inequality is due to $\delta\leq 1$. This
certainly implies that
$\mathcal{L}[\kappa](\lambda^{-1})\lesssim\lambda\kappa(\lambda)$. The
proposition is proved. ∎

###  2.2. Volterra-type equations associated with Bernstein functions

In this subsection, we give a connection between a kernel $\kappa$ of non-
local derivative in the evolution equation (1.2) and a Bernstein function in
probability theory. Moreover, we use properties of Bernstein functions to
construct $u$ and $f$ satisfying (1.10).

Let $\kappa:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ be a right-continuous
decreasing function with $\kappa(0+)=\lim_{t\downarrow 0}\kappa(t)=\infty$ and
$\kappa(\infty)=0$. Then there exists a unique non-negative measure $\mu$ on
$\mathbb{R}_{+}$ such that

$\displaystyle\kappa(s)-\kappa(t)=\mu\left((s,t]\right),$ (2.10)

for $0<s<t<\infty$. Note that

$\displaystyle\int_{0}^{\infty}(1\wedge
t)\mu(\mathrm{d}t)=\int_{0}^{1}\kappa(s)\mathrm{d}s\quad\textrm{and}\quad\mu(\mathbb{R}_{+})=\kappa(0+)=\infty.$
(2.11)

We will consider an evolution equation

$I^{\kappa}u(t):=\int_{0}^{t}\kappa(t-s)\left(u(s)-u(0)\right)\mathrm{d}s=\int_{0}^{t}f(s)\mathrm{d}s\,,$
(2.12)

or for simplicity, we just denote (2.12) by $\partial_{t}^{\kappa}u=f$.

######  Remark 2.5.

If $\kappa\in\mathcal{I}_{o}(-1,0)$, then by Lemma 2.3-($vi$), there exists
$\varepsilon>0$ such that

$s^{-\varepsilon}\kappa(1)\lesssim\kappa(s)\lesssim
s^{-1+\varepsilon}\kappa(1)$

for $s\leq 1$ and

$\kappa(r)\lesssim r^{-\varepsilon}\kappa(1)$

for $r\geq 1$ (see (2.3)). Therefore, we have

$\int_{0}^{1}\kappa(s)\,\mathrm{d}s<\infty,\quad\lim_{t\rightarrow
0}\kappa(t)=\infty,\quad\textrm{and}\quad\lim_{t\rightarrow\infty}\kappa(t)=0.$

######  Definition 2.6.

An infinitely differentiable function $\phi:(0,\infty)\to[0,\infty)$ is called
a Bernstein function if

$(-1)^{n}D^{n}\phi(\lambda)\leq 0,\quad\forall
n\in\mathbb{N},\,\lambda\in(0,\infty).$

It is well known (e.g. [24, Theorem 3.2]) that Bernstein function $\phi$ has a
unique representation

$\displaystyle\phi(\lambda)=a+b\lambda+\int_{0}^{\infty}\left(1-\mathrm{e}^{-\lambda
t}\right)\,\mu(\mathrm{d}t),$ (2.13)

where $a,b\geq 0$ and $\mu$ is a nonnegative measure on $\mathbb{R}_{+}$
satisfying

$\displaystyle\int_{0}^{\infty}\left(1\wedge t\right)\mu(\mathrm{d}t)<\infty.$
(2.14)

In (2.13), $a$, $b$, and $\mu$ are called the killing term, drift, and Lévy
measure of $\phi$, respectively. Furthermore, the triplet $(a,b,\mu)$, which
determines a Bernstein function $\phi$ uniquely, is called the Lévy triplet of
$\phi$. In this article, we only consider Bernstein functions without killing
term, i.e. $\phi(0+)=0$. Indeed, it is well-known the connection between
Bernstein functions without killing term and subordinators. Before its
description, we first give definitions of subordinators and convolution
semigroup of probability measures.

######  Definition 2.7.

Let $S=(S_{t})_{t\geq 0}$ be a real-valued Lévy process defined on a
probability space $(\Omega,\mathcal{F},\mathbb{P})$. We say $S=(S_{t})_{t\geq
0}$ is a subordinator if $\mathbb{P}(S_{t}\geq 0,\,\forall t\geq 0)=1$. A
convolution semigroup of probability measures on $[0,\infty)$ is a family of
probability measures $\\{\mu_{t}\\}_{t\geq 0}$ satisfying the following
properties;

  1. (i)

For all $t\geq 0$, $\mu_{t}([0,\infty))=1$.

  2. (ii)

For all $t,s\geq 0$, $\mu_{t}\ast\mu_{s}=\mu_{t+s}$, where

$\mu_{t}\ast\mu_{s}(B):=\int_{0}^{\infty}\int_{0}^{\infty}\mathbbm{1}_{B}(t^{\prime}+s^{\prime})\mu_{t}(\mathrm{d}t^{\prime})\mu_{s}(\mathrm{d}s^{\prime}).$

  3. (iii)

For every compactly supported continuous functions
$f:[0,\infty)\to\mathbb{R}$,

$\lim_{t\downarrow
0}\int_{0}^{\infty}f(t^{\prime})\mu_{t}(\mathrm{d}t^{\prime})=\int_{0}^{\infty}f(t^{\prime})\varepsilon_{0}(\mathrm{d}t^{\prime}),$

where $\varepsilon_{0}$ is the centered Dirac measure on $\mathbb{R}$.

For the definition and properties of Lévy processes, see e.g. [23]. Now we
give a short description for the connection between Bernstein functions
without killing term and subordinators. [24, Theorem 5.2] provides one-to-one
correspondence between Bernstein functions and convolution semigroups of sub-
probability measures in the sense that

$\int_{[0,\infty)}\mathrm{e}^{-\lambda
r}\,\mu_{t}(\mathrm{d}r)=\mathrm{e}^{-t\phi(\lambda)},\quad\forall(t,\lambda)\in[0,\infty)\times\mathbb{R}_{+}.$

Here, convolution semigroups of sub-probability measures implies Definition
2.7 with $\mu_{t}([0,\infty))=1$ replaced by $\mu_{t}([0,\infty))\leq 1$.
However, the proof of [24, Theorem 5.2] includes that there is one-to-one
correspondence between Bernstein functions without killing term and
convolution semigroups of probability measures. Similarly, from [24,
Proposition 5.5], we also obtain the fact that every law of subordinators is a
convolution semigroup of probability measures on $[0,\infty)$. Combining these
results, for a given subordinator $S=(S_{t})_{t\geq 0}$, there exists a
Bernstein function $\phi$ such that $\phi(0+)=0$ and

$\left(\int_{\Omega}\mathrm{e}^{-\lambda
S_{t}(\omega)}\,\mathbb{P}(\mathrm{d}\omega)=:\right)\mathbb{E}[\mathrm{e}^{-\lambda
S_{t}}]=\mathrm{e}^{-t\phi(\lambda)},\quad\forall(t,\lambda)\in[0,\infty)\times\mathbb{R}_{+}.$
(2.15)

For the converse, we observe that any convolution semigroups of probability
measures on $[0,\infty)$ is infinitely divisible (see e.g. [23, Definition
7.1]). Then by [23, Theorem 7.10], there exists a real-valued Lévy process
$S=(S_{t})_{t\geq 0}$ such that
$(\mu_{1})^{t}(\cdot)=\mathbb{P}(S_{t}\in\cdot)$, where $(\mu_{1})^{t}$ is a
probability measure satisfying

$\int_{-\infty}^{\infty}\mathrm{e}^{iz\cdot
r}(\mu_{1})^{t}(\mathrm{d}r)=\left(\int_{-\infty}^{\infty}\mathrm{e}^{iz\cdot
r}\mu_{1}(\mathrm{d}r)\right)^{t},\quad\forall z\in\mathbb{R}.$

If $t$ is a positive rational number, then by Definition 2.7 $(ii)$,
$(\mu_{1})^{t}=\mu_{t}$. Using the density argument and Definition 2.7
$(iii)$, $(\mu_{1})^{t}=\mu_{t}$ for all $t\geq 0$. Hence
$\mu_{t}(\cdot)=\mathbb{P}(S_{t}\in\cdot)$. Since the support of $\mu_{t}$ is
contained in $[0,\infty)$, the state space of $S$ is $[0,\infty)$. This
certainly implies that $S$ is a subordinator. Therefore for a given Bernstein
function $\phi$ without killing term, there exists a subordinator
$S=(S_{t})_{t\geq 0}$ satisfying (2.15). Summarizing these facts,

Subordinators $\displaystyle\rightleftharpoons\text{Convolution semigroups of
probability measures on }[0,\infty)$
$\displaystyle\rightleftharpoons\text{Bernstein functions without killing
term}.$

Since there is a one-to-one correspondence between Bernstein functions without
killing term and subordinators, we say a subordinator $S=(S_{t})_{t\geq 0}$ is
a subordinator with Laplace exponent $\phi$ if (2.15) holds.

The following proposition gives a relation between a kernel $\kappa$ and a
Bernstein function $\phi$.

######  Proposition 2.8.

Let $\mu$ be a nonnegative measure on $\mathbb{R}_{+}$ with (2.14). Then

$\kappa(x):=\int_{0}^{\infty}\mathbbm{1}_{(x,\infty)}(t)\mu(\mathrm{d}t)\in\mathcal{I}_{o}(-1,0)$

if and only if

$\phi(\lambda):=\int_{0}^{\infty}(1-\mathrm{e}^{-\lambda
t})\mu(\mathrm{d}t)\in\mathcal{I}_{o}(0,1).$

Moreover if $\kappa\in\mathcal{I}_{o}(-1,0)$ or $\phi\in\mathcal{I}_{o}(0,1)$,
then for all $\lambda\in\mathbb{R}_{+}$, we have

$\phi(\lambda)\simeq\kappa(\lambda^{-1}).$ (2.16)

###### Proof.

Observe that

$\displaystyle\phi(\lambda)=\int_{0}^{\infty}\left(\int_{0}^{t}\lambda\mathrm{e}^{-\lambda
s}\mathrm{d}s\right)\mu(\mathrm{d}t)=\lambda\int_{0}^{\infty}\left(\int_{s}^{\infty}\mu(\mathrm{d}t)\right)\mathrm{e}^{-\lambda
s}\mathrm{d}s=\lambda\mathcal{L}[\kappa](\lambda).$ (2.17)

By Lemma 2.3-$(ix)$ and Proposition 2.4, $\kappa\in\mathcal{I}_{o}(-1,0)$ is
equivalent to $\mathcal{L}[\kappa]\in\mathcal{I}_{o}(-1,0)$, and to
$\phi\in\mathcal{I}_{o}(0,1)$. The equivalence (2.16) is also obtained from
Proposition 2.4 with the fact that $\kappa$ is a decreasing function. The
proposition is proved. ∎

######  Remark 2.9.

From (2.16), we also obtain

$\displaystyle\phi^{-1}\simeq\frac{1}{\kappa^{-1}},$ (2.18)

where $\phi^{-1}$ is the inverse of $\phi$ and $\kappa^{-1}$ is a generalized
inverse of $\kappa$ defined by

$\displaystyle\kappa^{-1}(\lambda):=\inf\\{s>0\,:\,\kappa(s)\leq\lambda\\}$
(2.19)

(for more detail on properties of the generalized inverse, see [9, Proposition
1] with $T=-\kappa$ in there). Indeed, due to (2.16), there exists a constant
$N\geq 1$ such that

$\displaystyle\\{s>0\,:\,\phi\left(1/s\right)\leq
N^{-1}\lambda\\}\subset\\{s>0\,:\,\kappa\left(s\right)\leq\lambda\\}\subset\\{s>0\,:\,\phi\left(1/s\right)\leq
N\lambda\\}\,,$

which implies that

$\frac{1}{\phi^{-1}(N\lambda)}\leq\kappa^{-1}(\lambda)\leq\frac{1}{\phi^{-1}(N^{-1}\lambda)}.$

Due to $\phi\in\mathcal{I}_{o}(0,1)$ and 2.3-$(iv)$ and $(v)$, we have

$\phi^{-1}(N^{-1}\,\cdot\,)\simeq\phi^{-1}\simeq\phi^{-1}(N\,\cdot\,).$

This proves (2.18).

The following lemma shows that the second assertion of (2.11) is equivalent to
a subordinator $S$ being strictly increasing.

######  Lemma 2.10.

Let $S=(S_{t})_{t\geq 0}$ be a subordinator with Laplace exponent

$\phi(\lambda):=\int_{0}^{\infty}(1-\mathrm{e}^{-\lambda t})\mu(\mathrm{d}t),$

i.e., (2.15) holds for $(t,\lambda)\in[0,\infty)\times\mathbb{R}_{+}$, where
$\mu$ is a nonnegative measure defined on $\mathbb{R}_{+}$ with (2.14). Then
$\mu(\mathbb{R}_{+})=\infty$ if and only if $S$ is strictly increasing almost
surely.

###### Proof.

We only give a sketch of the proof. By the definition of subordinator, $S$ is
strictly increasing almost surely if and only if

$\displaystyle\mathbb{P}\left(\cup_{\begin{subarray}{c}p,q\in\mathbb{Q}_{+}\\\
p>q\end{subarray}}\\{S_{p}-S_{q}=0\\}\right)=\sum_{\begin{subarray}{c}p,q\in\mathbb{Q}_{+}\\\
p>q\end{subarray}}\mathbb{P}\left(S_{p}-S_{q}=0\right)=\sum_{\begin{subarray}{c}p,q\in\mathbb{Q}_{+}\\\
p>q\end{subarray}}\mathbb{P}\left(S_{p-q}=0\right)=0.$ (2.20)

Also (2.20) is equivalent to $\mathbb{P}(S_{t}=0)=0$ for any $t>0$, which
means that

$\lim_{\lambda\to\infty}\mathrm{e}^{-t\phi(\lambda)}=\lim_{\lambda\to\infty}\mathbb{E}[\mathrm{e}^{-\lambda
S_{t}}]=\mathbb{P}(S_{t}=0)+\lim_{\lambda\to\infty}\mathbb{E}[\mathrm{e}^{-\lambda
S_{t}}\mathbbm{1}_{\\{S_{t}>0\\}}]$

for any $t>0$. Since $\lim_{\lambda\to\infty}\mathbb{E}[\mathrm{e}^{-\lambda
S_{t}}\mathbbm{1}_{\\{S_{t}>0\\}}]=0$ by the Lebesgue’s dominated convergence
theorem, $\mathbb{P}(S_{t}=0)=0$ is equivalent to
$\lim_{\lambda\to\infty}\mathrm{e}^{-t\phi(\lambda)}=0$. This proves that $S$
is strictly increasing almost surely if and only if
$\lim_{\lambda\to\infty}\phi(\lambda)=\infty$, that is,
$\mu(\mathbb{R}_{+})=\infty$. ∎

Below are properties of subordinators in the literature, which we use
frequently and crucially in this paper.

######  Proposition 2.11.

Let $S=(S_{t})_{t\geq 0}$ be a subordinator with Laplace exponent
$\phi(\lambda):=\int_{0}^{\infty}(1-\mathrm{e}^{-\lambda s})\mu(\mathrm{d}s)$,
where $\mu$ satisfies (2.10). Suppose that $\phi\in\mathcal{I}_{o}(0,1)$, and
put $K(t)=\int_{0}^{t}\kappa(s)\mathrm{d}s$.

  1. (i)

([3, Lemma 2.1]) There exists a null set $\mathcal{N}\subset(0,\infty)$ (under
the Lebesgue measure) such that

$\mathbb{P}(S_{s}\geq
t)=\int_{0}^{s}\mathbb{E}\left[\kappa(t-S_{r})\mathbbm{1}_{\\{t\geq
S_{r}\\}}\right]\mathrm{d}r$

for all $s>0$ and $t\in(0,\infty)\setminus\mathcal{N}$. In particular,

$\displaystyle\int_{0}^{l}\mathbb{P}(S_{s}\geq
t)\mathrm{d}t=\int_{0}^{s}\mathbb{E}\left[K(l-S_{r})\mathbbm{1}_{\\{l\geq
S_{r}\\}}\right]\mathrm{d}r$ (2.21)

  2. (ii)

([3, (2.5)]) For any $t>0$,

$\displaystyle\int_{0}^{t}\kappa(t-r)\mathbb{P}(S_{s}>r)\mathrm{d}r=K(t)-\mathbb{E}\left[K(t-S_{s})\mathbbm{1}_{\\{t\geq
S_{s}\\}}\right].$ (2.22)

  3. (iii)

([4, Lemma 3.1]) Let $\phi\in\mathcal{I}_{o}(0,1)$. Then there exists $N\geq
1$ such that

$\displaystyle\lambda\phi^{\prime}(\lambda)\leq\phi(\lambda)\leq
N\lambda\phi^{\prime}(\lambda).$ (2.23)

In particular, $\phi^{\prime}\in\mathcal{I}_{o}(-1,0)$.

  4. (iv)

([4, Proposition 3.3.(i)]) Suppose that $\phi\in\mathcal{I}_{o}(0,1)$. Then
there are constant $c_{1},\,c_{2}>0$ such that for all $r,t\geq 0$,

$\displaystyle\mathbb{P}\Big{(}S_{r}\geq
t\big{(}1+\mathrm{e}^{r\phi(1/t)}\big{)}\Big{)}\leq c_{1}r\phi(1/t)$ (2.24)

and

$\displaystyle\mathbb{P}(S_{r}\geq t)\geq 1-\mathrm{e}^{-c_{2}r\phi(1/t)}.$
(2.25)

In particular, for each $L$, there exist constants $c_{1,L},c_{2,L}>0$ such
that for all $r\phi(1/t)\leq L$,

$\displaystyle c_{1,L}r\phi(1/t)\leq\mathbb{P}(S_{r}\geq t)\leq
c_{2,L}r\phi(1/t).$ (2.26)

The constants $c_{1},c_{2}$, $c_{1,L}$ and $c_{2,L}$ in (2.24)–(2.26) depends
only on $K_{\phi}$ in Remark 2.2.

The following lemma helps us to prove Theorem 1.6 by constructing appropriate
$u$ and $f$ satisfying (1.2).

######  Proposition 2.12.

Let $\kappa:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ be a right-continuous
decreasing function with $\kappa(0+)=\infty$, $\kappa(\infty)=0$, and
$\kappa\in\mathcal{I}_{o}(-1,0)$. We denote $\mu$ a nonnegative measure on
$\mathbb{R}_{+}$ defined by $\kappa$ such that $\mu$ satisfies
$\kappa(s)-\kappa(t)=\mu\left(\left[s,t\right)\right)$ for $0<s<t<\infty$, and

$\Theta(t,\lambda):=\lambda\int_{0}^{\infty}\mathrm{e}^{-\lambda
r}\mathbb{P}(S_{r}\geq t)\,\mathrm{d}r,$

where $S=(S_{t})_{t\geq 0}$ is a subordinator with Laplace exponent

$\phi(\lambda):=\int_{0}^{\infty}(1-\mathrm{e}^{-\lambda t})\mu(\mathrm{d}t).$

  1. (i)

For $t,\lambda\in\mathbb{R}_{+}$, the function $\Theta$ satisfies

$\int_{0}^{t}\kappa(t-s)\big{(}\Theta(s,\lambda)-\Theta(0,\lambda)\big{)}\,\mathrm{d}s=:I^{\kappa}\Theta(t,\lambda)=-\int_{0}^{t}\lambda\Theta(s,\lambda)\,\mathrm{d}s,$

that is,

$\partial_{t}^{\kappa}\Theta(t,\lambda)=-\lambda\Theta(t,\lambda),$

where $\kappa(x)=\int_{0}^{\infty}\mathbbm{1}_{(x,\infty)}\mu(\mathrm{d}t)$.

  2. (ii)

For $t,\lambda\in\mathbb{R}_{+}$,

$\Theta(t,\lambda)\simeq 1\wedge\frac{\phi(t^{-1})}{\lambda}\,,$

where the equivalence depends only on $K_{\phi}$ in Remark 2.2.

###### Proof.

First, note that for fixed $t\in\mathbb{R}_{+}$, $\mathbb{P}(S_{r}\geq t)$ is
(Borel) measurable with respect to $r\in\mathbb{R}_{+}$ by the definition of
subordinator, so $\Theta(t,\lambda)$ is well-defined.

$(i)$ It is clear that $\kappa$ and $\mu$ satisfy
$\kappa(x)=\int_{0}^{\infty}\mathbbm{1}_{(x,\infty)}\mu(\mathrm{d}t)$ and
$\Theta(0,\lambda)=1$ for all $\lambda\in\mathbb{R}_{+}$. Then by Fubini’s
theorem,

$\displaystyle I^{\kappa}\big{(}\Theta(\cdot,\lambda)\big{)}(t)$
$\displaystyle=\int_{0}^{t}\kappa(t-s)\big{(}\Theta(s,\lambda)-1\big{)}\,\mathrm{d}s$
$\displaystyle=\lambda\int_{0}^{\infty}\mathrm{e}^{-\lambda
r}\left(\int_{0}^{t}\kappa(t-s)\mathbb{P}(S_{r}\geq
s)\mathrm{d}s\right)\,\mathrm{d}r-K(t),$

where $K(t)=\int_{0}^{t}\kappa(s)\mathrm{d}s$. From (2.22), it is easily seen
that

$I^{\kappa}\big{(}\Theta(\cdot,\lambda)\big{)}(t)=-\lambda\int_{0}^{\infty}\mathrm{e}^{-\lambda
r}\mathbb{E}[K(t-S_{r})\mathbbm{1}_{\\{t\geq S_{r}\\}}]\mathrm{d}r.$

On the other hand, (2.21) and Fubini’s theorem implies that

$\int_{0}^{t}\Theta(s,\lambda)\mathrm{d}s=\lambda\int_{0}^{\infty}\mathrm{e}^{-\lambda
r}\left(\int_{0}^{t}\mathbb{P}(S_{r}\geq
s)\mathrm{d}s\right)\mathrm{d}r=\int_{0}^{\infty}\mathrm{e}^{-\lambda
r}\mathbb{E}[K(t-S_{r})\mathbbm{1}_{\\{t\geq S_{r}\\}}]\mathrm{d}r.$

Therefore, we prove
$\partial_{t}^{\kappa}\Theta(t,\lambda)=-\lambda\Theta(t,\lambda)$ for
$t,\lambda\in\mathbb{R}_{+}$.

$(ii)$ Due to Proposition 2.8, $\phi\in\mathcal{I}_{o}(0,1)$. Then by (2.25),
there exists a constant $c_{2}=c_{2}(K_{\phi})=c_{2}(\kappa)>0$ such that
$\mathbb{P}(S_{r}\geq t)\geq 1-\mathrm{e}^{-c_{2}r\phi(t^{-1})}$ and then

$\Theta(t,\lambda)\geq\lambda\int_{0}^{\infty}\mathrm{e}^{-\lambda
r}\left(1-\mathrm{e}^{-c_{2}r\phi(t^{-1})}\right)\mathrm{d}r=1-\frac{\lambda}{\lambda+c_{2}\phi(t^{-1})}.$

Obviously, one can take $c_{2}<1$ in the above estimate. Then since $1\wedge
a\simeq a/(1+a)$ for $a>0$ and

$\left(1-\frac{\lambda}{\lambda+c_{2}\phi(t^{-1})}\right)^{-1}=\frac{\lambda}{c_{2}\phi(t^{-1})}+1\lesssim_{c_{2}}\frac{\lambda}{\phi(t^{-1})}+1,$

we obtain
$\Theta(t,\lambda)\gtrsim_{c_{2}}1\wedge\frac{\phi(t^{-1})}{\lambda}$.

Now, we prove the opposite inequality. Divide $\Theta$ into two parts;

$\Theta(t,\lambda)=\lambda\int_{0}^{\infty}\mathrm{e}^{-\lambda
r}\mathbb{P}(S_{r}\geq
t)\mathrm{d}r=\int_{0}^{2/\phi(t^{-1})}\cdots+\int_{2/\phi(t^{-1})}^{\infty}\cdots=:I_{1}(t,\lambda)+I_{2}(t,\lambda).$

By taking $L=2$ in (2.26), there exists a constant
$c_{2,L}=c_{2,L}(K_{\phi})=c_{2,L}(\kappa)>0$ such that $\mathbb{P}(S_{r}\geq
t)\leq c_{2,L}r\phi(t^{-1})$ for any $r$ and $t$ satisfying $r\phi(t^{-1})\leq
2(=L)$. Then,

$I_{1}(t,\lambda)\leq
c_{2,L}\phi(t^{-1})\lambda\int_{0}^{2/\phi(t^{-1})}r\mathrm{e}^{-\lambda
r}\,\mathrm{d}r.$

By direct calculation, one can check that

$\lambda\int_{0}^{2/\phi(t^{-1})}r\mathrm{e}^{-\lambda
r}\,\mathrm{d}r\leq\frac{1}{\lambda}\wedge\frac{2}{\phi(t^{-1})},$

and hence
$I_{1}(t,\lambda)\lesssim_{c_{2,L}}1\wedge\frac{\phi(t^{-1})}{\lambda}$. For
$I_{2}$, since $\mathbb{P}(S_{r}\geq t)\leq 1$ for all $r,t\in\mathbb{R}_{+}$,

$I_{2}(t,\lambda)\leq\lambda\int_{2/\phi(t^{-1})}^{\infty}\mathrm{e}^{-\lambda
r}\mathrm{d}r=\mathrm{e}^{-2\lambda/\phi(t^{-1})}\lesssim
1\wedge\frac{\phi(t^{-1})}{\lambda},$

where we use the fact that $\mathrm{e}^{-t}\lesssim 1\wedge t^{-1}$ for
$t\in\mathbb{R}_{+}$. We obtain
$\Theta(t,\lambda)\lesssim_{c_{2,L}}1\wedge\frac{\phi(t^{-1})}{\lambda}$. The
proposition is proved. ∎

We end this section by showing the equivalence of two equations:

$\displaystyle\int_{0}^{t}\kappa(t-s)(u(s)-u_{0})\mathrm{d}s=\int_{0}^{t}f(s)\mathrm{d}s\Longleftrightarrow\int_{0}^{t}(u(s)-u_{0})\mathrm{d}s=\int_{0}^{t}\varkappa(t-s)f(s)\mathrm{d}s,$
(2.27)

where $\varkappa(t):=\int_{0}^{\infty}\mathbb{P}(S_{r}\leq t)\,\mathrm{d}r$
and $S_{r}$ is a subordinator introduced in Proposition 2.11. By Proposition
2.11,

$\int_{0}^{t}\kappa(t-s)\varkappa(\mathrm{d}s)=\int_{0}^{\infty}\mathbb{E}[\kappa(t-S_{r})\mathbbm{1}_{t\geq
S_{r}}]\mathrm{d}r=\lim_{r\to{\infty}}\mathbb{P}(S_{r}\geq t)=1,$ (2.28)

and

$\displaystyle\int_{0}^{t}\kappa(t-s)\varkappa(s)\mathrm{d}s$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{t}\kappa(t-s)\mathbb{P}(S_{r}\leq
s)\mathrm{d}s\mathrm{d}r$ (2.29)
$\displaystyle=\int_{0}^{\infty}\mathbb{E}[K(t-S_{r})\mathbbm{1}_{\\{t\geq
S_{r}\\}}]\mathrm{d}r=\int_{0}^{t}\lim_{r\to\infty}\mathbb{P}(S_{r}\geq
l)\mathrm{d}l=t,$

where $K(t)=\int_{0}^{t}\kappa(s)\mathrm{d}s$. Taking the integration to the
left equation in (2.27) with respect to $\varkappa(\mathrm{d}s)$, and making
use of Fubini’s theorem and (2.28), we have the right equation in (2.27):

$\displaystyle\int_{0}^{t}(u(l)-u_{0})\mathrm{d}l$
$\displaystyle=\int_{0}^{t}\left(\int_{0}^{t-l}\kappa(t-l-s)\varkappa(\mathrm{d}s)\right)(u(l)-u_{0})\mathrm{d}l$
$\displaystyle=\int_{0}^{t}\int_{0}^{t-s}\kappa(t-s-l)(u(l)-u_{0})\mathrm{d}l\,\varkappa(\mathrm{d}s)$
$\displaystyle=\int_{0}^{t}\int_{0}^{t-s}f(l)\mathrm{d}l\,\varkappa(\mathrm{d}s)=\int_{0}^{t}\varkappa(t-l)f(l)\mathrm{d}l.$

Similarly, taking the non-local derivative $\partial_{t}^{\kappa}$ to the
right equation in (2.27), and making use of Fubini’s theorem and (2.29), we
have the left equation in (2.27):

$\displaystyle\int_{0}^{t}\kappa(t-l)(u(l)-u_{0})\mathrm{d}l$
$\displaystyle=\partial_{t}\left(\int_{0}^{t}K(t-l)(u(l)-u_{0})\mathrm{d}l\right)$
$\displaystyle=\partial_{t}^{\kappa}\left(\int_{0}^{t}(u(s)-u_{0})\mathrm{d}s\right)$
$\displaystyle=\partial_{t}^{\kappa}\left(\int_{0}^{t}\varkappa(t-s)f(s)\mathrm{d}s\right)$
$\displaystyle=\partial_{t}\left(\int_{0}^{t}\kappa(t-s)\left(\int_{0}^{s}\varkappa(s-l)f(l)\mathrm{d}l\right)\mathrm{d}s\right)$
$\displaystyle=\partial_{t}\left(\int_{0}^{t}(t-l)f(l)\mathrm{d}l\right)=\int_{0}^{t}f(l)\mathrm{d}l.$

##  3\. Generalized real interpolation

In this section, we present a generalized real interpolation in the sense that
we put $\phi(t^{-1})$ rather than $t^{-\theta}$ in (1.5). The function $\phi$
is chosen to be of class $\mathcal{I}_{o}(0,1)$ introduced in the first part
of Section 2. Among various interpolation methods, $K$ and $J$ methods are our
main focus.

######  Definition 3.1.

Given two Banach spaces $A_{0}$ and $A_{1}$, we say that an ordered pair
$(A_{0},A_{1})$ is an interpolation couple if both $A_{0}$ and $A_{1}$ are
continuously embedded in the same Hausdorff topological vector space $Z$.

It can be easily checked that the two subspaces of $Z$

$\begin{gathered}A_{0}\cap A_{1}:=\\{a\in Z:a\in A_{0},\,a\in A_{1}\\},\\\
A_{1}+A_{1}:=\\{a\in Z:a=a_{0}+a_{1},\,a_{0}\in A_{0},\,a_{1}\in
A_{1}\\}\end{gathered}$

are Banach spaces with the norm

$\begin{gathered}\|a\|_{A_{0}\cap
A_{1}}:=\max(\|a\|_{A_{0}},\|a\|_{A_{1}}),\\\
\|a\|_{A_{0}+A_{1}}:=\inf\\{\|a_{0}\|_{A_{0}}+\|a_{1}\|_{A_{1}}:a=a_{0}+a_{1},\,a_{0}\in
A_{0},\,a_{1}\in A_{1}\\}.\end{gathered}$

Through this section, we assume that $(A_{0},A_{1})$ is an interpolation
couple, $\phi\in\mathcal{I}_{o}(0,1)$, and $p\in[1,\infty]$.

For $t>0$ we define

$\displaystyle
K(t,a;A_{0},A_{1}):=\inf\\{\|a_{0}\|_{A_{0}}+t\|a_{1}\|_{A_{1}}:a=a_{0}+a_{1},a_{0}\in
A_{0},a_{1}\in A_{1}\\},$ $\displaystyle J(t,a;A_{0},A_{1}):=\max_{a\in
A_{0}\cap A_{1}}(\|a\|_{A_{0}},t\|a\|_{A_{1}}).$

Note that $K$ and $J$ are merely the functional used in $K$-method and
$J$-method of the classical interpolation theory, respectively. For measurable
functions $F:\mathbb{R}_{+}\rightarrow[0,\infty]$, the functional
$\Phi^{\phi}_{p}(F)$ is defined by

$\Phi^{\phi}_{p}(F):=\begin{cases}\left(\int_{0}^{\infty}\left(\phi(t^{-1})F(t)\right)^{p}\frac{\mathrm{d}t}{t}\right)^{1/p}\quad&\textrm{if}\quad
p\in[1,\infty),\\\ \sup_{t>0}\phi(t^{-1})F(t)\quad&\textrm{if}\quad
p=\infty.\end{cases}$

The space $(A_{0},A_{1})_{\phi,p}^{K}$ is defined by

$\begin{gathered}(A_{0},A_{1})_{\phi,p}^{K}:=\\{a\in
A_{0}+A_{1}:\|a\|_{(A_{0},A_{1})_{\phi,p}^{K}}:=\Phi^{\phi}_{p}(K(\cdot,a;A_{0},A_{1}))<\infty\\}.\end{gathered}$

The space $(A_{0},A_{1})_{\phi,p}^{J}$ is defined by

$\begin{gathered}(A_{0},A_{1})_{\phi,p}^{J}:=\\{a\in
A_{0}+A_{1}:\|a\|_{(A_{1},A_{1})_{\phi,p}^{J}}:=\inf_{v}\Phi^{\phi}_{p}(J(\cdot,v(\cdot);A_{0},A_{1}))<\infty\\},\end{gathered}$

where the infimum is over all measurable functions $v:\mathbb{R}_{+}\to
A_{0}\cap A_{1}$ satisfying

$a=\int_{0}^{\infty}v(t)\frac{\mathrm{d}t}{t}.$

We provide concrete examples of $(A_{0},A_{1})_{\phi,p}^{K}$ and
$(A_{0},A_{1})_{\phi,p}^{J}$ in Appendix A.

By $\ell_{p}$, we denote the set of all real sequences
$a=(a_{j})_{j\in\mathbb{Z}}$ satisfying $\|a\|_{\ell_{p}}<\infty$, where

$\|a\|_{\ell_{p}}:=\begin{cases}\left(\sum_{j\in\mathbb{Z}}|a_{j}|^{p}\right)^{1/p}\quad&\textrm{for}\quad
p\in[1,\infty),\\\ \sup_{j\in\mathbb{Z}}|a_{j}|\quad&\textrm{for}\quad
p=\infty.\end{cases}$

The following proposition suggests sequential expression for interpolation
norms:

######  Proposition 3.2.

  1. (i)

For $a\in(A_{0},A_{1})_{\phi,p}^{K}$,

$\|a\|_{(A_{0},A_{1})_{\phi,p}^{K}}\simeq_{\phi}\|\alpha^{\phi}_{K}(a)\|_{\ell_{p}},$

where
$\alpha^{\phi}_{K}(a):=\big{(}\alpha^{\phi}_{K,j}(a)\big{)}_{j\in\mathbb{Z}}:=\big{(}\phi(2^{-j})K(2^{j},a;A_{0},A_{1})\big{)}_{j\in\mathbb{Z}}$.

  2. (ii)

For $a\in(A_{0},A_{1})_{\phi,p}^{J}$,

$\|a\|_{(A_{0},A_{1})_{\phi,p}^{J}}\simeq_{\phi}\inf_{u}\|\alpha^{\phi}_{J}(u)\|_{\ell_{p}},$

where
$\alpha^{\phi}_{J}(u):=(\alpha^{\phi}_{J,j})_{j\in\mathbb{Z}}:=\big{(}\phi(2^{-j})J(2^{j},u_{j};A_{0},A_{1})\big{)}_{j\in\mathbb{Z}}$
and the infimum is over all sequences $(u_{j})_{j\in\mathbb{Z}}\subset
A_{0}\cap A_{1}$ satisfying

$a=\sum_{j\in\mathbb{Z}}u_{j}\,\,\textrm{(convergence in }A_{0}+A_{1}).$

###### Proof.

$(i)$ It can be easily checked that for $t\in[2^{j},2^{j+1}]$,
$j\in\mathbb{Z}$,

$K(2^{j},a;A_{0},A_{1})\leq K(t,a;A_{0},A_{1})\leq 2K(2^{j},a;A_{0},A_{1}).$

Since $\phi\in\mathcal{I}_{o}(0,1)$, by Lemma 2.3-$(vi)$, we have

$1\leq(2^{-j}t)^{\varepsilon}\lesssim\frac{\phi(2^{-j})}{\phi(t^{-1})}\lesssim(2^{-j}t)^{1-\varepsilon}\leq
2^{1-\varepsilon}\quad\textrm{for all}\,\,t\in[2^{j},2^{j+1}]\,,$

i.e.,

$\phi(t^{-1})\simeq\phi(2^{-j})\quad\textrm{for all}\,\,t\in[2^{j},2^{j+1}].$
(3.1)

This certainly implies that

$\|a\|_{(A_{0},A_{1})_{\phi,p}^{K}}\simeq_{\phi}\|\alpha^{\phi}_{K}(a)\|_{\ell_{p}}.$

$(ii)$ We only prove for $p\in[1,\infty)$ since the proof of the case
$p=\infty$ is similar. Since $a\in(A_{0},A_{1})_{\phi,p}^{J}$, there exists a
measurable function $u:\mathbb{R}_{+}\to A_{0}\cap A_{1}$ such that
$a=\int_{0}^{\infty}u(t)\frac{\mathrm{d}t}{t}$. Put

$u_{j}:=\int_{2^{j}}^{2^{j+1}}u(t)\frac{\mathrm{d}t}{t}\in A_{0}\cap A_{1},$

then $a=\sum_{j\in\mathbb{Z}}u_{j}$. By the definition of $J$-functional and
Minkowski’s inequality,

$\alpha_{J,j}^{\phi}\leq\phi(2^{-j})\int_{2^{j}}^{2^{j+1}}\max(\|u\|_{A_{0}},t\|u\|_{A_{1}})\,\frac{\mathrm{d}t}{t}\lesssim\int_{2^{j}}^{2^{j+1}}\phi(t^{-1})J(t,u(t);A_{0},A_{1})\frac{\mathrm{d}t}{t},$

where we use (3.1) for $t\in[2^{j},2^{j+1}]$. This implies that
$\|\alpha^{\phi}_{J}(u)\|_{\ell_{p}}\lesssim_{\phi}\Phi^{\phi}_{p}\big{(}J(\cdot,u;A_{0},A_{1})\big{)}$,
and then,

$\inf_{u}\|\alpha^{\phi}_{J}(u)\|_{\ell_{p}}\lesssim_{\phi}\|a\|_{(A_{0},A_{1})_{\phi,p}^{J}}.$

For the converse, assume that there exists
$u=(u_{j})_{j\in\mathbb{Z}}\subseteq A_{0}\cap A_{1}$ such that

$\alpha_{J}^{\phi}(u)\in\ell_{p},\quad a=\sum_{j\in\mathbb{Z}}u_{j}.$

Set

$v(t):=\frac{1}{\log
2}\sum_{j\in\mathbb{Z}}u_{j}\mathbbm{1}_{(2^{j},2^{j+1}]}(t)\in A_{0}\cap
A_{1},$

then $a=\int_{0}^{\infty}v(t)\frac{\mathrm{d}t}{t}$. Clearly, $v$ is (Bochner)
measurable. Therefore, by (3.1) for $t\in[2^{j},2^{j+1}]$ again, we have

$\displaystyle\|a\|_{(A_{0},A_{1})_{\phi,p}^{J}}^{p}\leq\Phi^{\phi}_{p}\big{(}J(\cdot,v;A_{0},A_{1})\big{)}^{p}$
$\displaystyle=\sum_{j\in\mathbb{Z}}\int_{2^{j}}^{2^{j+1}}\big{(}\phi(t^{-1})J(t,v(t);A_{0},A_{1})\big{)}^{p}\frac{\mathrm{d}t}{t}$
$\displaystyle\lesssim\sum_{j\in\mathbb{Z}}(\phi(2^{-j})J(2^{j},u_{j};A_{0},A_{1}))^{p}=\|\alpha_{J}^{\phi}(u)\|_{\ell_{p}}^{p},$

and taking the infimum in $u$, we have the result. The proposition is proved.
∎

######  Lemma 3.3.

For $a\in(A_{0},A_{1})_{\phi,p}^{J}$, there exists an infinitely
differentiable function $v:\mathbb{R}_{+}\to A_{0}\cap A_{1}$ such that
$a=\int_{0}^{\infty}v(t)\frac{\mathrm{d}t}{t}$.

###### Proof.

Let $\varphi$ be a nonnegative infinitely differentiable function with compact
support in $(0,\infty)$,

$\int_{0}^{\infty}\varphi(s^{-1})\frac{\mathrm{d}s}{s}=1.$ (3.2)

Since $a\in(A_{0},A_{1})_{\phi,p}^{J}$, there exists measurable
$u:\mathbb{R}_{+}\to A_{0}\cap A_{1}$ such that
$a=\int_{0}^{\infty}u(t)\frac{\mathrm{d}t}{t}$. Put

$v(t):=\int_{0}^{\infty}\varphi\left(\frac{t}{s}\right)u\left(s\right)\frac{\mathrm{d}s}{s}.$

Clearly, $v$ is an infinitely differentiable function from $\mathbb{R}_{+}$ to
$A_{0}\cap A_{1}$. By Fubini’s theorem (e.g. [13, Proposition 1.2.7.]) and
(3.2),

$a=\int_{0}^{\infty}v(t)\frac{\mathrm{d}t}{t}.$

The lemma is proved. ∎

######  Lemma 3.4.

For any $a\in A_{0}+A_{1}$,

$K(t,a;A_{0},A_{1})\lesssim_{\phi,p}\frac{\|a\|_{(A_{0},A_{1})_{\phi,p}^{K}}}{\phi(t^{-1})}$

###### Proof.

The case $p=\infty$ is trivial, so we only prove the case $p\in[1,\infty)$. By
the definition of $K$-functional, for $t,s>0$,

$K(t,a;A_{0},A_{1})\leq\max(1,t/s)K(s,a:A_{0},A_{1}).$

This also implies that

$\min(1,s/t)K(t,a;A_{0},A_{1})\leq K(s,a;A_{0},A_{1}).$

Taking $\Phi^{\phi}_{p}$ both sides with respect to $s$,

$\Phi^{\phi}_{p}(\min(1,\cdot/t))K(t,a,;A_{0},A_{1})\leq\|a\|_{(A_{0},A_{1})_{\phi,p}^{K}}.$

Therefore it suffices to prove that
$\Phi^{\phi}_{p}(\min(1,\cdot/t))\gtrsim\phi(t^{-1})$, and this can be
verified by Lemma 2.3-$(vi)$ since

$\frac{\Phi^{\phi}_{p}(\min(1,\cdot/t))}{\phi(t^{-1})}=\left(\int_{0}^{\infty}\left|\frac{\phi(s^{-1})}{\phi(t^{-1})}\left(1\wedge\frac{s}{t}\right)\right|^{p}\,\frac{\mathrm{d}s}{s}\right)^{1/p}.$

The lemma is proved. ∎

We present two lemmas to show equivalence between $K$ and $J$ methods (Theorem
3.5) and further properties of our generalized interpolation (Proposition
3.6).

######  Theorem 3.5 (Equivalence theorem).

$(A_{0},A_{1})_{\phi,p}^{K}$ and $(A_{0},A_{1})_{\phi,p}^{J}$ have same
elements. In addition, for any
$a\in(A_{0},A_{1})_{\phi,p}^{K}\big{(}=(A_{0},A_{1})_{\phi,p}^{J}\big{)}$,

$\|a\|_{(A_{0},A_{1})_{\phi,p}^{K}}\simeq\|a\|_{(A_{0},A_{1})_{\phi,p}^{J}}.$

###### Proof.

The proof is comparable to the proof of classical real interpolation. However,
for the sake of completeness, we present proof. Let
$a\in(A_{0},A_{1})_{\phi,p}^{J}$. Then by Lemma 3.3, there exists an
infinitely differentiable $v:\mathbb{R}_{+}\to A_{0}\cap A_{1}$ such that
$a=\int_{0}^{\infty}v(t)\frac{\mathrm{d}t}{t}$. Then by [2, Lemma 3.2.1.],

$\displaystyle K(t,a;A_{0},A_{1})$
$\displaystyle\leq\int_{0}^{\infty}K(t,v(s);A_{0},A_{1})\frac{\mathrm{d}s}{s}\leq\int_{0}^{\infty}\min(1,t/s)J(s,v(s);A_{0},A_{1})\frac{\mathrm{d}s}{s}$
$\displaystyle=\int_{0}^{\infty}\min(1,s^{-1})J(ts,v(ts);A_{0},A_{1})\frac{\mathrm{d}s}{s}.$

Since $\phi\in\mathcal{I}_{o}(0,1)$, by Lemma 2.3-$(vi)$, there exists
$\varepsilon>0$ such that

$\frac{\phi(t^{-1}s)}{\phi(t^{-1})}\lesssim s^{\varepsilon}\mathbbm{1}_{s\leq
1}+s^{1-\varepsilon}\mathbbm{1}_{s>1}.$ (3.3)

Then by (3.3), for $p\in[1,\infty)$,

$\displaystyle\|a\|_{(A_{0},A_{1})_{\phi,p}^{K}}$
$\displaystyle\leq\int_{0}^{\infty}\left(\int_{0}^{\infty}\big{(}\phi(t^{-1})J(ts,v(ts);A_{0},A_{1})\big{)}^{p}\frac{\mathrm{d}t}{t}\right)^{1/p}\min(1,s^{-1})\frac{\mathrm{d}s}{s}$
$\displaystyle=\int_{0}^{\infty}\left(\int_{0}^{\infty}\big{(}\phi(t^{-1}s)J(t,v(t);A_{0},A_{1})\big{)}^{p}\frac{\mathrm{d}t}{t}\right)^{1/p}\min(1,s^{-1})\frac{\mathrm{d}s}{s}$
$\displaystyle\lesssim\Phi^{\phi}_{p}(v)\int_{0}^{\infty}(s^{\varepsilon}\mathbbm{1}_{s\leq
1}+s^{1-\varepsilon}\mathbbm{1}_{s>1})\min(1,s^{-1})\frac{\mathrm{d}s}{s}\lesssim\Phi^{\phi}_{p}(v).$

Taking the infimum in $v$, we have

$(A_{0},A_{1})_{\phi,p}^{J}\subseteq(A_{0},A_{1})_{\phi,p}^{K}.$

The case $p=\infty$ is almost the same.

For the converse, note that by Lemma 2.3-($vi$) and Lemma 3.4,

$\lim_{t\to\infty}\min(1,t^{-1})K(t,a;A_{0},A_{1})=\lim_{t\downarrow
0}\min(1,t^{-1})K(t,a;A_{0},A_{1})=0,$

for all $a\in(A_{0},A_{1})_{\phi,p}^{K}$. Since the assumption in [2, Lemma
3.3.2.] is satisfied, by this lemma, there exists
$u=(u_{j})_{j\in\mathbb{Z}}\subseteq A_{0}\cap A_{1}$ such that

$a=\sum_{j\in\mathbb{Z}}u_{j},\quad J(2^{j},u_{j};A_{0},A_{1})\leq
4K(2^{j},a;A_{0},A_{1}).$

Then by Proposition 3.2, we have
$(A_{0},A_{1})_{\phi,p}^{K}\subseteq(A_{0},A_{1})_{\phi,p}^{J}$. The theorem
is proved. ∎

Note that the inequality (3.3) actually depends on $\phi$ itself. Thus, if we
put $\phi=W\circ\psi$, then we have

$\|a\|_{(A_{0},A_{1})_{W\circ\psi,p}^{K}}\simeq_{W\circ\psi}\|a\|_{(A_{0},A_{1})_{W\circ\psi,p}^{J}}$

Due to the equivalence theorem, we now denote

$(A_{0},A_{1})_{\phi,p}:=(A_{0},A_{1})_{\phi,p}^{K}\big{(}=(A_{0},A_{1})_{\phi,p}^{J}\big{)}\quad\textrm{and}\quad\|a\|_{(A_{0},A_{1})_{\phi,p}}:=\|a\|_{(A_{0},A_{1})^{K}_{\phi,p}}.$
(3.4)

The space $(A_{0},A_{1})_{\phi,p}$ satisfies the following properties.

######  Proposition 3.6.

  1. (i)

For $1\leq p_{0}\leq p_{1}\leq\infty$, we have

$A_{0}\cap
A_{1}\subset(A_{0},A_{1})_{\phi,p_{0}}\subset(A_{0},A_{1})_{\phi,p_{1}}\subset
A_{0}+A_{1},$

that is,

$\|a\|_{A_{0}+A_{1}}\lesssim\|a\|_{(A_{0},A_{1})_{\phi,p_{1}}}\lesssim\|a\|_{(A_{0},A_{1})_{\phi,p_{0}}}\lesssim\|a\|_{A_{0}\cap
A_{1}}.$

  2. (ii)

Endowed with the norm $\|\cdot\|_{(A_{0},A_{1})_{\phi,p}}$ the space
$(A_{0},A_{1})_{\phi,p}$ is a Banach space.

  3. (iii)

For $\tilde{\phi}(t):=t\phi(t^{-1})$,
$(A_{0},A_{1})_{\phi,p}=(A_{1},A_{0})_{\tilde{\phi},p}$.

  4. (iv)

Suppose that that $A_{0}$ be continuously embedded into $A_{1}$. If
$\phi,\,\psi\in\mathcal{I}_{o}(0,1)$ satisfies that

$\displaystyle N_{1}^{-1}\phi(s)\leq\psi(s)\leq
N_{1}\phi(s)\quad\forall\,\,0<s\leq c\,,$ (3.5)

for some fixed $c>0$, then $(A_{0},A_{1})_{\phi,p}=(A_{0},A_{1})_{\psi,p}$.
Moreover, for any $a\in(A_{0},A_{1})_{\phi,p}\,(=(A_{0},A_{1})_{\psi,p})$,

$\displaystyle\|a\|_{(A_{0},A_{1})_{\phi,p}}\simeq\|a\|_{(A_{0},A_{1})_{\psi,p}}.$
(3.6)

  5. (v)

If $a\in A_{0}\cap A_{1}$ and $a\neq 0$, then

$\|a\|_{(A_{0},A_{1})_{\phi,p}}\lesssim\|a\|_{A_{0}}\,\phi\left(\|a\|_{A_{1}}/\|a\|_{A_{0}}\right).$

###### Proof.

$(i)$ The first and third embeddings follow from

$\min(1,t)\|a\|_{A_{0}+A_{1}}\leq
K(t,a;A_{0},A_{1})\leq\min(1,t)\|a\|_{A_{0}\cap A_{1}}$

and Lemma 2.3-$(vi)$. For the second embedding, by Lemma 3.4,
($\frac{\infty}{\infty}:=1$)

$\displaystyle\|a\|_{(A_{0},A_{1})_{\phi,p_{1}}}$
$\displaystyle\leq\Phi_{p_{0}}^{\phi}(K(\cdot,a;A_{0},A_{1}))^{p_{0}/p_{1}}\Phi_{\infty}^{\phi}(K(\cdot,a;A_{0},A_{1}))^{1-p_{0}/p_{1}}$
$\displaystyle\lesssim\|a\|_{(A_{0},A_{1})_{\phi,p_{0}}}^{p_{0}/p_{1}}\|a\|_{(A_{0},A_{1})_{\phi,p_{0}}}^{1-p_{0}/p_{1}}=\|a\|_{(A_{0},A_{1})_{\phi,p_{0}}}.$

$(ii)$ The case $p=\infty$ is clear, thus we prove completeness only for
$p\in[1,\infty)$. Let $\\{a_{n}\\}_{n=1}^{\infty}$ be a Cauchy sequence in
$(A_{0},A_{1})_{\phi,p}$. By $(i)$, $\\{a_{n}\\}_{n=1}^{\infty}$ is also a
Cauchy sequence in $A_{0}+A_{1}$, hence there exists
$\lim_{n\rightarrow\infty}a_{n}=:a$ in $A_{0}+A_{1}$. For given
$\varepsilon>0$, there is $N>0$ such that if $n,m\geq N$, then
$\|a_{n}-a_{m}\|_{(A_{0},A_{1})_{\phi,p}}<\varepsilon$. Therefore for $0<r\leq
R<\infty$, we have

$\displaystyle\left(\int_{r}^{R}\big{(}\phi(t^{-1})K(t,a-a_{n};A_{0},A_{1})\big{)}^{p}\frac{\mathrm{d}t}{t}\right)^{1/p}$
$\displaystyle<\varepsilon+\left(\int_{r}^{R}\big{(}\phi(t^{-1})K(t,a-a_{m};A_{0},A_{1})\big{)}^{p}\frac{\mathrm{d}t}{t}\right)^{1/p}.$

Using $K(t,a-a_{m};A_{0},A_{1})\leq\max(1,t)\|a-a_{m}\|_{A_{0}+A_{1}}$ and
letting $m\rightarrow\infty$, we have

$\left(\int_{r}^{R}\big{(}\phi(t^{-1})K(t,a-a_{n};A_{0},A_{1})\big{)}^{p}\frac{\mathrm{d}t}{t}\right)^{1/p}<\varepsilon.$

Since $r,\,R$ are arbitrary constant, by letting $r\rightarrow
0,\,R\rightarrow\infty$, we obtain that
$\|a_{n}-a\|_{(A_{0},A_{1})_{\phi,p}}\leq\varepsilon$. Therefore we conclude
that $a\in(A_{0},A_{1})_{\phi,p}$ and $a_{n}$ converges to $a$ in
$(A_{0},A_{1})_{\phi,p}$.

$(iii)$ By Lemma 2.3 $(ix)$, $\tilde{\phi}\in\mathcal{I}_{o}(0,1)$. Since

$K(t,a;A_{1},A_{0})=tK(t^{-1},a;A_{0},A_{1}),$

we have

$\displaystyle\|a\|_{(A_{1},A_{0})_{\tilde{\phi},p}}^{p}$
$\displaystyle=\int_{0}^{\infty}\left(\tilde{\phi}(t^{-1})K(t,a;A_{1},A_{0})\right)^{p}\frac{\mathrm{d}t}{t}$
$\displaystyle=\int_{0}^{\infty}\left(\phi(t)K(t^{-1},a;A_{0},A_{1})\right)^{p}\frac{\mathrm{d}t}{t}=\|a\|_{(A_{0},A_{1})_{\phi,p}}^{p}.$

$(iv)$ Due to the definition of the interpolation spaces, we only need to
prove that for any $a\in A_{0}+A_{1}$, (3.6) holds. Let us fix $a\in
A_{0}+A_{1}\,(\subseteq A_{1})$. Since (3.5) implies that

$\int_{c^{-1}}^{\infty}\big{|}\phi(t^{-1})K(t,a;A_{0},A_{1})\big{|}^{p}\,\frac{\mathrm{d}t}{t}\simeq\int_{c^{-1}}^{\infty}\big{|}\psi(t^{-1})K(t,a;A_{0},A_{1})\big{|}^{p}\,\frac{\mathrm{d}t}{t}\,,$

it is sufficient to prove that

$\displaystyle\int_{0}^{c^{-1}}\big{|}\phi(t^{-1})K(t,a;A_{0},A_{1})\big{|}^{p}\,\frac{\mathrm{d}t}{t}$
$\displaystyle\lesssim\int_{0}^{\infty}\big{|}\psi(t^{-1})K(t,a;A_{0},A_{1})\big{|}^{p}\,\frac{\mathrm{d}t}{t},$
(3.7)
$\displaystyle\int_{0}^{c^{-1}}\big{|}\psi(t^{-1})K(t,a;A_{0},A_{1})\big{|}^{p}\,\frac{\mathrm{d}t}{t}$
$\displaystyle\lesssim\int_{0}^{\infty}\big{|}\phi(t^{-1})K(t,a;A_{0},A_{1})\big{|}^{p}\,\frac{\mathrm{d}t}{t}.$
(3.8)

Due to the similarity, we only prove (3.7). Since $a\in A_{1}$, we have

$K(t,a;A_{0},A_{1})\leq t\|a\|_{A_{1}}\quad\forall t>0\,,$

which implies

$\displaystyle\int_{0}^{c^{-1}}\big{|}\phi(t^{-1})K(t,a;A_{0},A_{1})\big{|}^{p}\,\frac{\mathrm{d}t}{t}\lesssim\|a\|_{A_{1}}$
(3.9)

(recall that $\phi\in\mathcal{I}_{o}(0,1)$). Since $A_{0}$ is continuously
embedded in $A_{1}$, we obtain that for any $a_{0}\in A_{0}\,(\subset A_{1})$
and $a_{1}\in A_{1}$ with $a=a_{0}+a_{1}$,

$\displaystyle\|a\|_{A_{1}}\leq\|a_{0}\|_{A_{1}}+\|a_{1}\|_{A_{1}}\lesssim\|a_{0}\|_{A_{0}}+\|a_{1}\|_{A_{1}}\lesssim\|a_{0}\|_{A_{0}}+t\|a_{1}\|_{A_{1}}\quad\forall\,\,t\geq
c^{-1}.$

Therefore

$\|a\|_{A_{1}}\lesssim K(a,t;A_{0},A_{1})\quad\forall t\geq c^{-1}\,,$

which implies

$\displaystyle\|a\|_{A_{1}}\lesssim\int_{c^{-1}}^{c^{-2}}\big{|}\psi(t^{-1})K(t,a;A_{0},A_{1})\big{|}^{p}\,\frac{\mathrm{d}t}{t}\leq\int_{c^{-1}}^{\infty}\big{|}\psi(t^{-1})K(t,a;A_{0},A_{1})\big{|}^{p}\,\frac{\mathrm{d}t}{t}$
(3.10)

(note that $\psi(t^{-1})\simeq\psi(1)$ for all $1\leq t\leq 2$). By combining
(3.9) and (3.10), (3.7) is obtained.

$(v)$ Put $\alpha:=\|a\|_{A_{0}}/\|a\|_{A_{1}}$. Then

$\displaystyle
K(t,a;A_{0},A_{1})\leq\min(\|a\|_{0},t\|a\|_{A_{1}})=\|a\|_{A_{0}}\min\Big{(}1,\frac{t}{\alpha}\Big{)}\,,$
(3.11)

and due to $\phi\in\mathcal{I}_{o}(0,1)$,

$\displaystyle\phi(t^{-1})\lesssim\phi(\alpha^{-1})\max\left(\left(\frac{\alpha}{t}\right)^{\varepsilon},\left(\frac{\alpha}{t}\right)^{1-\varepsilon}\right)$
(3.12)

for some $\varepsilon\in(0,1/2)$. (3.11) and (3.11) imply that

$\phi(t^{-1})K(t,a;A_{0},A_{1})\lesssim\|a\|_{A_{0}}\phi(\alpha^{-1})\min\Big{(}\big{(}\frac{\alpha}{t}\big{)}^{\varepsilon},\big{(}\frac{t}{\alpha}\big{)}^{\varepsilon}\Big{)}.$

Therefore, we have

$\displaystyle\|a\|_{(A_{0},A_{1})_{\phi,p}}:=$
$\displaystyle\left(\int_{0}^{\infty}\left|\phi\left(t^{-1}\right)K\left(t,a;A_{0},A_{1}\right)\right|^{p}\,\frac{\mathrm{d}t}{t}\right)^{1/p}$
$\displaystyle\lesssim$
$\displaystyle\|a\|_{A_{0}}\phi(\alpha^{-1})\left(\int_{0}^{\infty}\left|\min\left(\frac{\alpha}{t},\frac{t}{\alpha}\right)\right|^{p\varepsilon}\,\frac{\mathrm{d}t}{t}\right)^{1/p}$
$\displaystyle\lesssim$
$\displaystyle\|a\|_{A_{0}}\phi\left(\|a\|_{A_{1}}/\|a\|_{A_{0}}\right).$

The lemma is proved. ∎

To compute interpolation norms of examples in Appendix, it might be helpful if
one has a stability theorem such as Theorem 1.10.2 in [25]. We first give a
definition for the stability theorem. Note that for $\theta\in(0,1)$ and
$p\in[1,\infty]$, if we take $\phi(t)=t^{\theta}\in\mathcal{I}_{o}(0,1)$, then
$(A_{0},A_{1})_{\phi,p}$ coincides with the classical real interpolation space
$(A_{0},A_{1})_{\theta,p}$.

######  Definition 3.7.

We say a Banach space $E$ belongs to the class $J(\theta,A_{0},A_{1})$ if

$(A_{0}\cap A_{1}\subset\,)\,\,\,(A_{0},A_{1})_{\theta,1}\subset E\subset
A_{0}+A_{1},$

and $E$ belongs to the class $K(\theta,A_{0},A_{1})$ if

$A_{0}\cap A_{1}\subset E\subset(A_{0},A_{1})_{\theta,\infty}\,\,\,(\,\subset
A_{0}+A_{1}).$

From the above notions, we have the following lemma:

######  Lemma 3.8 ([25], Lemma 1.10.1).

Let $E$ be a Banach space such that $A_{0}\cap A_{1}\subset E\subset
A_{0}+A_{1}$ and $\theta\in(0,1)$.

  1. (i)

The following statements are equivalent.

    1. (a)

$E$ is of class $J(\theta)$.

    2. (b)

There exists a positive constant $c$ such that for all $a\in A_{0}\cap A_{1}$,

$\|a\|_{E}\leq c\|a\|_{A_{0}}^{1-\theta}\|a\|_{A_{1}}^{\theta}.$

    3. (c)

There exists a positive constant $c$ such that for all $a\in A_{0}\cap A_{1}$
and $t\in(0,\infty)$,

$\|a\|_{E}\leq ct^{-\theta}J(t,a;A_{0},A_{1}).$

  2. (ii)

$E$ is of class $K(\theta)$ if and only if there exists a positive constant
$c$ such that for all $a\in E$ and $t\in(0,\infty)$, $K(t,a;A_{0},A_{1})\leq
ct^{\theta}\|a\|_{E}$.

Using Lemma 3.8, we state a $\phi$-analogue of the stability theorem (Theorem
1.10.2 in [25]), and it will be used to compute examples in Appendix .

######  Theorem 3.9 (Stability theorem).

Let $\theta_{0},\,\theta_{1}\in[0,1]$ with $\theta_{0}\neq\theta_{1}$,
$E_{j}\in K(\theta_{j})\cap J(\theta_{j})$, $j=0,1$, and $p\in[1,\infty]$.
Then

$(E_{0},E_{1})_{\phi,p}=(A_{0},A_{1})_{\psi,p},\quad\psi(s)=s^{\theta_{0}}\phi(s^{\theta_{1}-\theta_{0}}).$

###### Proof.

Due to Proposition 3.6.(iii), it suffices to prove for the case
$0\leq\theta_{0}<\theta_{1}\leq 1$. Note that
$\psi\in\mathcal{I}_{o}(\theta_{0},\theta_{1})$, and observe that

$\phi(s)=s^{-\frac{\theta_{0}}{\theta_{1}-\theta_{0}}}\psi(s^{\frac{1}{\theta_{1}-\theta_{0}}}),$

We borrow the argument in [25, Theorem 1.10.2] and only deal with the case
$p\in[1,\infty)$ since proof for the case $p=\infty$ is much simpler.

We first prove $(E_{0},E_{1})_{\phi,p}\subset(A_{0},A_{1})_{\psi,p}$. For
$a\in(E_{0},E_{1})_{\phi,p}$, take arbitrary $e_{0}\in E_{0}$ and $e_{1}\in
E_{1}$ satisfying $a=e_{0}+e_{1}$. By Lemma 3.8-($ii$),

$\displaystyle K(t,a;A_{0},A_{1})$ $\displaystyle\leq
K(t,e_{0};A_{0},A_{1})+K(t,e_{1};A_{0},A_{1})$ $\displaystyle\lesssim
t^{\theta_{0}}\|e_{0}\|_{E_{0}}+t^{\theta_{1}}\|e_{1}\|_{E_{1}}$
$\displaystyle=t^{\theta_{0}}\left(\|e_{0}\|_{E_{0}}+t^{\theta_{1}-\theta_{0}}\|e_{1}\|_{E_{1}}\right).$

Taking the infimum for $e_{0},\,e_{1}$ gives

$K(t,a;A_{0},A_{1})\lesssim
t^{\theta_{0}}K(t^{\theta_{1}-\theta_{0}},a;E_{0},E_{1}).$

Then it follows that

$\displaystyle\|a\|_{(A_{0},A_{1})_{\psi,p}}$
$\displaystyle=\left\|\psi(t^{-1})K(t,a;A_{0},A_{1})\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}$
$\displaystyle\lesssim\left\|\psi(t^{-1})t^{\theta_{0}}K\big{(}t^{\theta_{1}-\theta_{0}},a;E_{0},E_{1}\big{)}\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}$
$\displaystyle\simeq\left\|\psi(t^{-\frac{1}{\theta_{1}-\theta_{0}}})t^{\frac{\theta_{0}}{\theta_{1}-\theta_{0}}}K(t,a;E_{0},E_{1})\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}$
$\displaystyle=\|a\|_{(E_{0},E_{1})_{\phi,p}}.$

Hence we have $(E_{0},E_{1})_{\phi,p}\subset(A_{0},A_{1})_{\psi,p}$.

We now prove $(E_{0},E_{1})_{\phi,p}\supset(A_{0},A_{1})_{\psi,p}$. For a
fixed $a\in(A_{0},A_{1})_{\psi,p}$, take $u:\mathbb{R}_{+}\rightarrow
A_{0}\cap A_{1}$ such that

$\displaystyle
a=\int_{0}^{\infty}u(t^{\theta_{1}-\theta_{0}})\frac{\mathrm{d}t}{t},\quad\big{\|}\psi(t^{-1})J(t,u(t^{\theta_{1}-\theta_{0}});A_{0},A_{1})\big{\|}_{L_{p}(\mathbb{R}_{+},\frac{\mathrm{d}t}{t})}\leq
2\|a\|_{(A_{0},A_{1})_{\psi,p}^{J}}.$ (3.13)

Since $A_{0}\cap A_{1}\subset E_{0}\cap E_{1}$, the integral in (3.13) holds
in $E_{0}\cap E_{1}$. By Lemma 3.8-($c$), we obtain that for any
$\tau\in(0,\infty)$

$\displaystyle J(t,u(t);E_{0},E_{1})$
$\displaystyle=\max\left(\|u(t)\|_{E_{0}},t\|u(t)\|_{E_{1}}\right)$
$\displaystyle\lesssim\max\left(\tau^{-\theta_{0}}J(\tau,u(t);A_{0},A_{1}),t\tau^{-\theta_{1}}J(\tau,u(t);A_{0},A_{1})\right).$

Take $\tau^{\theta_{1}-\theta_{0}}=t$ so that $J(t,u(t);E_{0},E_{1})\lesssim
t^{-\frac{\theta_{0}}{\theta_{1}-\theta_{0}}}J(t^{\frac{1}{\theta_{1}-\theta_{0}}},u(t);A_{0},A_{1})$.
Finally, it follows that

$\displaystyle\|a\|_{(E_{0},E_{1})_{\phi,p}^{J}}\,$
$\displaystyle\leq\left\|\phi(t^{-1})J(t,u(t);E_{0},E_{1})\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}$
$\displaystyle\lesssim\left\|t^{\frac{\theta_{0}}{\theta_{1}-\theta_{0}}}\psi(t^{-\frac{1}{\theta_{1}-\theta_{0}}})t^{-\frac{\theta_{0}}{\theta_{1}-\theta_{0}}}J(t^{\frac{1}{\theta_{1}-\theta_{0}}},u(t);A_{0},A_{1})\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}$
$\displaystyle\simeq\left\|\psi(t^{-1})J\big{(}t,u(t^{\theta_{1}-\theta_{0}});A_{0},A_{1}\big{)}\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}\leq
2\|a\|_{(A_{0},A_{1})_{\psi,p}^{J}}$

Hence we have $(A_{0},A_{1})_{\psi,p}\subset(E_{0},E_{1})_{\phi,p}$. The
theorem is proved. ∎

##  4\. Proof of Main Results

We begin with definitions and lemmas, which are crucially used in our proof.
Proofs of Theorems 1.1, 1.2, 1.5 and 1.6 are given in Subsections 4.1 and 4.2.
We first prove the results with local derivative (Theorems 1.1 and 1.5) in
Section 4.1. The non-local results for half line case (Theorems 1.2 and 1.6)
are proved in Section 4.2.

######  Definition 4.1.

Let $p\in[1,\infty]$, $v:\mathbb{R}\to[0,\infty)$ be a measurable function,
and $X$ be a Banach space. For a $X$-valued measurable function $f=f(t)$, we
say $f\in L_{p}(\mathcal{O},v\,\mathrm{d}t;X)$ if

$\|f\|_{L_{p}(\mathcal{O},v\,\mathrm{d}t;X)}:=\begin{cases}\left(\int_{\mathcal{O}}\|f(t)\|_{X}^{p}v(t)\,\mathrm{d}t\right)^{1/p}<\infty\quad&\textrm{for}\quad
p\in[1,\infty),\\\
\sup_{t\in\mathcal{O}}\|f(t)\|_{X}<\infty\quad&\textrm{for}\quad
p=\infty.\end{cases}$

Here $\mathcal{O}$ is an open subset of $\mathbb{R}_{+}$. If $X=\mathbb{R}$,
we simply set
$L_{p}(\mathcal{O},v\,\mathrm{d}t;X)=L_{p}(\mathcal{O},v\,\mathrm{d}t)$.

######  Definition 4.2.

Let $p\in(1,\infty)$. For a nonnegative measurable function
$w:\mathbb{R}\to[0,\infty)$, we say $w\in A_{p}(\mathbb{R})(=A_{p})$
(Muckenhoupt $A_{p}$-weight) if

$[w]_{A_{p}(\mathbb{R})}=[w]_{A_{p}}:=\sup_{[a,b]\subset\mathbb{R}}\left(\frac{1}{b-a}\int_{a}^{b}w(t)\,\mathrm{d}t\right)\left(\frac{1}{b-a}\int_{a}^{b}w(t)^{-\frac{1}{p-1}}\,\mathrm{d}t\right)^{p-1}<\infty.$

For $p=1$, we define

$[w]_{A_{1}(\mathbb{R})}=[w]_{A_{1}}:=\sup_{[a,b]\subset\mathbb{R}}\left[\left(\frac{1}{b-a}\int_{a}^{b}w\left(t\right)\,\mathrm{d}t\right)\sup_{t\in[a,b]}\left(w\left(t\right)^{-1}\right)\right]<\infty.$

For given $w\in A_{p}$, we denote $W(t):=\int_{0}^{t}w(s)\,\mathrm{d}s$.

While this paper focuses on evolution equations in $\mathbb{R}_{+}$, for
convenience, we consider Muckenhoupt $A_{p}$-weight on $\mathbb{R}$. It is
worth noting that if a function $w_{0}$, defined on $[0,\infty)$, is a
Muckenhoupt $A\big{(}[0,\infty)\big{)}$-weight (as defined by Definition 4.2
with $\mathbb{R}$ replaced by $[0,\infty)$), then the even extension of
$w_{0}$ is a Muckenhoupt $A_{p}(\mathbb{R})$-weight.

######  Remark 4.3.

Let $p\in(1,\infty)$ and $w\in A_{p}$. Then we claim that
$W^{1/p}\in\mathcal{I}_{o}(0,1)$. It suffices to prove that there exists
$\varepsilon=\varepsilon(p,[w]_{A_{p}})>0$ such that

$\lambda^{\varepsilon}\lesssim_{p,[w]_{A_{p}}}\frac{W(\lambda
t)}{W(t)}\lesssim_{p,[w]_{A_{p}}}\lambda^{p-\varepsilon}$ (4.1)

for $t\in\mathbb{R}_{+}$ and $\lambda\geq 1$. By [10, Corollary 7.2.8.], there
exists $\delta=\delta(p,[w]_{A_{p}})\in(0,1)$ such that for $\lambda\in[0,1]$,

$s_{W}(\lambda):=\sup_{t>0}\frac{W(\lambda t)}{W(t)}\leq C\lambda^{\delta},$
(4.2)

where $C=C(p,[w]_{A_{p}})>0$. On the other hand, by [10, Corollary 7.2.6.],
there exists $\delta_{0}=\delta_{0}(p,[w]_{A_{p}})\in(0,1)$ such that
$p-\delta_{0}\geq 1$ and $w\in A_{p-\delta_{0}}$. In the virtue of [10,
Proposition 7.1.5.-(8)] (by taking $f=\mathbbm{1}_{(0,t)}$ and $Q=(0,\lambda
t)$ in there), for $\lambda\in[1,\infty)$,

$W(\lambda
t)\leq\lambda^{p-\delta_{0}}[w]_{A_{p-\delta_{0}}}W(t)\lesssim_{p,[w]_{A_{p}}}\lambda^{p-\delta_{0}}W(t).$
(4.3)

By combining (4.2), (4.3), and by taking $\varepsilon=\delta\wedge\delta_{0}$,
we obtain (4.1), thus $W^{1/p}\in\mathcal{I}_{o}(0,1)$ is also.

######  Definition 4.4.

For $h:\mathbb{R}\to\mathbb{R}$, the (uncentered) Hardy-Littlewood maximal
operator $\mathcal{M}$ is defined by

$\mathcal{M}h(t):=\sup_{[a,b]\ni
t}\frac{1}{b-a}\int_{a}^{b}|h(s)|\mathrm{d}s.$

It is well-known that for $p\in(1,\infty)$,
$\mathcal{M}:L_{p}(\mathbb{R},w\,\mathrm{d}t)\to
L_{p}(\mathbb{R},w\,\mathrm{d}t)$ is bounded if and only if $w\in A_{p}$.
Moreover,

$\|\mathcal{M}\|_{L_{p}(\mathbb{R},w\,\mathrm{d}t)\to
L_{p}(\mathbb{R},w\,\mathrm{d}t)}\lesssim_{p}[w]_{A_{p}}^{\frac{1}{p-1}}$

For more detail, see [10, Chapter 7].

######  Lemma 4.5 ([20], Theorems 1 and 2).

Let $p\in[1,\infty]$ and

$F_{0}(\lambda):=\int_{0}^{\lambda}f(s)\mathrm{d}s,\quad
F_{\infty}(\lambda):=\int_{\lambda}^{\infty}f(s)\mathrm{d}s.$

Then

  1. (i)

there exists a constant $C_{0}>0$ such that

$\|F_{0}U\|_{L_{p}(\mathbb{R}_{+})}\leq C_{0}\|fV\|_{L_{p}(\mathbb{R}_{+})}$

if and only if

$\displaystyle
B_{0}:=\sup_{r\in\mathbb{R}_{+}}\|U\|_{L_{p}((r,\infty))}\|1/V\|_{L_{p^{\prime}}((0,r))}<\infty.$
(4.4)

Moreover, if (4.4) is true, then one can take $C_{0}$ such that
$C_{0}\lesssim_{p}B_{0}$.

  2. (ii)

there exists a constant $C_{\infty}>0$ such that

$\|F_{\infty}U\|_{L_{p}(\mathbb{R}_{+})}\leq
C_{\infty}\|fV\|_{L_{p}(\mathbb{R}_{+})}$

if and only if

$\displaystyle
B_{\infty}:=\sup_{r\in\mathbb{R}_{+}}\|U\|_{L_{p}((0,r))}\|1/V\|_{L_{p^{\prime}}((r,\infty))}<\infty.$
(4.5)

Moreover, if (4.5) is true, one can take $C_{\infty}$ such that
$C_{\infty}\lesssim_{p}B_{\infty}$.

######  Lemma 4.6.

For any measurable function $h:\mathbb{R}_{+}\rightarrow[0,\infty]$ and $s>0$,

$g(s):=\int_{0}^{\infty}te^{-t}h(st)\mathrm{d}t\lesssim\mathcal{M}(h\mathbbm{1}_{\mathbb{R}_{+}})(s).$

Here $\mathcal{M}$ is the (uncentered) Hardy-Littlewood maximal operator
defined in Definition 4.4, and the inequality is independent of $h$ and $s$.

###### Proof.

Observe that $g(s)=\int_{0}^{1}\ldots+\int_{1}^{\infty}\ldots$ and

$\int_{0}^{1}te^{-t}h(st)\,\mathrm{d}t=\frac{1}{s}\int_{0}^{s}\frac{t}{s}e^{-t/s}h(t)\,\mathrm{d}t\lesssim\frac{1}{2s}\int_{0}^{2s}h(t)\,\mathrm{d}t\leq\mathcal{M}(h\mathbbm{1}_{\mathbb{R}_{+}})(s).$

On the other hand,

$\int_{1}^{\infty}te^{-t}h(st)\,\mathrm{d}t\lesssim\int_{1}^{\infty}e^{-t/2}h(st)\,\mathrm{d}t=2\int_{1}^{\infty}\left(\int_{t}^{\infty}e^{-x/2}\,\mathrm{d}x\right)h(st)\,\mathrm{d}t$
$\simeq\int_{1}^{\infty}\left(\int_{1}^{x}h(st)\,\mathrm{d}t\right)e^{-x/2}\,\mathrm{d}x=\int_{1}^{\infty}\left(\int_{s}^{xs}h(t)\,\mathrm{d}t\right)\frac{e^{-x/2}}{s}\,\mathrm{d}x$
$\leq\int_{1}^{\infty}\frac{xs}{s}\left(\frac{1}{xs}\int_{0}^{xs}h(t)\,\mathrm{d}t\right)e^{-x/2}\,\mathrm{d}x\leq\mathcal{M}(h\mathbbm{1}_{\mathbb{R}_{+}})(s)\int_{1}^{\infty}xe^{-x/2}\,\mathrm{d}x\simeq\mathcal{M}(h\mathbbm{1}_{\mathbb{R}_{+}})(s).$

The lemma is proved. ∎

In the following subsections, we identify the (optimal) initial trace spaces
for evolutionary equations such as (1.1) and (1.2). More precisely, for $u\in
L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})$ such that $\partial_{t}u$ or
$\partial_{t}^{\kappa}u\in L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})$ in a
proper sense, we prove which space $u(0)$ belongs to in a trace sense by using
the tools from Sections 2 and 3.

###  4.1. Proof of Theorems 1.1 and 1.5: local derivative case

In this subsection, we prove the trace theorem (Theorem 1.1) for functions
$u\in L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})$ with the following
expression

$u(t)=u_{0}+\int_{0}^{t}f(s)\,\mathrm{d}s$

for some $u_{0}\in X_{0}+X_{1}$ and $f\in
L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})$. We also prove the corresponding
extension theorem (Theorem 1.5). Note that in the proofs of Theorems 1.1 and
1.5, we do not need the results from Section 2.2.

####  4.1.1. Proof of Theorem 1.1

Observe that taking the Laplace transform in (1.7) yields

$\displaystyle u_{0}=\lambda\mathcal{L}[u](\lambda)-\mathcal{L}[f](\lambda).$
(4.6)

By (4.6) and (3.4), we have

$\displaystyle
K(\lambda,u_{0};X_{0},X_{1})\leq\lambda\mathcal{L}[\|u\|_{X_{0}}](\lambda)+\lambda\mathcal{L}[\|f\|_{X_{1}}](\lambda).$
(4.7)

Then it follows that

$\displaystyle\|u_{0}\|_{(X_{0},X_{1})_{W^{1/p},p}}^{p}$
$\displaystyle=\int_{0}^{\infty}W(\lambda^{-1})K(\lambda,u_{0};X_{0},X_{1})^{p}~{}\frac{\mathrm{d}\lambda}{\lambda}$
(4.8)
$\displaystyle\lesssim\int_{0}^{\infty}W(\lambda^{-1})|\lambda\mathcal{L}[\|u\|_{X_{0}}](\lambda)|^{p}~{}\frac{\mathrm{d}\lambda}{\lambda}$
$\displaystyle\quad+\int_{0}^{\infty}W(\lambda^{-1})|\lambda\mathcal{L}[\|f\|_{X_{1}}](\lambda)|^{p}~{}\frac{\mathrm{d}\lambda}{\lambda}.$

Due to (4.8), it suffices for (1.8) to show that for nonnegative $h\in
L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t)$,

$\displaystyle\left(\int_{0}^{\infty}W(\lambda^{-1})|\lambda\mathcal{L}[h](\lambda)|^{p}\frac{\mathrm{d}\lambda}{\lambda}\right)^{1/p}\lesssim\|h\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t)},$
(4.9)

where $\mathcal{L}[h](\lambda)=\int_{0}^{\infty}e^{-\lambda
t}h(t)\,\mathrm{d}t$. Note that (4.9) also verifies a well-definedness of
(4.7) in $X_{0}+X_{1}$.

To prove (4.9), we use Lemma 4.6 and assume the following lemma for a moment:

######  Lemma 4.7.

Let $p\in(1,\infty)$ and $w\in A_{p}$. Then for $f\in
L_{p}(\mathbb{R},w\,\mathrm{d}t)$, we have

$\displaystyle\int_{0}^{\infty}|F(t)|^{p}\frac{W(t)}{t^{p+1}}\,\mathrm{d}t\lesssim_{p,[w]_{A_{p}}}\int_{0}^{\infty}|f(t)|^{p}w(t)\,\mathrm{d}t,$

where $F(t):=\int_{0}^{t}f(s)\,\mathrm{d}s$.

By changing variables $\lambda\to\lambda^{-1}$,

$\int_{0}^{\infty}W(\lambda^{-1})|\lambda\mathcal{L}[h](\lambda)|^{p}\frac{\mathrm{d}\lambda}{\lambda}=\int_{0}^{\infty}|\mathcal{L}[h](\lambda^{-1})|^{p}\frac{W(\lambda)}{\lambda^{p+1}}\mathrm{d}\lambda.$

Since

$\mathcal{L}[h](\lambda^{-1})=\int_{0}^{\lambda}g(s)\mathrm{d}s,\quad
g(s):=\int_{0}^{\infty}te^{-t}h(st)\mathrm{d}t,$

by Lemma 4.6, we have

$\int_{0}^{\infty}W(\lambda^{-1})|\lambda\mathcal{L}[h](\lambda)|^{p}\frac{\mathrm{d}\lambda}{\lambda}\lesssim\int_{0}^{\infty}\left|\int_{0}^{\lambda}\mathcal{M}(h\mathbbm{1}_{\mathbb{R}_{+}})(s)\mathrm{d}s\right|^{p}\frac{W(\lambda)}{\lambda^{p+1}}\mathrm{d}s.$

On the other hand, by Lemma 4.7 and $L_{p}$ estimates for Hardy-Littlewood
maximal function with $A_{p}$-weights,

$\displaystyle\int_{0}^{\infty}\left|\int_{0}^{\lambda}\mathcal{M}(h\mathbbm{1}_{\mathbb{R}_{+}})(s)\mathrm{d}s\right|^{p}\frac{W(\lambda)}{\lambda^{p+1}}\mathrm{d}\lambda$
$\displaystyle\lesssim_{p,[w]_{A_{p}}}\int_{0}^{\infty}|\mathcal{M}(h\mathbbm{1}_{\mathbb{R}_{+}})(s)|^{p}w(s)\mathrm{d}s$
$\displaystyle\lesssim_{p,[w]_{A_{p}}}\|h\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t)}^{p}.$

This certainly implies (4.9).

We prove Lemma 4.7 and complete the proof of Theorem 1.1

###### Proof of Lemma 4.7.

With the help of Theorem 4.5-($i$), it is enough to check

$\sup_{r>0}\left(\int_{r}^{\infty}\frac{W(t)}{t^{p+1}}\mathrm{d}t\right)^{1/p}\left(\int_{0}^{r}w(t)^{-\frac{1}{p-1}}\mathrm{d}t\right)^{1/p^{\prime}}<\infty.$

By Lemma 2.3-($vi$) and (4.1) of Lemma 4.3,

$\int_{r}^{\infty}\frac{W(t)}{t^{p+1}}\mathrm{d}t\lesssim_{p,[w]_{A_{p}}}r^{-p}W(r)=r^{-p}\int_{0}^{r}w(s)\,\mathrm{d}s.$

Therefore,

$\left(\int_{r}^{\infty}\frac{W(t)}{t^{p+1}}\mathrm{d}t\right)^{1/p}\left(\int_{0}^{r}w(t)^{-\frac{1}{p-1}}\mathrm{d}t\right)^{1/p^{\prime}}\lesssim\left(\frac{1}{r}\int_{0}^{r}w(s)\,\mathrm{d}s\right)^{1/p}\left(\frac{1}{r}\int_{0}^{r}w(t)^{-\frac{1}{p-1}}\mathrm{d}t\right)^{1/p^{\prime}},$

and then,

$\sup_{r>0}\left(\int_{r}^{\infty}\frac{W(t)}{t^{p+1}}\mathrm{d}t\right)^{1/p}\left(\int_{0}^{r}w(t)^{-\frac{1}{p-1}}\mathrm{d}t\right)^{1/p^{\prime}}\lesssim[w]_{A_{p}}^{1/p}.$

The lemma is proved. ∎

####  4.1.2. Proof of Theorem 1.5

Since

$a\in(X_{0},X_{1})_{W^{1/p},p}=(X_{0},X_{1})_{W^{1/p},p}^{J},$

there exists a measurable function $v:\mathbb{R}_{+}\to X_{0}\cap X_{1}$ such
that

$a=\int_{0}^{\infty}v(\lambda)\frac{\mathrm{d}\lambda}{\lambda}\quad\textrm{and}\quad\Phi_{p}^{W^{1/p}}\left(J\left(\cdot,v\left(\cdot\right);X_{0},X_{1}\right)\right)\leq
2\|a\|_{(X_{0},X_{1})_{W^{1/p},p}^{J}}\big{(}\simeq\|a\|_{(X_{0},X_{1})_{W^{1/p},p}}\big{)}.$

Let

$u(t):=\int_{0}^{\infty}\mathrm{e}^{-t\lambda}v(\lambda)\frac{\mathrm{d}\lambda}{\lambda},\quad
f(t):=-\int_{0}^{\infty}\lambda\mathrm{e}^{-t\lambda}v(\lambda)\frac{\mathrm{d}\lambda}{\lambda}$

for $t\in[0,\infty)$ and $t\in\mathbb{R}_{+}$, respectively. Then
$u(t)-\int_{0}^{t}f(s)\mathrm{d}s=a$ for all $t\geq 0$, since

$\int_{0}^{t}f(s)~{}\mathrm{d}s=-\int_{0}^{\infty}(1-\mathrm{e}^{-t\lambda})v(\lambda)\frac{\mathrm{d}\lambda}{\lambda}.$

It is clear that $u(0)=a$ and

$\displaystyle\|u(t)\|_{X_{0}}+\|f(t)\|_{X_{1}}\lesssim\int_{0}^{\infty}\mathrm{e}^{-t\lambda}J(\lambda,v(\lambda);X_{0},X_{1})\frac{\mathrm{d}\lambda}{\lambda}$
(4.10)

for $t\in\mathbb{R}_{+}$. Note that the right-hand side of (4.10) is a form of
$\int_{0}^{\infty}\mathrm{e}^{-t\lambda}f(\lambda)\frac{\mathrm{d}\lambda}{\lambda}$.
For this type of quantity, we assume the following lemma:

######  Lemma 4.8.

Let $p\in(1,\infty)$ and $w\in A_{p}$. Then for $f\in
L_{p}(\mathbb{R}_{+},W(t^{-1})\frac{\mathrm{d}t}{t})$, we have

$\displaystyle\int_{0}^{\infty}|F(t)|^{p}w(t)\mathrm{d}t\lesssim_{p,[w]_{A_{p}}}\int_{0}^{\infty}|f(t)|^{p}W(t^{-1})\frac{\mathrm{d}t}{t},$

where
$F(t):=\int_{0}^{\infty}\mathrm{e}^{-t\lambda}f(\lambda)\frac{\mathrm{d}\lambda}{\lambda}$.

Then by Lemma 4.8 with $J(\lambda,v(\lambda);X_{0},X_{1})$ in place of
$f(\lambda)$,

$\displaystyle\|u\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})}+\|f\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})}$
$\displaystyle\lesssim\left\|\int_{0}^{\infty}\exp(-\cdot\lambda)J(\lambda,v(\lambda);X_{0},X_{1})\frac{\mathrm{d}\lambda}{\lambda}\right\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t)}$
$\displaystyle\lesssim_{p,[w]_{A_{p}}}\Phi_{p}^{W^{1/p}}\left(J\left(\cdot,v\left(\cdot\right);X_{0},X_{1}\right)\right)\lesssim\|a\|_{(X_{0},X_{1})_{W^{1/p},p}}.$

Thus $u$ and $f$ are well-defined in
$L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})$ and
$L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})$, respectively. Also, by Fubini’s
theorem (e.g. [13, Proposition 1.2.7.]), we have

$u(t)=a+\int_{0}^{t}f(s)\,\mathrm{d}s,$

for $t\in\mathbb{R}_{+}$.

We prove Lemma 4.8 and complete the proof of Theorem 1.5.

###### Proof of Lemma 4.8.

Without loss of generality, we assume that $f$ is nonnegative. First we divide
$F$ into two parts;

$F(t)=\int_{0}^{t}\mathrm{e}^{-t/\lambda}f(\lambda^{-1})\frac{\mathrm{d}\lambda}{\lambda}+\int_{t}^{\infty}\mathrm{e}^{-t/\lambda}f(\lambda^{-1})\frac{\mathrm{d}\lambda}{\lambda}=:F_{1}(t)+F_{2}(t).$

Since $e^{-x}\leq 1\wedge x^{-1}$ for all $x\in\mathbb{R}_{+}$, we have

$F_{1}(t)\leq\frac{1}{t}\int_{0}^{t}f(\lambda^{-1})\mathrm{d}\lambda,\quad
F_{2}(t)\leq\int_{t}^{\infty}f(\lambda^{-1})\frac{\mathrm{d}\lambda}{\lambda}$

Estimate of $F_{1}$: We claim that

$\displaystyle\int_{0}^{\infty}|F_{1}(t)|^{p}w(t)\mathrm{d}t\leq
C\int_{0}^{\infty}|f(t)|^{p}W(t^{-1})\frac{\mathrm{d}t}{t}$ (4.11)

for some $C=C(p,[w]_{A_{p}})>0$. To obtain (4.11), it is enough to verify that
(4.4) holds for

$U(t):=\frac{w(t)^{1/p}}{t},\quad\textrm{and}\quad
V(t):=\frac{W(t)^{1/p}}{t^{1/p}}$

with the help of Lemma 4.5-($i$). (Take $tF_{1}(t)$ and $f(t^{-1})$ in place
of $F_{0}$ and $f$, respectively.) Since
$t^{-p}=\int_{0}^{t^{-1}}ps^{p-1}\,\mathrm{d}s$, we have

$\displaystyle\int_{r}^{\infty}|U(t)|^{p}\mathrm{d}t=p\int_{0}^{r^{-1}}\left(\int_{r}^{s^{-1}}w(t)\mathrm{d}t\right)s^{p-1}\mathrm{d}s\leq
p\int_{0}^{r^{-1}}W(s^{-1})s^{p-1}\mathrm{d}s.$

Note that by Lemma 2.3-($vi$) and (4.1) of Lemma 4.3, for $s\in(0,r^{-1})$ we
have

$\frac{W(s^{-1})}{W(r)}=\frac{W(s^{-1}r^{-1}r)}{W(r)}\lesssim_{p,[w]_{A_{p}}}(s^{-1}r^{-1})^{p-\varepsilon},$

for some $\varepsilon=\varepsilon(p,[w]_{A_{p}})>0$ and then,
$\int_{0}^{r^{-1}}W(s^{-1})s^{p-1}\mathrm{d}s\lesssim_{\varepsilon,p}r^{-p}W(r)$.
This implies that

$\displaystyle\|U\|_{L_{p}((r,\infty))}\lesssim_{p,[w]_{A_{p}}}r^{-1}W(r)^{1/p}.$
(4.12)

For $0<t<r$, by Lemma 2.3-($vi$) and (4.1) of Lemma 4.3 again,

$\left|\frac{1}{V(t)}\right|^{p}\lesssim_{p,[w]_{A_{p}}}W(r)^{-1}r^{p-\varepsilon}t^{1+\varepsilon-p},$

and then,

$\displaystyle\|1/V\|_{L_{p^{\prime}}((0,r))}\lesssim_{p,[w]_{A_{p}}}rW(r)^{-1/p}.$
(4.13)

Combining (4.12) and (4.13), we have (4.4).

Estimate of $F_{2}$: Now we prove that

$\displaystyle\int_{0}^{\infty}|F_{2}(t)|^{p}w(t)\mathrm{d}t\leq
C\int_{0}^{\infty}|f(t)|^{p}W(t^{-1})\frac{\mathrm{d}t}{t}$ (4.14)

for some $C=C(p,[w]_{A_{p}})>0$. If we prove (4.5) for

$U(t):=w(t)^{1/p},\quad V(t):=W(t)^{1/p}t^{1-\frac{1}{p}},$

then by Theorem 4.5-($ii$), we obtain (4.14). Clearly,

$\|U\|_{L_{p}((0,r))}=W(r)^{1/p}.$ (4.15)

For $0<r<t$, by Lemma 2.3-($vi$) and (4.1) of Lemma 4.3 again,

$\left|\frac{1}{V(t)}\right|^{p}\lesssim_{p,[w]_{A_{p}}}W(r)^{-1}r^{\varepsilon}t^{1-\varepsilon-p},$

and then,

$\|1/V\|_{L_{p^{\prime}}((r,\infty))}\lesssim_{p,[w]_{A_{p}}}W(r)^{-1/p}.$
(4.16)

Combining (4.15) and (4.16), we obtain (4.5). The lemma is proved. ∎

###  4.2. Proof of Theorems 1.2 and 1.6: non-local derivative case

In this subsection, we prove the trace theorem (Theorems 1.2) for functions
$u$, roughly speaking, involving a non-local derivative. We also provide the
corresponding extension theorem (Theorem 1.6).

We briefly recall the essential tools from Section 2.2.

$\bullet$ $\kappa:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a right-
continuous decreasing function with $\kappa(0+)=\infty$, $\kappa(\infty)=0$,
and $\int_{0}^{1}\kappa(s)\,\mathrm{d}s<\infty$. For the remainder of this
section, we fix a $\kappa$ that satisfies the above assumptions. Also, see
Remark 2.5 for a sufficient condition for the above assumptions.

$\bullet$ There is a nonnegative measure $\mu$ defined on $\mathbb{R}_{+}$
such that
$\kappa(x)=\int_{0}^{\infty}\mathbbm{1}_{(x,\infty)}\mu(\mathrm{d}t)$. By the
choice of $\kappa$, this $\mu$ satisfies (2.14).

$\bullet$ Since $\mu$ satisfies (2.14),
$\phi(\lambda)=\int_{0}^{\infty}\left(1-e^{-\lambda t}\right)\mu(\mathrm{d}t)$
is a Bernstein function.

$\bullet$ There is a subordinator $S=(S_{t})_{t\geq 0}$ defined on an
probability space $(\Omega,\mathcal{F},\mathbb{P})$ with Laplace exponent
$\phi$, i.e., $\mathbb{E}\big{[}\mathrm{e}^{-\lambda
S_{t}}\big{]}=\mathrm{e}^{-t\phi(\lambda)}$ for any $t,\,\lambda\geq 0$.

$\bullet$ For $(t,\lambda)\in[0,\infty)\times\mathbb{R}_{+}$,
$\Theta(t,\lambda):=\lambda\int_{0}^{\infty}\mathrm{e}^{-\lambda
r}\mathbb{P}(S_{r}\geq t)\mathrm{d}r$ satisfies $\Theta(0,\lambda)=1$ and
$\partial_{t}^{\kappa}\Theta(t,\lambda)=\lambda\Theta(t,\lambda)$.

It is given in Proposition 2.8 that for $\kappa$ mentioned above, if
$\kappa\in\mathcal{I}_{o}(-1,0)$, then
$\phi(\lambda):=\lambda\mathcal{L}[\kappa](\lambda)$ is a Bernstein function
and $\phi(\lambda)\simeq\kappa(\lambda^{-1})$. For this $\phi$, we denote

$\displaystyle\psi(t):=\frac{1}{\phi^{-1}(t^{-1})},$ (4.17)

where $\phi^{-1}$, the inverse function of $\phi$, is well-defined since
Bernstein functions are continuous and strictly increasing. It is provided in
Remark 2.9 that $\psi\simeq\kappa^{*}$, where $\kappa^{*}$ denotes the
function defined as (1.9). Due to
$W(t):=\int_{0}^{t}w(s)\,\mathrm{d}s\in\mathcal{I}_{o}(0,p)$, we have
$W\circ\psi\simeq W\circ\kappa^{*}$, so that

$(X_{0},X_{1})_{(W\circ\kappa^{*})^{1/p},p}=(X_{0},X_{1})_{(W\circ\psi)^{1/p},p}.$

Therefore, we prove Theorems 1.2 and 1.6 for
$(X_{0},X_{1})_{(W\circ\psi)^{1/p},p}$ instead of
$(X_{0},X_{1})_{(W\circ\kappa^{*})^{1/p},p}$.

We first prove a non-local counterpart of Theorem 1.1.

####  4.2.1. Proof of Theorem 1.2

Note that the space $(X_{0},X_{1})_{(W\circ\psi)^{1/p},p}$ is well-defined due
to the assumption $(W\circ\psi)^{1/p}\in\mathcal{I}_{o}(0,1)$.

Similar to the proof of Theorem 1.1, we take $\mathcal{L}$ to the both sides
of (1.10) and use Fubini’s theorem (e.g. [13, Proposition 1.2.7.]) to obtain

$\mathcal{L}[\kappa](\lambda)\left(\mathcal{L}[u](\lambda)-\frac{u_{0}}{\lambda}\right)=\frac{\phi(\lambda)}{\lambda}\left(\mathcal{L}[u](\lambda)-\frac{u_{0}}{\lambda}\right)=\frac{1}{\lambda}\mathcal{L}[f](\lambda),\quad\lambda\in\mathbb{R}_{+},$

where we use (2.17) for the first equality. That is, we have

$\displaystyle
u_{0}=\lambda\mathcal{L}[u](\lambda)+\frac{\lambda\mathcal{L}[f](\lambda)}{\phi(\lambda)},\quad\lambda\in\mathbb{R}_{+},$
(4.18)

or, equivalently, (Put $\lambda=\phi^{-1}(\tau)(=1/\psi(\tau^{-1}))$.)

$\displaystyle
u_{0}=\frac{\mathcal{L}[u](1/\psi(\tau^{-1}))}{\psi(\tau^{-1})}+\frac{\mathcal{L}[f](1/\psi(\tau^{-1}))}{\tau\psi(\tau^{-1})},\quad\tau\in\mathbb{R}_{+}.$
(4.19)

Then by the definition of $K$-functional,

$\displaystyle
K(\tau,u_{0};X_{0},X_{1})\leq\frac{\mathcal{L}[\|u\|_{X_{0}}+\|f\|_{X_{1}}](1/\psi(\tau^{-1}))}{\psi(\tau^{-1})},$
(4.20)

and it follows from (4.20) that

$\displaystyle\|u_{0}\|_{(X_{0},X_{1})_{(W\circ\psi)^{1/p},p}}^{p}$
$\displaystyle=\int_{0}^{\infty}(W\circ\psi)(\tau^{-1})\left|K(\tau,u_{0};X_{0},X_{1})\right|^{p}~{}\frac{\mathrm{d}\tau}{\tau}$
(4.21)
$\displaystyle\lesssim\int_{0}^{\infty}(W\circ\psi)(\tau^{-1})\left|\frac{\mathcal{L}[\|u\|_{X_{0}}+\|f\|_{X_{1}}](1/\psi(\tau^{-1}))}{\psi(\tau^{-1})}\right|^{p}~{}\frac{\mathrm{d}\tau}{\tau}$

Thus it suffices for (1.11) to show that for nonnegative $h$,

$\displaystyle\left(\int_{0}^{\infty}\left|(W\circ\psi)(\tau^{-1})\frac{\mathcal{L}[h](1/\psi(\tau^{-1}))}{\psi(\tau^{-1})}\right|^{p}\frac{\mathrm{d}\tau}{\tau}\right)^{1/p}\lesssim_{p,[w]_{A_{p}}}\|h\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t)},$
(4.22)

where $\mathcal{L}[h](\lambda)=\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}h(t)\,\mathrm{d}t$. Note that (4.18) and (4.19) are well-defined in
$X_{0}+X_{1}$ due to (4.22).

By changing variable $\tau\to\phi(\lambda^{-1})$ with the help of (2.23), we
have

$\int_{0}^{\infty}\left|(W\circ\psi)(\tau^{-1})\frac{\mathcal{L}[h](1/\psi(\tau^{-1}))}{\psi(\tau^{-1})}\right|^{p}\,\frac{\mathrm{d}\tau}{\tau}\simeq_{\kappa}\int_{0}^{\infty}|\mathcal{L}[h](\lambda^{-1})|^{p}\frac{W(\lambda)}{\lambda^{p+1}}\,\mathrm{d}\lambda.$

We also have

$\mathcal{L}[h](\lambda^{-1})=\int_{0}^{\lambda}\left(\int_{0}^{\infty}t\mathrm{e}^{-t}h(st)\,\mathrm{d}t\right)\mathrm{d}s:=\int_{0}^{\lambda}g(s)\,\mathrm{d}s$

and $g(s)\lesssim\mathcal{M}(h\mathbbm{1}_{\mathbb{R}_{+}})(s)$ for
$s\in\mathbb{R}_{+}$. Then it follows by Lemma 4.6 that

$\int_{0}^{\infty}|\mathcal{L}[h](\lambda^{-1})|^{p}\frac{W(\lambda)}{\lambda^{p+1}}\mathrm{d}\lambda\lesssim\int_{0}^{\infty}\left|\int_{0}^{\lambda}\mathcal{M}(h\mathbbm{1}_{\mathbb{R}_{+}})(s)\mathrm{d}s\right|^{p}\frac{W(\lambda)}{\lambda^{p+1}}\mathrm{d}s.$

Also, by Lemma 4.7 and weighted $L_{p}$ estimates for the Hardy-Littlewood
maximal functions,

$\displaystyle\int_{0}^{\infty}\left|\int_{0}^{\lambda}\mathcal{M}(h\mathbbm{1}_{\mathbb{R}_{+}})(s)\mathrm{d}s\right|^{p}\frac{W(\lambda)}{\lambda^{p+1}}\mathrm{d}\lambda$
$\displaystyle\lesssim_{p,[w]_{A_{p}}}\int_{0}^{\infty}|\mathcal{M}(h\mathbbm{1}_{\mathbb{R}_{+}})(s)|^{p}w(s)\mathrm{d}s$
$\displaystyle\lesssim_{p,[w]_{A_{p}}}\|h\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t)}^{p},$

and this implies (4.22). Hence the theorem is proved.

####  4.2.2. Proof of Theorem 1.6

As noted in the proof of Theorem 1.2, the space
$(X_{0},X_{1})_{(W\circ\psi)^{1/p},p}$ is well-defined. Since

$a\in(X_{0},X_{1})_{(W\circ\psi)^{1/p},p}=(X_{0},X_{1})_{(W\circ\psi)^{1/p},p}^{J}\,,$

there exists a measurable function $v:\mathbb{R}_{+}\to X_{0}\cap X_{1}$ such
that

$a=\int_{0}^{\infty}v(\lambda)\frac{\mathrm{d}\lambda}{\lambda}\quad\textrm{and}\quad\Phi_{p}^{(W\circ\psi)^{1/p}}\left(J\left(\cdot,v\left(\cdot\right);X_{0},X_{1}\right)\right)\leq
2\|a\|_{(X_{0},X_{1})_{(W\circ\psi)^{1/p},p}}.$

Recall that $\Theta(t,\lambda)=\lambda\int_{0}^{\infty}\mathrm{e}^{-\lambda
r}\mathbb{P}(S_{r}\geq t)\mathrm{d}r$ for
$(t,\lambda)\in[0,\infty)\times\mathbb{R}_{+}$. Let

$u(t):=\int_{0}^{\infty}\Theta(t,\lambda)v(\lambda)\frac{\mathrm{d}\lambda}{\lambda},\quad
f(t):=-\int_{0}^{\infty}\lambda\Theta(t,\lambda)v(\lambda)\frac{\mathrm{d}\lambda}{\lambda}$

for $t\in[0,\infty)$ and $t\in\mathbb{R}_{+}$, respectively. Since
$\Theta(0,\lambda)=1$ for $\lambda\in\mathbb{R}_{+}$, it is clear that
$u(0)=a$. We also have

$\displaystyle\|u(t)\|_{X_{0}}+\|f(t)\|_{X_{1}}\lesssim\int_{0}^{\infty}\Theta(t,\lambda)J(\lambda,v(\lambda);X_{0},X_{1})\frac{\mathrm{d}\lambda}{\lambda}.$
(4.23)

Note that the right-hand side of (4.23) is a form of
$\int_{0}^{\infty}\Theta(t,\lambda)f(\lambda)\frac{\mathrm{d}\lambda}{\lambda}$.
For this quantity, we assume the following lemma for a moment:

######  Lemma 4.9.

Let $p\in(1,\infty)$ and $w\in A_{p}$. Suppose that
$\phi\in\mathcal{I}_{o}(0,1)$ and $W\circ\psi\in\mathcal{I}_{o}(0,p)$. Then
for $f\in L_{p}(\mathbb{R}_{+},(W\circ\psi)(t^{-1})\frac{\mathrm{d}t}{t})$, we
have

$\displaystyle\int_{0}^{\infty}|F(t)|^{p}w(t)\mathrm{d}t\lesssim_{N}\int_{0}^{\infty}|f(t)|^{p}(W\circ\psi)(t^{-1})\frac{\mathrm{d}t}{t},$

where
$F(t)=\int_{0}^{\infty}\Theta(t,\lambda)f(\lambda)\,\frac{\mathrm{d}\lambda}{\lambda}$
and $N=N(p,[w]_{A_{p}},W\circ\psi,\kappa)$.

Then by Lemma 4.9 with $J(\lambda,v(\lambda);X_{0},X_{1})$ in place of
$f(\lambda)$,

$\displaystyle\|u\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})}+\|f\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})}$
$\displaystyle\lesssim\left\|\int_{0}^{\infty}\Theta(\cdot,\lambda)J(\lambda,v(\lambda);X_{0},X_{1})\frac{\mathrm{d}\lambda}{\lambda}\right\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t)}$
$\displaystyle\lesssim\Phi_{p}^{(W\circ\psi)^{1/p}}\left(J\left(\cdot,v\left(\cdot\right);X_{0},X_{1}\right)\right)\lesssim\|a\|_{(X_{0},X_{1})_{(W\circ\psi)^{1/p},p}}.$

Thus $u$ and $f$ are well-defined in
$L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})$ and
$L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})$, respectively. Finally, by
Fubini’s theorem (e.g. [13, Proposition 1.2.7.]) and Proposition 2.12-($ii$),
it directly follows that (1.12).

We prove Lemma 4.9 and complete the proof of Theorem 1.6. Note that it is a
variant of Lemma 4.8 in the sense that one can recover Lemma 4.8 by choosing
$\phi(t)=\psi(t)=t$ and $\Theta(t,\lambda)=e^{-t\lambda}$ in Lemma 4.9.

###### Proof of Lemma 4.9.

Without loss of generality, we assume that $f$ is nonnegative. First we divide
$F$ into two parts; As in the proof of Theorem 1.2, by using (2.23),

$F(t)\simeq_{\kappa}\int_{0}^{\infty}\Theta(t,\phi(\lambda^{-1}))(f\circ\phi)(\lambda^{-1})\frac{\mathrm{d}\lambda}{\lambda}=\int_{0}^{t}\cdots+\int_{t}^{\infty}\cdots=:F_{1}(t)+F_{2}(t).$

By Proposition 2.12, we know that $\Theta(t,\phi(\lambda^{-1}))\simeq
1\wedge\frac{\phi(t^{-1})}{\phi(\lambda^{-1})}$ and then

$F_{1}(t)\lesssim_{\kappa}\phi(t^{-1})\int_{0}^{t}\frac{(f\circ\phi)(\lambda^{-1})}{\phi(\lambda^{-1})}\frac{\mathrm{d}\lambda}{\lambda},\quad
F_{2}(t)\lesssim_{\kappa}\int_{t}^{\infty}(f\circ\phi)(\lambda^{-1})\frac{\mathrm{d}\lambda}{\lambda}.$

On the other hand, by changing variable $t\to\phi(t)$ with the help of (2.23)
again, we have

$\int_{0}^{\infty}|f(t)|^{p}(W\circ\psi)(t^{-1})\frac{\mathrm{d}t}{t}\simeq_{\kappa}\int_{0}^{\infty}|(f\circ\phi)(t)|^{p}\frac{W(t^{-1})}{t}\mathrm{d}t.$

Estimate of $F_{1}$: We claim that

$\displaystyle\int_{0}^{\infty}|F_{1}(t)|^{p}w(t)\mathrm{d}t$
$\displaystyle\leq
C\int_{0}^{\infty}|(f\circ\phi)(t)|^{p}W(t^{-1})\,\frac{\mathrm{d}t}{t}$
$\displaystyle=C\int_{0}^{\infty}|(f\circ\phi)(t^{-1})|^{p}W(t)\,\frac{\mathrm{d}t}{t}$
(4.24)

for some $C=C(p,W\circ\psi,[w]_{A_{p}})>0$, and by Theorem 4.5-($i$), it is
enough to obtain (4.4) for

$U(t):=\phi(t^{-1})w(t)^{1/p}\quad\textrm{and}\quad
V(t):=W(t)^{1/p}\phi(t^{-1})t^{1-1/p}.$

Since $\phi(t^{-1})^{p}=p\int_{0}^{\phi(t^{-1})}s^{p-1}\,\mathrm{d}s$, we have

$\displaystyle\int_{r}^{\infty}|U(t)|^{p}~{}\mathrm{d}t$
$\displaystyle=p\int_{0}^{\phi(r^{-1})}\left(\int_{r}^{\psi(s^{-1})}w(t)\mathrm{d}t\right)s^{p-1}~{}\mathrm{d}s$
$\displaystyle\leq
p\int_{0}^{\phi(r^{-1})}(W\circ\psi)(s^{-1})s^{p-1}~{}\mathrm{d}s.$

By the assumption that $W\circ\psi\in\mathcal{I}_{o}(0,p)$, we have

$\frac{\left(W\circ\psi\right)\left(s^{-1}\right)}{\left(W\circ\psi\right)\left(\frac{1}{\phi\left(r^{-1}\right)}\right)}\lesssim_{W\circ\psi}\left(\frac{\phi(r^{-1})}{s}\right)^{p-\varepsilon}$

for $r\in\mathbb{R}_{+}$, $s\in(0,\phi(r^{-1}))$ and
$\varepsilon=\varepsilon(W\circ\psi)>0$. Then

$\int_{0}^{\phi(r^{-1})}(W\circ\psi)(s^{-1})s^{p-1}~{}\mathrm{d}s\lesssim_{W\circ\psi}\phi(r^{-1})^{p}W(r)$

and this implies

$\displaystyle\|U\|_{L_{p}((r,\infty))}\lesssim_{p,W\circ\psi}\phi(r^{-1})W(r)^{1/p}.$
(4.25)

Similarly, by the assumption that $W\circ\psi\in\mathcal{I}_{o}(0,p)$, for
$0<t<r$,

$\displaystyle\frac{1}{V(t)}$
$\displaystyle=\frac{t^{\frac{1}{p}-1}}{\left(\left(W\circ\psi\right)\left(\frac{1}{\phi(t^{-1})}\right)\right)^{1/p}\phi(t^{-1})}$
$\displaystyle\lesssim_{p,[w]_{A_{p}}}\left(\phi\left(r^{-1}\right)\right)^{-1+\frac{\varepsilon}{p}}W(r)^{-1/p}\phi(t^{-1})^{-\frac{\varepsilon}{p}}t^{\frac{1}{p}-1}$

where $\varepsilon=\varepsilon(p,[w]_{A_{p}})>0$. From this,

$\displaystyle\|1/V\|_{L_{p^{\prime}}((0,r))}\lesssim_{p,[w]_{A_{p}}}\phi(r^{-1})^{-1}W(r)^{-1/p}.$
(4.26)

Combining (4.25) and (4.26), we have (4.4), and thus (4.2.2).

Estimate $F_{2}$: Now we prove that

$\displaystyle\int_{0}^{\infty}|F_{2}(t)|^{p}w(t)\mathrm{d}t\leq
C\int_{0}^{\infty}|(f\circ\phi)(t)|^{p}\frac{W(t^{-1})}{t}\mathrm{d}t$ (4.27)

for some $C=C(p,[w]_{A_{p}})>0$, and by Theorem 4.5-($ii$), it is enough to
obtain (4.5) for

$U(t):=w(t)^{1/p}\quad\textrm{and}\quad V(t):=W(t)^{1/p}t^{1-\frac{1}{p}}.$

Clearly, $\|U\|_{L_{p}((0,r))}=W(r)^{1/p}$. For $0<r<t$, by Lemmas 2.3-($ix$)
and 4.3,

$\frac{1}{V(t)}\lesssim_{p,[w]_{A_{p}}}W(r)^{-1/p}r^{\varepsilon/p}t^{\frac{(1-\varepsilon)}{p}-1},$

where $\varepsilon=\varepsilon(p,[w]_{A_{p}})>0$, and then

$\|U\|_{L_{p}((0,r))}\|1/V\|_{L_{p^{\prime}}((r,\infty))}\lesssim_{p,[w]_{A_{p}}}1.$

Now we obtain (4.27) and also desired inequality. The lemma is proved. ∎

######  Remark 4.10.

By keep tracking the proof of Theorem 1.2, one verify that the trace estimate
(1.11) depend only on $p$, $M$, $\nu_{l}$, $\nu_{u}$, and $K_{\kappa}$, where
$[w]_{A_{p}}\leq M$, $\nu_{l}\leq\kappa(1)\leq\nu_{u}$ and $K_{\kappa}$
($\kappa$ in place of $\phi$ in Remark 2.2). Similarly, the inequality (1.13)
in Theorem 1.6 depend only on $p$, $M$, $\nu_{l}$, $\nu_{u}$, $K_{\kappa}$,
and $K_{W\circ\kappa^{*}}$.

##  5\. Trace theorems for finite time interval

In this section, we assume that $X_{0}$ is continuously embedded into $X_{1}$,
i.e., $X_{0}\subset X_{1}$ so that there exists a constant $N_{X_{0},X_{1}}>0$
such that

$\displaystyle\|a\|_{X_{1}}\leq
N_{X_{0},X_{1}}\|a\|_{X_{0}}\quad\forall\,\,a\in X_{0}.$ (5.1)

A representative example of this situation is when
$X_{0}=W_{p}^{2}(\mathbb{R}^{d})$ and $X_{1}=L_{p}(\mathbb{R}^{d})$ with
$N_{X_{0},X_{1}}=1$.

As a corollary of Theorem 1.1, we obtain a version of Theorem 1.1 for finite
time intervals $(0,T)$, $T<\infty$, instead of $\mathbb{R}_{+}$.

######  Corollary 5.1 (Trace theorem with local derivative; finite interval).

Let $T\in(0,\infty)$, $p\in(1,\infty)$ and $w\in A_{p}$. Assume that $X_{0}$
is continuously embedded into $X_{1}$, and $u\in
L_{p}((0,T),w\,\mathrm{d}t;X_{0})$, $f\in L_{p}((0,T),w\,\mathrm{d}t;X_{1})$
and $u_{0}\in X_{0}+X_{1}$ satisfy that

$u(t)=u_{0}+\int_{0}^{t}f(s)\,\mathrm{d}s\quad\forall t\in(0,T).$

Then $u_{0}\in(X_{0},X_{1})_{W^{1/p},p}$ and

$\|u_{0}\|_{(X_{0},X_{1})_{W^{1/p},p}}\lesssim_{p,[w]_{A_{p}},N_{X_{0},X_{1}},T}\|u\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{0})}+\|f\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{1})}\,,$

where $W(t):=\int_{0}^{t}w(s)\,\mathrm{d}s$.

###### Proof.

Take a nonnegative decreasing $\zeta\in C^{\infty}(\mathbb{R})$ such that
$\zeta(s)=1$ on $s\leq\frac{T}{2}$ and $\zeta(s)=0$ for all $s\geq T$. Put
$\widetilde{u}:=\zeta u$ and $\widetilde{f}:=\zeta f+\zeta^{\prime}u$. Since
$X_{0}$ is continuously embedded into $X_{1}$,

$\begin{gathered}\|\widetilde{u}\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{0})}\leq\|u\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{0})},\\\
\|\widetilde{f}\|_{L_{p}(\mathbb{R}_{+},w\,\mathrm{d}t;X_{1})}\leq\|f\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{1})}+N(N_{X_{0},X_{1}},T)\|u\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{0})}.\end{gathered}$

Moreover

$\widetilde{u}(t)=u_{0}+\int_{0}^{t}\widetilde{f}(s)\,\mathrm{d}s\quad\forall
t\in\mathbb{R}_{+}.$

By applying Theorem 1.1 for $(u,f)$ replaced by
$(\widetilde{u},\widetilde{f})$, the corollary is proved. ∎

Similar to Corollary 5.1, in the situation that $X_{0}$ is continuously
embedded into $X_{1}$, we can also consider a version of Theorem 1.2 for
finite time intervals $(0,T)$ instead of $\mathbb{R}_{+}$. However, the non-
local derivative $\partial_{t}^{\kappa}$ makes the proof more intricate than
the straightforward computation seen in Corollary 5.1. In particular, for
finite time interval $(0,T)$ cases, it is natural to consider a kernel
$\kappa^{\circ}$ and a weight function $w^{\circ}$ defined on $(0,T)$. It
turns out that if $w^{\circ}\in A_{p}\left(\left(0,T\right)\right)$ (i.e.
$w^{\circ}$ satisfies (B.8)) and $\kappa^{\circ}:(0,T)\to\mathbb{R}_{+}$
satisfy

$\lambda^{-1+\varepsilon}\lesssim\frac{\kappa^{\circ}(\lambda
t)}{\kappa^{\circ}(t)}\lesssim\lambda^{-\varepsilon}\quad\forall\lambda\in[1,\infty),\,0<t\leq\lambda
t<T.$

and

$\lambda^{\delta}\lesssim\frac{W^{\circ}\big{(}{\kappa^{\circ}}^{\ast}(\lambda
t)\big{)}}{W^{\circ}\big{(}{\kappa^{\circ}}^{\ast}(t)\big{)}}\lesssim\lambda^{p-\delta}\quad\forall\,0<t<\lambda
t<\frac{1}{\kappa(T)}$

for some $\varepsilon$, $\delta>0$, then there are proper extensions $w\in
A_{p}(\mathbb{R})$ and $\kappa:\mathbb{R}_{+}\to\mathbb{R}_{+}$ of
$w^{\circ}\in A_{p}\left(\left(0,T\right)\right)$ and of $\kappa^{\circ}$,
repectively. In Proposition B.4, we provide detailed descriptions of such
extensions. For simplicity, in the remaining of this section, we use $w\in
A_{p}(\mathbb{R})$ and $\kappa:\mathbb{R}_{+}\to\mathbb{R}_{+}$ instead of
$w^{\circ}\in A_{p}\left(\left(0,T\right)\right)$ and $\kappa^{\circ}$,
respectively.

We give a lemma required for the proof of Theorem 5.3.

######  Lemma 5.2.

Let $X_{0}$ be continuously embedded into $X_{1}$ and let $p\in(1,\infty)$,
$w\in A_{p}$, $\kappa\in\mathcal{I}_{o}(-1,0)$, and
$W\circ\kappa^{*}\in\mathcal{I}_{o}(0,p)$. Suppose $u\in
L_{p}((0,T),w\,\mathrm{d}t;X_{0})$, $f\in L_{p}((0,T),w\,\mathrm{d}t;X_{1})$
and for some $u_{0}\in X_{1}\,(=X_{0}+X_{1})$, we have

$\int_{0}^{t}\kappa(t-s)\left(u(s)-u_{0}\right)\,\mathrm{d}s=\int_{0}^{t}f(s)\,\mathrm{d}s,$
(5.2)

for $t\in(0,T)$. Then

$\displaystyle\|u_{0}\|_{X_{1}}\lesssim_{N}\|u\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{0})}+\|f\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{1})},$
(5.3)

where $N=N(p,[w]_{A_{p}},\kappa,N_{X_{0},X_{1}},T)$.

###### Proof.

By (5.2),

$\left(\int_{0}^{t}\kappa(s)\,\mathrm{d}s\right)\|u_{0}\|_{X_{1}}\leq\int_{0}^{t}\kappa(t-s)\|u(s)\|_{X_{1}}\,\mathrm{d}s+\int_{0}^{t}\|f(s)\|_{X_{1}}\,\mathrm{d}s$

for $t\in(0,T)$. Then,

$\displaystyle\begin{split}&\|u_{0}\|_{X_{1}}\int_{0}^{T}\mathrm{e}^{-t}\left(\int_{0}^{t}\kappa(s)\,\mathrm{d}s\right)\mathrm{d}t\\\
\leq&\int_{0}^{\infty}\mathrm{e}^{-t}\left(\kappa*U\right)\left(t\right)\,\mathrm{d}t+\int_{0}^{\infty}\mathrm{e}^{-t}\left(\int_{0}^{t}F(s)\,\mathrm{d}s\right)\mathrm{d}t\\\
=&\mathcal{L}[\kappa](1)\int_{0}^{\infty}\mathrm{e}^{-s}U(s)\,\mathrm{d}s+\int_{0}^{\infty}\mathrm{e}^{-s}F(s)\,\mathrm{d}s,\end{split}$
(5.4)

where $U(s)=\|u(s)\|_{X_{1}}1_{(0,T)}(s)$ and
$F(s)=\|f(s)\|_{X_{1}}1_{(0,T)}(s)$ for $s\in\mathbb{R}_{+}$. For

$w^{*}:=w^{-1/(p-1)}\in A_{p/(p-1)},$

we have

$\displaystyle\left|\int_{0}^{\infty}\mathrm{e}^{-s}U(s)\,\mathrm{d}s\right|\leq\left(\int_{0}^{\infty}\mathrm{e}^{-sp/(p-1)}w^{*}(s)\,\mathrm{d}s\right)^{(p-1)/p}\left(\int_{0}^{\infty}\left|U(s)\right|^{p}w(s)\,\mathrm{d}s\right)^{1/p}.$
(5.5)

Note that

$\displaystyle\int_{0}^{\infty}\mathrm{e}^{-sp/(p-1)}w^{*}(s)\,\mathrm{d}s$
$\displaystyle\simeq_{p}\int_{0}^{\infty}\int_{s}^{\infty}\mathrm{e}^{-rp/(p-1)}w^{*}(s)\,\mathrm{d}r\,\mathrm{d}s$
$\displaystyle=\int_{0}^{\infty}\left(\int_{0}^{r}w^{*}(s)\,\mathrm{d}s\right)\mathrm{e}^{-rp/(p-1)}\,\mathrm{d}r$
$\displaystyle=\int_{0}^{\infty}W^{*}(r)\mathrm{e}^{-rp/(p-1)}\,\mathrm{d}r\lesssim_{[w]_{A_{p}}}1,$

where

$W^{*}(r):=\int_{0}^{r}w^{*}(s)\,\mathrm{d}s.$

The last inequality is due to Lemma 4.3, in particular,
$W^{*}\in\mathcal{I}_{o}(0,p/(p-1))$. Due to (5.1), we have

$\int_{0}^{\infty}\mathrm{e}^{-s}U(s)\,\mathrm{d}s\lesssim_{N}\|u\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{0})}\,,$

where $N=N(p,[w]_{A_{p}},N_{X_{0},X_{1}})$. Similarly,

$\displaystyle\int_{0}^{\infty}\mathrm{e}^{-s}F(s)\,\mathrm{d}s\lesssim_{p,[w]_{A_{p}}}\|f\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{1})}.$
(5.6)

Finally, by the assumption $\kappa\in\mathcal{I}_{o}(-1,0)$ and (2.3),

$\displaystyle\|u_{0}\|_{X_{1}}\lesssim_{\kappa,T}\int_{0}^{T}\mathrm{e}^{-t}\left(\int_{0}^{t}\kappa(s)\,\mathrm{d}s\right)\mathrm{d}t\,\|u_{0}\|_{X_{1}}$
(5.7)

and then, by combining (5.4) - (5.7), we obtain (5.3). The lemma is proved. ∎

######  Theorem 5.3 (Trace theorem with non-local derivative; finite
interval).

Let $X_{0}$ be continuously embedded into $X_{1}$ and let $p\in(1,\infty)$,
$w\in A_{p}$, $\kappa\in\mathcal{I}_{o}(-1,0)$, and
$W\circ\kappa^{*}\in\mathcal{I}_{o}(0,p)$. Suppose $u\in
L_{p}((0,T),w\,\mathrm{d}t;X_{0})$, $f\in L_{p}((0,T),w\,\mathrm{d}t;X_{1})$
and for some $u_{0}\in X_{1}\,(=X_{0}+X_{1})$, we have

$\int_{0}^{t}\kappa(t-s)\left(u(s)-u_{0}\right)\,\mathrm{d}s=\int_{0}^{t}f(s)\,\mathrm{d}s,$
(5.8)

for $t\in(0,T)$. Then we have
$u_{0}\in(X_{0},X_{1})_{(W\circ\kappa^{*})^{1/p},p}$ and

$\displaystyle\|u_{0}\|_{(X_{0},X_{1})_{(W\circ\kappa^{*})^{1/p},p}}\lesssim_{N}\|u\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{0})}+\|f\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{1})},$
(5.9)

where $N=N(p,[w]_{A_{p}},\kappa,N_{X_{0},X_{1}},T)$.

###### Proof.

Thanks to Proposition B.2, we have an extension
$\kappa\in\mathcal{I}_{o}(-1,0)$ of $\kappa^{\circ}$. Through zero-extension,
we consider $u$ and $f$ as functions defined on $\mathbb{R}_{+}$, so that

$u(t)=f(t)=0\quad\forall\,\,t\geq T.$

From (5.8), we have

$\left(\int_{0}^{t}\kappa(s)\,\mathrm{d}s\right)u_{0}=\int_{0}^{t}\kappa(t-s)u(s)\,\mathrm{d}s-\int_{0}^{t}f(s)\,\mathrm{d}s$

for $t\in(0,T)$. Then by integration by parts with the fact that
$\mathcal{L}[\kappa](\lambda)=\phi(\lambda)/\lambda$,

$\displaystyle u_{0}\frac{\phi(\lambda)}{\lambda^{2}}$
$\displaystyle=\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}\left(\int_{0}^{t}\kappa(s)\,\mathrm{d}s\right)u_{0}\,\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\mathrm{e}^{-\lambda
t}\left(\int_{0}^{t}\kappa(t-s)u(s)\,\mathrm{d}s\right)\,\mathrm{d}t$
$\displaystyle\quad-\int_{0}^{T}\mathrm{e}^{-\lambda
t}\left(\int_{0}^{t}f(s)\,\mathrm{d}s\right)\,\mathrm{d}t+\left(\int_{T}^{\infty}\mathrm{e}^{-\lambda
t}\left(\int_{0}^{t}\kappa(s)\,\mathrm{d}s\right)\,\mathrm{d}t\right)u_{0}$
$\displaystyle:=I_{1}(\lambda)+I_{2}(\lambda)+I_{3}(\lambda).$

By the definition of $K$-functional,

$K(\tau,u_{0};X_{0},X_{1})\leq\frac{\lambda^{2}}{\phi(\lambda)}\left(\left\|I_{1}(\lambda)\right\|_{X_{0}}+\tau\left\|I_{2}(\lambda)\right\|_{X_{1}}+\tau\left\|I_{3}(\lambda)\right\|_{X_{1}}\right)\quad\forall\tau,\lambda\in(0,\infty).$

If we take $0<\tau:=\phi(\lambda)$, i.e.,

$\lambda=\phi^{-1}(\tau)=\frac{1}{\psi(\tau^{-1})}.$ (5.10)

Then

$\displaystyle\|u_{0}\|_{(X_{0},X_{1})_{(W\circ\psi)^{1/p},p}}^{p}$ (5.11)
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}(W\circ\psi)(\tau^{-1})\left|K(\tau,u_{0};X_{0},X_{1})\right|^{p}~{}\frac{\mathrm{d}\tau}{\tau}$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{\infty}(W\circ\psi)(\tau^{-1})\left|\frac{\tau^{-1}}{\psi(\tau^{-1})^{2}}\left(\left\|I_{1}(1/\psi(\tau^{-1}))\right\|_{X_{0}}+\tau\left\|I_{2}(1/\psi(\tau^{-1}))\right\|_{X_{1}}\right)\right|^{p}~{}\frac{\mathrm{d}\tau}{\tau}$
$\displaystyle+\int_{0}^{\infty}(W\circ\psi)(\tau^{-1})\left|\frac{1}{\psi(\tau^{-1})^{2}}\left\|I_{3}(1/\psi(\tau^{-1}))\right\|_{X_{1}}\right|^{p}~{}\frac{\mathrm{d}\tau}{\tau}$
$\displaystyle:=$ $\displaystyle A+B.$

Estimate of $A$: Note that

$\displaystyle\|I_{1}(\lambda)\|_{X_{0}}$
$\displaystyle\leq\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}\left(\int_{0}^{t}\kappa(t-s)\|u(s)\|_{X_{0}}\,\mathrm{d}s\right)\,\mathrm{d}t$
$\displaystyle=\mathcal{L}[\kappa](\lambda)\mathcal{L}[\|u\|_{X_{0}}](\lambda)=\frac{\phi(\lambda)}{\lambda}\mathcal{L}[\|u\|_{X_{0}}](\lambda)$

and

$\|I_{2}(\lambda)\|_{X_{1}}\leq\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}\left(\int_{0}^{t}\|f(s)\|_{X_{1}}\,\mathrm{d}s\right)\,\mathrm{d}t=\lambda^{-1}\mathcal{L}[\|f\|_{X_{1}}](\lambda).$

Therefore,

$\displaystyle
A\leq\int_{0}^{\infty}(W\circ\psi)(\tau^{-1})\left|\frac{\mathcal{L}[\|u\|_{X_{0}}+\|f\|_{X_{1}}](1/\psi(\tau^{-1}))}{\psi(\tau^{-1})}\right|^{p}\frac{\mathrm{d}\tau}{\tau}$

Then by (4.22), we obtain

$A\lesssim_{p,[w]_{A_{p}},\kappa}\|u\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{0})}+\|f\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{1})}.$

Estimate of $B$: We prove only

$\displaystyle
J^{p}:=\int_{0}^{\infty}(W\circ\psi)(\tau^{-1})\left|\frac{1}{\psi(\tau^{-1})^{2}}\left(\int_{T}^{\infty}\mathrm{e}^{-\frac{t}{\psi(\tau^{-1})}}\left(\int_{0}^{t}\kappa(s)\,\mathrm{d}s\right)\,\mathrm{d}t\right)\right|^{p}~{}\frac{\mathrm{d}\tau}{\tau}\leq
N,$ (5.12)

where $N=N(p,[w]_{A_{p}},\kappa)$. If we have (5.12), then by Lemma 5.2,

$B\lesssim_{p,[w]_{A_{p}},\kappa}\|u_{0}\|_{X_{1}}\lesssim_{N}\|u\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{0})}+\|f\|_{L_{p}((0,T),w\,\mathrm{d}t;X_{1})}\,,$

where $N=N(p,[w]_{A_{p}},\kappa,N_{X_{0},X_{1}},T)$. Note that

$\int_{0}^{t}\kappa(s)\,\mathrm{d}s\leq\mathrm{e}\int_{0}^{\infty}\mathrm{e}^{-s/t}\kappa(s)\,\mathrm{d}s=\mathrm{e}\,t\phi(t^{-1}).$

Then by Minkowski’s integral inequality, we have

$\displaystyle J$
$\displaystyle\lesssim\left(\int_{0}^{\infty}(W\circ\psi)(\tau^{-1})\left|\frac{1}{\psi(\tau^{-1})^{2}}\left(\int_{T}^{\infty}\mathrm{e}^{-\frac{t}{\psi(\tau^{-1})}}t\phi(t^{-1})\,\mathrm{d}t\right)\right|^{p}~{}\frac{\mathrm{d}\tau}{\tau}\right)^{1/p}$
$\displaystyle\leq\int_{T}^{\infty}\left(\int_{0}^{\infty}\frac{\mathrm{e}^{-\frac{tp}{\psi(\tau^{-1})}}(W\circ\psi)(\tau^{-1})}{\psi(\tau^{-1})^{2p}}\,\frac{\mathrm{d}\tau}{\tau}\right)^{1/p}t\phi(t^{-1})\,\mathrm{d}t.$

By changing variable $\tau\to\phi(\lambda)$ with the help of (2.23),

$\int_{0}^{\infty}\frac{\mathrm{e}^{-\frac{tp}{\psi(\tau^{-1})}}(W\circ\psi)(\tau^{-1})}{\psi(\tau^{-1})^{2p}}\,\frac{\mathrm{d}\tau}{\tau}\lesssim\int_{0}^{\infty}\lambda^{2p-1}\mathrm{e}^{-\lambda
tp}W(\lambda^{-1})\,\mathrm{d}\lambda\lesssim_{p,[w]_{A_{p}}}t^{-2p}W(t),$

where the last inequality is due to Lemma 4.3. Thus, by changing variable
$\lambda\to\phi^{-1}(\tau)$ again,

$\displaystyle J$
$\displaystyle\lesssim\int_{T}^{\infty}\phi(t^{-1})\left(W\left(t\right)\right)^{1/p}\,\frac{\mathrm{d}t}{t}\simeq\int_{0}^{\phi(T^{-1})}\left((W\circ\psi)\left(\tau^{-1}\right)\right)^{1/p}\,\mathrm{d}\tau\leq
N(p,[w]_{A_{p}},\kappa,T),$

where the last inequality is due to
$\left((W\circ\psi)\left(\tau^{-1}\right)\right)^{1/p}\in\mathcal{I}_{o}(-1,0)$.
By combining the above estimates for $A$ and $B$, we obtain (5.9). The theorem
is proved. ∎

######  Remark 5.4.

Note that Theorem 5.3 is independent of the choice of the extension $\kappa$.
More precisely, for two extensions $\kappa$ and $\tilde{\kappa}$ of
$\kappa^{\circ}$ satisfying the assumptions in Theorem 5.3,

$(X_{0},X_{1})_{(W\circ\kappa^{*})^{1/p},p}=(X_{0},X_{1})_{(W\circ\tilde{\kappa}^{*})^{1/p},p}.$

This is an easy consequence of Proposition 3.6-($iv$).

Acknowledgements. J.-H. Choi has been supported by the National Research
Foundation of Korea(NRF) grant funded by the Korea government(ME)
(No.RS-2023-00237315). J. B. Lee has been supported by the NRF grants funded
by the Korea government(MSIT) (No.2021R1C1C2008252 and No. 2022R1A4A1018904).
J. Seo has been supported by a KIAS Individual Grant (MG095801) at Korea
Institute for Advanced Study. K. Woo has been supported by the NRF grant
funded by the Korea government(MSIT) (No.2019R1A2C1084683 and
No.2022R1A4A1032094).

##  Appendix A Examples of generalized real interpolation spaces

In this appendix, we present concrete examples of spaces
$(X_{0},X_{1})_{\phi,p}$ when $X_{0}$ and $X_{1}$ are Sobolev or Besov spaces.
First, we recall the definitions of Sobolev and Besov spaces.

######  Definition A.1.

Let $p\in(1,\infty)$, $q\in[1,\infty]$, $\gamma\in\mathbb{R}$ and $w\in
A_{p}$. The sets $H_{p}^{\gamma}(w)$ and $B_{p,q}^{\gamma}(w)$ are Banach
spaces equipped with norms given by

$\displaystyle\|f\|_{H_{p}^{\gamma}(w)}:=\|(1-\Delta)^{\gamma/2}f\|_{L_{p}(w)},$
$\displaystyle\|f\|_{B_{p,q}^{\gamma}(w)}:=\|S_{0}f\|_{L_{p}(w)}+\left\|\left\\{(2^{j\gamma}\|\Delta_{j}f\|_{L_{p}(w)}\right\\}_{j\in\mathbb{N}}\right\|_{\ell_{q}(\mathbb{N})},$

where

$\displaystyle(1-\Delta)^{\gamma/2}f:=\mathcal{F}^{-1}[(1+|\cdot|^{2})^{\gamma/2}\mathcal{F}[f]],\quad\Delta_{j}f:=\mathcal{F}^{-1}[\Psi(2^{-j}\cdot)\mathcal{F}[f]],\quad
S_{0}:=1-\sum_{j\in\mathbb{N}}\Delta_{j}.$

Here $\mathcal{F}$ and $\mathcal{F}^{-1}$ are Fourier and inverse Fourier
transforms and $\Psi$ is a nonnegaive infinitely smooth function satisfying
$supp(\Psi)\subseteq\\{\xi\in\mathbb{R}^{d}:1/2\leq|\xi|\leq 2\\}$ and

$\sum_{k\in\mathbb{Z}}\Psi(2^{-j}\xi)=1\quad\forall\xi\in\mathbb{R}^{d}\setminus\\{0\\}.$

For $\gamma\in\mathbb{R}_{+}$, we also denote the homogeneous spaces
$\mathring{H}_{p}^{\gamma}(w)$ and $\mathring{B}_{p,q}^{\gamma}(w)$

$\displaystyle\|f\|_{\mathring{H}_{p}^{\gamma}(w)}:=\|(-\Delta)^{\gamma/2}f\|_{L_{p}(w)},\quad\|f\|_{\mathring{B}_{p,q}^{\gamma}(w)}:=\left\|\left\\{(2^{j\gamma}\|\Delta_{j}f\|_{L_{p}(w)}\right\\}_{j\in\mathbb{Z}}\right\|_{\ell_{q}(\mathbb{Z})}.$

We also introduce a result of the generalized interpolation on $L_{p}$ spaces
which yields Orlicz-type spaces $L_{\phi,p}$. To describe the Orlicz-type
spaces, we need a definition of decreasing rearrangements of $f$.

######  Definition A.2.

Let $X$ be a measure space with a $\sigma$-finite positive measure $\mu$.

  1. (i)

For a measurable function $f$ on $X$, $d_{f}(\lambda):=\mu\big{(}\\{x\in
X:|f(x)|>\lambda\\}\big{)}$ is called the distribution function of $f$.

  2. (ii)

For a measurable function $f$ on $X$, the function

$f^{*}:[0,\infty)\rightarrow[0,\infty),\quad
f^{\ast}(t):=\inf\\{s>0:d_{f}(s)\leq t\\},\quad t\in[0,\infty).$

is called the decreasing rearrangement of $f$.

  3. (iii)

For $p\in[1,\infty]$ and $\phi\in\mathcal{I}_{o}(0,1)$, $L_{\phi,p}$ denotes
the set of all measurable functions satisfying

$\|f\|_{L_{\phi,p}}:=\|\phi
f^{*}\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}<\infty.$

We state the main result of this appendix.

######  Proposition A.3.

Let $p,p_{0},p_{1}\in[1,\infty]$, $q_{0},q_{1},q\in[1,\infty]$,
$s,s_{0},s_{1}\in\mathbb{R}$, $w\in A_{p}$ and $\phi\in\mathcal{I}_{o}(0,1)$.

  1. (i)

If $s_{0}\neq s_{1}$, then

$\displaystyle(B_{p,q_{0}}^{s_{0}}(w),B_{p,q_{1}}^{s_{1}}(w))_{\phi,q}$
$\displaystyle=B_{p,q}^{\phi(s_{0},s_{1})}(w),$
$\displaystyle(\mathring{B}_{p,q_{0}}^{s_{0}}(w),\mathring{B}_{p,q_{1}}^{s_{1}}(w))_{\phi,q}$
$\displaystyle=\mathring{B}_{p,q}^{\phi(s_{0},s_{1})}(w),$

where $B_{p,q}^{\phi(s_{0},s_{1})}(w)$ and
$\mathring{B}_{p,q}^{\phi(s_{0},s_{1})}(w)$ are spaces equipped with norms
given by

$\displaystyle\|f\|_{B_{p,q}^{\phi(s_{0},s_{1})}(w)}$
$\displaystyle=\|S_{0}f\|_{L_{p}(\mathbb{R}^{d},w\,\mathrm{d}x)}+\left\|\left\\{2^{js_{0}}\phi(2^{-j(s_{0}-s_{1})})\|\Delta_{j}f\|_{L_{p}(w)}\right\\}_{j\in\mathbb{N}}\right\|_{\ell_{q}(\mathbb{N})}\,,$
$\displaystyle\|f\|_{\dot{B}_{p,q}^{\phi(s_{0},s_{1})}(w)}$
$\displaystyle=\left\|\left\\{2^{js_{0}}\phi(2^{-j(s_{0}-s_{1})})\|\Delta_{j}f\|_{L_{p}(w)}\right\\}_{j\in\mathbb{Z}}\right\|_{\ell_{q}(\mathbb{Z})}.$

  2. (ii)

If $s_{0}\neq s_{1}$, then

$\displaystyle(H_{p}^{s_{0}}(w),H_{p}^{s_{1}}(w))_{\phi,q}$
$\displaystyle=B_{p,q}^{\phi(s_{0},s_{1})}(w),$
$\displaystyle(\mathring{H}_{p}^{s_{0}}(w),\mathring{H}_{p}^{s_{1}}(w))_{\phi,q}$
$\displaystyle=\mathring{B}_{p,q}^{\phi(s_{0},s_{1})}(w).$

  3. (iii)

If we put $\widetilde{\phi}(t):=t\phi(t^{-1})$, then

$\displaystyle(L_{1},L_{\infty})_{\phi,q}=L_{\widetilde{\phi},q},\quad\widetilde{\phi}(t):=t\phi(t^{-1}).$
(A.1)

Moreover for any $p_{0},\,p_{1}\in[1,\infty]$ with $p_{1}\neq p_{1}$,

$\displaystyle(L_{p_{0}},L_{p_{1}})_{\phi,q}=(L_{1},L_{\infty})_{\psi,q},\quad\psi(s):=s^{\theta_{0}}\phi(s^{\theta_{1}-\theta_{0}}),\quad\theta_{i}:=1-\frac{1}{p_{i}},\,i=0,1.$
(A.2)

For verifying $(i)$, we need the following proposition:

######  Proposition A.4.

Let $1\leq q,q_{0},q_{1}\leq\infty$ and $\sigma_{0},\sigma_{1}\in\mathbb{R}$
with $\sigma_{0}\neq\sigma_{1}$. If $A$ is a Banach space, then we have

$\displaystyle(\ell_{q_{0}}^{\sigma_{0}}(A),\ell_{q_{1}}^{\sigma_{1}}(A))_{\phi,q}=\ell_{q}^{\phi(\sigma_{0},\sigma_{1})}(A),$

where

$\displaystyle\|a\|_{\ell_{q}^{\phi(\sigma_{0},\sigma_{1})}(A)}:=\begin{cases}\bigg{(}\sum_{j\in\mathbb{Z}}\left(2^{j\sigma_{0}}\phi\left(2^{-j\left(\sigma_{0}-\sigma_{1}\right)}\right)\|a_{j}\|_{A}\right)^{q}\bigg{)}^{1/q}\quad&\textrm{if}\,\,\,\,q\in[1,\infty)\\\\[11.38109pt]
\sup_{j\in\mathbb{Z}}\left(2^{j\sigma_{0}}\phi\left(2^{-j\left(\sigma_{0}-\sigma_{1}\right)}\right)\|a_{j}\|_{A}\right)\quad&\textrm{if}\,\,\,\,q=\infty\end{cases}$

###### Proof of Proposition A.3.

($i$) This is a direct consequence of Proposition A.4 with $A=L_{p}(w)$. For
detailed arguments, see the proof of Theorem [25, Theorem 2.4.2/(a)] with [16,
Theorem 1].

($ii$) First, we prove the homogeneous case,

$(\mathring{H}_{p}^{s_{0}}(w),\mathring{H}_{p}^{s_{1}}(w))_{\phi,q}=\mathring{B}_{p,q}^{\phi(s_{0},s_{1})}(w).$

For $f=f_{0}+f_{1}$, $f_{j}\in\mathring{H}_{p}^{s_{j}}(w)$ with $j=0,1$, one
can observe that

$\displaystyle\|\Delta_{k}f\|_{L_{p}(w)}$
$\displaystyle\leq\|\Delta_{k}f_{0}\|_{L_{p}(w)}+\|\Delta_{k}f_{1}\|_{L_{p}(w)}$
(A.3)
$\displaystyle=2^{-s_{0}k}\|M_{k,s_{0}}((-\Delta)^{s_{0}/2}f_{0})\|_{L_{p}(w)}+2^{-s_{1}k}\|M_{k,s_{1}}((-\Delta)^{s_{1}/2}f_{1})\|_{L_{p}(w)},$

where $k\in\mathbb{Z}$ and $M_{k,s}$ is defined by

$\mathcal{F}[M_{k,s}g](\xi):=m_{s}(2^{-k}\xi)\mathcal{F}[g](\xi):=\frac{\mathcal{F}[\varphi](2^{-k}\xi)}{|2^{-k}\xi|^{s}}\mathcal{F}[g](\xi).$

Since for all multi-index $\alpha$,

$|D^{\alpha}m_{s}(\xi)|\lesssim_{\alpha,s}\mathbbm{1}_{B_{2}\setminus
B_{1/2}}(\xi),$

we have

$|D^{\alpha}(m_{s}(2^{-k}\cdot))(\xi)|\lesssim_{\alpha,s}2^{-k|\alpha|}\mathbbm{1}_{B_{2^{k+1}}\setminus
B_{2^{k-1}}}(\xi)\lesssim|\xi|^{-|\alpha|}.$

Therefore by the weighted Mikhlin multiplier theorem (e.g. [16, Theorem 1]),
(A.3) implies that

$\|\Delta_{k}f\|_{L_{p}(w)}\lesssim
2^{-s_{0}k}\|f_{0}\|_{\mathring{H}_{p}^{s_{0}}(w)}+2^{-s_{1}k}\|f_{1}\|_{\mathring{H}_{p}^{s_{1}}(w)}$

which yields

$\displaystyle\|\Delta_{k}f\|_{L_{p}(w)}$ $\displaystyle\lesssim
2^{-s_{0}k}K(2^{k(s_{0}-s_{1})},f;\mathring{H}_{p}^{s_{0}}(w),\mathring{H}_{p}^{s_{1}}(w)).$
(A.4)

With the help of (A.4) and Proposition 3.2-$(i)$, we have

$\displaystyle\|f\|_{\mathring{B}_{p,q}^{\phi(s_{0},s_{1})}(w)}$
$\displaystyle:=\left\|\left\\{2^{s_{0}k}\phi(2^{-(s_{0}-s_{1})k})\|\Delta_{k}f\|_{L_{p}(w)}\right\\}_{k\in\mathbb{Z}}\right\|_{\ell_{q}(\mathbb{Z})}$
(A.5)
$\displaystyle\lesssim\left\|\left\\{\phi\big{(}2^{-(s_{0}-s_{1})k}\big{)}K\left(2^{\left(s_{0}-s_{1}\right)k},f;\mathring{H}_{p}^{s_{0}}\left(w\right),\mathring{H}_{p}^{s_{1}}\left(w\right)\right)\right\\}_{k\in\mathbb{Z}}\right\|_{\ell_{q}(\mathbb{Z})}$
$\displaystyle\simeq\|f\|_{\left(\mathring{H}_{p}^{s_{0}}\left(w\right),\mathring{H}_{p}^{s_{1}}\left(w\right)\right)_{\phi,q}}.$

For the converse, we make use of $J$-method. Since

$\|\Delta_{k}f\|_{\mathring{H}_{p}^{s}(w)}=\|(2^{s(k-1)}M_{k-1,-s_{i}}+2^{sk}M_{k,-s_{i}}+2^{s(k+1)}M_{k+1,-s})(\Delta_{k}f)\|_{L_{p}(w)}\,,$

we have

$\displaystyle
J\left(2^{(s_{0}-s_{1})k},\Delta_{k}f;\mathring{H}_{p}^{s_{0}}(w),\mathring{H}_{p}^{s_{1}}(w)\right)\lesssim
2^{s_{0}k}\|\Delta_{k}f\|_{L_{p}(w)}.$ (A.6)

By (A.6) and Proposition 3.2-$(ii)$, we have

$\displaystyle\|f\|_{(\mathring{H}_{p}^{s_{0}}(w),\mathring{H}_{p}^{s_{1}}(w))_{\phi,q}}$
(A.7)
$\displaystyle\lesssim\left\|\left\\{\phi(2^{-(s_{0}-s_{1})k})J\left(2^{(s_{0}-s_{1})k},\Delta_{k}f;\mathring{H}_{p}^{s_{0}}(w),\mathring{H}_{p}^{s_{1}}(w)\right)\right\\}_{k\in\mathbb{Z}}\right\|_{\ell_{q}(\mathbb{Z})}$
$\displaystyle\lesssim\left\|\left\\{2^{s_{0}k}\phi(2^{-(s_{0}-s_{1})k})\|\Delta_{k}f\|_{L_{p}(w)}\right\\}_{k\in\mathbb{Z}}\right\|_{\ell_{q}(\mathbb{Z})}\simeq\|f\|_{B_{p,q}^{\phi(s_{0},s_{1})}(w)}.$

The homogeneous case is proved by (A.5) and (A.7). For the inhomogeneous case,
based on the proof for the homogeneous case, it suffices to observe that
additionally

$\displaystyle J(1,S_{0}f;H_{p}^{s_{0}}(w),H_{p}^{s_{1}}(w))$
$\displaystyle\lesssim\|S_{0}f\|_{L_{p}(\mathbb{R}^{d},w\,\mathrm{d}x)}$
$\displaystyle\lesssim K(1,f;H_{p}^{s_{0}}(w),H_{p}^{s_{1}}(w))$
$\displaystyle\lesssim\|f\|_{(H_{p}^{s_{0}}(w),H_{p}^{s_{1}}(w))_{\phi,q}}.$

This follows from a modification of the proof for the homogeneous case.

($iii$) It is clear that (A.2) follows from (A.1) and Theorem 3.9; note that
for $A_{0}=L_{1}$ and $A_{1}=L_{\infty}$, $L_{p_{i}}\in K(\theta_{i})\cap
J(\theta_{i})$ (see [25, Theorem 1.10.3/1] for the case $0<\theta_{i}<1$).
Therefore, we only prove (A.1). We borrow the argument in [25, Theorem
1.18.6]. Let $f\in L_{1}+L_{\infty}$. By [25, 1.18.6/(9)], we have

$K(t,f;L_{1},L_{\infty})=\int_{0}^{t}f^{*}(\tau)\mathrm{d}\tau\geq
tf^{\ast}(t),$

which yields

$\displaystyle\|f\|_{(L_{1},L_{\infty})_{\phi,p}}$
$\displaystyle:=\left\|\phi(\cdot^{-1})K(\cdot,f;L_{1},L_{\infty})\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}\geq\left\|\widetilde{\phi}f^{\ast}\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}=\|f\|_{L_{\widetilde{\phi},p}}.$

For the converse inequality,

$\displaystyle\left\|\phi(\cdot^{-1})K(\cdot,f;L_{1},L_{\infty})\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}$
$\displaystyle=\left\|\tilde{\phi}\cdot\left(\int_{0}^{1}f^{\ast}(\cdot\tau)\,\mathrm{d}\tau\right)\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}$
$\displaystyle\leq\int_{0}^{1}\left\|\tilde{\phi}f^{\ast}(\cdot\tau)\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}\mathrm{d}\tau$
$\displaystyle=\int_{0}^{1}\left\|\tilde{\phi}(\cdot\tau^{-1})f^{\ast}\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}\mathrm{d}\tau$
$\displaystyle\lesssim\int_{0}^{1}\tau^{-1+\varepsilon}\left\|\tilde{\phi}f^{\ast}\right\|_{L_{p}\left(\mathbb{R}_{+},\frac{\mathrm{d}t}{t}\right)}\mathrm{d}\tau\simeq\|f\|_{L_{\widetilde{\phi},p}}.$

Note that the last inequalities hold due to Proposition 2.3.-($vi$). The
proposition is proved. ∎

We end this appendix by proving Proposition A.4.

###### Proof of Proposition A.4.

Without loss of generality, we assume that $\sigma_{0}>\sigma_{1}$ (see
Proposition 3.6-($iii$)).

Step 1) We first prove that

$\displaystyle(\ell_{\infty}^{\sigma_{0}}(A),\ell_{\infty}^{\sigma_{1}}(A))_{\phi,q}\subset\ell_{q}^{\phi(\sigma_{0},\sigma_{1})}(A)\quad\forall
q\,\,\in[1,\infty].$ (A.8)

For the notational convenience, put $\ell_{\infty}^{\sigma_{0}}(A)=X_{0}$ and
$\ell_{\infty}^{\sigma_{1}}(A)=X_{1}$. Note that

$\displaystyle
K(t,a;X_{0},X_{1})\simeq\sup_{k\in\mathbb{Z}}\min(2^{k\sigma_{0}},t2^{k\sigma_{1}})\|a_{k}\|_{A}$
(A.9)

(see, e.g., [25, 1.18.2/(3)]). Then for $q\in[1,\infty)$, it follows that