# Uniqueness for $2$D Euler and transport equations via extrapolation

Óscar Domínguez  Departamento de Análisis Matemático y Matemática Aplicada,
Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias
3, 28040 Madrid, Spain<EMAIL_ADDRESS>and  Mario Milman  Instituto
Argentino de Matematica, Buenos Aires, Argentina<EMAIL_ADDRESS>https://sites.google.com/site/mariomilman

###### Abstract.

Using extrapolation theory, we develop a new framework to prove the uniqueness
of solutions for transport equations. We apply our methodology to unify and
extend the classical results of Yudovich and Vishik for $2$D Euler equations.
In particular, we establish the uniqueness for the Euler flow whose vorticity
belongs to new scales of function spaces that contain both Yudovich spaces and
BMO. We give a self contained presentation.

###### Key words and phrases:

$2$D incompressible Euler equations, transport equations, Yudovich’s
uniqueness theorem, Vishik’s uniqueness theorem, BMO, extrapolation

###### 1991 Mathematics Subject Classification:

Primary 76B03, 46M35. Secondary 46E30, 46E35.

_Acknowledgements._ Part of this research was carried out while the first-
named author was a postdoc at the Institut Camille Jordan, Lyon, supported by
the Labex Milyon and the French National Research Agency (ANR-10-LABX-0070),
(ANR-11-IDEX-0007). It is our pleasure to thank Prof. Petru Mironescu for some
helpful comments that helped to improve the presentation of the paper.

###### Contents

  1. 1 Introduction
    1. 1.1 Uniqueness of Euler flows
    2. 1.2 A novel methodology via extrapolation
    3. 1.3 Extrapolation and the role of the Yudovich functions: a preview
    4. 1.4 A priori estimates via extrapolation
    5. 1.5 Uniqueness of Lagrangian weak solutions
    6. 1.6 Summary of our approach to uniqueness
    7. 1.7 Extending Yudovich’s uniqueness theorem beyond BMO
    8. 1.8 Vishik’s uniqueness theorem
    9. 1.9 Active scalar equations
    10. 1.10 Road map
  2. 2 The spaces $Y^{\\#\Theta}_{p_{0}}$
    1. 2.1 Characterizations
    2. 2.2 Proof of Theorem 1.2
    3. 2.3 Examples of vorticities in $Y^{\\#\Theta}_{p_{0}}$ that are not in $Y^{\Theta}_{p_{0}}$
  3. 3 The spaces $\dot{B}^{\beta}_{\Pi}$
    1. 3.1 Characterizations
    2. 3.2 Uniqueness for a family of active scalar equations
    3. 3.3 Proof of Theorems 1.3 and 1.4
    4. 3.4 Comparison of Theorem 1.4 with the uniqueness result of Vishik
  4. A Interpolation and Extrapolation: An Atlas
    1. A.1 Interpolation of Besov spaces via retraction revisited
    2. A.2 Results from Section 1.3
      1. A.2.1 Characterizations of $\Delta$-extrapolation spaces
      2. A.2.2 Reiteration
    3. A.3 Computability of $K$-functionals

##  1\. Introduction

###  1.1. Uniqueness of Euler flows

The uniqueness of weak solutions of the Cauchy problem for the Euler equations
of an ideal incompressible fluid on $\Omega=\mathbb{R}^{2}$ or
$\mathbb{T}^{2}$ still presents challenging open questions. In $2$D the Euler
equations can be formulated in terms of a transport equation

(1.1)
$\left\\{\begin{array}[c]{c}\omega_{t}+v\cdot\nabla\omega=0,\qquad\text{on
}(0,\infty)\times\Omega,\\\ v=k\ast\omega,\qquad\text{on
}(0,\infty)\times\Omega,\\\ \omega(0,x)=\omega_{0}(x),\qquad
x\in\Omega,\end{array}\right.$

where $v:[0,\infty)\times\Omega\rightarrow\mathbb{R}^{2}$ is the _velocity_ of
the flow,
$\omega=\operatorname{curl}(v)=\partial_{x_{1}}v_{2}-\partial_{x_{2}}v_{1}$ is
the _vorticity_ , and $k$ is standard Biot-Savart kernel111The expression of
$k$ is particularly simple in $\mathbb{R}^{2},$ namely,
$k(x)=\frac{x^{\perp}}{2\pi\left|x\right|^{2}}$ for
$x=(x_{1},x_{2})\neq\mathbf{0}$. Here $x^{\perp}=(-x_{2},x_{1})$.. The
equation $v=k\ast\omega$ is the so-called _Biot-Savart law_ , allowing to
recover $v$ from $\omega$. It follows (cf. [35, 26, 9]) that

(1.2) $\nabla v=\mathcal{K}\omega,$

where $\mathcal{K}$ is a certain Calderón–Zygmund operator (in short CZO).

The uniqueness of weak solutions to (1.1) with bounded vorticity was
established in a celebrated paper by Yudovich [35], via a clever argument
involving the energy method combined with sharp $L^{p}$-inequalities for CZO.
To extend the uniqueness result to unbounded vorticities, Yudovich [36]
proposed new function spaces, nowadays called _Yudovich spaces_ , that are
somewhat bigger than $L^{\infty}$ (or $L^{p_{0}}\cap L^{\infty}$ in the case
of unbounded domains $\Omega$). More precisely, these spaces are constructed
by means of fixing a non-decreasing doubling function
$\Theta:(1,\infty)\rightarrow(0,\infty)$ (“a growth function”) and $p_{0}\geq
1,$ and letting $Y_{p_{0}}^{\Theta}(\Omega)$ to be the set of all
$\omega\in\cap_{p>p_{0}}L^{p}(\Omega)$ such that

(1.3)
$\left\|\omega\right\|_{Y_{p_{0}}^{\Theta}(\Omega)}:=\sup_{p>p_{0}}\frac{\left\|\omega\right\|_{L^{p}(\Omega)}}{\Theta(p)}<\infty.$

In particular, if222Given two non-negative quantities $A$ and $B$, the
notation $A\lesssim B$ means that there exists a constant $C$, independent of
all essential parameters, such that $A\leq CB$. We write $A\approx B$ if
$A\lesssim B\lesssim A$. $\Theta(p)\approx 1,$ it is easy to see that (cf.
Appendix A.3, Proposition 3)

(1.4) $Y_{p_{0}}^{\Theta}(\Omega)=L^{p_{0}}(\Omega)\cap L^{\infty}(\Omega),$

while if $\Theta(p)\rightarrow\infty,$ as $p\rightarrow\infty,$
$Y_{p_{0}}^{\Theta}(\Omega)$ contains unbounded functions. Although the
definitions do not give explicit descriptions of the $Y_{p_{0}}^{\Theta}$
spaces, it is relatively easy to construct examples of elements belonging to
them. It is worthwhile to mention that the definition of
$Y_{p_{0}}^{\Theta}(\Omega)$ is independent of $p_{0}$ if
$\Omega=\mathbb{T}^{2}$ (or more generally, if $\Omega$ has finite measure),
but if $\Omega=\mathbb{R}^{2}$ (cf. (1.4)) we only have the trivial
embeddings333Given two normed spaces $X$ and $Y$, we write $X\hookrightarrow
Y$ if $X\subset Y$ and the natural embedding from $X$ to $Y$ is continuous.

$Y_{p_{0}}^{\Theta}(\Omega)\hookrightarrow
Y_{p_{1}}^{\Theta}(\Omega),\qquad\text{if}\qquad p_{1}>p_{0}.$

The uniqueness results for $Y^{\Theta}_{p_{0}}$ vorticities are formulated in
terms of functions associated with a growth function $\Theta$, which we shall
term _Yudovich functions_ 444In [36], the function $y_{\Theta}$ is defined in
slightly different way for $r\in(0,1)$. However, this minor modification will
not play a role in what follows, cf. (1.7). $y_{\Theta},$

(1.5) $y_{\Theta}(r):=\inf_{p>p_{0}}\\{\Theta(p)r^{1/p}\\},\qquad r>0.$

In the theory of [36], Yudovich functions appear naturally in the crucial
differential inequality connecting $Y^{\Theta}_{p_{0}}$ solutions with the
energy method. Indeed, let $(v_{i},\omega_{i}),i=1,2,$ be two solutions of
(1.1) and let
$E(t)=\left\|v_{1}(t,\cdot)-v_{2}(t,\cdot)\right\|_{L^{2}(\Omega)}^{2},$ from
the definitions of weak solution and the norm of $Y^{\Theta}_{p_{0}}(\Omega)$,
combined with the Biot-Savart law (1.2) together with the sharp $L^{p}$-norm
estimates for CZO, one has

$\frac{dE(t)}{dt}\leq
c\left\|\omega_{1}\right\|_{Y^{\Theta}_{p_{0}}(\Omega)}E(t)\,y_{\Theta_{1}}(E(t)^{-1}),$

where $y_{\Theta_{1}}$ is the Yudovich function associated with the growth
function $\Theta_{1}$,

(1.6) $\Theta_{1}(p):=p\,\Theta(p).$

Uniqueness (i.e., $v_{1}=v_{2}$) is then achieved under the following Osgood
condition on $y_{\Theta_{1}}$(cf. [36])

(1.7) $\int_{0}^{1}\frac{dr}{ry_{\Theta_{1}}(\frac{1}{r})}=\infty.$

It is easy to verify that (1.7) holds for $\Theta(p)\approx 1$, therefore from
(1.4) we obtain the classical uniqueness result of [35]. It also holds for
$\Theta(p)\approx\log p,$ whose corresponding space $Y^{\Theta}_{p_{0}}$ can
be seen to include unbounded vorticities of the form
$\omega(x)\approx\left|\log\left|\log\left|x\right|\right|\right|$. However,
(1.7) places a severe restriction on $\Theta$ and, indeed, it fails for linear
growth $\Theta(p)\approx p,$ which corresponds to vorticities $\omega$ in the
Orlicz space $e^{L}$ of exponentially integrable functions. It follows that,
by this method, uniqueness cannot be guaranteed for vorticities of the form
$\omega(x)\approx\left|\log\left|x\right|\right|,$ the prototype of an
unbounded function in BMO (cf. [21]). We mention that an alternative and
elementary approach to the well-posedness of $2$D Euler equations in
$Y^{\Theta}_{p_{0}}$ has been recently proposed by Crippa and Stefani [12].

Working in the full plane, the expected uniqueness result for vorticities with
logarithmic singularities was obtained somewhat later by Vishik [32], using a
different method. Interestingly, Vishik’s method is also constructive and
relies on the introduction of the “Vishik spaces” $B_{\Pi}(\mathbb{R}^{d})$
associated with “growth functions” $\Pi$, that control the growth of partial
sums of the $L^{\infty}$-norm of dyadic frequency localizations
$\\{\Delta_{j}f\\}_{j\geq 0}$ of $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$.
Specifically,

(1.8) $\|f\|_{B_{\Pi}(\mathbb{R}^{d})}:=\sup_{N\geq
0}\frac{1}{\Pi(N)}\,\sum\limits_{j=0}^{N}\left\|\Delta_{j}f\right\|_{L^{\infty}(\mathbb{R}^{d})}<\infty.$

Note that $\Pi(N)\approx 1$ gives
$B_{\Pi}(\mathbb{R}^{d})=B_{\infty,1}^{0}(\mathbb{R}^{d})$, a classical Besov
space.

In this approach the Osgood condition on $\Pi,$ controlling uniqueness, is
given by

(1.9) $\int_{1}^{\infty}\frac{dr}{\Pi(r)}=\infty.$

Under this assumption, uniqueness of the Euler flow of is guaranteed provided
that555The result holds for $d$ arbitrary, under the assumption
$p_{0}\in(1,d)$.

(1.10) $\omega\in L^{\infty}([0,T];B_{\Pi}(\mathbb{R}^{2})\cap
L^{p_{0}}(\mathbb{R}^{2}))$

for some $p_{0}\in(1,2)$.

In particular, $\Pi(p)\approx p$ satisfies (1.9) and uniqueness for
vorticities satisfying666$\text{bmo}(\mathbb{R}^{d})$ refers to the local
(a.k.a. inhomogeneous) version of $\text{BMO}(\mathbb{R}^{d})$. Recall that
$\text{bmo}(\mathbb{R}^{d})\subsetneq\text{BMO}(\mathbb{R}^{d})$.

(1.11) $\omega\in L^{\infty}([0,T];\text{bmo}(\mathbb{R}^{2})\cap
L^{p_{0}}(\mathbb{R}^{2})),\qquad p_{0}\in(1,2),$

can be deduced from the embeddings

(1.12) $\text{bmo}(\mathbb{R}^{2})\hookrightarrow
B_{\infty,\infty}^{0}(\mathbb{R}^{2})\hookrightarrow B_{\Pi}(\mathbb{R}^{2}).$

Another approach to uniqueness in BMO is due to Azzam and Bedrossian [3]. In
fact, these authors deal with a large class of active scalar equations,
including Euler and Navier–Stokes equations and modified SQG equations. Their
method (built on $H^{-1}$ norms) may be considered as a further refinement of
the energy method of [35], that takes into account not only the integrability,
but also the inherited regularity properties of BMO functions.

In what concerns larger classes of vorticities, a challenging open problem in
the theory is to decide whether uniqueness of solutions for (1.1) can be
achieved for vorticities in $L^{p}.$ In recent remarkable work [33, 34],
Vishik (cf. also the lecture notes [1]) shows that, for any $2<p<\infty,$
there exist $\omega_{0}\in L^{1}(\mathbb{R}^{2})\cap L^{p}(\mathbb{R}^{2}),$
and a force $f$, such that there exist infinitely many weak solutions of

$\omega_{t}+v\cdot\nabla\omega=f,$

with $\omega\in L^{1}(\mathbb{R}^{2})\cap L^{p}(\mathbb{R}^{2})$, uniformly in
time.

In view of the above discussion, an outstanding question in the area is to
obtain sufficiently large classes of vorticities between $L^{p}$ and BMO that
still guarantee uniqueness for the $2$D Euler flow. In this paper, we address
such a question by introducing the _sharp Yudovich space_
$Y^{\\#\Theta}_{p_{0}}(\Omega)$ (the precise definition will be postponed to
Section 1.7, Definition 1). This space has some remarkable features. Indeed,
we show that the Yudovich functions $y_{\Theta_{1}}$ (cf. (1.5) and (1.6))
associated with $Y_{p_{0}}^{\Theta}$ and $Y_{p_{0}}^{\\#\Theta}$ are the same,
but now, for every growth777The space $\text{BMO}(\Omega)$ in (1.13) should be
replaced by $\text{BMO}(\Omega)\cap L^{p_{0}}(\Omega)$ if
$\Omega=\mathbb{R}^{2}$. $\Theta$,

(1.13) $Y^{\Theta}_{p_{0}}(\Omega)\cup\text{BMO}(\Omega)\subset
Y^{\\#\Theta}_{p_{0}}(\Omega).$

As a consequence, we establish the following uniqueness assertion for
$Y^{\\#\Theta}_{p_{0}}$.

######  Theorem 1.1.

Let $\Omega=\mathbb{R}^{2},\mathbb{T}^{2}$. Assume that the growth function
$\Theta$ satisfies the Osgood type condition (1.7). Then a (Lagrangian) weak
solution $\omega$ of (1.1), such that

(1.14) $\omega\in
L^{\infty}([0,\infty);Y^{\\#\Theta}_{p_{0}}(\Omega))\qquad\text{for
some}\qquad p_{0}\in(2,\infty),$

is uniquely determined by its initial value $\omega_{0}$.

The previous result tells us that the uniqueness vorticity classes
$Y^{\Theta}_{p_{0}}$ and BMO considered in [36] and [32] can be considerably
enlarged. Note that (1.14) holds provided that $\omega_{0}\in
Y_{p_{0}}^{\Theta}(\Omega)$ (cf. (1.13)), while Theorem 1.1 with
$\Theta(p)\approx 1$ recovers uniqueness in

(1.15)
$Y_{p_{0}}^{\\#\Theta}(\Omega)=L^{p_{0}}(\Omega)\cap\text{BMO}(\Omega).$

Hence, we improve the classical result (1.4) in the sense that
$L^{\infty}(\Omega)$ is now replaced by $\text{BMO}(\Omega)$. However, the
applicability of Theorem 1.1 goes further beyond $Y^{\Theta}_{p_{0}}$ and BMO.
Indeed, the admissible growth $\Theta(p)\approx\log p$ in Theorem 1.1 provides
uniqueness for vorticities of type
$\omega(x)\approx(1+|\log|x||)\log(1+|\log|x||)$ (cf. Section 2.3, Example 4).
Note that these vorticities do not belong to $Y_{p_{0}}^{\Theta}$ nor to BMO
(they do not even belong to the larger space888Recall that, as a consequence
of the John–Nirenberg inequality [21], BMO is (locally) contained in $e^{L}$.
$e^{L}$). In fact, they grow to infinity faster than both $\log(1+|\log|x||)$
and $1+|\log|x||$, which are the prototypes of vorticity in Yudovich’s and
Vishik’s methods, respectively. Furthermore, for a general growth $\Theta$, we
propose a simple approach to construct elements in
$Y^{\\#\Theta}_{p_{0}}(\Omega)$ that are not in the classical scale
$Y^{\Theta}_{p_{0}}(\Omega)$; cf. Section 2.3. The limiting case for the
$Y_{p_{0}}^{\\#\Theta}$ scale is once again the linear growth
$\Theta(p)\approx p.$ This suggests a possible route to settle the $L^{p}$
uniqueness problem by means of showing a counterexample belonging to
$Y_{p_{0}}^{\\#\Theta}$ with $\Theta(p)\approx p$.

In the forthcoming sections, we will adequately motivate and explain how the
spaces $Y^{\\#\Theta}_{p_{0}}(\Omega)$ arise naturally in the study of Euler
equations via techniques coming from extrapolation theory.

###  1.2. A novel methodology via extrapolation

In this paper, we apply the extrapolation theory of Jawerth–Milman [19] to
develop a new methodology999Our presentation was inspired by the Colombian
writer García Márquez and his novel “Chronicle of a death foretold”, vintage
publishers, 2003, that starts with a murder and then develops the plot
backwards. In particular, further explanations and documentation on
extrapolation theory will be given in due time. that allows us to
significantly enlarge the known uniqueness classes of vorticities for Euler
equations and, at the same time, establish new uniqueness results for a large
class of active scalar equations, including SQG equations and their
generalizations. Furthermore, our framework includes the Yudovich and Vishik
spaces as particular examples.

Indeed, using extrapolation we will construct new uniqueness spaces that
contain BMO. Our new understanding of Yudovich spaces (cf. Section 1.3)
motivated the construction of extrapolation spaces, where the role played by
$(L^{p_{0}},L^{\infty})$ in Yudovich’s theory is now replaced by the larger
interpolation pair $(L^{p_{0}},\text{BMO}).$ In concrete terms, new spaces
$Y_{p_{0}}^{\\#\Theta}(\Omega)$ (cf. Definition 1 below) are introduced by
means of replacing $\left\|\omega\right\|_{L^{p}(\Omega)}$ in (1.3) by
$\|M_{\Omega}^{\\#}\omega\|_{L^{p}(\Omega)},$ where $M_{\Omega}^{\\#}\omega$
denotes the _Strömberg–Jawerth–Torchinsky maximal operator_ [20]

(1.16) $M_{\Omega}^{\\#}\omega(x):=\sup_{\Omega\supset Q\ni
x}\inf_{c\in\mathbb{R}}((f-c)\chi_{Q})^{\ast}(|Q|/\alpha).$

Here, $f^{\ast}$ is the _non-increasing rearrangement_ of $f$, $Q$ is a cube
with sides parallel to the axes of coordinates, and $\alpha>0$ is a
sufficiently small fixed parameter. While this change worsens the $L^{p}$ norm
by a factor of101010Recall that the classical Fefferman–Stein inequality [16]
says, loosely speaking, that $L^{p}$-norms of $\omega$ and
$M^{\\#}_{\Omega}\omega$ are comparable, but the equivalence constant
deteriorates as $p$ when $p\to\infty$.  $``p"$, this is compensated by the
fact that CZO are bounded on BMO. However, there are obstacles when trying to
implement this idea using the delicate energy method of [36]. We therefore
avoid the energy method and apply extrapolation directly, exploiting the
operators involved in the Biot-Savart law, to obtain a priori estimates for
the modulus of continuity of the flow. This strategy naturally leads to the
proof of uniqueness, in the Lagrangian formulation, under the assumption that
a certain Osgood condition (cf. (1.7) and (1.9)) is satisfied. In our
approach, the interplay between two extrapolation spaces arises. On the one
hand, the extrapolation space related to the vorticities (say
$Y_{p_{0}}^{\\#\Theta}$) and, on the other hand, the extrapolation space in
the Besov scale (e.g. associated with the pair
$(L^{\infty},\dot{W}_{\infty}^{1}))$. The latter controls the modulus of
continuity of the flow and informs the corresponding Osgood condition.
Breaking up the argument in this fashion allows to treat uniqueness not only
for Euler equations, but also for a large class of active scalar equations,
for which SQG equations and their generalizations are distinguished examples.

###  1.3. Extrapolation and the role of the Yudovich functions: a preview

To explain the connection of $Y_{p_{0}}^{\\#\Theta}$, BMO, and CZO with
extrapolation we need to develop some background information. This subsection
contains some basic definitions with more details and documentation to follow
(cf. Appendix A). Two important aspects regarding the level of generality that
we require should be emphasized here:

  * •

Since it is crucial for us to consider a variety of scales of function spaces
(e.g., $L^{p}$, BMO, Besov, Sobolev, Yudovich, Vishik) it will be necessary to
formulate the definitions in a sufficiently general context.

  * •

To deal with different types of scales of function spaces, that measure
different characteristics of their elements (smoothness, integrability,
oscillations, etc.), while at the same achieving a unified description, the
Peetre $K$-functional _associated with each scale_ (cf. (1.17) below) will be
an invaluable tool. Indeed, using the $K$-functional the format of the
formulae for the interpolation norms is the same for _all_ the scales under
consideration, since it is the particular $K$-functional that contains the
quantitative information associated with the given scale. This explains the
“universal” characterization of extrapolation spaces (cf. (1.21) below).

In an informal manner, in interpolation we start with a pair
$\bar{A}=(A_{0},A_{1})$ of compatible111111Informally speaking, this means
that $A_{0}+A_{1}$ “makes sense”. Banach spaces and we wish to extract as
information as possible on intermediate spaces from the end-points $A_{0}$ and
$A_{1}$. Let $(\theta,p)\in(0,1)\times[1,\infty],$ the _real interpolation
space_ $\bar{A}_{\theta,p}$ is the set of all $f\in A_{0}+A_{1}$ such that

$\left\|f\right\|_{\bar{A}_{\theta,p}}:=\bigg{\\{}\int_{0}^{\infty}[t^{-\theta}K(t,f;A_{0},A_{1})]^{p}\frac{dt}{t}\bigg{\\}}^{1/p}<\infty$

(with the usual modification if $p=\infty$), where

(1.17)
$K(t,f;A_{0},A_{1}):=\|f\|_{A_{0}+tA_{1}}=\inf\,\\{\left\|f_{0}\right\|_{A_{0}}+t\left\|f_{1}\right\|_{A_{1}}:f=f_{0}+f_{1},\quad
f_{i}\in A_{i},\quad i=0,1\\}$

is the _Peetre $K$-functional_ (cf. [5, 6]). It is convenient to normalize the
spaces in order to have a continuous scale with respect to $\theta$. This is
achieved by letting
$\bar{A}_{\theta,p}^{\blacktriangleleft}=c_{\theta,p}\,\bar{A}_{\theta,p}$,
where121212If $p=\infty,$ we let $c_{\theta,p}=1.$ Then
$\bar{A}_{\theta,\infty}^{\blacktriangleleft}=\bar{A}_{\theta,\infty}$ with
equality of norms. $c_{\theta,p}:=(\theta(1-\theta)p)^{1/p}$ (cf. [19]), and

(1.18)
$\|f\|_{\bar{A}_{\theta,p}^{\blacktriangleleft}}:=c_{\theta,p}\,\|f\|_{\bar{A}_{\theta,p}}.$

The philosophy behind extrapolation may be viewed as the converse of
interpolation, i.e., in extrapolation we start with a family of intermediate
spaces and we wish to extract as information as possible on their end-points.
The rigorous definition is as follows. Given a growth function $\Theta$, the
$\Delta$-_extrapolation space_
$\Delta_{\theta\in(0,1)}\big{\\{}\frac{\bar{A}_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\big{\\}}$
is defined as the set of all
$f\in\cap_{\theta\in(0,1)}\bar{A}_{\theta,p(\theta)}^{\blacktriangleleft}$
such that (cf. [18, 19])

(1.19)
$\left\|f\right\|_{\Delta_{\theta\in(0,1)}\big{\\{}\frac{\bar{A}_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\big{\\}}}:=\sup_{\theta\in(0,1)}\frac{\left\|f\right\|_{\bar{A}_{\theta,p(\theta)}^{\blacktriangleleft}}}{\Theta(\frac{1}{1-\theta})}<\infty.$

The characterization of these spaces hinges upon the fact that the second
index $p(\theta)$ can be replaced by $\infty,$ in other words the second index
is not important at the level of normalized norms (cf. [19])

(1.20)
$\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\bigg{\\}}=\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(A_{0},A_{1})_{\theta,\infty}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\bigg{\\}}.$

Thus, the commutation of the underlying suprema (“Fubini”!) yields131313See
the discussion in Appendix A.2.

(1.21)
$\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\bigg{\\}}=\bigg{\\{}f:\sup_{t\in(0,\infty)}\frac{K(t,f;A_{0},A_{1})}{t\varphi_{\Theta}\big{(}\frac{1}{t}\big{)}}<\infty\bigg{\\}},$

where

(1.22)
$\varphi_{\Theta}(t):=\inf_{\theta\in(0,1)}\bigg{\\{}\Theta\bigg{(}\frac{1}{1-\theta}\bigg{)}\,t^{1-\theta}\bigg{\\}}.$

For the remaining of this section, we shall place ourselves under the
conditions of [36] and focus our attention on the spaces
$Y_{p_{0}}^{\Theta}(\Omega),\,\Omega=\mathbb{T}^{2}$ (although there are
analogous statements for $\Omega=\mathbb{R}^{2}$ or even general domains
$\Omega\subset\mathbb{R}^{d}$). It is known that, with equivalence of norms
independent of $\theta$,

(1.23)
$(L^{p_{0}}(\Omega),L^{\infty}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}=L^{p(\theta)}(\Omega),\qquad\frac{1}{p(\theta)}=\frac{1-\theta}{p_{0}}$

(cf. [23, eq. (31), p. 61]). Using this fact, together with the monotonicity
properties of the Lebesgue scale and (1.20), yields

(1.24)
$Y_{p_{0}}^{\Theta}(\Omega)=\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(L^{p_{0}}(\Omega),L^{\infty}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{p_{0}}{1-\theta})}\bigg{\\}}=\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(L^{p_{0}}(\Omega),L^{\infty}(\Omega))_{\theta,\infty}^{\blacktriangleleft}}{\Theta(\frac{p_{0}}{1-\theta})}\bigg{\\}}.$

Furthermore, comparing (1.5) and (1.22) via the change
$p\leftrightarrow\frac{p_{0}}{1-\theta},$ we find that, with constants
depending only on $p_{0}$,

(1.25) $\varphi_{\Theta}(t)\approx y_{\Theta}(t).$

To give a full characterization of $Y_{p_{0}}^{\Theta}$ we recall that (cf.
(A.9))

$K(t,f;L^{1}(\Omega),L^{\infty}(\Omega))=\int_{0}^{t}f^{\ast}(s)\,ds.$

Consider the maximal function
$f^{\ast\ast}(t):=\frac{1}{t}\int_{0}^{t}f^{\ast}(s)\,ds$. This information
combined with (1.20), (1.21), (1.24) and (1.25) yields141414Since
$\left|\Omega\right|<\infty,$ it is not hard to see that we can restrict to
$t$ $\in(0,1).$ Indeed, observe that
$\int_{0}^{t}f^{*}(s)\,ds=\int_{0}^{1}f^{*}(s)\,ds$ if $t>1$ and
$\sup_{t>1}\frac{1}{t\varphi_{\Theta}(\frac{1}{t})}<\infty$ because $\Theta$
is a non-decreasing function.

(1.26)
$\left\|f\right\|_{Y_{p_{0}}^{\Theta}(\Omega)}\approx\sup_{t\in(0,\infty)}\,\frac{\int_{0}^{t}f^{\ast}(s)\,ds}{t\varphi_{\Theta}(\frac{1}{t})}\approx\sup_{t\in(0,1)}\,\frac{f^{\ast\ast}(t)}{y_{\Theta}(\frac{1}{t})}\approx\sup_{t\in(0,1)}\,\frac{f^{\ast}(t)}{y_{\Theta}(\frac{1}{t})},$

where the last equivalence follows from (2.16) below. In this way Yudovich
spaces are identified with the more familiar Marcinkiewicz spaces, that have
been extensively studied in the literature (cf. [25, 5]).

######  Example 1.

Let $\Theta(p)\approx p,$ then
$Y^{\Theta}_{p_{0}}(\mathbb{T}^{2})=e^{L}(\mathbb{T}^{2})$ (cf. [18, 19]) with

$\left\|f\right\|_{Y^{\Theta}_{p_{0}}(\mathbb{T}^{2})}\approx\sup_{t\in(0,1)}\,\frac{f^{\ast\ast}(t)}{1-\log
t}\approx\sup_{t\in(0,1)}\,\frac{f^{\ast}(t)}{1-\log t}.$

######  Example 2.

More generally, suppose that $\Theta$ is a growth function such that
$p\in(p_{0},\infty)\mapsto e^{p_{0}/p}\Theta(p)$ is a quasi-decreasing
function151515Some examples are $\Theta(p)\approx p^{\alpha}(\log
p)^{\alpha_{1}}(\log_{2}p)^{\alpha_{2}}\cdots(\log_{m}p)^{\alpha_{m}}$, where
$\alpha,\alpha_{i}\in\mathbb{R}$ and
$\log_{m}p=\underbrace{\log\ldots\log}_{\text{m times}}p$ for $m\geq 2$.
(i.e., equivalent to a decreasing function), then

$\|f\|_{Y_{p_{0}}^{\Theta}(\mathbb{T}^{2})}\approx\sup_{t\in(0,1)}\,\frac{f^{\ast\ast}(t)}{\Theta(1-\log
t)}\approx\sup_{t\in(0,1)}\,\frac{f^{\ast}(t)}{\Theta(1-\log t)}.$

This is a consequence of (1.26) and Lemma 2 below.

###  1.4. A priori estimates via extrapolation

It will be instructive to revisit the main uniqueness result of [36] using our
method. As a first step we use the Biot-Savart law to obtain a priori
estimates for the modulus of continuity of the solutions.

To simplify the exposition, we work with $\Omega=\mathbb{T}^{2}$ (but similar
results hold for $\Omega=\mathbb{R}^{2}$ or $\Omega$ a smooth domain in
$\mathbb{R}^{2}$). Fix $p_{0}>2$. By (1.23), (1.2), the sharp $L^{p}$ norm
inequalities for CZO, and the definition of $Y_{p_{0}}^{\Theta}(\Omega),$ we
derive

(1.27) $\left\|\nabla
v\right\|_{(L^{p_{0}}(\Omega),L^{\infty}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}\leq
c_{p_{0}}\,\frac{1}{1-\theta}\,\Theta\bigg{(}\frac{1}{1-\theta}\bigg{)}\left\|\omega\right\|_{Y_{p_{0}}^{\Theta}(\Omega)}.$

The interpolation theory of Sobolev spaces (cf. [13]) allows us to rewrite the
left-hand side as

(1.28) $\left\|\nabla
v\right\|_{(L^{p_{0}}(\Omega),L^{\infty}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}\approx\left\|v\right\|_{(\dot{W}_{p_{0}}^{1}(\Omega),\dot{W}_{\infty}^{1}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}.$

Then, inserting this information in (1.27), and rewriting the right-hand side
using the definition of $\Theta_{1}$ (cf. (1.6)), we obtain

$\left\|v\right\|_{(\dot{W}_{p_{0}}^{1}(\Omega),\dot{W}_{\infty}^{1}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}\leq
c_{p_{0}}\Theta_{1}\bigg{(}\frac{1}{1-\theta}\bigg{)}\left\|\omega\right\|_{Y_{p_{0}}^{\Theta}(\Omega)}.$

Consequently,

(1.29)
$\left\|v\right\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(\dot{W}_{p_{0}}^{1}(\Omega),\dot{W}_{\infty}^{1}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta_{1}(\frac{1}{1-\theta})}\Big{\\}}}=\sup_{\theta\in(0,1)}\frac{\left\|v\right\|_{(\dot{W}_{p_{0}}^{1}(\Omega),\dot{W}_{\infty}^{1}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}}{\Theta_{1}(\frac{1}{1-\theta})}\lesssim\left\|\omega\right\|_{Y_{p_{0}}^{\Theta}(\Omega)}.$

The Sobolev embedding theorem (recall $p_{0}>d=2$, where $d$ denotes the
dimension of the ambient space) combined with (1.29) yields

$\left\|v\right\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{\infty}(\Omega),\dot{W}_{\infty}^{1}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta_{1}(\frac{1}{1-\theta})}\Big{\\}}}\lesssim\left\|\omega\right\|_{Y_{p_{0}}^{\Theta}(\Omega)}.$

The extrapolation norm that appears on the left-hand side can be computed
explicitly using (1.21), (1.25) and the well-known fact that (cf. (A.11))

(1.30)
$K(t,v;L^{\infty}(\Omega),\dot{W}_{\infty}^{1}(\Omega))\approx\sup_{\left|x-y\right|\leq
t}\left|v(x)-v(y)\right|.$

It then follows that

(1.31)
$\left|v(x)-v(y)\right|\lesssim\left|x-y\right|y_{\Theta_{1}}\bigg{(}\frac{1}{\left|x-y\right|}\bigg{)}\left\|\omega\right\|_{Y_{p_{0}}^{\Theta}(\Omega)},$

where

(1.32)
$y_{\Theta_{1}}(t)=\inf_{\theta\in(0,1)}\,\bigg{\\{}\Theta_{1}\bigg{(}\frac{1}{1-\theta}\bigg{)}\,t^{1-\theta}\bigg{\\}}.$

This is the main estimate of the modulus of continuity in [36, Theorem 2]. To
obtain the corresponding result of [35], let $\Theta(p)\approx 1.$ Then,
$Y_{p_{0}}^{\Theta}(\Omega)=L^{p_{0}}(\Omega)\cap L^{\infty}(\Omega)$ (cf.
(1.4)), and $y_{\Theta_{1}}(t)\approx\log t$ if $t>1,$ yielding

$\left|v(x)-v(y)\right|\lesssim\left|x-y\right||\log|x-y||\left\|\omega\right\|_{L^{\infty}(\Omega)}$

if $|x-y|<1$. See also [26, Section 4.1].

###  1.5. Uniqueness of Lagrangian weak solutions

The proof of uniqueness now appeals to the Lagrangian formulation, cf. [12].
Recall that $(\omega,v)$ is said to be a _Lagrangian weak solution_ to (1.1)
if $\omega$ is obtained via the usual _push-forward_

$\omega(t,x)=\omega_{0}(\phi^{-1}(t,x)),$

where $\phi$ is the flow map relative to $v$ according to

(1.33) $\frac{d}{dt}\phi(t,x)=v(t,\phi(t,x)),\qquad\phi(0,x)=x.$

To avoid unnecessary technical issues, throughout this paper, we shall
restrict our attention to this class of solutions. However, this is not
restrictive at all since, following [2], [10] and [7], every integrable weak
solution of (1.1) is Lagrangian.

The uniqueness result then follows from a well-known

######  Lemma 1.

Let $X$ be a function space and assume that there exists a continuous
nondecreasing function $L:(0,\varepsilon_{L})\rightarrow(0,\infty)$, where
$\varepsilon_{L}\in(0,\infty]$, such that

$|v(t,x)-v(t,y)|\leq L(|x-y|)\,\|\omega(t)\|_{X}\qquad\text{for a.e.}\quad
t\in(0,\varepsilon_{L}),$

where

$\|\omega(t)\|_{X}\in L_{\text{loc}}^{1}(0,\varepsilon_{L}),$

and $L$ satisfies the Osgood condition

$\int_{0}^{\varepsilon_{L}}\frac{dr}{L(r)}=\infty.$

Then, the solution $\phi$ to (1.33) must be unique.

###### Proof.

Indeed, if $\phi_{1}$ and $\phi_{2}$ are two flows related to (1.33), then

$\displaystyle|\phi_{1}(t,x)-\phi_{2}(t,x)|$
$\displaystyle\leq\int_{0}^{t}|v(s,\phi_{1}(s,x))-v(s,\phi_{2}(s,x))|\,ds$
$\displaystyle\leq\int_{0}^{t}L(|\phi_{1}(s,x)-\phi_{2}(s,x)|)\,\|\omega(s)\|_{X}\,ds$

and the desired uniqueness follows immediately as an application of Osgood’s
lemma in the form stated in [4, Lemma 3.4]. ∎

###  1.6. Summary of our approach to uniqueness

The method described above can be summarized as follows.

Step 1: Compute the extrapolation spaces involved.

Step 2: Use the Biot-Savart laws to obtain a priori estimates of the
smoothness of solutions via extrapolation.

Step 3: Prove uniqueness via Osgood conditions.

###  1.7. Extending Yudovich’s uniqueness theorem beyond BMO

As already announced in Theorem 1.1 (and, in particular, the subsequent
discussion), in this paper we extend BMO uniqueness to spaces that contain
_functions of unbounded mean oscillation_. This will be done through the
introduction of the new (extrapolation) spaces $Y_{p_{0}}^{\\#\Theta}$.

######  Definition 1 (Sharp Yudovich spaces).

Let161616The definition can obviously be given in a more general context.
$\Omega=\mathbb{R}^{2},\,\mathbb{T}^{2}$ and let $p_{0}\in[1,\infty)$. Given a
growth function $\Theta$, the _sharp Yudovich space_
$Y_{p_{0}}^{\\#\Theta}(\Omega)$ is defined to be the set of
all171717$\|f\|_{(L^{p})^{\\#}(\Omega)}:=\|M^{\\#}_{\Omega}f\|_{L^{p}(\Omega)}.$
In particular, $(L^{\infty})^{\\#}(\Omega)=\text{BMO}(\Omega)$ and
$(L^{p})^{\\#}(\Omega)=L^{p}(\Omega),\,p\in(1,\infty),$ suitably interpreted
modulo constants; cf. [20, Corollaries 2.5 and 2.6].
$f\in\cap_{p>p_{0}}(L^{p})^{\\#}(\Omega)$ such that

$\left\|f\right\|_{Y_{p_{0}}^{\\#\Theta}(\Omega)}:=\sup_{p>p_{0}}\frac{\left\|f\right\|_{(L^{p})^{\\#}(\Omega)}}{\Theta(p)}<\infty.$

As in the case of Yudovich spaces, the definition of
$Y_{p_{0}}^{\\#\Theta}(\Omega),$ with $\Omega=\mathbb{T}^{2},$ does not depend
on $p_{0}$. When $\Omega=\mathbb{R}^{2}$, we only have the trivial embeddings

$Y_{p_{0}}^{\\#\Theta}(\Omega)\hookrightarrow
Y_{p_{1}}^{\\#\Theta}(\Omega),\qquad\text{if}\qquad p_{1}>p_{0}.$

For example, when $\Theta(p)\approx 1$,

$Y_{p_{0}}^{\\#\Theta}(\mathbb{T}^{2})=\text{BMO}(\mathbb{T}^{2})\qquad\text{and}\qquad
Y_{p_{0}}^{\\#\Theta}(\mathbb{R}^{2})=\text{BMO}(\mathbb{R}^{2})\cap
L^{p_{0}}(\mathbb{R}^{2}),$

cf. (1.15). On the other hand, since
$\|f\|_{(L^{p})^{\\#}(\Omega)}\lesssim\|f\|_{L^{p}(\Omega)},\,1<p_{0}<p\leq\infty$,
we have

(1.34) $Y_{p_{0}}^{\Theta}(\Omega)\hookrightarrow
Y_{p_{0}}^{\\#\Theta}(\Omega).$

In Section 2.1 we show that $Y_{p_{0}}^{\\#\Theta}$ fits into the abstract
extrapolation framework proposed above. In this context, BMO plays the same
role as $L^{\infty}$ in connection with $Y_{p_{0}}^{\Theta}$. In particular,
we obtain characterizations of $Y_{p_{0}}^{\\#\Theta}$ corresponding to those
for $Y_{p_{0}}^{\Theta}$ (cf. Table 1).

$Y^{\Theta}_{p_{0}}$ | $Y^{\\#\Theta}_{p_{0}}$
---|---
$\Big{\\{}f\in\bigcap_{p>p_{0}}L^{p}:\sup_{p>p_{0}}\frac{\left\|f\right\|_{L^{p}}}{\Theta(p)}<\infty\Big{\\}}$ | $\Big{\\{}f\in\bigcap_{p>p_{0}}(L^{p})^{\\#}:\sup_{p>p_{0}}\frac{\left\|f\right\|_{(L^{p})^{\\#}}}{\Theta(p)}<\infty\Big{\\}}$
$\Delta_{\theta\in(0,1)}\,\Big{\\{}\frac{(L^{p_{0}},L^{\infty})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{p_{0}}{1-\theta})}\Big{\\}}$ | $\Delta_{\theta\in(0,1)}\,\Big{\\{}\frac{(L^{p_{0}},\text{BMO})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{p_{0}}{1-\theta})}\Big{\\}}$
$\Big{\\{}f:\sup_{t\in(0,\infty)}\frac{K(t^{1/p_{0}},f;L^{p_{0}},L^{\infty})}{t^{1/p_{0}}y_{\Theta}(\frac{1}{t})}<\infty\Big{\\}}$ | $\Big{\\{}f:\sup_{t\in(0,\infty)}\frac{K(t^{1/p_{0}},f;L^{p_{0}},\text{BMO})}{t^{1/p_{0}}y_{\Theta}(\frac{1}{t})}<\infty\Big{\\}}$
$\Big{\\{}f:\sup_{t\in(0,\infty)}\frac{(|f|^{p_{0}})^{\ast\ast}(t)^{1/p_{0}}}{y_{\Theta}(\frac{1}{t})}<\infty\Big{\\}}$ | $\Big{\\{}f:\sup_{t\in(0,\infty)}\frac{(|M^{\\#}f|^{p_{0}})^{**}(t)^{1/p_{0}}}{y_{\Theta}(\frac{1}{t})}<\infty\Big{\\}}$
Table 1. Yudovich vs. sharp Yudovich

The usefulness of $Y_{p_{0}}^{\\#\Theta}$ emerges when establishing a priori
estimates for the modulus of continuity of the velocity.

######  Theorem 1.2.

Assume that $\omega\in Y_{p_{0}}^{\\#\Theta}(\Omega)$ for some
$p_{0}\in(2,\infty)$. Then

(1.35)
$|v(x)-v(y)|\lesssim|x-y|\,y_{\Theta_{1}}\bigg{(}\frac{1}{|x-y|}\bigg{)}\,\|\omega\|_{Y_{p_{0}}^{\\#\Theta}(\Omega)},$

where $y_{\Theta_{1}}$ is given by (1.32).

The proof of this result will be given in Section 2.2. In particular, (1.35)
extends the classical Yudovich’s estimate (1.31) from $Y^{\Theta}_{p_{0}}$ to
$Y^{\\#\Theta}_{p_{0}}$. As a consequence (cf. Section 1.5), we achieve the
desired uniqueness result for $Y^{\\#\Theta}_{p_{0}}$ stated in Theorem 1.1.

###  1.8. Vishik’s uniqueness theorem

As already claimed in Section 1.2, we show that Vishik spaces
$B_{\Pi}(\Omega)$ (cf. (1.8)) are also special examples of extrapolation
constructions in the sense of (1.19). In fact, we obtain several
characterizations of Vishik spaces in terms of a variety of means
(extrapolation, interpolation, Yudovich functions, and growths of classical
Besov norms) showing the full analogy between $B_{\Pi}$ and
$Y^{\Theta}_{p_{0}}$. The results are collected in Table 2 (where, for
simplicity again, we let $p_{0}=1$ in the definition of $Y_{p_{0}}^{\Theta}$)
and the proofs may be found in Section 3.1 below.

$Y^{\Theta}_{1}$ | $B_{\Pi}$
---|---
$\Big{\\{}f\in\bigcap_{p>1}L^{p}:\sup_{p>1}\frac{\left\|f\right\|_{L^{p}}}{\Theta(p)}<\infty\Big{\\}}$ | $\Big{\\{}f\in\bigcap_{\alpha\in(-1,0)}B^{\alpha}_{\infty,1}:\sup_{\alpha\in(-1,0)}\frac{\|f\|_{B^{\alpha}_{\infty,1}}}{\Pi(-\frac{1}{\alpha})}<\infty\Big{\\}}$
$\Delta_{\theta\in(0,1)}\,\bigg{\\{}\frac{(L^{1},L^{\infty})_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\bigg{\\}}$ | $\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(B^{-1}_{\infty,1},B^{0}_{\infty,1})_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}{\Pi(\frac{1}{1-\theta})}\bigg{\\}}$
$\Big{\\{}f:\sup_{t\in(0,\infty)}\frac{K(t,f;L^{1},L^{\infty})}{ty_{\Theta}(\frac{1}{t})}<\infty\Big{\\}}$ | $\Big{\\{}f:\sup_{t\in(0,\infty)}\frac{K(t,f;B^{-1}_{\infty,1},B^{0}_{\infty,1})}{ty_{\Pi}(\frac{1}{t})}<\infty\Big{\\}}$
$\Big{\\{}f:\sup_{0<t<\infty}\frac{f^{\ast\ast}(t)}{y_{\Theta}(\frac{1}{t})}<\infty\Big{\\}}$ | $\Big{\\{}f:\sup_{N\geq 0}\frac{1}{\Pi(N)}\,\sum\limits_{j=0}^{N}\left\|\Delta_{j}f\right\|_{L^{\infty}}<\infty\Big{\\}}$
Table 2. Yudovich vs. Vishik

Having at hand the information contained in Table 2, we are in a position to
apply the extrapolation approach to uniqueness developed in Section 1.6. Our
results are formulated in terms of homogeneous function spaces, rather than
their inhomogeneous counterparts. As we will see later, this will result in
several improvements.

######  Theorem 1.3.

Assume that the growth function $\Pi\in\mathcal{P}_{1}$ (cf. Definition 3) and
$\omega\in\dot{B}_{\Pi}(\mathbb{R}^{2})\cap\dot{B}_{\infty,1}^{-1}(\mathbb{R}^{2})$.
Then

$|v(x)-v(y)|\lesssim|x-y|\,y_{\Pi}\bigg{(}\frac{1}{|x-y|}\bigg{)}\,\|\omega\|_{\dot{B}_{\Pi}(\mathbb{R}^{2})\cap\dot{B}_{\infty,1}^{-1}(\mathbb{R}^{2})},$

where $y_{\Pi}$ is the Yudovich function associated to $\Pi$ (cf. (1.5)).

As a by-product, we arrive at the following uniqueness statement for Vishik
spaces.

######  Theorem 1.4.

Assume that the growth function $\Pi\in\mathcal{P}_{1}$ satisfies the Osgood
condition

(1.36) $\int_{1}^{\infty}\frac{dr}{ry_{\Pi}(r)}=\infty.$

Then, a Lagrangian weak solution $\omega$ of (1.1), such that

(1.37) $\omega\in
L^{\infty}([0,T];\dot{B}_{\Pi}(\mathbb{R}^{2})\cap\dot{B}_{\infty,1}^{-1}(\mathbb{R}^{2}))$

is uniquely determined by its initial value $\omega_{0}$.

######  Remark 1.5.

It follows from the trivial estimate $y_{\Pi}(r)\lesssim\Pi(\log r)$ that

$\int_{1}^{\infty}\frac{dr}{\Pi(r)}\lesssim\int_{1}^{\infty}\frac{dr}{ry_{\Pi}(r)}.$

Therefore, the validity of the Osgood condition (1.9) implies181818In fact,
under natural assumptions on $\Pi$, we have $y_{\Pi}(r)\approx\Pi(\log r)$;
cf. Lemma 2 below. Then, a simple change of variables gives
$\int_{1}^{\infty}\frac{dr}{\Pi(r)}\approx\int_{1}^{\infty}\frac{dr}{ry_{\Pi}(r)}.$
(1.36). Furthermore, the formulation (1.36) shows a connection with (1.7) via
the exchange $\Theta_{1}\leftrightarrow\Pi$.

In Section 3.4 we will show that Theorem 1.4 gives an improvement of Vishik’s
uniqueness theorem [32, Theorem 7.1] (cf. also (1.10)) in the following sense

$B_{\Pi}(\mathbb{R}^{2})\cap
L^{p_{0}}(\mathbb{R}^{2})\hookrightarrow\dot{B}_{\Pi}(\mathbb{R}^{2})\cap\dot{B}_{\infty,1}^{-1}(\mathbb{R}^{2}),$

for every growth $\Pi$ and $p_{0}\in(1,2)$; cf. Proposition 1. In particular,
if $\Pi(r)\approx r$ (so that $y_{\Pi}(r)\approx\left|\log r\right|$ and thus
(1.36) holds) then

$\text{BMO}(\mathbb{R}^{2})\hookrightarrow\dot{B}_{\infty,\infty}^{0}(\mathbb{R}^{2})\hookrightarrow\dot{B}_{\Pi}(\mathbb{R}^{2});$

compare with the inhomogeneous statement (1.12). According to Theorem 1.4,
uniqueness of the Euler flow is guaranteed provided that

(1.38) $\omega\in
L^{\infty}([0,T];\text{BMO}(\mathbb{R}^{2})\cap\dot{B}_{\infty,1}^{-1}(\mathbb{R}^{2})).$

As a consequence, we are able to extend Vishik’s uniqueness condition (1.11)
from $\text{bmo}(\mathbb{R}^{2})$ to $\text{BMO}(\mathbb{R}^{2})$. Moreover,
we remove (at least, working with Lagrangian solutions) the decay at infinity
of the vorticity inherited to the $L^{p_{0}}(\mathbb{R}^{2})$ assumption.
Specifically, in Proposition 2 we prove the following embedding

$\text{bmo}(\mathbb{R}^{2})\cap
L^{p_{0}}(\mathbb{R}^{2})\hookrightarrow\text{BMO}(\mathbb{R}^{2})\cap\dot{B}_{\infty,1}^{-1}(\mathbb{R}^{2}),\qquad
p_{0}\in(1,2).$

###  1.9. Active scalar equations

So far, we have discussed the role played by a variety of function spaces in
connection with $2$D Euler equations. However, it is also important to deal
with the more general class of active scalar equations modelled by

(1.39) $\omega_{t}+TS\omega\cdot\nabla\omega=0,$

and equipped with the abstract Biot-Savart law described (at least, formally)
as

(1.40) $TS:\omega\mapsto v=TS\omega.$

Here, $\omega=\omega(t,x),\,x\in\mathbb{R}^{d},t>0,$ is a scalar function and
$T$ and $S$ are (linear) operators such that $TS=ST$.

Distinguished examples of (1.39)-(1.40) in $2$D are given by

(1.41) $T=R=(R_{l})_{l=1,\ldots,d}\quad\text{(Riesz-type transforms)},\qquad
S=(-\Delta)^{\frac{\beta-1}{2}},$

for191919Recall that $R=\nabla^{\perp}(-\Delta)^{-\frac{1}{2}}.$ Then
$v=\nabla^{\perp}(-\Delta)^{\frac{\beta}{2}-1}\omega$. $\beta\in\mathbb{R}$.
In particular, if $\beta=0$ then one recovers the classical $2$D Euler
equations in its vorticity form (cf. (1.1)), while $\beta=1$ corresponds to
_surface quasi-geostrophic (SQG) equations_ , and the range $\beta\in(0,1)$
refers to the so-called _intermediate SQG equations_. The SQG equations and
their generalizations arise from applications in atmospheric science and have
attracted a big deal of attention in recent times. In particular, Constantin,
Majda and Tabak [11] established a remarkable connection between SQG equation
and the $3$D-Euler equation. We also refer the reader to the survey paper [24]
(and the references therein), where the outstanding issue of singularity
formation in SQG equations is discussed. We mention that singular202020Note
that $TS=\nabla^{\perp}(-\Delta)^{\frac{\beta}{2}-1}$ and
$\lim_{|\xi|\rightarrow\infty}\widehat{(-\Delta)^{\frac{\beta}{2}-1}}(\xi)=\infty$
if $\beta>2$. regimes $\beta>2$ are also of interest, e.g. $\beta=3$ is
connected to Hall-magnetohydrodynamics (see e.g. [8]).

One of the advantages of the extrapolation method, described in previous
sections for Euler equations, is its flexibility. Indeed, it allows us to
consider in a unified fashion a variety of function spaces, and also works for
the class of active scalar equations (1.39)–(1.41). In this regard, the basic
observation is that $S^{-1}=(-\Delta)^{\frac{1-\beta}{2}}$ plays the
role212121Recall the informal assertion $\nabla``="(-\Delta)^{1/2}$. of
$\nabla$ when applying the Sobolev embedding theorem with $\beta=0$ (i.e., the
Eulerian setting) in Section 1.4.

Based on the above considerations, in Section 3.2 we show that Theorem 1.4 is
a special case (for the $2$D Euler equations) of a more general phenomenon
related to generalized SQG equations (or more generally, (1.39)-(1.40)). Our
results will be formulated in terms of the Vishik spaces
$\dot{B}_{\Pi}^{\beta},$ which are constructed in the same way as
$\dot{B}_{\Pi}$ (cf. (1.8)), but now with $\dot{B}_{\infty,1}^{\beta}$ playing
the role previously assigned to $\dot{B}_{\infty,1}^{0}$ (cf. Definition 2).

In order to facilitate the reading for non experts, we close the paper with an
Atlas on Interpolation and Extrapolation (cf. Appendix A), where we collect
documentation and supplementary material.

We believe that the techniques developed in this paper could be useful in
other related contexts.

###  1.10. Road map

We simply remark that the previous discussion and the table of contents show
the local organization of the paper.

##  2\. The spaces $Y^{\\#\Theta}_{p_{0}}$

###  2.1. Characterizations

We establish several characterizations of $Y^{\\#\Theta}_{p_{0}}$ (cf.
Definition 1) in terms of extrapolation of interpolation scales and
$K$-functionals involving BMO, and maximal functions.

######  Theorem 2.1 (Characterization via extrapolation).

Let $p_{0}\in[1,\infty),\,\theta\in(0,1),$ and let
$\Omega=\mathbb{R}^{2},\mathbb{T}^{2}$. We have

(2.1)
$(L^{p_{0}}(\Omega),\emph{BMO}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}=(L^{p(\theta)})^{\\#}(\Omega),\qquad\frac{1}{p(\theta)}=\frac{1-\theta}{p_{0}},$

with equivalence constants independent of $\theta$. As a consequence,

(2.2)
$Y_{p_{0}}^{\\#\Theta}(\Omega)=\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(L^{p_{0}}(\Omega),\emph{BMO}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{p_{0}}{1-\theta})}\bigg{\\}}.$

###### Proof.

Using the Jawerth–Torchinsky formula (cf. (A.10))

(2.3)
$K(t,f;L^{p_{0}}(\Omega),\text{BMO}(\Omega))\approx\bigg{(}\int_{0}^{t^{p_{0}}}[(M^{\\#}_{\Omega}f)^{*}(\xi)]^{p_{0}}\,d\xi\bigg{)}^{1/p_{0}},$

where the maximal function $M^{\\#}_{\Omega}$ is defined by (1.16), we have

(2.4)
$\|f\|_{(L^{p_{0}}(\Omega),\text{BMO}(\Omega))_{\theta,p(\theta)}}\approx\bigg{\\{}\int_{0}^{\infty}\bigg{(}\frac{1}{t}\int_{0}^{t}[(M_{\Omega}^{\\#}f)^{*}(\xi)]^{p_{0}}\,d\xi\bigg{)}^{p(\theta)/p_{0}}\,dt\bigg{\\}}^{1/p(\theta)}.$

Applying the sharp version of Hardy’s inequality (note that
$p(\theta)=\frac{p_{0}}{1-\theta}>p_{0}$) stated in [31, Appendix A.4, page
272], we can estimate (2.4) as follows

(2.5)
$\|f\|_{(L^{p_{0}}(\Omega),\text{BMO}(\Omega))_{\theta,p(\theta)}}\lesssim\theta^{-1/p_{0}}\|f\|_{(L^{p(\theta)})^{\\#}(\Omega)}.$

Conversely, one can invoke the reverse Hardy inequality (cf. [28] and the
references therein). Indeed

$\displaystyle\|f\|_{(L^{p(\theta)})^{\\#}(\Omega)}$
$\displaystyle=\|(M^{\\#}f)^{\ast}\|_{L^{p(\theta)}(0,\infty)}=\|(M^{\\#}f)^{p_{0}\ast}\|_{L^{p(\theta)/p_{0}}(0,\infty)}^{1/p_{0}}$
(2.6)
$\displaystyle\lesssim\theta^{1/p(\theta)}\,\|[(M^{\\#}f)^{p_{0}}]^{\ast\ast}\|_{L^{p(\theta)/p_{0}}(0,\infty)}^{1/p_{0}}\approx\theta^{1/p(\theta)}\,\|f\|_{(L^{p_{0}}(\Omega),\text{BMO}(\Omega))_{\theta,p(\theta)}},$

where in the last estimate we have used (2.4). Combining (2.5) and (2.6) we
obtain

$\displaystyle\|f\|_{(L^{p_{0}}(\Omega),\text{BMO}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}$
$\displaystyle\lesssim c_{\theta
p(\theta)}\theta^{-1/p_{0}}\|f\|_{(L^{p(\theta)})^{\\#}(\Omega)}$
$\displaystyle\lesssim\theta^{-\theta/p_{0}}\,\|f\|_{(L^{p_{0}}(\Omega),\text{BMO}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}\approx\|f\|_{(L^{p_{0}}(\Omega),\text{BMO}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}.$

Now, (2.1) follows readily since $c_{\theta
p(\theta)}=(\theta(1-\theta)p(\theta))^{1/p(\theta)}\approx\theta^{1/p(\theta)}\approx\theta^{1/p_{0}}$.
∎

######  Theorem 2.2 (Characterization via $K$-functional and maximal
function).

Let $p_{0}\in[1,\infty)$ and $\Omega=\mathbb{R}^{2},\mathbb{T}^{2}$.

  1. (i)

We have

(2.7)
$\|f\|_{Y_{p_{0}}^{\\#\Theta}(\Omega)}\approx\sup_{t\in(0,\infty)}\frac{K(t^{1/p_{0}},f;L^{p_{0}}(\Omega),\emph{BMO}(\Omega))}{t^{1/p_{0}}y_{\Theta}(\frac{1}{t})}\approx\sup_{t\in(0,\infty)}\frac{[(M_{\Omega}^{\\#}f)^{p_{0}}]^{\ast\ast}(t)^{1/p_{0}}}{y_{\Theta}(\frac{1}{t})},$

where $y_{\Theta}$ is given by (1.5).

  2. (ii)

Assume that the map

(2.8) $p\in(p_{0},\infty)\mapsto e^{p_{0}/p}\Theta(p)\qquad\text{is quasi-
decreasing}.$

Then

(2.9) $\displaystyle\|f\|_{Y_{p_{0}}^{\\#\Theta}(\Omega)}$
$\displaystyle\approx\|f\|_{(L^{p_{0}})^{\\#}(\Omega)}+\sup_{t\in(0,e^{-1})}\frac{K(t,f;L^{p_{0}}(\Omega),\emph{BMO}(\Omega))}{t\,\Theta(-\log
t)}$
$\displaystyle\approx\|f\|_{(L^{p_{0}})^{\\#}(\Omega)}+\sup_{t\in(0,e^{-1})}\frac{[(M_{\Omega}^{\\#}f)^{p_{0}}]^{\ast\ast}(t)^{1/p_{0}}}{\Theta(-\log
t)}.$

In the proof of Theorem 2.2 we will make use of the following lemma, which
provides an explicit characterization of $y_{\Theta}$ in terms of $\Theta$.

######  Lemma 2.

Let $y_{\Theta}$ be the Yudovich function relative to $\Theta$ (cf. (1.5)).

  1. (i)

We have

(2.10) $y_{\Theta}(r)=\Theta(p_{0})r^{1/p_{0}},\qquad\text{if}\qquad
r\in(0,1).$

  2. (ii)

Suppose that $\Theta$ satisfies (2.8). Then

$y_{\Theta}(r)\approx\Theta(\log r),\qquad\text{if}\qquad r>e^{p_{0}}.$

###### Proof.

(i) The formula (2.10) is an immediate consequence of the fact that for each
$r\in(0,1),$ the map $p\in(p_{0},\infty)\mapsto\Theta(p)r^{1/p}$ is
increasing.

(ii) Note that $y_{\Theta}(r)\leq\Theta(\log r)\,r^{1/\log
r}\approx\Theta(\log r)$. Conversely, given any $p>p_{0}$,

$\Theta(p)r^{1/p}\geq\Theta(\log r)\qquad\text{if}\qquad p>\log r.$

If $p\in(p_{0},\log r)$, then

$\Theta(p)r^{1/p}=e^{p_{0}/p}\Theta(p)\bigg{(}\frac{r}{e^{p_{0}}}\bigg{)}^{1/p}\gtrsim
e^{p_{0}/\log r}\Theta(\log r)\bigg{(}\frac{r}{e^{p_{0}}}\bigg{)}^{1/\log
r}\approx\Theta(\log r).$

Hence,

$y_{\Theta}(r)=\inf_{p>p_{0}}\\{\Theta(p)r^{1/p}\\}\gtrsim\Theta(\log r).$

∎

###### Proof of Theorem 2.2.

(i) By the reiteration property of the $\Delta$-extrapolation method (cf.
(A.3)), we have

(2.11)
$\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(L^{p_{0}}(\Omega),\text{BMO}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{p_{0}}{1-\theta})}\bigg{\\}}=\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(L^{p_{0}}(\Omega),\text{BMO}(\Omega))_{\theta,\infty}^{\blacktriangleleft}}{\Theta(\frac{p_{0}}{1-\theta})}\bigg{\\}}.$

Then (cf. (A.4) and (A.5))

(2.12)
$\|f\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{p_{0}}(\Omega),\text{BMO}(\Omega))_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{p_{0}}{1-\theta})}\Big{\\}}}\approx\sup_{t\in(0,\infty)}\frac{K(t^{1/p_{0}},f;L^{p_{0}}(\Omega),\text{BMO}(\Omega))}{t^{1/p_{0}}y_{\Theta}(\frac{1}{t})}.$

Combining with (2.2) we obtain the first equivalence in (2.7). The second
equivalence now follows from (2.3).

(ii) Suppose now that $\Theta$ satisfies (2.8). By Lemma 2 and (2.3), we have

$\displaystyle\sup_{t\in(0,\infty)}\frac{K(t^{1/p_{0}},f;L^{p_{0}}(\Omega),\text{BMO}(\Omega))}{t^{1/p_{0}}y_{\Theta}(\frac{1}{t})}$
$\displaystyle\approx$
$\displaystyle\sup_{t\in(0,e^{-p_{0}})}\frac{K(t^{1/p_{0}},f;L^{p_{0}}(\Omega),\text{BMO}(\Omega))}{t^{1/p_{0}}y_{\Theta}(\frac{1}{t})}+\sup_{t\in(1,\infty)}\frac{K(t^{1/p_{0}},f;L^{p_{0}}(\Omega),\text{BMO}(\Omega))}{t^{1/p_{0}}y_{\Theta}(\frac{1}{t})}$
$\displaystyle\approx\sup_{t\in(0,e^{-p_{0}})}\frac{K(t^{1/p_{0}},f;L^{p_{0}}(\Omega),\text{BMO}(\Omega))}{t^{1/p_{0}}\Theta(-\log
t)}+\sup_{t\in(1,\infty)}K(t^{1/p_{0}},f;L^{p_{0}}(\Omega),\text{BMO}(\Omega))$
(2.13)
$\displaystyle\approx\sup_{t\in(0,e^{-1})}\frac{K(t,f;L^{p_{0}}(\Omega),\text{BMO}(\Omega))}{t\Theta(-\log
t)}+\|M^{\\#}_{\Omega}f\|_{L^{p_{0}}(\Omega)}.$

Similarly, one can show that

(2.14)
$\sup_{t\in(0,\infty)}\frac{[(M_{\Omega}^{\\#}f)^{p_{0}}]^{\ast\ast}(t)^{1/p_{0}}}{y_{\Theta}(\frac{1}{t})}\approx\sup_{t\in(0,e^{-1})}\frac{[(M_{\Omega}^{\\#}f)^{p_{0}}]^{\ast\ast}(t)^{1/p_{0}}}{\Theta(-\log
t)}+\|M^{\\#}_{\Omega}f\|_{L^{p_{0}}(\Omega)}.$

Combining (2.7) with (2.13) and (2.14), we complete the proof of (2.9). ∎

When $\Omega=\mathbb{T}^{2}$, the characterizations provided by Theorem 2.2
can be supplemented as follows.

######  Theorem 2.3.

Let $p_{0}\in[1,\infty)$ and suppose that $\Theta$ satisfies (2.8). Then

$\displaystyle\|f\|_{Y_{p_{0}}^{\\#\Theta}(\mathbb{T}^{2})}$
$\displaystyle\approx\sup_{t\in(0,e^{-1})}\frac{K(t,f;L^{p_{0}}(\mathbb{T}^{2}),\emph{BMO}(\mathbb{T}^{2}))}{t\Theta(-\log
t)}$ (2.15)
$\displaystyle\approx\sup_{t\in(0,e^{-1})}\frac{[(M_{\mathbb{T}^{2}}^{\\#}f)^{p_{0}}]^{\ast\ast}(t)^{1/p_{0}}}{\Theta(-\log
t)}\approx\sup_{t\in(0,e^{-1})}\frac{(M_{\mathbb{T}^{2}}^{\\#}f)^{\ast}(t)}{\Theta(-\log
t)}.$

######  Remark 2.4.

Note that the last expression in (2.15) is independent of $p_{0}$. This is in
accord with the fact that the definition of
$Y_{p_{0}}^{\\#\Theta}(\mathbb{T}^{2})$ does not depend on $p_{0}$.

###### Proof of Theorem 2.3.

Using (2.3) we can write

$K(e^{-1},f;L^{p_{0}}(\mathbb{T}^{2}),\text{BMO}(\mathbb{T}^{2}))\approx[(M_{\mathbb{T}^{2}}^{\\#}f)^{p_{0}}]^{\ast\ast}(e^{-1})\approx\|f\|_{(L^{p_{0}})^{\\#}(\mathbb{T}^{2})}.$

Accordingly, the first and second equivalences in (2.15) are consequences of
(2.9).

We now prove the last equivalence in (2.15). We claim that

(2.16) $\int_{0}^{t}y_{\Theta}\bigg{(}\frac{1}{s}\bigg{)}^{p_{0}}\,ds\lesssim
ty_{\Theta}\bigg{(}\frac{1}{t}\bigg{)}^{p_{0}},\qquad t\in(0,1).$

Assuming momentarily the validity of (2.16), we have

$\displaystyle[(M_{\mathbb{T}^{2}}^{\\#}f)^{p_{0}}]^{\ast\ast}(t)^{1/p_{0}}$
$\displaystyle=\bigg{[}\frac{1}{t}\int_{0}^{t}\bigg{[}\frac{(M_{\mathbb{T}^{2}}^{\\#}f)^{\ast}(s)}{y_{\Theta}(\frac{1}{s})}\,\,y_{\Theta}\bigg{(}\frac{1}{s}\bigg{)}\bigg{]}^{p_{0}}\,ds\bigg{]}^{1/p_{0}}$
$\displaystyle\leq\bigg{[}\frac{1}{t}\int_{0}^{t}y_{\Theta}\bigg{(}\frac{1}{s}\bigg{)}^{p_{0}}\,ds\bigg{]}^{1/p_{0}}\,\sup_{t\in(0,1)}\frac{(M_{\mathbb{T}^{2}}^{\\#}f)^{\ast}(t)}{y_{\Theta}(\frac{1}{t})}$
$\displaystyle\lesssim
y_{\Theta}\bigg{(}\frac{1}{t}\bigg{)}\,\sup_{t\in(0,1)}\frac{(M_{\mathbb{T}^{2}}^{\\#}f)^{\ast}(t)}{y_{\Theta}(\frac{1}{t})}.$

Consequently,

$\sup_{t\in(0,1)}\frac{[(M_{\mathbb{T}^{2}}^{\\#}f)^{p_{0}}]^{\ast\ast}(t)^{1/p_{0}}}{y_{\Theta}(\frac{1}{t})}\lesssim\sup_{t\in(0,1)}\frac{(M_{\mathbb{T}^{2}}^{\\#}f)^{\ast}(t)}{y_{\Theta}(\frac{1}{t})}.$

The converse estimate is a simple consequence of the fact that
$(M_{\mathbb{T}^{2}}^{\\#}f)^{\ast}$ is decreasing. This completes the proof
of the last equivalence in (2.15), under the assumption that (2.16) holds
true.

Next we turn to the proof of (2.16). For this purpose we need the following
easily established fact:

(2.17)
$y_{\Theta}(r)\approx\widetilde{y}_{\Theta}(r):=\inf_{p>2p_{0}}\\{\Theta(p)r^{1/p}\\},\qquad\text{for}\qquad
r>1.$

Indeed, the estimate $y_{\theta}(r)\leq\widetilde{y}_{\Theta}(r)$ is obvious.
Conversely, let $p>p_{0},$ then $2p>2p_{0},$ and using the doubling property
of $\Theta$ we see that, for $r>1,$

$\Theta(p)r^{\frac{1}{p}}\approx\Theta(2p)r^{\frac{1}{p}}\geq\Theta(2p)r^{\frac{1}{2p}}\geq\widetilde{y}_{\Theta}(r).$

Therefore, taking the infimum over all $p>p_{0}$, we obtain
$y_{\Theta}(r)\gtrsim\widetilde{y}_{\Theta}(r)$, completing the proof of
(2.17).

Let $\varepsilon\in(\frac{1}{2p_{0}},\frac{1}{p_{0}})$. Observe that

(2.18) $r\in(1,\infty)\mapsto
r^{-\varepsilon}\,\widetilde{y}_{\Theta}(r)=\inf_{p>2p_{0}}\\{\Theta(p)r^{-\varepsilon+1/p}\\}\quad\text{is
a decreasing function}.$

Let $t\in(0,1)$, then by (2.17) and (2.18), we have

$\displaystyle\int_{0}^{t}y_{\Theta}\bigg{(}\frac{1}{s}\bigg{)}^{p_{0}}\,ds$
$\displaystyle\approx\int_{0}^{t}\widetilde{y}_{\Theta}\bigg{(}\frac{1}{s}\bigg{)}^{p_{0}}\,ds=\int_{0}^{t}s^{-\varepsilon
p_{0}}\bigg{[}\bigg{(}\frac{1}{s}\bigg{)}^{-\varepsilon}\widetilde{y}_{\Theta}\bigg{(}\frac{1}{s}\bigg{)}\bigg{]}^{p_{0}}\,ds$
$\displaystyle\leq\bigg{[}\bigg{(}\frac{1}{t}\bigg{)}^{-\varepsilon}\widetilde{y}_{\Theta}\bigg{(}\frac{1}{t}\bigg{)}\bigg{]}^{p_{0}}\int_{0}^{t}s^{-\varepsilon
p_{0}}\,ds$ $\displaystyle\approx t^{1-\varepsilon
p_{0}}\bigg{[}\bigg{(}\frac{1}{t}\bigg{)}^{-\varepsilon}\widetilde{y}_{\Theta}\bigg{(}\frac{1}{t}\bigg{)}\bigg{]}^{p_{0}}\approx
ty_{\Theta}\bigg{(}\frac{1}{t}\bigg{)}^{p_{0}},$

this concludes the proof of (2.16) and consequently the theorem is proved. ∎

###  2.2. Proof of Theorem 1.2

We shall assume that all function spaces are defined on
$\Omega=\mathbb{R}^{2}$ (cf. Remark 2.5 below for the modifications needed to
deal with $\Omega=\mathbb{T}^{2}$).

Let $p_{0}\in(1,\infty).$ Recall that CZOs act boundedly on $L^{p_{0}}\text{
and BMO}$ (cf. [30]), in particular, for $\mathcal{K}$ given by (1.2) we have,

$\mathcal{K}:L^{p_{0}}\rightarrow
L^{p_{0}}\qquad\text{and}\qquad\mathcal{K}:\text{BMO}\rightarrow\text{BMO}.$

By interpolation we find that, with norm independent of $\theta$,

$\mathcal{K}:(L^{p_{0}},\text{BMO})_{\theta,p(\theta)}^{\blacktriangleleft}\rightarrow(L^{p_{0}},\text{BMO})_{\theta,p(\theta)}^{\blacktriangleleft},$

where $p(\theta)=p_{0}/(1-\theta)$. This implies (cf. (1.2))

(2.19) $\|\nabla
v\|_{(L^{p_{0}},\text{BMO})_{\theta,p(\theta)}^{\blacktriangleleft}}\lesssim\|\omega\|_{(L^{p_{0}},\text{BMO})_{\theta,p(\theta)}^{\blacktriangleleft}}$

with constant independent of $\theta$.

Next we show that

(2.20)
$\|f-f_{\infty}\|_{(L^{p_{0}},L^{\infty})_{\theta,p(\theta)}^{\blacktriangleleft}}\lesssim(1-\theta)^{-1}\|f\|_{(L^{p_{0}},\text{BMO})_{\theta,p(\theta)}^{\blacktriangleleft}},$

where $f_{\infty}:=\lim_{|Q|\rightarrow\infty}f_{Q}$ and $f_{Q}$ denotes the
_integral average_ of $f$ related to the cube $Q$, i.e.,
$f_{Q}:=\frac{1}{|Q|}\int_{Q}f$.

We start by reformulating the following quantitative version of the
John–Nirenberg embedding [21], which asserts that BMO is locally embedded into
$e^{L}$ (cf. [5, Proposition 8.10, p. 398])

(2.21)
$(f-f_{\infty})^{\ast\ast}(t)\lesssim\int_{t}^{\infty}f^{\\#\ast}(s)\,\frac{ds}{s}.$

Here $f^{\\#}$ is the _sharp maximal function of Fefferman–Stein_ 222222Recall
that $f^{\\#}(x)=\sup_{Q\ni x}\frac{1}{|Q|}\int_{Q}|f-f_{Q}|$. [16]. Indeed,
for our purposes, it is convenient to rewrite (2.21) in terms of $M^{\\#}f$
rather than $f^{\\#}.$ The connection between these maximal functions is given
by (cf. [20, Lemma 3.4])

(2.22) $f^{\\#\ast}(t)\approx(M^{\\#}f)^{\ast\ast}(t).$

Therefore, (2.21) can be expressed as

(2.23)
$(f-f_{\infty})^{\ast\ast}(t)\lesssim\int_{t}^{\infty}(M^{\\#}f)^{\ast\ast}(s)\,\frac{ds}{s}.$

Applying $L^{p}$-norms on both sides of (2.23) and estimating the right-hand
side using the pair of Hardy inequalities in [31, Appendix A.4, page 272], we
arrive at

$\displaystyle\bigg{\\{}\int_{0}^{\infty}[(f-f_{\infty})^{\ast\ast}(t)]^{p}\,dt\bigg{\\}}^{1/p}$
$\displaystyle\lesssim\bigg{\\{}\int_{0}^{\infty}\bigg{[}\int_{t}^{\infty}(M^{\\#}f)^{\ast\ast}(s)\,\frac{ds}{s}\bigg{]}^{p}\,dt\bigg{\\}}^{1/p}$
$\displaystyle\leq
p\,\bigg{\\{}\int_{0}^{\infty}[(M^{\\#}f)^{\ast\ast}(t)]^{p}\,dt\bigg{\\}}^{1/p}$
$\displaystyle=p\,\bigg{\\{}\int_{0}^{\infty}\bigg{[}\frac{1}{t}\int_{0}^{t}(M^{\\#}f)^{\ast}(s)\,ds\bigg{]}^{p}\,dt\bigg{\\}}^{1/p}$
$\displaystyle\leq\frac{p^{2}}{p-1}\,\|f\|_{(L^{p})^{\\#}}.$

Consequently, we have the following variant of the Fefferman–Stein inequality
(cf. [5])

$\|f-f_{\infty}\|_{L^{p}}=\bigg{\\{}\int_{0}^{\infty}[(f-f_{\infty})^{\ast}(t)]^{p}\,dt\bigg{\\}}^{1/p}\lesssim\frac{p^{2}}{p-1}\,\|f\|_{(L^{p})^{\\#}}.$

In particular, since $p_{0}>1,$ and
$\frac{1}{p(\theta)}=\frac{1-\theta}{p_{0}},$

$\|f-f_{\infty}\|_{L^{p(\theta)}}\lesssim\frac{(1-\theta)^{-1}}{p_{0}-1+\theta}\,\|f\|_{(L^{p(\theta)})^{\\#}}\lesssim(1-\theta)^{-1}\|f\|_{(L^{p(\theta)})^{\\#}}.$

In view of (1.23) and (2.1), the previous estimate can be rewritten as

$\|f-f_{\infty}\|_{(L^{p_{0}},L^{\infty})_{\theta,p(\theta)}^{\blacktriangleleft}}\lesssim(1-\theta)^{-1}\|f\|_{(L^{p_{0}},\text{BMO})_{\theta,p(\theta)}^{\blacktriangleleft}},$

proving that (2.20) holds.

Applying (2.20) to $f=\nabla v$ (and noting that $(\nabla v)_{\infty}=0$),
combined with (2.19), yields

$\|\nabla
v\|_{(L^{p_{0}},L^{\infty})_{\theta,p(\theta)}^{\blacktriangleleft}}\lesssim(1-\theta)^{-1}\|\omega\|_{(L^{p_{0}},\text{BMO})_{\theta,p(\theta)}^{\blacktriangleleft}}$

uniformly with respect to $\theta\in(0,1)$. Multiplying both sides of the
above estimate by the factor
$(1-\theta)\Theta\big{(}\frac{p_{0}}{1-\theta}\big{)}^{-1}$ and taking the
supremum over all $\theta\in(0,1)$, we find

(2.24) $\|\nabla
v\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{p_{0}},L^{\infty})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\Big{\\}}}\lesssim\left\|\omega\right\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{p_{0}},\text{BMO})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{p_{0}}{1-\theta})}\Big{\\}}}\approx\|\omega\|_{Y_{p_{0}}^{\\#\Theta}},$

where $\Theta_{1}$ was introduced in (1.6) and we have also used (2.2) in the
last step.

Now using (1.28), we rewrite the left-hand side of (2.24) as

$\|\nabla
v\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{p_{0}},L^{\infty})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\Big{\\}}}\approx\|v\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(\dot{W}_{p_{0}}^{1},\dot{W}_{\infty}^{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\Big{\\}}}.$

Furthermore, the Sobolev embedding theorem $\dot{W}_{p_{0}}^{1}\hookrightarrow
L^{\infty}$ (recall that $p_{0}>2$), and the reiteration property of
$\Delta$-extrapolation (cf. Appendix A.2, (A.3)) yield

$\displaystyle\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(\dot{W}_{p_{0}}^{1},\dot{W}_{\infty}^{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\bigg{\\}}$
$\displaystyle\hookrightarrow\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(L^{\infty},\dot{W}_{\infty}^{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\bigg{\\}}$
$\displaystyle=\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(L^{\infty},\dot{W}_{\infty}^{1})_{\theta,\infty}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\bigg{\\}}.$

Consequently,

(2.25)
$\left\|v\right\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{\infty},\dot{W}_{\infty}^{1})_{\theta,\infty}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\Big{\\}}}\lesssim\|v\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(\dot{W}_{p_{0}}^{1},\dot{W}_{\infty}^{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\Big{\\}}}.$

Updating (2.24) via (2.25) we arrive at

(2.26)
$\left\|v\right\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{\infty},\dot{W}_{\infty}^{1})_{\theta,\infty}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\Big{\\}}}\lesssim\|\omega\|_{Y_{p_{0}}^{\\#\Theta}}.$

Next we compute the norm of the extrapolation space on the left-hand side of
(2.26) and show that

(2.27)
$\|v\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{\infty},\dot{W}_{\infty}^{1})_{\theta,\infty}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\Big{\\}}}\approx\sup_{x,y}\,\frac{|v(x)-v(y)|}{\inf_{t>|x-y|}ty_{\Theta_{1}}(\frac{1}{t})}.$

The argument was already outlined in Section 1.4. For the sake of
completeness, next we give full details. Indeed, in light of (1.5), we have
(cf. (A.4) and (A.5))

(2.28)
$\left\|v\right\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{\infty},\dot{W}_{\infty}^{1})_{\theta,\infty}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\Big{\\}}}\approx\sup_{t\in(0,\infty)}\frac{K(t,v;L^{\infty},\dot{W}_{\infty}^{1})}{ty_{\Theta_{1}}(\frac{1}{t})}.$

Applying the characterization of the $K$-functional for
$(L^{\infty},\dot{W}_{\infty}^{1})$ (cf. (1.30)) we find

$\displaystyle\left\|v\right\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{\infty},\dot{W}_{\infty}^{1})_{\theta,\infty}^{\blacktriangleleft}}{\Theta_{1}(\frac{p_{0}}{1-\theta})}\Big{\\}}}$
$\displaystyle\approx\sup_{t\in(0,\infty)}\frac{\sup_{|x-y|<t}|v(x)-v(y)|}{ty_{\Theta_{1}}(\frac{1}{t})}$
$\displaystyle=\sup_{x,y}|v(x)-v(y)|\,\sup_{t>|x-y|}\frac{1}{ty_{\Theta_{1}}(\frac{1}{t})}.$

Hence, the desired result (2.27) follows.

Putting together (2.26) and (2.27), we obtain

$\sup_{x,y\in\Omega}\,\frac{|v(x)-v(y)|}{\inf_{t>|x-y|}ty_{\Theta_{1}}(\frac{1}{t})}\lesssim\|\omega\|_{Y_{p_{0}}^{\\#\Theta}}.$

In particular,

$|v(x)-v(y)|\lesssim|x-y|\,y_{\Theta_{1}}\bigg{(}\frac{1}{|x-y|}\bigg{)}\|\omega\|_{Y_{p_{0}}^{\\#\Theta}},$

thus completing the proof of Theorem 1.2. ∎

######  Remark 2.5.

The above proof can be easily adapted to deal with $\Omega=\mathbb{T}^{2}$. In
particular, the periodic counterpart of (2.21) is given by (cf. [5, Corollary
7.4, p. 379]), for $t\in(0,\frac{1}{6})$,

$(f-f_{\mathbb{T}^{2}})^{\ast\ast}(t)\lesssim\int_{t}^{1}f^{\\#\ast}(s)\,\frac{ds}{s},$

where $f_{\mathbb{T}^{2}}$ denotes the integral mean of $f$. Accordingly, the
fact that $(\nabla v)_{\infty}=0$ in the above proof is replaced by232323Using
the Fundamental Theorem of Calculus together with the continuity of $v$
(recall that $\omega\in L^{p}(\mathbb{T}^{2})$ with $p>2$), we have $(\nabla
v)_{\mathbb{T}^{2}}=0$. $(\nabla v)_{\mathbb{T}^{2}}=0$.

###  2.3. Examples of vorticities in $Y^{\\#\Theta}_{p_{0}}$ that are not in
$Y^{\Theta}_{p_{0}}$

To simplify the exposition, throughout this section we assume that
$\Omega=\mathbb{T}^{2}$. Recall that by construction, given a growth function
$\Theta$, $Y_{p_{0}}^{\\#\Theta}$ is a bigger space than $Y_{p_{0}}^{\Theta}$
(cf. (1.34)). Furthermore, in the special case $\Theta(p)\approx 1$, we have
$Y_{p_{0}}^{\Theta}=L^{\infty}\subsetneq\text{BMO}=Y_{p_{0}}^{\\#\Theta}$. It
is of interest to understand better the relationship between these spaces. In
this section, we provide a method to construct explicit examples of functions
$\omega\in Y_{p_{0}}^{\\#\Theta}\backslash Y_{p_{0}}^{\Theta},$ for a variety
of growths.

######  Example 3 (The case $\Theta(p)\approx p^{\alpha},\,\alpha>0$).

Let

$\omega(x)=|\log|x||^{\alpha+1}.$

We will show that

(2.29) $\omega\in Y^{\\#\Theta},\qquad\omega\not\in Y^{\Theta}.$

Indeed, basic computations lead to

(2.30) $\omega^{*}(t)\approx(-\log t)^{\alpha+1}$

and (recall that $\Omega=\mathbb{T}^{2}$)

(2.31) $|\nabla\omega|^{*}(t)\approx t^{-1/2}(-\log t)^{\alpha}.$

Next we are able to bound $\omega^{\\#\ast}$ via the following pointwise
estimate obtained in [15, Theorem 2.6]:

(2.32)
$\omega^{\\#\ast}(t)\lesssim\sum_{l=0}^{1}\bigg{[}t^{-1/r_{0}}\bigg{(}\int_{0}^{t}(\xi^{1/r}\,|\nabla^{l}\omega|^{\ast}(\xi))^{r_{0}}\,\frac{d\xi}{\xi}\bigg{)}^{1/r_{0}}+\sup_{t<\xi<1}\xi^{1/2}\,|\nabla^{l}\omega|^{\ast}(\xi)\bigg{]},$

where $r_{0}>2$ and $r=\frac{2r_{0}}{2+r_{0}}$. We treat the four terms
appearing on the right-hand side of the previous estimate. Assume first that
$l=0$ (i.e., $|\nabla^{0}\omega|^{\ast}=\omega^{\ast}$). By (2.30), we have

$\bigg{(}\int_{0}^{t}(\xi^{1/r}\,\omega^{\ast}(\xi))^{r_{0}}\,\frac{d\xi}{\xi}\bigg{)}^{1/r_{0}}\approx\bigg{(}\int_{0}^{t}[\xi^{1/r}(-\log\xi)^{\alpha+1}]^{r_{0}}\,\frac{d\xi}{\xi}\bigg{)}^{1/r_{0}}\approx
t^{1/r}(-\log t)^{\alpha+1}$

and

$\sup_{t<\xi<1}\xi^{1/2}\,\omega^{\ast}(\xi)\approx\sup_{t<\xi<1}\xi^{1/2}\,(-\log\xi)^{\alpha+1}\approx
1.$

Putting these estimates together, we obtain

$\displaystyle
t^{-1/r_{0}}\bigg{(}\int_{0}^{t}(\xi^{1/r}\,\omega^{\ast}(\xi))^{r_{0}}\,\frac{d\xi}{\xi}\bigg{)}^{1/r_{0}}+\sup_{t<\xi<1}\xi^{1/2}\,\omega^{\ast}(\xi)$
(2.33) $\displaystyle\approx t^{1/2}(-\log t)^{\alpha+1}+1\approx 1.$

Next we deal with the term on the right-hand side of (2.32) that corresponds
to $l=1$. Using (2.31), we get the following estimates

$\displaystyle\bigg{(}\int_{0}^{t}(\xi^{1/r}\,|\nabla\omega|^{\ast}(\xi))^{r_{0}}\,\frac{d\xi}{\xi}\bigg{)}^{1/r_{0}}$
$\displaystyle\approx\bigg{(}\int_{0}^{t}(\xi^{1/r_{0}}(-\log\xi)^{\alpha})^{r_{0}}\,\frac{d\xi}{\xi}\bigg{)}^{1/r_{0}}$
$\displaystyle\approx t^{1/r_{0}}(-\log t)^{\alpha}$

and (since $\alpha>0$)

$\sup_{t<\xi<1}\xi^{1/2}\,|\nabla\omega|^{\ast}(\xi)\approx\sup_{t<\xi<1}(-\log\xi)^{\alpha}=(-\log
t)^{\alpha}.$

Hence

(2.34)
$t^{-1/r_{0}}\bigg{(}\int_{0}^{t}(\xi^{1/r}\,|\nabla\omega|^{\ast}(\xi))^{r_{0}}\,\frac{d\xi}{\xi}\bigg{)}^{1/r_{0}}+\sup_{t<\xi<1}\xi^{1/2}\,|\nabla\omega|^{\ast}(\xi)\approx(-\log
t)^{\alpha}.$

Inserting (3) and (2.34) into (2.32), we obtain the following upper estimate
for $\omega^{\\#*}$,

$\omega^{\\#*}(t)\lesssim(-\log t)^{\alpha},$

or equivalently (cf. (2.22))

(2.35) $(M^{\\#}\omega)^{**}(t)\lesssim(-\log t)^{\alpha}.$

Since we are working on $\Omega=\mathbb{T}^{2},$ without loss of generality we
may assume that $p_{0}=1$. Applying Theorem 2.3 and Example 2 (with
$\Theta(p)\approx p^{\alpha}$) and using (2.30) and (2.35), we compute

$\|\omega\|_{Y^{\Theta}}\approx\sup_{t\in(0,e^{-1})}\frac{\omega^{\ast\ast}(t)}{\Theta(-\log
t)}\approx\sup_{t\in(0,e^{-1})}\frac{(-\log t)^{\alpha+1}}{(-\log
t)^{\alpha}}=\infty$

and

$\|\omega\|_{Y^{\\#\Theta}}\approx\sup_{t\in(0,e^{-1})}\frac{(M^{\\#}\omega)^{\ast\ast}(t)}{\Theta(-\log
t)}\lesssim\sup_{t\in(0,e^{-1})}\frac{(-\log t)^{\alpha}}{(-\log
t)^{\alpha}}=1.$

This concludes the proof of (2.29).

######  Example 4 (The case $\Theta(p)\approx\log p$).

This example is motivated by the fact that $\log(1+|\log|x||)$ is a prototype
of a function in $Y^{\Theta},$ while $|\log|x||$ is a prototype of a function
in BMO. We consider their product, namely,

$\omega(x)=(1+|\log|x||)\log(1+|\log|x||),$

and show that

(2.36) $\omega\in Y^{\\#\Theta},\qquad\omega\not\in Y^{\Theta}.$

We follow closely the method of Example 3. Specifically, by elementary
manipulations we find

(2.37) $\omega^{\ast}(t)\approx(1-\log t)\log(1-\log t)$

and

$|\nabla\omega|^{\ast}(t)\approx t^{-1/2}\,(1+\log(1-\log t)).$

Recall that $r=\frac{2r_{0}}{2+r_{0}}$, where $r_{0}>2$. Therefore

$t^{-1/r_{0}}\bigg{(}\int_{0}^{t}(\xi^{1/r}\,\omega^{\ast}(\xi))^{r_{0}}\,\frac{d\xi}{\xi}\bigg{)}^{1/r_{0}}+\sup_{t<\xi<1}\xi^{1/2}\,\omega^{\ast}(\xi)\approx
t^{1/2}(1-\log t)\log(1-\log t)+1\approx 1$

and

$t^{-1/r_{0}}\bigg{(}\int_{0}^{t}(\xi^{1/r}\,|\nabla\omega|^{\ast}(\xi))^{r_{0}}\,\frac{d\xi}{\xi}\bigg{)}^{1/r_{0}}+\sup_{t<\xi<1}\xi^{1/2}\,|\nabla\omega|^{\ast}(\xi)\approx\log(1-\log
t).$

Inserting these two estimates into (2.32), we obtain

(2.38) $\omega^{\\#\ast}(t)\lesssim\log(1-\log t).$

Invoking Theorem 2.3 and Example 2 (with $\Theta(p)\approx\log p$) together
with (2.37) and (2.38), we get

$\|\omega\|_{Y^{\Theta}}\approx\sup_{t\in(0,e^{-1})}(-\log t)=\infty$

and

$\|\omega\|_{Y^{\\#\Theta}}\lesssim\sup_{t\in(0,e^{-1})}\frac{\log(-\log
t)}{\log(-\log t)}=1.$

Hence $\omega$ fulfils (2.36).

######  Remark 2.6.

It is possible to extend the methodology applied in Examples 3 and 4 to deal
with more general growths $\Theta$ of logarithmic type. Further details are
left to the reader.

##  3\. The spaces $\dot{B}^{\beta}_{\Pi}$

Let $\mathcal{S}(\mathbb{R}^{d})$ denote the Schwartz space and
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ the space of tempered distributions. We
consider the space $\dot{\mathcal{S}}(\mathbb{R}^{d})$ formed by
$\varphi\in\mathcal{S}(\mathbb{R}^{d})$ with
$(D^{\alpha}\widehat{\varphi})(0)=0$ for any multi-index
$\alpha\in\mathbb{N}_{0}^{d}$, where $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$;
this space carries the natural Fréchet topology inherited from
$\mathcal{S}(\mathbb{R}^{d})$. Let
$\dot{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ be its dual space, which can be
identified with $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ modulo polynomials.

Let $\varphi\in C_{c}^{\infty}(\mathbb{R}^{d})$ be a radial function with

$\text{supp
}\varphi\subset\bigg{\\{}\xi:\frac{3}{4}<|\xi|<\frac{7}{4}\bigg{\\}},$
$\varphi(\xi)=1\quad\text{for}\quad\frac{7}{8}<|\xi|<\frac{9}{8},$
$\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1\quad\text{for all}\quad\xi\neq 0.$

Then

$\widehat{\dot{\Delta}_{j}f}(\xi):=\varphi(2^{-j}\xi)\widehat{f}(\xi),$

and

$\Delta_{j}:=\dot{\Delta}_{j}\quad\text{if}\quad
j>0,\qquad\Delta_{0}:=\text{Id}-\sum_{j>0}\Delta_{j}.$

######  Definition 2.

Let $\beta\in\mathbb{R}$, and let $\Pi$ be a growth function. Thus
$\dot{B}_{\Pi}^{\beta}(\mathbb{R}^{d})$ will denote the (homogeneous) _Vishik
space_ formed by all $f\in\dot{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ such
that

$\|f\|_{\dot{B}_{\Pi}^{\beta}(\mathbb{R}^{d})}:=\sup_{N\geq
0}\,\frac{1}{\Pi(N)}\,\sum_{j=-\infty}^{N}2^{j\beta}\,\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}<\infty.$

The inhomogeneous counterpart, $B_{\Pi}^{\beta}(\mathbb{R}^{d}),$ is the set
of all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ such that

$\|f\|_{B_{\Pi}^{\beta}(\mathbb{R}^{d})}:=\sup_{N\geq
0}\,\frac{1}{\Pi(N)}\,\sum_{j=0}^{N}2^{j\beta}\,\|\Delta_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}<\infty.$

######  Remark 3.1.

  1. (i)

Standard properties of multipliers can be used to show that Vishik spaces do
not depend (up to equivalence of norms) on the chosen generator $\varphi$.

  2. (ii)

In the special case $\beta=0,$ $B_{\Pi}^{\beta}(\mathbb{R}^{d})$ coincides
with the classical space $B_{\Pi}(\mathbb{R}^{d})$ (cf. (1.8)).

  3. (iii)

Assume $\Pi(N)\approx 1$. Then
$\dot{B}^{\beta}_{\Pi}(\mathbb{R}^{d})=\dot{B}^{\beta}_{\infty,1}(\mathbb{R}^{d})$,
the classical Besov space (cf. [6]) defined by

(3.1)
$\|f\|_{\dot{B}^{\beta}_{\infty,1}(\mathbb{R}^{d})}:=\sum_{j\in\mathbb{Z}}2^{j\beta}\,\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}.$

Analogously,
$B^{\beta}_{\Pi}(\mathbb{R}^{d})=B^{\beta}_{\infty,1}(\mathbb{R}^{d})$ with

$\|f\|_{B^{\beta}_{\infty,1}(\mathbb{R}^{d})}:=\sum_{j\in\mathbb{N}_{0}}2^{j\beta}\,\|\Delta_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}.$

###  3.1. Characterizations

The goal of this section is to provide characterizations of the Vishik spaces
as extrapolation spaces. We show analogs of the results obtained earlier for
the $Y_{p_{0}}^{\Theta}$ and $Y_{p_{0}}^{\\#\Theta}$ spaces (cf. Section 2.1).

We need to impose some natural restrictions on the growth functions used in
this section.

######  Definition 3.

Let $\kappa>0.$ We shall denote by $\mathcal{P}_{\kappa}$, the set of all
growth functions $\Pi$ satisfying the following conditions:

  1. (i)

$\Pi:[0,\infty)\to(0,\infty)$ is non-decreasing,

  2. (ii)

$\Pi$ is doubling,

  3. (iii)

$e^{1/p}\,\Pi(p)$ is quasi-decreasing,

  4. (iv)

$\sum_{j=N}^{\infty}2^{-j\kappa}\Pi(j)\lesssim 2^{-N\kappa}\Pi(N)$ for every
$N\geq 0$.

Clearly,
$\Pi(p)=(p+1)^{\alpha}(\log(p+e))^{\alpha_{1}}(\log_{2}(p+e))^{\alpha_{2}}\cdots(\log_{m}(p+e))^{\alpha_{m}}$,
where $\alpha,\alpha_{i}\geq 0$, are examples of growth functions in
$\mathcal{P}_{\kappa}.$

In dealing with interpolation of Besov spaces, we make use of the fact that
the computation of $K$-functionals for the Besov pair
$(\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d})$
can be reduced, via the method of retracts (cf. Appendix A.1 for further
details), to the computation of $K$-functionals for the vector-valued sequence
spaces
$(\ell_{1}^{\beta-\kappa}(L^{\infty}(\mathbb{R}^{d}),\ell_{1}^{\beta}(L^{\infty}(\mathbb{R}^{d}))$
(cf. (A.1)).

Using the retract technique we proved Theorem A.1 below which, in particular,
implies

(3.2)
$(\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,1}^{\blacktriangleleft}=\dot{B}_{\infty,1}^{\alpha}(\mathbb{R}^{d})$

with $\alpha=(1-\theta)(\beta-\kappa)+\theta\beta$. Here, the equivalence
constant is independent of $\theta$.

######  Theorem 3.2.

Let $\Pi\in\mathcal{P}_{\kappa}$ and $\beta\in\mathbb{R}$. Then

$\displaystyle\|f\|_{\dot{B}_{\Pi}^{\beta}(\mathbb{R}^{d})\cap\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d})}$
$\displaystyle\approx\sup_{t\in(0,\infty)}\frac{K(t,f;\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))}{ty_{\Pi}(\frac{1}{t})}$
(3.3)
$\displaystyle\approx\|f\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}{\Pi(\frac{1}{1-\theta})}\Big{\\}}}$
$\displaystyle\approx\sup_{\alpha\in(\beta-\kappa,\beta)}\frac{\|f\|_{\dot{B}_{\infty,1}^{\alpha}(\mathbb{R}^{d})}}{\Pi(\frac{1}{\beta-\alpha})}.$

Here $y_{\Pi}$ is given by (1.5) (with $p_{0}=1$).

###### Proof.

According to (A.3) and (A.4) (with $p_{0}=1$),

(3.4)
$\|f\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}{\Pi(\frac{1}{1-\theta})}\Big{\\}}}\approx\sup_{t\in(0,\infty)}\frac{K(t,f;\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))}{ty_{\Pi}(\frac{1}{t})}.$

This shows the second equivalence in (3.3).

We now prove the third equivalence in (3.3). We do this using the reiteration
property for the $\Delta$-extrapolation method given in (A.3) and (A.6) (with
$p=1$)

$\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}{\Pi(\frac{1}{1-\theta})}\bigg{\\}}=\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,1}^{\blacktriangleleft}}{\Pi(\frac{1}{1-\theta})}\bigg{\\}}$

combined with (3.2), and a change of variables, to obtain

$\left\|f\right\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}{\Pi(\frac{1}{1-\theta})}\Big{\\}}}\approx\sup_{\alpha\in(\beta-\kappa,\beta)}\frac{\|f\|_{\dot{B}_{\infty,1}^{\alpha}(\mathbb{R}^{d})}}{\Pi(\frac{1}{\beta-\alpha})}.$

Finally, we prove the first equivalence in (3.3). Note that in view of the
properties of $y_{\Pi},$ proved in Lemma 2, and the monotonicity properties of
$K$-functionals, we have

$\displaystyle\sup_{t\in(0,\infty)}\frac{K(t,f;\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))}{ty_{\Pi}(\frac{1}{t})}$
$\displaystyle\approx\sup_{t\in(0,1/e)}\,\frac{K(t,f;\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))}{t\Pi(-\log
t)}$ $\displaystyle\hskip
28.45274pt+\sup_{t\in(1/e,1)}\frac{K(t,f;\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))}{ty_{\Pi}(\frac{1}{t})}$
$\displaystyle\hskip
28.45274pt+\sup_{t\in(1,\infty)}\,K(t,f;\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))$
(3.5) $\displaystyle\approx I+II,$

where

$I:=\sup_{t\in(0,1/e)}\,\frac{K(t,f;\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))}{t\Pi(-\log
t)}$

and

$II:=\sup_{t\in(1,\infty)}K(t,f;\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d})).$

The desired equivalence in (3.3) will then follow if we show the following

(3.6) $I\approx\|f\|_{\dot{B}_{\Pi}^{\beta}(\mathbb{R}^{d})}$

and

(3.7) $II\approx\|f\|_{\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d})}.$

To prove these claims we revert to sequences of vector-valued functions using
the method of retracts. Then we see that

$\displaystyle I$ $\displaystyle\approx\sup_{t\in(0,1/e)}\,\frac{1}{t\Pi(-\log
t)}\,K(t,\\{\dot{\Delta}_{j}f\\}_{j\in\mathbb{Z}};\ell_{1}^{\beta-\kappa}(L^{\infty}(\mathbb{R}^{d})),\ell_{1}^{\beta}(L^{\infty}(\mathbb{R}^{d})))$
$\displaystyle\approx\sup_{N\geq
0}\,\frac{1}{2^{-N\kappa}\,\Pi(N)}\,K(2^{-N\kappa},\\{\dot{\Delta}_{j}f\\}_{j\in\mathbb{Z}};\ell_{1}^{\beta-\kappa}(L^{\infty}(\mathbb{R}^{d})),\ell_{1}^{\beta}(L^{\infty}(\mathbb{R}^{d}))),$

where in the last step we have used the monotonicity properties of $\Pi$ and
$K$-functionals. Applying known characterizations for $K$-functionals (cf.
(A.2)), we derive

$\displaystyle I$ $\displaystyle\approx\sup_{N\geq
0}\,\frac{1}{2^{-N\kappa}\,\Pi(N)}\,\sum_{j=-\infty}^{\infty}\min\\{1,2^{(j-N)\kappa}\\}\,2^{j(\beta-\kappa)}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\approx\sup_{N\geq
0}\,\frac{1}{\Pi(N)}\,\sum_{j=-\infty}^{N}2^{j\beta}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\hskip 28.45274pt+\sup_{N\geq
0}\,\frac{1}{2^{-N\kappa}\,\Pi(N)}\,\sum_{j=N}^{\infty}2^{j(\beta-\kappa)}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle=:I_{1}+I_{2}.$

We claim that

(3.8) $I_{2}\lesssim I_{1},$

from where it follows that (cf. Definition 2)

$I\approx I_{1}=\|f\|_{\dot{B}_{\Pi}^{\beta}(\mathbb{R}^{d})},$

i.e., (3.6) holds.

It remains to prove (3.8). Let $N\geq 0$. We use the assumption (iv) in the
definition of $\mathcal{P}_{\kappa}$ (cf. Definition 3), to estimate

$\displaystyle\sum_{j=N}^{\infty}2^{j(\beta-\kappa)}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\leq\sup_{j\geq
0}\,\\{\Pi(j)^{-1}2^{j\beta}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}\\}\,\sum_{j=N}^{\infty}2^{-j\kappa}\Pi(j)$
$\displaystyle\approx 2^{-N\kappa}\Pi(N)\,\sup_{j\geq
0}\,\\{\Pi(j)^{-1}2^{j\beta}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}\\}$
$\displaystyle\leq 2^{-N\kappa}\Pi(N)\,I_{1}.$

Taking the supremum over all $N\geq 0$, we arrive at the desired estimate
(3.8).

We turn to the proof of (3.7). The estimate $\lesssim$ follows trivially from
the very definition of $K$-functional. The converse estimate can be obtained
from (A.2). To be more precise, let $N\in\mathbb{N}$, then

$\displaystyle
K(2^{N\kappa},\\{f_{j}\\}_{j\in\mathbb{Z}};\ell_{1}^{\beta-\kappa}(L^{\infty}(\mathbb{R}^{d})),\ell_{1}^{\beta}(L^{\infty}(\mathbb{R}^{d})))$
$\displaystyle\approx\sum_{j=-\infty}^{\infty}\min\\{1,2^{(j+N)\kappa}\\}\,2^{j(\beta-\kappa)}\|f_{j}\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\geq\sum_{j=-N}^{\infty}2^{j(\beta-\kappa)}\|f_{j}\|_{L^{\infty}(\mathbb{R}^{d})}.$

Hence

$\sup_{N\in\mathbb{N}}\,K(2^{N\kappa},\\{f_{j}\\}_{j\in\mathbb{Z}};\ell_{1}^{\beta-\kappa}(L^{\infty}(\mathbb{R}^{d})),\ell_{1}^{\beta}(L^{\infty}(\mathbb{R}^{d})))\gtrsim\sum_{j=-\infty}^{\infty}2^{j(\beta-\kappa)}\|f_{j}\|_{L^{\infty}(\mathbb{R}^{d})}.$

Using once again the retraction method, we obtain (cf. (3.1))

$II\gtrsim\|f\|_{\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d})}.$

This ends the proof of (3.7).

Finally, putting together (3.5), (3.6) and (3.7),

$\sup_{t\in(0,\infty)}\frac{K(t,f;\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))}{ty_{\Pi}(\frac{1}{t})}\approx\|f\|_{\dot{B}_{\Pi}^{\beta}(\mathbb{R}^{d})\cap\dot{B}_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d})}.$

This concludes the proof of the theorem. ∎

######  Remark 3.3.

The inhomogeneous counterpart of Theorem 3.2 also holds. Note that, in this
case, the outcome is independent of $\kappa>0$ since

$B_{\Pi}^{\beta}(\mathbb{R}^{d})\cap
B_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d})=B_{\Pi}^{\beta}(\mathbb{R}^{d}).$

Indeed, for any fixed $N\in\mathbb{N}$, we have

$\displaystyle\|f\|_{B_{\infty,1}^{\beta-\kappa}(\mathbb{R}^{d})}$
$\displaystyle=\sum_{j=0}^{N}2^{j(\beta-\kappa)}\|\Delta_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}+\sum_{j=N+1}^{\infty}2^{j(\beta-\kappa)}\|\Delta_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\leq\sum_{j=0}^{N}2^{j\beta}\|\Delta_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}+\sum_{j=N+1}^{\infty}2^{j(\beta-\kappa)}\|\Delta_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\leq\Pi(N)\|f\|_{B_{\Pi}^{\beta}(\mathbb{R}^{d})}+\bigg{(}\sum_{j=N+1}^{\infty}2^{-j\kappa}\Pi(j)\bigg{)}\,\|f\|_{B_{\Pi}^{\beta}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\Pi(N)(1+2^{-N\kappa})\,\|f\|_{B_{\Pi}^{\beta}(\mathbb{R}^{d})},$

where the last step follows from the property (iv) in the definition of
$\mathcal{P}_{\kappa}$ (cf. Definition 3).

One may also show that the characterizations provided by Theorem 3.2 in the
inhomogoneous setting are independent of $\kappa$ via

$\sup_{\alpha\in(\beta-\kappa,\beta)}\frac{\|f\|_{B_{\infty,1}^{\alpha}(\mathbb{R}^{d})}}{\Pi(\frac{1}{\beta-\alpha})}\approx\sup_{\alpha\in(\beta-\kappa_{0},\beta)}\frac{\|f\|_{B_{\infty,1}^{\alpha}(\mathbb{R}^{d})}}{\Pi(\frac{1}{\beta-\alpha})}$

for $\kappa,\kappa_{0}>0$. The latter is an immediate consequence of the
trivial embedding $B^{\alpha}_{\infty,1}(\mathbb{R}^{d})\hookrightarrow
B^{\alpha-\varepsilon}_{\infty,1}(\mathbb{R}^{d})$ for any $\varepsilon>0$.

A third explanation to the independence of $\kappa$ in Theorem 3.2, now from
an extrapolation point of view, may be found in (A.7) below.

###  3.2. Uniqueness for a family of active scalar equations

Consider the class of active scalar equations on $\mathbb{R}^{d}$ described in
Section 1.9, namely,

(3.9) $\left\\{\begin{array}[c]{c}\omega_{t}+v\cdot\nabla\omega=0,\\\
v=R(-\Delta)^{\frac{\beta-1}{2}}\omega,\end{array}\right.$

where $\beta\in\mathbb{R}$. The main goal of this section is to establish the
uniqueness result for (3.9) in the class of Vishik spaces. To do this, we will
follow the extrapolation approach outlined in Section 1.6 adequately adapted
to (3.9) and Vishik spaces. The corresponding Step 1 was already carried out
in Section 3.1. Next, we turn our attention to Step 2, i.e., estimates for the
modulus of continuity of $v$.

######  Theorem 3.4.

Assume that the growth function $\Pi\in\mathcal{P}_{1},$ and let
$\omega\in\dot{B}_{\Pi}^{\beta}(\mathbb{R}^{d})\cap\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d})$.
Then

$|v(x)-v(y)|\lesssim|x-y|\,y_{\Pi}\bigg{(}\frac{1}{|x-y|}\bigg{)}\,\|\omega\|_{\dot{B}_{\Pi}^{\beta}(\mathbb{R}^{d})\cap\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d})},$

where $y_{\Pi}$ is the Yudovich function associated to $\Pi$ (cf. (1.5) with
$p_{0}=1$).

###### Proof.

Recall the well-known fact that $R$ (Riesz-type transforms) acts boundedly on
the Besov space $\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d})$ for all
$\beta\in\mathbb{R}$ (see e.g. [17, Proposition 4.7]). By interpolation, it
follows that

$\|R\omega\|_{(\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}\lesssim\|\omega\|_{(\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}$

uniformly with respect to $\theta\in(0,1)$. Using the Biot–Savart law given in
(3.9), we can rewrite the previous estimate as

(3.10)
$\|(-\Delta)^{\frac{-\beta+1}{2}}v\|_{(\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}\lesssim\|\omega\|_{(\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}.$

Furthermore, as a consequence of basic multiplier assertions (cf. [6, Lemma
6.2.1]), the operator $(-\Delta)^{\frac{-\beta+1}{2}}$ acts as an isomorphism
from $\dot{B}_{\infty,1}^{0}(\mathbb{R}^{d})$ onto
$\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d})$ and from
$\dot{B}_{\infty,1}^{1}(\mathbb{R}^{d})$ onto
$\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d})$. Once again by interpolation, we
derive

$\|(-\Delta)^{\frac{-\beta+1}{2}}v\|_{(\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}\approx\|v\|_{(\dot{B}_{\infty,1}^{0}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{1}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}.$

Therefore we can update (3.10) as follows

(3.11)
$\|v\|_{(\dot{B}_{\infty,1}^{0}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{1}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}\lesssim\|\omega\|_{(\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}.$

It follows from the trivial embedding

(3.12) $\dot{B}_{\infty,1}^{0}(\mathbb{R}^{d})\hookrightarrow
L^{\infty}(\mathbb{R}^{d})$

and the classical Bernstein inequality for entire functions of exponential
type (cf. [29, p. 116]) that

$\displaystyle\|\nabla v\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\|\nabla
v\|_{\dot{B}_{\infty,1}^{0}(\mathbb{R}^{d})}=\sum_{j=-\infty}^{\infty}\|\nabla\dot{\Delta}_{j}v\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\sum_{j=-\infty}^{\infty}2^{j}\,\|\dot{\Delta}_{j}v\|_{L^{\infty}(\mathbb{R}^{d})}=\|v\|_{\dot{B}_{\infty,1}^{1}(\mathbb{R}^{d})}.$

In other words,

(3.13)
$\dot{B}_{\infty,1}^{1}(\mathbb{R}^{d})\hookrightarrow\dot{W}_{\infty}^{1}(\mathbb{R}^{d}).$

According to (3.12) and (3.13),

$\|v\|_{(L^{\infty}(\mathbb{R}^{d}),\dot{W}_{\infty}^{1}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}\lesssim\|v\|_{(\dot{B}_{\infty,1}^{0}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{1}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}},$

which implies (cf. (3.11))

$\|v\|_{(L^{\infty}(\mathbb{R}^{d}),\dot{W}_{\infty}^{1}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}\lesssim\|\omega\|_{(\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}.$

Multiplying this inequality by $1/\Pi((1-\theta)^{-1})$ and taking supremum
over all $\theta\in(0,1)$, we arrive at the extrapolation inequality

(3.14)
$\|v\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{\infty}(\mathbb{R}^{d}),\dot{W}_{\infty}^{1}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}{\Pi(\frac{1}{1-\theta})}\Big{\\}}}\lesssim\|\omega\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}{\Pi(\frac{1}{1-\theta})}\Big{\\}}}.$

The right-hand side of (3.14) was computed in Theorem 3.2:

(3.15)
$\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}{\Pi(\frac{1}{1-\theta})}\bigg{\\}}=\dot{B}_{\Pi}^{\beta}(\mathbb{R}^{d})\cap\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d}).$

While the left-hand side of (3.14) was already characterized in (2.27)
(replace $\Theta_{1}$ by $\Pi$ and let $p_{0}=1$) as

(3.16)
$\|v\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(L^{\infty}(\mathbb{R}^{d}),\dot{W}_{\infty}^{1}(\mathbb{R}^{d}))_{\theta,\frac{1}{1-\theta}}^{\blacktriangleleft}}{\Pi(\frac{1}{1-\theta})}\Big{\\}}}\approx\sup_{x,y\in\mathbb{R}^{d}}\,\frac{|v(x)-v(y)|}{\inf_{t>|x-y|}ty_{\Pi}(\frac{1}{t})}.$

Inserting (3.15) and (3.16) into (3.14), we have

$|v(x)-v(y)|\lesssim|x-y|y_{\Pi}\bigg{(}\frac{1}{|x-y|}\bigg{)}\,\|\omega\|_{\dot{B}_{\Pi}^{\beta}(\mathbb{R}^{d})\cap\dot{B}_{\infty,1}^{\beta-1}(\mathbb{R}^{d})}.$

∎

To obtain a uniqueness result for transport equations (3.9) in Vishik spaces
we impose an Osgood condition on the growth.

######  Theorem 3.5.

Assume that the growth function $\Pi\in\mathcal{P}_{1}$ satisfies the Osgood
type condition

$\int_{1}^{\infty}\frac{dr}{ry_{\Pi}(r)}=\infty.$

Then a Lagrangian weak solution $\omega$ of (3.9), such that

$\omega\in
L^{\infty}([0,T];\dot{B}^{\beta}_{\Pi}(\mathbb{R}^{d})\cap\dot{B}^{\beta-1}_{\infty,1}(\mathbb{R}^{d}))$

is uniquely determined by its initial value $\omega_{0}$.

###### Proof.

Apply the methodology developed in Section 1.5 together with Theorem 3.4. ∎

###  3.3. Proof of Theorems 1.3 and 1.4

Invoke Theorems 3.4 and 3.5 with $d=2$ and $\beta=0$ (cf. Remark 3.1(ii)).

###  3.4. Comparison of Theorem 1.4 with the uniqueness result of Vishik

As already mentioned in Remark 1.5, both conditions (1.9) and (1.36) are
equivalent for $\Pi\in\mathcal{P}_{1}$. In this section, we investigate the
relationships between the function spaces involved in (1.10) and (1.37),
namely,

$B_{\Pi}(\mathbb{R}^{2})\cap L^{p_{0}}(\mathbb{R}^{2}),\qquad p_{0}\in(1,2),$

and

$\dot{B}_{\Pi}(\mathbb{R}^{2})\cap\dot{B}_{\infty,1}^{-1}(\mathbb{R}^{2}),$

respectively.

######  Proposition 1.

Let $p_{0}\in(1,d)$. Then

$B_{\Pi}(\mathbb{R}^{d})\cap
L^{p_{0}}(\mathbb{R}^{d})\hookrightarrow\dot{B}_{\Pi}(\mathbb{R}^{d})\cap\dot{B}^{-1}_{\infty,1}(\mathbb{R}^{d}).$

###### Proof.

Assume $f\in B_{\Pi}(\mathbb{R}^{d})\cap L^{p_{0}}(\mathbb{R}^{d})$. The norm
of $f$ in $\dot{B}_{\Pi}(\mathbb{R}^{d})$ can be estimated as follows. For
each $N\in\mathbb{N}$, we can apply the classical Nikolskii’s inequality for
entire functions of exponential type (cf. [29, Theorem 3.3.5, p. 126]) and
basic multiplier assertions in order to get

$\displaystyle\sum_{j=-\infty}^{N}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle=\sum_{j=-\infty}^{0}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}+\sum_{j=1}^{N}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\sum_{j=-\infty}^{0}2^{jd/p_{0}}\|\dot{\Delta}_{j}f\|_{L^{p_{0}}(\mathbb{R}^{d})}+\Pi(N)\|f\|_{B_{\Pi}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\|f\|_{L^{p_{0}}(\mathbb{R}^{d})}+\Pi(N)\|f\|_{B_{\Pi}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\Pi(N)\|f\|_{B_{\Pi}(\mathbb{R}^{d})\cap
L^{p_{0}}(\mathbb{R}^{d})}.$

Hence

(3.17) $B_{\Pi}(\mathbb{R}^{d})\cap
L^{p_{0}}(\mathbb{R}^{d})\hookrightarrow\dot{B}_{\Pi}(\mathbb{R}^{d})$

for any $p_{0}\in(1,\infty)$.

On the other hand, the $\dot{B}_{\infty,1}^{-1}$-norm of $f$ can be split into

(3.18)
$\|f\|_{\dot{B}_{\infty,1}^{-1}(\mathbb{R}^{d})}=\sum_{j=-\infty}^{\infty}2^{-j}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}=I+II,$

where

(3.19)
$I:=\sum_{j=-\infty}^{0}2^{-j}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}$

and

(3.20)
$II:=\sum_{j=1}^{\infty}2^{-j}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}.$

We can estimate $I$ applying Nikolskii’s inequality:

(3.21)
$I\lesssim\sum_{j=-\infty}^{0}2^{-j(1-\frac{d}{p_{0}})}\|\dot{\Delta}_{j}f\|_{L^{p_{0}}(\mathbb{R}^{d})}\lesssim\|f\|_{L^{p_{0}}(\mathbb{R}^{d})}\,\sum_{j=0}^{\infty}2^{j(1-\frac{d}{p_{0}})}\lesssim\|f\|_{L^{p_{0}}(\mathbb{R}^{d})},$

where the last step follows from the assumption $p_{0}<d$.

To estimate $II$, we can argue as follows. The fact that $f\in
B_{\Pi}(\mathbb{R}^{d})$ implies that

$\|\Delta_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}\leq\Pi(j)\|f\|_{B_{\Pi}(\mathbb{R}^{d})},\qquad
j\in\mathbb{N}.$

Therefore, taking into account that $\Pi\in\mathcal{P}_{1}$ (cf. item (iv) in
Definition 3), we obtain

(3.22)
$II=\sum_{j=1}^{\infty}2^{-j}\|\Delta_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}\leq\|f\|_{B_{\Pi}(\mathbb{R}^{d})}\,\sum_{j=1}^{\infty}2^{-j}\Pi(j)\lesssim\|f\|_{B_{\Pi}(\mathbb{R}^{d})}.$

Inserting (3.21) and (3.22) into (3.18), one achieves

(3.23) $B_{\Pi}(\mathbb{R}^{d})\cap
L^{p_{0}}(\mathbb{R}^{d})\hookrightarrow\dot{B}_{\infty,1}^{-1}(\mathbb{R}^{d}).$

The combination of (3.17) and (3.23) yields the desired embedding

$B_{\Pi}(\mathbb{R}^{d})\cap
L^{p_{0}}(\mathbb{R}^{d})\hookrightarrow\dot{B}_{\Pi}(\mathbb{R}^{d})\cap\dot{B}^{-1}_{\infty,1}(\mathbb{R}^{d}).$

∎

We close this section by showing that Theorem 1.4 with $\Pi(p)\approx p$ (cf.
(1.38)) also improves the classical Vishik’s uniqueness result formulated in
terms of (1.11).

######  Proposition 2.

Let $p_{0}\in(1,d)$. Then

(3.24) $\emph{bmo}(\mathbb{R}^{d})\cap
L^{p_{0}}(\mathbb{R}^{d})\hookrightarrow\emph{BMO}(\mathbb{R}^{d})\cap\dot{B}^{-1}_{\infty,1}(\mathbb{R}^{d}).$

###### Proof.

Assume $f\in\text{bmo}(\mathbb{R}^{d})\cap L^{p_{0}}(\mathbb{R}^{d})$. The
Besov $\dot{B}^{-1}_{\infty,1}$-norm can be expressed as in (3.18)–(3.20),
with $I$ satisfying (3.21). Hence

(3.25)
$\|f\|_{\dot{B}^{-1}_{\infty,1}(\mathbb{R}^{d})}=I+II\lesssim\|f\|_{L^{p_{0}}(\mathbb{R}^{d})}+II.$

Next we estimate $II$. The well-known embedding
$\text{bmo}(\mathbb{R}^{d})\hookrightarrow
B^{0}_{\infty,\infty}(\mathbb{R}^{d})$ (cf. (1.12)) implies

(3.26)
$II=\sum_{j=1}^{\infty}2^{-j}\|\Delta_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}\lesssim\|f\|_{B^{0}_{\infty,\infty}(\mathbb{R}^{d})}\lesssim\|f\|_{\text{bmo}(\mathbb{R}^{d})}.$

According to (3.25) and (3.26),

$\text{bmo}(\mathbb{R}^{d})\cap
L^{p_{0}}(\mathbb{R}^{d})\hookrightarrow\dot{B}^{-1}_{\infty,1}(\mathbb{R}^{d}).$

This together with the trivial embedding
$\text{bmo}(\mathbb{R}^{d})\hookrightarrow\text{BMO}(\mathbb{R}^{d})$ implies
the desired result (3.24). ∎

##  Appendix A Interpolation and Extrapolation: An Atlas

In order to help non experts in extrapolation, in this appendix we give a
summary of results used in the paper, with documentation, commentaries and
examples. We keep the notation and assumptions laid out in the previous
sections. For the sake of convenience we also recall the location of some
basic definitions.

###  A.1. Interpolation of Besov spaces via retraction revisited

A common technique used in interpolation theory is to translate interpolation
of function spaces into equivalent interpolation problems for sequence spaces
(“the method of retracts”). In particular, in this paper the
$\ell_{1}^{\beta}(L^{\infty}(\mathbb{R}^{d}))$ spaces of sequences of vector-
valued functions play an important role. We say that
$\\{f_{j}\\}_{j\in\mathbb{Z}}\in\ell_{1}^{\beta}(L^{\infty}(\mathbb{R}^{d}))$
if

(A.1)
$\|\\{f_{j}\\}_{j\in\mathbb{Z}}\|_{\ell_{1}^{\beta}(L^{\infty}(\mathbb{R}^{d}))}:=\sum_{j\in\mathbb{Z}}2^{j\beta}\|f_{j}\|_{L^{\infty}(\mathbb{R}^{d})}<\infty.$

Their usefulness for us is that we can translate the interpolation of
$\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d})$ spaces into interpolation of
$\ell_{1}^{\beta}(L^{\infty}(\mathbb{R}^{d})).$ This is effected by showing
that the map

$f\mapsto\\{\dot{\Delta}_{j}f\\}_{j\in\mathbb{Z}}$

defines a retract from $\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d})$ onto
$\ell_{1}^{\beta}(L^{\infty}(\mathbb{R}^{d}))$ (cf. [6, Definition 6.4.1 and
Theorem 6.4.3, pages 150-152]). In particular, $K$-functionals relative to
vector-valued sequence spaces can be explicitly computed, cf. [6, p. 120].

######  Example 5.

We have

(A.2)
$K(t,\\{f_{j}\\}_{j\in\mathbb{Z}};\ell_{1}^{\beta-\kappa}(L^{\infty}(\mathbb{R}^{d})),\ell_{1}^{\beta}(L^{\infty}(\mathbb{R}^{d})))\approx\sum_{j=-\infty}^{\infty}\min\\{1,2^{j\kappa}t\\}\,2^{j(\beta-\kappa)}\,\|f_{j}\|_{L^{\infty}(\mathbb{R}^{d})}.$

In fact, more general statements are available in the literature. For example,
(A.2) holds true with $L^{\infty}(\mathbb{R}^{d})$ replaced by any Banach
space $X$. However, for our purposes, it is enough to restrict attention to
(A.2).

The next result is a (possible) slight improvement of a known result (cf. [6,
Section 6.4]) since we provide sharp constants.

######  Theorem A.1.

Let $-\infty<s_{0}<s_{1}<\infty,\,\theta\in(0,1)$, and
$s=(1-\theta)s_{0}+\theta s_{1}$. Then

$(\dot{B}_{\infty,1}^{s_{0}}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{s_{1}}(\mathbb{R}^{d}))_{\theta,1}^{\blacktriangleleft}=\dot{B}_{\infty,1}^{s}(\mathbb{R}^{d})$

with underlying equivalence constants independent of $\theta$.

###### Proof.

We use the retraction method of interpolation, which implies that

$\|f\|_{(\dot{B}_{\infty,1}^{s_{0}}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{s_{1}}(\mathbb{R}^{d}))_{\theta,1}}\approx\|\\{\dot{\Delta}_{j}f\\}_{j\in\mathbb{Z}}\|_{(\ell_{1}^{s_{0}}(L^{\infty}(\mathbb{R}^{d})),\ell_{1}^{s_{1}}(L^{\infty}(\mathbb{R}^{d})))_{\theta,1}}$

with constants independent of $\theta$. On the other hand, it follows from
(A.2) that, for
$\\{f_{j}\\}_{j\in\mathbb{Z}}\in\ell_{1}^{s_{0}}(L^{\infty}(\mathbb{R}^{d}))+\ell_{1}^{s_{1}}(L^{\infty}(\mathbb{R}^{d}))$,

$\displaystyle\|\\{f_{j}\\}_{j\in\mathbb{Z}}\|_{(\ell_{1}^{s_{0}}(L^{\infty}(\mathbb{R}^{d})),\ell_{1}^{s_{1}}(L^{\infty}(\mathbb{R}^{d})))_{\theta,1}}$
$\displaystyle\approx\sum_{\nu=-\infty}^{\infty}2^{-\theta\nu(s_{1}-s_{0})}K(2^{\nu(s_{1}-s_{0})},\\{f_{j}\\}_{j\in\mathbb{Z}};\ell_{1}^{s_{0}}(L^{\infty}(\mathbb{R}^{d})),\ell_{1}^{s_{1}}(L^{\infty}(\mathbb{R}^{d})))$
$\displaystyle\approx\sum_{\nu=-\infty}^{\infty}2^{-\theta\nu(s_{1}-s_{0})}\sum_{j=-\infty}^{\infty}\min\\{2^{js_{0}},2^{js_{1}}2^{\nu(s_{1}-s_{0})}\\}\,\|f_{j}\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle=\sum_{j=-\infty}^{\infty}2^{js_{0}}\|f_{j}\|_{L^{\infty}(\mathbb{R}^{d})}\bigg{(}2^{j(s_{1}-s_{0})}\sum_{\nu=-\infty}^{-j}2^{\nu(s_{1}-s_{0})(1-\theta)}+\sum_{\nu=-j}^{\infty}2^{-\theta\nu(s_{1}-s_{0})}\bigg{)}$
$\displaystyle\approx(\theta(1-\theta))^{-1}\sum_{j=-\infty}^{\infty}2^{js}\|f_{j}\|_{L^{\infty}(\mathbb{R}^{d})}.$

As a consequence,

$\displaystyle\|f\|_{(\dot{B}_{\infty,1}^{s_{0}}(\mathbb{R}^{d}),\dot{B}_{\infty,1}^{s_{1}}(\mathbb{R}^{d}))_{\theta,1}^{\blacktriangleleft}}$
$\displaystyle\approx(\theta(1-\theta))\|\\{\dot{\Delta}_{j}f\\}_{j\in\mathbb{Z}}\|_{(\ell_{1}^{s_{0}}(L^{\infty}(\mathbb{R}^{d})),\ell_{1}^{s_{1}}(L^{\infty}(\mathbb{R}^{d})))_{\theta,1}}$
$\displaystyle\approx\sum_{j=-\infty}^{\infty}2^{js}\|\dot{\Delta}_{j}f\|_{L^{\infty}(\mathbb{R}^{d})}=\left\|f\right\|_{\dot{B}_{\infty,1}^{s}(\mathbb{R}^{d})},$

as we wished to show. ∎

######  Remark A.2.

The above proof can be easily adapted to deal with the general scale of the
Besov spaces $\dot{B}^{s}_{p,q}(\mathbb{R}^{d})$. Namely, if $1\leq
p,q\leq\infty,-\infty<s_{0}<s_{1}<\infty,\theta\in(0,1)$, and
$s=(1-\theta)s_{0}+\theta s_{1}$, then

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

with equivalence constants independent of $\theta$.

###  A.2. Results from Section 1.3

####  A.2.1. Characterizations of $\Delta$-extrapolation spaces

A very useful result for the computation of $\Delta$-extrapolation spaces is
the formula (1.20), which for convenience we reproduce here

(A.3)
$\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\bigg{\\}}=\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(A_{0},A_{1})_{\theta,\infty}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\bigg{\\}}.$

The result is used implicitly in [19] and a detailed proof can be found in
[27, Theorem 21, page 44]. Further results and generalizations can be found in
[23]. The import of (A.3) relies on the fact that the norm of the space on the
right-hand side is a double supremum, which allows us to apply “Fubini” as
follows

$\displaystyle\left\|f\right\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(A_{0},A_{1})_{\theta,\infty}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\Big{\\}}}$
$\displaystyle=\sup_{\theta\in(0,1)}\sup_{t>0}\frac{t^{-\theta}K(t,f;A_{0},A_{1})}{\Theta(\frac{1}{1-\theta})}$
$\displaystyle=\sup_{t>0}\frac{K(t,f;A_{0},A_{1})}{t}\sup_{\theta\in(0,1)}\frac{t^{1-\theta}}{\Theta(\frac{1}{1-\theta})}$
$\displaystyle=\sup_{t>0}\frac{K(t,f;A_{0},A_{1})}{t\inf_{\theta\in(0,1)}\\{\Theta(\frac{1}{1-\theta})t^{-(1-\theta)}\\}}$
(A.4)
$\displaystyle=\sup_{t>0}\frac{K(t^{1/p_{0}},f;A_{0},A_{1})}{t^{1/p_{0}}\varphi_{\Theta}(\frac{1}{t})},$

where

$\varphi_{\Theta}(t)=\inf_{\theta\in(0,1)}\bigg{\\{}\Theta\bigg{(}\frac{1}{1-\theta}\bigg{)}\,t^{\frac{1-\theta}{p_{0}}}\bigg{\\}},\qquad
p_{0}>0.$

The $\varphi_{\Theta}$ functions thus played a major role in [19]. Remarkably,
as we pointed out in Section 1.3, although coming from completely different
considerations they essentially coincide with the original Yudovich functions
$y_{\Theta}$ (cf. (1.5))

(A.5) $\varphi_{\Theta}(t)\approx y_{\Theta}(t);$

see also (1.25).

In this paper we make a strong use of both (A.3) and (A.4). In particular, to
give an explicit characterization of the Yudovich spaces $Y^{\Theta}$ (cf.
(1.24) and (1.26)) and the sharp Yudovich spaces $Y^{\\#\Theta}$ (cf. Theorem
2.2, specially (2.11) and (2.12)), to establish various characterizations of
the Vishik spaces $B^{\beta}_{\Pi}$ (cf. Theorem 3.2, in particular, (3.4)),
and to get estimates for the modulus of smoothness (cf. (2.28)).

####  A.2.2. Reiteration

The scale $\\{(A_{0},A_{1})_{\theta,p(\theta)}\\}_{\theta\in(0,1)}$ is an
example of _interpolation scale of exact order $\theta$_. We refer to [19, p.
7] for the precise definition. Another important example is given by
$\\{(A_{0},A_{1})^{\blacktriangleleft}_{\theta,p}\\}_{\theta\in(0,1)}$ for a
fixed $p\in[1,\infty]$ (cf. (1.18)). It turns out that the extrapolation
formula (A.3) is only a special case of a more general phenomenon based on
interpolation scales of exact order $\theta$. In particular, the formula is
still valid when $\\{(A_{0},A_{1})_{\theta,p(\theta)}\\}_{\theta\in(0,1)}$ is
replaced by
$\\{(A_{0},A_{1})^{\blacktriangleleft}_{\theta,p}\\}_{\theta\in(0,1)}$,
namely,

(A.6)
$\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(A_{0},A_{1})_{\theta,p}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\bigg{\\}}=\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(A_{0},A_{1})_{\theta,\infty}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\bigg{\\}}.$

See [27, Theorem 21, page 44].

_The ordered case $A_{1}\hookrightarrow A_{0}$_. In this case we can achieve
further simplifications for the computation of the $\Delta$-extrapolation
spaces. In particular, we claim that

(A.7)
$\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\bigg{\\}}=\Delta_{\theta\in(\theta_{0},1)}\bigg{\\{}\frac{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\bigg{\\}}$

for every $\theta_{0}\in(0,1)$. This result may be considered as the
extrapolation version of the fact that the definitions of
$Y^{\Theta}_{p_{0}}(\Omega)$ and $Y^{\\#\Theta}_{p_{0}}(\Omega)$ with
$|\Omega|<\infty$ are independent of $p_{0}\in[1,\infty)$. Indeed, recall that
$Y^{\Theta}_{p_{0}}(\Omega)$ and $Y^{\\#\Theta}_{p_{0}}(\Omega)$ are the
$\Delta$-extrapolation spaces relative to the pairs
$\left(L^{p_{0}}(\Omega),L^{\infty}(\Omega)\right)$ and
$(L^{p_{0}}(\Omega),\text{BMO}(\Omega))$, respectively; cf. (1.23)-(1.24) and
(2.1)-(2.2).

###### Proof of (A.7).

The non trivial part of the statement is the embedding $\hookleftarrow$. This
can be derived as follows. We have

$\left\|f\right\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\Big{\\}}}\leq\left\|f\right\|_{\Delta_{\theta\in(0,\theta_{0})}\Big{\\{}\frac{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\Big{\\}}}+\left\|f\right\|_{\Delta_{\theta\in(\theta_{0},1)}\Big{\\{}\frac{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\Big{\\}}}.$

To estimate the first term on the right-hand side of the previous inequality,
we will make use of the fact that

(A.8)
$(A_{0},A_{1})_{\theta_{0},p(\theta_{0})}\hookrightarrow(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}\qquad\text{if}\qquad\theta<\theta_{0},$

where the embedding constant is independent of $\theta$. This result can be
shown by using similar techniques as in [14, Lemma 4.5(ii)]. To make the
presentation self-contained, we next provide full details. Using that
$A_{1}\hookrightarrow A_{0}$, it is plain to see that

$K(t,f;A_{0},A_{1})\approx\|f\|_{A_{0}}\approx
K(1,f;A_{0},A_{1}),\qquad\text{for}\qquad t>1.$

Then, by Hölder’s inequality (noting that
$p(\theta)=\frac{1}{1-\theta}<\frac{1}{1-\theta_{0}}=p(\theta_{0})$) and
monotonicity properties of $K$-functionals,

$\displaystyle\|f\|_{(A_{0},A_{1})_{\theta,p(\theta)}}^{p(\theta)}$
$\displaystyle=\int_{0}^{1}[t^{-\theta}K(t,f;A_{0},A_{1})]^{p(\theta)}\,\frac{dt}{t}+\int_{1}^{\infty}[t^{-\theta}K(t,f;A_{0},A_{1})]^{p(\theta)}\,\frac{dt}{t}$
$\displaystyle\approx\int_{0}^{1}[t^{-\theta}K(t,f;A_{0},A_{1})]^{p(\theta)}\,\frac{dt}{t}+\frac{1}{\theta
p(\theta)}\,\|f\|_{A_{0}}^{p(\theta)}$
$\displaystyle\leq\bigg{(}\int_{0}^{1}[t^{-\theta_{0}}K(t,f;A_{0},A_{1})]^{p(\theta_{0})}\,\frac{dt}{t}\bigg{)}^{p(\theta)/p(\theta_{0})}+\frac{1}{\theta
p(\theta)}\,\|f\|_{A_{0}}^{p(\theta)}$
$\displaystyle\lesssim\frac{1}{\theta}\,\bigg{(}\int_{0}^{1}[t^{-\theta_{0}}K(t,f;A_{0},A_{1})]^{p(\theta_{0})}\,\frac{dt}{t}\bigg{)}^{p(\theta)/p(\theta_{0})}$
$\displaystyle\leq\frac{1}{\theta}\,\|f\|_{(A_{0},A_{1})_{\theta_{0},p(\theta_{0})}}^{p(\theta)},$

where we have also used that $\theta\in(0,\theta_{0})$ (and so
$p(\theta)\approx 1$) in the penultimate estimate. The proof of (A.8) is
finished.

It follows from (A.8) that

$\displaystyle\sup_{\theta\in(0,\theta_{0})}\frac{\left\|f\right\|_{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}}{\Theta(\frac{1}{1-\theta})}$
$\displaystyle\lesssim\left\|f\right\|_{(A_{0},A_{1})_{\theta_{0},p(\theta_{0})}}\sup_{\theta\in(0,\theta_{0})}\frac{1}{\Theta(\frac{1}{1-\theta})}$
$\displaystyle=\frac{\left\|f\right\|_{(A_{0},A_{1})_{\theta_{0},p(\theta_{0})}}}{\Theta(1)}$
$\displaystyle\lesssim\sup_{\theta\in(\theta_{0},1)}\frac{\left\|f\right\|_{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}}{\Theta(\frac{1}{1-\theta})}.$

This completes the proof of (A.7).

∎

###  A.3. Computability of $K$-functionals

A central issue in interpolation theory is to find explicit expressions for
$K$-functionals. Next we list some well-known examples of characterizations
for $K$-functionals (see also Example 5) and we refer the interested reader to
[5, 6] for further examples.

######  Example 6.

Let $(L^{p_{0}}(\Omega),L^{\infty}(\Omega)),\,p_{0}\in(0,\infty)$. Then (cf.
[6, Theorem 5.2.1, page 109] and [5, Theorem 1.6, page 298])

(A.9)
$K(t,f;L^{p_{0}}(\Omega),L^{\infty}(\Omega))\approx\bigg{(}\int_{0}^{t^{p_{0}}}(f^{\ast}(\xi))^{p_{0}}\,d\xi\bigg{)}^{1/p_{0}}.$

Equality holds in (A.9) if $p_{0}=1$.

######  Example 7.

Let
$(L^{p_{0}}(\mathbb{R}^{d}),\text{BMO}(\mathbb{R}^{d})),\,p_{0}\in(0,\infty)$.
Then (cf. [20, Corollary 3.3])

(A.10)
$K(t,f;L^{p_{0}}(\mathbb{R}^{d}),\text{BMO}(\mathbb{R}^{d}))\approx\bigg{(}\int_{0}^{t^{p_{0}}}[(M^{\\#}_{\mathbb{R}^{d}}f)^{*}(\xi)]^{p_{0}}\,d\xi\bigg{)}^{1/p_{0}},$

where $M^{\\#}_{\mathbb{R}^{d}}$ is the maximal function given in (1.16). The
local counterpart for function spaces defined on cubes also holds true. In
order to suitably interpret
$(L^{p_{0}}(\mathbb{R}^{d}),\text{BMO}(\mathbb{R}^{d}))$ as an interpolation
pair, it is necessary to factor out constant functions. Then
$L^{p_{0}}(\mathbb{R}^{d})$ can be identified with
$L^{p_{0}}(\mathbb{R}^{d})/\mathbb{C}$, where
$\|f\|_{L^{p_{0}}(\mathbb{R}^{d})/\mathbb{C}}=\inf_{c\in\mathbb{C}}\|f-c\|_{L^{p_{0}}(\mathbb{R}^{d})}$.

######  Example 8.

The $K$-functional for the pair
$(L^{\infty}(\mathbb{R}^{d}),\dot{W}^{1}_{\infty}(\mathbb{R}^{d}))$ plays an
important role in our work (cf. (1.30)) and can be characterized as (cf. [5,
(4.42), p. 341] and [22, Theorem 1])

(A.11)
$K(t,f;L^{\infty}(\mathbb{R}^{d}),\dot{W}_{\infty}^{1}(\mathbb{R}^{d}))\approx\sup_{\left|x-y\right|\leq
t}\left|f(x)-f(y)\right|.$

The corresponding formula for periodic functions is also true.

A natural assumption for $(A_{0},A_{1})$ is to be _Gagliardo closed_ in the
sense that

$\left\|f\right\|_{A_{0}}\approx\sup_{t>0}K(t,f;A_{0},A_{1})\qquad\text{and}\qquad\left\|f\right\|_{A_{1}}\approx\sup_{t>0}\frac{K(t,f;A_{0},A_{1})}{t}.$

See [5, p. 320]. This condition is easily verified for many classical pairs of
spaces, in particular, for the pairs given in Examples 6–8.

######  Proposition 3.

Suppose that $(A_{0},A_{1})$ is Gagliardo closed, and let $\Theta(p)\approx
1$. Then

$\Delta_{\theta\in(0,1)}\bigg{\\{}\frac{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\bigg{\\}}=A_{0}\cap
A_{1}.$

###### Proof.

Using (A.3) and the monotonicity properties of $K$-functionals (note that
$K(t,f;A_{0},A_{1})$ increases and $\frac{K(t,f;A_{0},A_{1})}{t}$ decreases),
we have

$\displaystyle\|f\|_{\Delta_{\theta\in(0,1)}\Big{\\{}\frac{(A_{0},A_{1})_{\theta,p(\theta)}^{\blacktriangleleft}}{\Theta(\frac{1}{1-\theta})}\Big{\\}}}$
$\displaystyle\approx\sup_{\theta\in(0,1)}\sup_{t>0}t^{-\theta}K(t,f;A_{0},A_{1})$
$\displaystyle\approx\sup_{\theta\in(0,1)}\sup_{t>1}t^{-\theta}K(t,f;A_{0},A_{1})+\sup_{\theta\in(0,1)}\sup_{t<1}t^{-\theta}K(t,f;A_{0},A_{1})$
$\displaystyle=\sup_{t>1}K(t,f;A_{0},A_{1})\sup_{\theta\in(0,1)}t^{-\theta}+\sup_{t<1}\frac{K(t,f;A_{0},A_{1})}{t}\sup_{\theta\in(0,1)}t^{1-\theta}$
$\displaystyle=\sup_{t>1}K(t,f;A_{0},A_{1})+\sup_{t<1}\frac{K(t,f;A_{0},A_{1})}{t}$
$\displaystyle\approx\left\|f\right\|_{A_{0}}+\left\|f\right\|_{A_{1}}.$

∎

The previous result can be used to readily justify the assertions made in
(1.4) and (1.15) for $Y^{\Theta}_{p_{0}}(\Omega)$ and
$Y^{\\#\Theta}_{p_{0}}(\Omega)$ via the extrapolation formulae (1.24) and
(2.2). For Vishik spaces, we have
$\dot{B}_{\Pi}^{\beta}(\mathbb{R}^{d})=\dot{B}_{\infty,1}^{\beta}(\mathbb{R}^{d}),$
$\Pi(p)\approx 1$, and one could apply Proposition 3 to obtain the
corresponding result that appears in (3.3).

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