# Fractional Operators inp-adic Variable Exponent Lebesgue Spaces and Application to p-adic Derivative

Authors: Leonardo Fabio Chacón-Cortés; Humberto Rafeiro
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in thep-adic variable exponent Lebesgue spaces. As an application, we show the existence and uniqueness of the solution for a nonhomogeneous Cauchy problem in the p-adic variable exponent Lebesgue spaces.


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## Body

## 1. Introduction

The field ofp-adic numbers are an interesting and useful tool to study phenomena in physics, biology, and medicine, among other sciences; see, e.g., [1–4] and references therein. For this reason, the study of operators that allows us to describe such phenomena is essential. Even more so when in the p-adic setting it is not possible to define the derivative in the classical sense.Variable exponent Lebesgue spaces generalize the notion ofq-integrability in the classical Lebesgue spaces, allowing the exponent to be a measurable function. These spaces were introduced in 1931 by Orlicz [5] but lay essentially dormant for more than 50 years. They received a thrust in the paper [6] and are now an active area of research having many known applications, e.g., in the modeling of thermorheological fluids [7] as well as electrorheological fluids [8–11], in differential equations with nonstandard growth [12, 13], and in the study of image processing [14–20]. For a thorough history, theory, and applications of variable exponent Lebesgue spaces, see [6, 21–24].In this article, we are interested in the boundedness of the fractional integral and maximal fractional operator on thep-adic Lebesgue spaces with a variable exponent. The corresponding result for classical p-adic Lebesgue space is known (cf. [25]). These operators play an important role in such areas such as Sobolev spaces, potential theory, PDEs, and integral geometry, to name a few.This work is divided as follows. Section2 contains a quick description of the preliminary on the topic of the p-adic analysis and variable exponent Lebesgue spaces on the p-adic numbers, necessary for the development of this work. In Section 3, the boundedness of the fractional maximal operator(1)Mαfx=supγ∈ℤ1pγn−α∫Bγnxfydy,x∈ℚpn,is studied in the framework of variable exponent p-adic Lebesgue spaces. The boundedness of the special case M0, the so-called Hardy-Littlewood maximal operator, was obtained in [26] under appropriate conditions on the exponent function. We prove, using a suitable pointwise estimate, the boundedness of the fractional maximal operator from Lq·ℚpn to Lq#·ℚpn, where q# is the Sobolev limiting exponent; see (31) for the corresponding definition. The boundedness of the fractional integral operator(2)Iαfx=∫ℚpnfyx−ypn−αdy,x∈ℚpn,is obtained from the boundedness of the fractional maximal operator and Welland’s pointwise inequality tailored for the p-adic setting; this approach is inspired from [27]. In the literature, it is customary to prove first the boundedness of the fractional potential operator and, as a corollary, the boundedness of the fractional maximal operator is obtained using the lattice property of the norm and the elementary estimate
(3)Mαfx≤Iαfx.As already mentioned, we will use a reverse approach.Finally, in Section4, we define the Taibleson operator in p-adic Lebesgue spaces with variable exponent, which is the analogue of the derivative in the spatial variable x (x∈ℚpn), and study the nonhomogeneous Cauchy problem (72) associated with this operator.The notationa≲b denotes the existence of a constant C for which a≤Cb, a≍b means that a≲b and b≲a.


## 2. Preliminaries

For an exposition on thep-adic analysis, see [25, 28].


### 2.1. The Field ofp-adic Numbers

Byp we denote a prime number. The field ℚp is given as the completion of ℚ with respect to the p-adic norm ·p, given by
(4)xp≔0,ifx=0,p−γ,ifx=pγab,where a,b are integers coprime with p. The integer γ≔ordxord0≔+∞ is denoted as the p-adic order of  x. This norm can be extended to ℚpn as
(5)xp≔max1≤i≤nxipforx=x1,⋯,xn∈ℚpn,and satisfies the so-called strong triangular inequality
(6)x+yp≤maxxp,yp,with equality when xp≠yp. If ordx≔min1≤i≤nordxi, it follows that xp=p−ordx. The set ℚpn,·p is a complete ultrametric space and, as a topological space, ℚp is homeomorphic to a Cantor-like subset of the real line. A p-adic number x≠0 has a unique series expansion, viz.,
(7)x=pordx∑j=0∞xjpj,where xj∈0,1,2,⋯,p−1 and x0≠0. For γ∈ℤ, we denote by
(8)Bγna=x∈ℚpn:x−ap≤pγthe ball of radius pγ with center at a=a1,⋯,an∈ℚpn and by
(9)Sγna=x∈ℚpn:x−ap=pγ=Bγna\Bγ−1na,the corresponding sphere. We denote
(10)Bγn0≔Bγn,Sγn0≔Sγn,and note that
(11)ℚpn\0=⋃γ∈ℤSγn.Note thatBγna=Bγa1×⋯×Bγan, where Bγai≔x∈ℚp:x−aip≤pγ is the one-dimensional ball. In ℚpn there exists the additive Haar measure dnx≔dx (by E we denote the Haar measure of the set E), since the field ℚpn is a locally compact commutative group with respect to addition. Normalizing the measure dx by ∫B0ndx=1, we get a unique measure. From here onwards, we use the normalized Haar measure; thus,
(12)VolBγna=pγn,VolSγna=pγn1−p−n,for any a∈ℚpn.Note that the collection of all disjoint balls of the same radiusγ forms a partition of ℚpn, since inequality (6) implies that any two balls in ℚpn with the same radius are either identical or disjoint.


### 2.2. Some Function Spaces

A complex-valued functionφ defined on ℚpn is called locally constant if for any x∈ℚpn, there exists an integer lx∈ℤ such that
(13)φx+x′=φxforx′∈Blxn.A functionφ:ℚpn⟶ℂ is called a Schwartz-Bruhat function (or a test function) if it is locally constant with the compact support. The ℂ-vector space of Schwartz-Bruhat functions is denoted by Sℚpn≔S.A measurable functionf:ℚpn⟶ℂ belongs to the Lebesgue space Lqℚpn, 1≤q<∞, when
(14)fLqℚpnq≔∫ℚpnfxqdx<∞,1≤q<∞,where
(15)∫ℚpnfxqdx≔limγ⟶∞∫Bγn0fxqdx,if the limit exists.We now introduce the notion ofp-adic Lebesgue spaces with a variable exponent and give some properties needed in the sequel; see [26] for the respective proofs.We say that a measurable function is a variable exponent ifq:ℚpn⟶1,∞. By Qℚpn we denote the set of all variable exponent satisfying q+<∞, where q+≔esssupx∈ℚpnqx and q−≔essinfx∈ℚpnqx.Forq∈Qℚpn by Lq·ℚpn, we denote the space of measurable functions f:ℚpn⟶ℝ such that
(16)fLq·ℚpn=infλ>0:ϱq·fλ≤1<∞,where ϱq·f≔∫ℚpnfxqxdx.For the Lebesgue space with a variable exponent, we have(17)fLq·ℚpn≤ϱq·f+1,(18)ϱq·f≤1+fLq·ℚpnq+,(19)fLq·ℚpn=fsLq·/sℚpn1/s,s∈0,q−.The Hölder inequality is valid, up to a multiplicative constant, in the framework of Lebesgue spaces with variable exponent, viz.,(20)∫ℚpnfxgxdx≤CfLq·ℚpngLq′·ℚpn,where q and q′ are conjugate exponents, viz., 1=1/qx+1/q′x.Forq∈Qℚpn, we say that q∈W0ℚpn when there is a positive constant C, for which
(21)γq−Bγnx−q+Bγnx≤C,for allγ∈ℤ and any x∈ℚpn. We say that q∈W∞ℚpn when there is a positive constant C, for which
(22)qx−qy≤C1logpp+minyp,xp,for anyx,y∈ℚpn.The classW0∞ℚpn is defined as W0∞ℚpn≔W0ℚpn∩W∞ℚpn. The importance of the class W0∞ℚpn stems from the fact that it is a sufficient condition for the boundedness of the maximal operator in Lq·ℚpn: if q∈W0∞ℚpn, then
(23)MfLq·ℚpn≤CfLq·ℚpn,see Theorem 5.2 of [26] for the corresponding proof.In the case whereΩpn is a bounded subset of ℚpn, we have the following: if q∈W0ℚpn, then,
(24)MfLq·Ωpn≤CfLq·Ωpn,see Theorem 5.1 of [26].Lemma 1.
Letq∈Qℚpn be a L-Lipschitz function, for some L≥0. Then, q∈W0ℚpn.Proof.
We give the proof only for the caseγ≤0 since the other case is immediate. Since q is a continuous function, for any ball Bγnx⊂ℚpn, there exists a maximum (respectively, minimum) point x¯∈Bγnx (respectively, x¯∈Bγnx). From the Lipschitzianity of q, we have
(25)−γq+Bγnx−q−Bγnx=−γqx¯−qx¯≤−Lγx¯−x¯p≤2Lγpγ≤C,which completes the proof.We now show an extension result via the well-known McShane extension technique (a similar approach was used in the Euclidean framework; see [23]).Lemma 2.
Letq∈W0Ωpn, where Ωpn is a bounded subset of ℚpn. Then, there exists an extension function q˘∈W0∞ℚpn which is constant outside some fixed ball.Proof.
The proof will be divided into two steps.
First step: we show that there exists an extension function q~∈W0ℚpn. Let us define q~ as
(26)q~x≔supξ∈Ωpnqξ−ωx−ξp,with
(27)ωt∶=Clogpp+1/t,t>0,0,t=0,where C comes from equation (21). Since ωt is an increasing and concave function for t≥0 and approaches zero with t, then from Theorem 2 of [29], we have that q~x−q~y≤ωx−yp. In order to prove that q~∈W0ℚpn, it suffices to check for γ<0. Since ω is an increasing function and taking x¯ and x¯ as in the proof of Lemma 1, we see that
(28)γq~−Bγnx−q~+Bγnx=γq~x¯−q~x¯=γq~x¯−q~x¯∣≤γωx¯−x¯p≤Cγlogpp+pγ≤C,which ends the first step.
Second step: we show that there exists an extension function q˘∈W0∞ℚpn. Since Ωpn is a bounded set, let us take γ>0 such that Ωpn⊂Bγn0. We define two Urysohn functions, u0 and u∞, as follows:
(29)u0x=dx,ℚpn\Bγ+1n0dx,ℚpn\Bγ+1n0+dx,Ωpn,u∞x=dx,Ωpndx,ℚpn\Bγ+1n0+dx,Ωpn.
The functionsu0 and u∞ are L-Lipschitz with L≤1/p, due to the fact that dΩpn,ℚpn\Bγ+1n0≥p, see Prop. 2.1.1 of [30]. Defining the exponent q˘ as,
(30)q˘x≔q~xu0x+q+u∞x,
we see thatq˘∈W0ℚpn since the class W0ℚpn is closed under addition and multiplication, and q˘∈W∞ℚpn because q˘x≡q+ in the exterior of the ball Bγ+1n0.


## 2.1. The Field ofp-adic Numbers

Byp we denote a prime number. The field ℚp is given as the completion of ℚ with respect to the p-adic norm ·p, given by
(4)xp≔0,ifx=0,p−γ,ifx=pγab,where a,b are integers coprime with p. The integer γ≔ordxord0≔+∞ is denoted as the p-adic order of  x. This norm can be extended to ℚpn as
(5)xp≔max1≤i≤nxipforx=x1,⋯,xn∈ℚpn,and satisfies the so-called strong triangular inequality
(6)x+yp≤maxxp,yp,with equality when xp≠yp. If ordx≔min1≤i≤nordxi, it follows that xp=p−ordx. The set ℚpn,·p is a complete ultrametric space and, as a topological space, ℚp is homeomorphic to a Cantor-like subset of the real line. A p-adic number x≠0 has a unique series expansion, viz.,
(7)x=pordx∑j=0∞xjpj,where xj∈0,1,2,⋯,p−1 and x0≠0. For γ∈ℤ, we denote by
(8)Bγna=x∈ℚpn:x−ap≤pγthe ball of radius pγ with center at a=a1,⋯,an∈ℚpn and by
(9)Sγna=x∈ℚpn:x−ap=pγ=Bγna\Bγ−1na,the corresponding sphere. We denote
(10)Bγn0≔Bγn,Sγn0≔Sγn,and note that
(11)ℚpn\0=⋃γ∈ℤSγn.Note thatBγna=Bγa1×⋯×Bγan, where Bγai≔x∈ℚp:x−aip≤pγ is the one-dimensional ball. In ℚpn there exists the additive Haar measure dnx≔dx (by E we denote the Haar measure of the set E), since the field ℚpn is a locally compact commutative group with respect to addition. Normalizing the measure dx by ∫B0ndx=1, we get a unique measure. From here onwards, we use the normalized Haar measure; thus,
(12)VolBγna=pγn,VolSγna=pγn1−p−n,for any a∈ℚpn.Note that the collection of all disjoint balls of the same radiusγ forms a partition of ℚpn, since inequality (6) implies that any two balls in ℚpn with the same radius are either identical or disjoint.


## 2.2. Some Function Spaces

A complex-valued functionφ defined on ℚpn is called locally constant if for any x∈ℚpn, there exists an integer lx∈ℤ such that
(13)φx+x′=φxforx′∈Blxn.A functionφ:ℚpn⟶ℂ is called a Schwartz-Bruhat function (or a test function) if it is locally constant with the compact support. The ℂ-vector space of Schwartz-Bruhat functions is denoted by Sℚpn≔S.A measurable functionf:ℚpn⟶ℂ belongs to the Lebesgue space Lqℚpn, 1≤q<∞, when
(14)fLqℚpnq≔∫ℚpnfxqdx<∞,1≤q<∞,where
(15)∫ℚpnfxqdx≔limγ⟶∞∫Bγn0fxqdx,if the limit exists.We now introduce the notion ofp-adic Lebesgue spaces with a variable exponent and give some properties needed in the sequel; see [26] for the respective proofs.We say that a measurable function is a variable exponent ifq:ℚpn⟶1,∞. By Qℚpn we denote the set of all variable exponent satisfying q+<∞, where q+≔esssupx∈ℚpnqx and q−≔essinfx∈ℚpnqx.Forq∈Qℚpn by Lq·ℚpn, we denote the space of measurable functions f:ℚpn⟶ℝ such that
(16)fLq·ℚpn=infλ>0:ϱq·fλ≤1<∞,where ϱq·f≔∫ℚpnfxqxdx.For the Lebesgue space with a variable exponent, we have(17)fLq·ℚpn≤ϱq·f+1,(18)ϱq·f≤1+fLq·ℚpnq+,(19)fLq·ℚpn=fsLq·/sℚpn1/s,s∈0,q−.The Hölder inequality is valid, up to a multiplicative constant, in the framework of Lebesgue spaces with variable exponent, viz.,(20)∫ℚpnfxgxdx≤CfLq·ℚpngLq′·ℚpn,where q and q′ are conjugate exponents, viz., 1=1/qx+1/q′x.Forq∈Qℚpn, we say that q∈W0ℚpn when there is a positive constant C, for which
(21)γq−Bγnx−q+Bγnx≤C,for allγ∈ℤ and any x∈ℚpn. We say that q∈W∞ℚpn when there is a positive constant C, for which
(22)qx−qy≤C1logpp+minyp,xp,for anyx,y∈ℚpn.The classW0∞ℚpn is defined as W0∞ℚpn≔W0ℚpn∩W∞ℚpn. The importance of the class W0∞ℚpn stems from the fact that it is a sufficient condition for the boundedness of the maximal operator in Lq·ℚpn: if q∈W0∞ℚpn, then
(23)MfLq·ℚpn≤CfLq·ℚpn,see Theorem 5.2 of [26] for the corresponding proof.In the case whereΩpn is a bounded subset of ℚpn, we have the following: if q∈W0ℚpn, then,
(24)MfLq·Ωpn≤CfLq·Ωpn,see Theorem 5.1 of [26].Lemma 1.
Letq∈Qℚpn be a L-Lipschitz function, for some L≥0. Then, q∈W0ℚpn.Proof.
We give the proof only for the caseγ≤0 since the other case is immediate. Since q is a continuous function, for any ball Bγnx⊂ℚpn, there exists a maximum (respectively, minimum) point x¯∈Bγnx (respectively, x¯∈Bγnx). From the Lipschitzianity of q, we have
(25)−γq+Bγnx−q−Bγnx=−γqx¯−qx¯≤−Lγx¯−x¯p≤2Lγpγ≤C,which completes the proof.We now show an extension result via the well-known McShane extension technique (a similar approach was used in the Euclidean framework; see [23]).Lemma 2.
Letq∈W0Ωpn, where Ωpn is a bounded subset of ℚpn. Then, there exists an extension function q˘∈W0∞ℚpn which is constant outside some fixed ball.Proof.
The proof will be divided into two steps.
First step: we show that there exists an extension function q~∈W0ℚpn. Let us define q~ as
(26)q~x≔supξ∈Ωpnqξ−ωx−ξp,with
(27)ωt∶=Clogpp+1/t,t>0,0,t=0,where C comes from equation (21). Since ωt is an increasing and concave function for t≥0 and approaches zero with t, then from Theorem 2 of [29], we have that q~x−q~y≤ωx−yp. In order to prove that q~∈W0ℚpn, it suffices to check for γ<0. Since ω is an increasing function and taking x¯ and x¯ as in the proof of Lemma 1, we see that
(28)γq~−Bγnx−q~+Bγnx=γq~x¯−q~x¯=γq~x¯−q~x¯∣≤γωx¯−x¯p≤Cγlogpp+pγ≤C,which ends the first step.
Second step: we show that there exists an extension function q˘∈W0∞ℚpn. Since Ωpn is a bounded set, let us take γ>0 such that Ωpn⊂Bγn0. We define two Urysohn functions, u0 and u∞, as follows:
(29)u0x=dx,ℚpn\Bγ+1n0dx,ℚpn\Bγ+1n0+dx,Ωpn,u∞x=dx,Ωpndx,ℚpn\Bγ+1n0+dx,Ωpn.
The functionsu0 and u∞ are L-Lipschitz with L≤1/p, due to the fact that dΩpn,ℚpn\Bγ+1n0≥p, see Prop. 2.1.1 of [30]. Defining the exponent q˘ as,
(30)q˘x≔q~xu0x+q+u∞x,
we see thatq˘∈W0ℚpn since the class W0ℚpn is closed under addition and multiplication, and q˘∈W∞ℚpn because q˘x≡q+ in the exterior of the ball Bγ+1n0.


## 3. Main Results

### 3.1. Boundedness inℚpn

In this section, we study the boundedness of the operators in the case ofℚpn.


#### 3.1.1. Fractional Maximal Operator

The classical result regarding boundedness of the fractional maximal operator says that if1<q<n/α and 1/s=1/q−α/n, then Mα:Lqℚpn⟶Lsℚpn is bounded (this follows at once from inequality (3) and the boundedness of the operator Iα:Lqℚpn⟶Lsℚpn, cf. [25]). For further goals, we need estimate (32).Lemma 3.
Let0<α<n, q be an exponent function such that 1<q−≤q+<n/α, and we define the Sobolev limiting exponent q# by
(31)1q#x≔1qx−αn.
Then,(32)Mαfx≤cMf·q·/q#·n/n−αx1−α/n∫ℚpnfyqydyα/n.Proof.
Fromqx/q#x+αqx/n=1 and Hölder’s inequality (20), we have
(33)1pγn−γα∫Bγnxfydy=1pγn−γα∫Bγnxfyqy/q#yfyαqy/ndy≤c1pγn−γα∫Bγnxfynqy/q#yα−ndy1−α/n∫Bγnxfyqydyα/n.
Taking the supremum over allγ∈ℤ, we establish the desired inequality.Theorem 4.
Let0<α<n and q be an exponent function such that 1<q−≤q+<n/α. Suppose, moreover, that q#∈W0∞ℚpn, where q# is the Sobolev limiting exponent (31). Then, the operator Mα:Lq·ℚpn⟶Lq#·ℚpn is bounded.Proof.
TakingfLq·ℚpn=1, the estimate (32), the relation (19), and the boundedness of the maximal operator M in Lq#·ℚpn, we have
(34)MαfxLq#·ℚpn≲Mf·q·/q#·n/n−αx1−α/nLq#·ℚpn=Mf·q·/q#·n/n−αxLq#·1−α/nℚpn1−α/n≲α,n,pf·q·/q#·n/n−αLq#·1−α/nℚpn1−α/n≲α,n,pf·q·/q#·Lq#·ℚpn≲α,n,p1.
The general case follows from homogeneity considerations.


#### 3.1.2. Fractional Potential Operator

The well-knownSobolev theorem states that the fractional potential operator (2), sometimes introduced with a normalizing factor, is bounded from Lqℚpn to Lq#ℚpn where 1/q#=1/q−α/n is the so-called Sobolev limiting exponent; see, for instance, [25].In order to obtain the boundedness result in the variable exponent framework, we first obtain the validity of a Welland-type estimate in thep-adic setting; see [31] for the Euclidean counterpart.Lemma 5.
Let0<α<n, 0<ε<maxα,n−α, and f∈Lloc1ℚpn. Then,
(35)Iαfx≲α,n,pMα−εfx1/2Mα+εfx1/2.Proof.
Letγ∈ℤ, then
(36)∫Bγnxfyx−ypn−αdy=∑k=−∞γ1pkn−α∫Sknxfydy≤∑k=−∞γpkε1pkn−α+ε∫Bknxfydy≤∑k=−∞γpkεMα−εfx≲α,n,ppεγ+11−p−εMα−εfx≲α,n,p,εpεγMα−εfx.
On the other hand,(37)∫BCγnxfyx−ypn−αdy=∑k=γ+1∞1pkn−α∫Sknxfydy≤∑k=γ+1∞p−kε1pkn−α−ε∫Bknxfydy≤∑k=γ+1∞p−kεMα+εfx≲α,n,ppε−γ+11−p−εMα+εfx≲α,n,ppε−γ+11−p−εMα+εfx≲α,n,pp−εγMα+εfx.
Taking the previous estimates into consideration, we have(38)Iαfx≲α,n,p,εpεγMα−εfx+p−εγMα+εfx.
The inequality (35) is obtained taking into account (38) with γ given by
(39)γ=logpMα+εfx/Mα−εfx1/2ε,since x≤x<x+1.Theorem 6.
Let1<q−≤q+≤n/α and q∈W0∞ℚpn. Then, the fractional potential operator,
(40)Iα:Lq·ℚpn⟶Lq#·ℚpn,is bounded, where q# is the Sobolev limiting exponent (31).Proof.
From the definition of the variable exponent Lebesgue norm and homogeneity, it suffices to show that(41)∫ℚpnIαfxq#xdx≤C,when fLq·ℚpn≤1, see Lemma 3.3 of [26] for more details.
From (35) and Hölder’s inequality, we obtain
(42)∫ℚpnIαfxq#xdx≲α,n,p∫ℚpnMα−εfxq#x/2Mα+εfxq#x/2dx≲α,n,pMα−εf·q#·/2q1·Mα+εf·q#·/2q2·≤cI1I2,where q1 and q2 are conjugate exponents and will be chosen appropriately. To estimate I1, from (17), we have
(43)I1≤ϱq1·Mα−εf·q#·/2+1=∫ℚpnMα−εfxq#xq1x/2dx+1.
Definingq1 by 1/qx−2/q#xq1x=α−ε/n, we have, by Lemma 3, that the operator Mα−ε:Lq·ℚpn⟶Lq~·ℚpn is bounded, where q~x=q#xq1x/2. From (43) and (18), it follows that
(44)Mα−εf·q#·/2q1·≤1+Mα−εfLq~·ℚpnq~++1≤c1+fLq·ℚpnq~++1≤c,which end the estimate for I1.
The estimate forI2 follows, mutatis mutandis, as I1 taking into account that defining q2 as 1/qx−2/q#xq2x=α+ε/n, we get that q1 and q2 are indeed conjugate exponents and q2 is the right exponent for the boundedness of Mα+ε; the details are left to the reader.


### 3.2. Boundedness inΩpn

The fractional integral operatorIα can be defined for an open set Ωpn⊂ℚpn in the following way:
(45)Iαfx=∫Ωpnfyx−ypn−αdy,x∈Ωpn.We are interested in proving theLq·Ωpn⟶Lq#·Ωpn boundedness for the operator Iα, where q# is defined by (31). We begin with two lemmas, which are important on their own.Although we need Lemma7 for bounded set Ωpn, we give the lemma for general measurable sets Ωpn which include, as a particular case, Ωpn=ℚpn.Lemma 7.
Letq∈W0∞Ωpn. Then,
(46)χBγnx·Lq·Ωpn≤Cpγn/qx,γ,where
(47)qx,γ=qx,γ<0,q∞,γ≥0.Proof.
We split the proof into three cases: (1)γ<0, (2) γ≥0 and pγ≤xp, and (3) γ≥0 and pγ>xp.Case 1.
γ<0. Since q∈W0Ωpn, we see that
(48)pγnqy/qx−1=pγnqy−qx/qx≥pγnq+−q−/q−≥p−C,
thus,p−γnqy/qx≤p−γnpC. Integrating the last inequality over Bγnx, we obtain
(49)∫Bγnx1pγnqy/qxdy≤∫Bγnxp−γnpCdy≤pC,from which it follows, using the definition of variable exponent norm and (49), that
(50)χBγnx·Lq·ℚpn≲pγn/qx.Case 2.
γ≥0 and pγ≤xp. In this case, we have yp≥xp−x−yp≥pγp−1 (because y∈Bγnx). Since q∈W∞ℚpn, then taking limit when xp⟶∞, we get
(51)q∞−qy≤Clogpp+yp,from which we obtain
(52)pγn1−qy/q∞=pγnq∞−qy/q∞≤pγnC/logpp+yp≤pC,consequently, p−γnqy/q∞≤p−γn+C. Integrating the last inequality over Bγnx, we have
(53)∫Bγnx1pγnqy/q∞dy≤∫Bγnxp−γn+Cdy≤pC,from which, by the definition of variable exponent norm, we get
(54)χBγnx·Lq·ℚpn≲pγn/q∞.Case 3.
γ≥0 and xp<pγ. By the ultrametricity and the condition on xp, we have Bγnx⊂Bγn0. Then,
(55)∫Bγnx1pγnqy/q∞dy≤∫Bγn01pγnqy/q∞dy≲∫Bγn01pγndy≲1pγn∫Bγn0dy≲1,from which, by the definition of variable exponent norm, we get
(56)χBγnx·Lq·ℚpn≲pγn/q∞,which completes the proof.Lemma 8.
Letq∈W0∞ℚpn. Then,
(57)χℚpn\Bγnx·x−·pβLq·ℚpn≤Cpγn/qx,γ−γβ,where qx,γ is defined in (47).Proof.
We first prove the theorem for the caseβ=n. When y∉Bγnx, by the ultrametricity condition, we have Bγnx⊂B−ordx−yny; thus,
(58)MχBγnx·pγny≥1p−ordx−yn∫B−ordx−ynxχBγnxzpγndz≥χℚpn\Bγnxyx−ypn.
From the pointwise estimate (58), the boundedness of the maximal operator M (see (23)), and (46), we see that
(59)χℚpn\Bγnx·∥x−·∥pnLq·ℚpn≤MχBγnx·pγnyLq·ℚpn≤C1pγnχBγnx·Lq·ℚpn≤Cpγn/qx,γ−γn,which proves the estimate for β=n. The case for general β follows from the case β=n and the identity fβLq·ℚpn=fnLβq·/nℚpnβ/n.Lemma 9.
Letf∈Lq·Ωpn with fLq·Ωpn≤1 and q∈W0Ωpn, where Ωpn is a bounded open subset of ℚpn. Then,
(60)Iαfx≤CMfxqx/q#x,x∈Ωpn,where q# is the Sobolev limiting exponent defined by (31) and the constant does not depend on f and x.Proof.
Assumingfx=0 for x∈ℚpn\Ωpn, we have
(61)∫Ωpnfyx−ypn−αdy≤∫Bγnxfyx−ypn−αdy+∫ℚpn\Bγnxfyx−ypn−αdy≲pγαMfx+∫ℚpn\Bγnxfyx−ypn−αdy,where the estimate in the second inequality follows taking ε=α in (36) (we are indeed allowed to take ε=α in that estimate).To estimate the integral over the exterior of the ball, using the extensionq˘ exponent (30), Hölder’s inequality, and the estimate (57), we have
(62)∫ℚpn\Bγnxfyx−ypn−αdy≲fLq·Ωpnχℚpn\Bγnx·x−·pn−αLq¯′·ℚpn≲pγn/q′x−γn+γα,where, in the last inequality, we use the fact thatq˘∞≥qx for x∈Ωpn. From the previous estimates, when fLq·Ωpn≤1, we have
(63)Iαfx≲pγαMfx+pγα−γn/qx.Choosing(64)γ=−logpMfxqx/n,and replacing it in (63), we obtain (60).Theorem 10.
Letq∈W0Ωpn with 1≤q−≤q+<α/n and Ωpn be a bounded open set in ℚpn. Then, the operator
(65)Iα:Lq·Ωpn⟶Lq#·Ωpn,is bounded, where q# is defined by (31).Proof.
Let us takefLq·Ωpn≤1. From (60), (18), and the boundedness of the maximal operator (24), we have
(66)∫ΩpnIαfxq#xdx≤∫ΩpnMfxqxdx≤1+MfLq·Ωpnq+≲1+fLq·Ωpnq+≲1.
The result now follows from homogeneity of the norm.


## 3.1. Boundedness inℚpn

In this section, we study the boundedness of the operators in the case ofℚpn.


### 3.1.1. Fractional Maximal Operator

The classical result regarding boundedness of the fractional maximal operator says that if1<q<n/α and 1/s=1/q−α/n, then Mα:Lqℚpn⟶Lsℚpn is bounded (this follows at once from inequality (3) and the boundedness of the operator Iα:Lqℚpn⟶Lsℚpn, cf. [25]). For further goals, we need estimate (32).Lemma 3.
Let0<α<n, q be an exponent function such that 1<q−≤q+<n/α, and we define the Sobolev limiting exponent q# by
(31)1q#x≔1qx−αn.
Then,(32)Mαfx≤cMf·q·/q#·n/n−αx1−α/n∫ℚpnfyqydyα/n.Proof.
Fromqx/q#x+αqx/n=1 and Hölder’s inequality (20), we have
(33)1pγn−γα∫Bγnxfydy=1pγn−γα∫Bγnxfyqy/q#yfyαqy/ndy≤c1pγn−γα∫Bγnxfynqy/q#yα−ndy1−α/n∫Bγnxfyqydyα/n.
Taking the supremum over allγ∈ℤ, we establish the desired inequality.Theorem 4.
Let0<α<n and q be an exponent function such that 1<q−≤q+<n/α. Suppose, moreover, that q#∈W0∞ℚpn, where q# is the Sobolev limiting exponent (31). Then, the operator Mα:Lq·ℚpn⟶Lq#·ℚpn is bounded.Proof.
TakingfLq·ℚpn=1, the estimate (32), the relation (19), and the boundedness of the maximal operator M in Lq#·ℚpn, we have
(34)MαfxLq#·ℚpn≲Mf·q·/q#·n/n−αx1−α/nLq#·ℚpn=Mf·q·/q#·n/n−αxLq#·1−α/nℚpn1−α/n≲α,n,pf·q·/q#·n/n−αLq#·1−α/nℚpn1−α/n≲α,n,pf·q·/q#·Lq#·ℚpn≲α,n,p1.
The general case follows from homogeneity considerations.


### 3.1.2. Fractional Potential Operator

The well-knownSobolev theorem states that the fractional potential operator (2), sometimes introduced with a normalizing factor, is bounded from Lqℚpn to Lq#ℚpn where 1/q#=1/q−α/n is the so-called Sobolev limiting exponent; see, for instance, [25].In order to obtain the boundedness result in the variable exponent framework, we first obtain the validity of a Welland-type estimate in thep-adic setting; see [31] for the Euclidean counterpart.Lemma 5.
Let0<α<n, 0<ε<maxα,n−α, and f∈Lloc1ℚpn. Then,
(35)Iαfx≲α,n,pMα−εfx1/2Mα+εfx1/2.Proof.
Letγ∈ℤ, then
(36)∫Bγnxfyx−ypn−αdy=∑k=−∞γ1pkn−α∫Sknxfydy≤∑k=−∞γpkε1pkn−α+ε∫Bknxfydy≤∑k=−∞γpkεMα−εfx≲α,n,ppεγ+11−p−εMα−εfx≲α,n,p,εpεγMα−εfx.
On the other hand,(37)∫BCγnxfyx−ypn−αdy=∑k=γ+1∞1pkn−α∫Sknxfydy≤∑k=γ+1∞p−kε1pkn−α−ε∫Bknxfydy≤∑k=γ+1∞p−kεMα+εfx≲α,n,ppε−γ+11−p−εMα+εfx≲α,n,ppε−γ+11−p−εMα+εfx≲α,n,pp−εγMα+εfx.
Taking the previous estimates into consideration, we have(38)Iαfx≲α,n,p,εpεγMα−εfx+p−εγMα+εfx.
The inequality (35) is obtained taking into account (38) with γ given by
(39)γ=logpMα+εfx/Mα−εfx1/2ε,since x≤x<x+1.Theorem 6.
Let1<q−≤q+≤n/α and q∈W0∞ℚpn. Then, the fractional potential operator,
(40)Iα:Lq·ℚpn⟶Lq#·ℚpn,is bounded, where q# is the Sobolev limiting exponent (31).Proof.
From the definition of the variable exponent Lebesgue norm and homogeneity, it suffices to show that(41)∫ℚpnIαfxq#xdx≤C,when fLq·ℚpn≤1, see Lemma 3.3 of [26] for more details.
From (35) and Hölder’s inequality, we obtain
(42)∫ℚpnIαfxq#xdx≲α,n,p∫ℚpnMα−εfxq#x/2Mα+εfxq#x/2dx≲α,n,pMα−εf·q#·/2q1·Mα+εf·q#·/2q2·≤cI1I2,where q1 and q2 are conjugate exponents and will be chosen appropriately. To estimate I1, from (17), we have
(43)I1≤ϱq1·Mα−εf·q#·/2+1=∫ℚpnMα−εfxq#xq1x/2dx+1.
Definingq1 by 1/qx−2/q#xq1x=α−ε/n, we have, by Lemma 3, that the operator Mα−ε:Lq·ℚpn⟶Lq~·ℚpn is bounded, where q~x=q#xq1x/2. From (43) and (18), it follows that
(44)Mα−εf·q#·/2q1·≤1+Mα−εfLq~·ℚpnq~++1≤c1+fLq·ℚpnq~++1≤c,which end the estimate for I1.
The estimate forI2 follows, mutatis mutandis, as I1 taking into account that defining q2 as 1/qx−2/q#xq2x=α+ε/n, we get that q1 and q2 are indeed conjugate exponents and q2 is the right exponent for the boundedness of Mα+ε; the details are left to the reader.


## 3.1.1. Fractional Maximal Operator

The classical result regarding boundedness of the fractional maximal operator says that if1<q<n/α and 1/s=1/q−α/n, then Mα:Lqℚpn⟶Lsℚpn is bounded (this follows at once from inequality (3) and the boundedness of the operator Iα:Lqℚpn⟶Lsℚpn, cf. [25]). For further goals, we need estimate (32).Lemma 3.
Let0<α<n, q be an exponent function such that 1<q−≤q+<n/α, and we define the Sobolev limiting exponent q# by
(31)1q#x≔1qx−αn.
Then,(32)Mαfx≤cMf·q·/q#·n/n−αx1−α/n∫ℚpnfyqydyα/n.Proof.
Fromqx/q#x+αqx/n=1 and Hölder’s inequality (20), we have
(33)1pγn−γα∫Bγnxfydy=1pγn−γα∫Bγnxfyqy/q#yfyαqy/ndy≤c1pγn−γα∫Bγnxfynqy/q#yα−ndy1−α/n∫Bγnxfyqydyα/n.
Taking the supremum over allγ∈ℤ, we establish the desired inequality.Theorem 4.
Let0<α<n and q be an exponent function such that 1<q−≤q+<n/α. Suppose, moreover, that q#∈W0∞ℚpn, where q# is the Sobolev limiting exponent (31). Then, the operator Mα:Lq·ℚpn⟶Lq#·ℚpn is bounded.Proof.
TakingfLq·ℚpn=1, the estimate (32), the relation (19), and the boundedness of the maximal operator M in Lq#·ℚpn, we have
(34)MαfxLq#·ℚpn≲Mf·q·/q#·n/n−αx1−α/nLq#·ℚpn=Mf·q·/q#·n/n−αxLq#·1−α/nℚpn1−α/n≲α,n,pf·q·/q#·n/n−αLq#·1−α/nℚpn1−α/n≲α,n,pf·q·/q#·Lq#·ℚpn≲α,n,p1.
The general case follows from homogeneity considerations.


## 3.1.2. Fractional Potential Operator

The well-knownSobolev theorem states that the fractional potential operator (2), sometimes introduced with a normalizing factor, is bounded from Lqℚpn to Lq#ℚpn where 1/q#=1/q−α/n is the so-called Sobolev limiting exponent; see, for instance, [25].In order to obtain the boundedness result in the variable exponent framework, we first obtain the validity of a Welland-type estimate in thep-adic setting; see [31] for the Euclidean counterpart.Lemma 5.
Let0<α<n, 0<ε<maxα,n−α, and f∈Lloc1ℚpn. Then,
(35)Iαfx≲α,n,pMα−εfx1/2Mα+εfx1/2.Proof.
Letγ∈ℤ, then
(36)∫Bγnxfyx−ypn−αdy=∑k=−∞γ1pkn−α∫Sknxfydy≤∑k=−∞γpkε1pkn−α+ε∫Bknxfydy≤∑k=−∞γpkεMα−εfx≲α,n,ppεγ+11−p−εMα−εfx≲α,n,p,εpεγMα−εfx.
On the other hand,(37)∫BCγnxfyx−ypn−αdy=∑k=γ+1∞1pkn−α∫Sknxfydy≤∑k=γ+1∞p−kε1pkn−α−ε∫Bknxfydy≤∑k=γ+1∞p−kεMα+εfx≲α,n,ppε−γ+11−p−εMα+εfx≲α,n,ppε−γ+11−p−εMα+εfx≲α,n,pp−εγMα+εfx.
Taking the previous estimates into consideration, we have(38)Iαfx≲α,n,p,εpεγMα−εfx+p−εγMα+εfx.
The inequality (35) is obtained taking into account (38) with γ given by
(39)γ=logpMα+εfx/Mα−εfx1/2ε,since x≤x<x+1.Theorem 6.
Let1<q−≤q+≤n/α and q∈W0∞ℚpn. Then, the fractional potential operator,
(40)Iα:Lq·ℚpn⟶Lq#·ℚpn,is bounded, where q# is the Sobolev limiting exponent (31).Proof.
From the definition of the variable exponent Lebesgue norm and homogeneity, it suffices to show that(41)∫ℚpnIαfxq#xdx≤C,when fLq·ℚpn≤1, see Lemma 3.3 of [26] for more details.
From (35) and Hölder’s inequality, we obtain
(42)∫ℚpnIαfxq#xdx≲α,n,p∫ℚpnMα−εfxq#x/2Mα+εfxq#x/2dx≲α,n,pMα−εf·q#·/2q1·Mα+εf·q#·/2q2·≤cI1I2,where q1 and q2 are conjugate exponents and will be chosen appropriately. To estimate I1, from (17), we have
(43)I1≤ϱq1·Mα−εf·q#·/2+1=∫ℚpnMα−εfxq#xq1x/2dx+1.
Definingq1 by 1/qx−2/q#xq1x=α−ε/n, we have, by Lemma 3, that the operator Mα−ε:Lq·ℚpn⟶Lq~·ℚpn is bounded, where q~x=q#xq1x/2. From (43) and (18), it follows that
(44)Mα−εf·q#·/2q1·≤1+Mα−εfLq~·ℚpnq~++1≤c1+fLq·ℚpnq~++1≤c,which end the estimate for I1.
The estimate forI2 follows, mutatis mutandis, as I1 taking into account that defining q2 as 1/qx−2/q#xq2x=α+ε/n, we get that q1 and q2 are indeed conjugate exponents and q2 is the right exponent for the boundedness of Mα+ε; the details are left to the reader.


## 3.2. Boundedness inΩpn

The fractional integral operatorIα can be defined for an open set Ωpn⊂ℚpn in the following way:
(45)Iαfx=∫Ωpnfyx−ypn−αdy,x∈Ωpn.We are interested in proving theLq·Ωpn⟶Lq#·Ωpn boundedness for the operator Iα, where q# is defined by (31). We begin with two lemmas, which are important on their own.Although we need Lemma7 for bounded set Ωpn, we give the lemma for general measurable sets Ωpn which include, as a particular case, Ωpn=ℚpn.Lemma 7.
Letq∈W0∞Ωpn. Then,
(46)χBγnx·Lq·Ωpn≤Cpγn/qx,γ,where
(47)qx,γ=qx,γ<0,q∞,γ≥0.Proof.
We split the proof into three cases: (1)γ<0, (2) γ≥0 and pγ≤xp, and (3) γ≥0 and pγ>xp.Case 1.
γ<0. Since q∈W0Ωpn, we see that
(48)pγnqy/qx−1=pγnqy−qx/qx≥pγnq+−q−/q−≥p−C,
thus,p−γnqy/qx≤p−γnpC. Integrating the last inequality over Bγnx, we obtain
(49)∫Bγnx1pγnqy/qxdy≤∫Bγnxp−γnpCdy≤pC,from which it follows, using the definition of variable exponent norm and (49), that
(50)χBγnx·Lq·ℚpn≲pγn/qx.Case 2.
γ≥0 and pγ≤xp. In this case, we have yp≥xp−x−yp≥pγp−1 (because y∈Bγnx). Since q∈W∞ℚpn, then taking limit when xp⟶∞, we get
(51)q∞−qy≤Clogpp+yp,from which we obtain
(52)pγn1−qy/q∞=pγnq∞−qy/q∞≤pγnC/logpp+yp≤pC,consequently, p−γnqy/q∞≤p−γn+C. Integrating the last inequality over Bγnx, we have
(53)∫Bγnx1pγnqy/q∞dy≤∫Bγnxp−γn+Cdy≤pC,from which, by the definition of variable exponent norm, we get
(54)χBγnx·Lq·ℚpn≲pγn/q∞.Case 3.
γ≥0 and xp<pγ. By the ultrametricity and the condition on xp, we have Bγnx⊂Bγn0. Then,
(55)∫Bγnx1pγnqy/q∞dy≤∫Bγn01pγnqy/q∞dy≲∫Bγn01pγndy≲1pγn∫Bγn0dy≲1,from which, by the definition of variable exponent norm, we get
(56)χBγnx·Lq·ℚpn≲pγn/q∞,which completes the proof.Lemma 8.
Letq∈W0∞ℚpn. Then,
(57)χℚpn\Bγnx·x−·pβLq·ℚpn≤Cpγn/qx,γ−γβ,where qx,γ is defined in (47).Proof.
We first prove the theorem for the caseβ=n. When y∉Bγnx, by the ultrametricity condition, we have Bγnx⊂B−ordx−yny; thus,
(58)MχBγnx·pγny≥1p−ordx−yn∫B−ordx−ynxχBγnxzpγndz≥χℚpn\Bγnxyx−ypn.
From the pointwise estimate (58), the boundedness of the maximal operator M (see (23)), and (46), we see that
(59)χℚpn\Bγnx·∥x−·∥pnLq·ℚpn≤MχBγnx·pγnyLq·ℚpn≤C1pγnχBγnx·Lq·ℚpn≤Cpγn/qx,γ−γn,which proves the estimate for β=n. The case for general β follows from the case β=n and the identity fβLq·ℚpn=fnLβq·/nℚpnβ/n.Lemma 9.
Letf∈Lq·Ωpn with fLq·Ωpn≤1 and q∈W0Ωpn, where Ωpn is a bounded open subset of ℚpn. Then,
(60)Iαfx≤CMfxqx/q#x,x∈Ωpn,where q# is the Sobolev limiting exponent defined by (31) and the constant does not depend on f and x.Proof.
Assumingfx=0 for x∈ℚpn\Ωpn, we have
(61)∫Ωpnfyx−ypn−αdy≤∫Bγnxfyx−ypn−αdy+∫ℚpn\Bγnxfyx−ypn−αdy≲pγαMfx+∫ℚpn\Bγnxfyx−ypn−αdy,where the estimate in the second inequality follows taking ε=α in (36) (we are indeed allowed to take ε=α in that estimate).To estimate the integral over the exterior of the ball, using the extensionq˘ exponent (30), Hölder’s inequality, and the estimate (57), we have
(62)∫ℚpn\Bγnxfyx−ypn−αdy≲fLq·Ωpnχℚpn\Bγnx·x−·pn−αLq¯′·ℚpn≲pγn/q′x−γn+γα,where, in the last inequality, we use the fact thatq˘∞≥qx for x∈Ωpn. From the previous estimates, when fLq·Ωpn≤1, we have
(63)Iαfx≲pγαMfx+pγα−γn/qx.Choosing(64)γ=−logpMfxqx/n,and replacing it in (63), we obtain (60).Theorem 10.
Letq∈W0Ωpn with 1≤q−≤q+<α/n and Ωpn be a bounded open set in ℚpn. Then, the operator
(65)Iα:Lq·Ωpn⟶Lq#·Ωpn,is bounded, where q# is defined by (31).Proof.
Let us takefLq·Ωpn≤1. From (60), (18), and the boundedness of the maximal operator (24), we have
(66)∫ΩpnIαfxq#xdx≤∫ΩpnMfxqxdx≤1+MfLq·Ωpnq+≲1+fLq·Ωpnq+≲1.
The result now follows from homogeneity of the norm.


## 4. Applications top-adic Derivative

When we work with functions of ap-adic variable, f:ℚpn⟶ℂ, it is not possible to define the derivative in the classical sense (as a limit); therefore, we must resort to the pseudodifferential operators to supply such need. The most popular operator in the p-adic numbers that plays the role of the derivative is the Taibleson operator (not local operator).First, we need the Fourier transform. We setχpy=exp2πiyp for y∈ℚp. The map χp· is an additive character on ℚp, i.e., a continuous map from ℚp into the unit circle satisfying χpy0+y1=χpy0χpy1, y0,y1∈ℚp.Givenξ=ξ1,⋯,ξn and x=x1,⋯,xn∈ℚpn, we set ξ·x≔∑j=1nξjxj. The Fourier transform of φ∈Sℚpn is defined as
(67)Fφξ=∫ℚpnχp−ξ·xφxdxforξ∈ℚpn,where dx=dnx is the Haar measure on ℚpn normalized by the condition VolB0n=1.The Fourier transform is a linear isomorphism fromSℚpn onto itself satisfying FFφξ=φ−ξ. We will also use the notation Fx⟶ξφ and φ^ for the Fourier transform of φ.Definition 11.
The Taibleson pseudodifferential is given by operator(68)DTαφx=F−1ξpαFφ,φ∈Sℚpn,α>0.This operator can be expressed in the following way:(69)DTαφx=R−α∗φx=1−pα1−p−α−n∫ℚpnφx−y−φxypn+αdy,whereRα is the p-adic Riesz kernel; see, e.g., [25]. The right-hand side of the last equation makes sense for a wider class of functions, for example, for locally constant functions φx satisfying
(70)∫xp≥1φxxpn+αdx<∞.Remark 12.
We note that the constant functionsφ∈DomDTα; furthermore, DTαφ=0. Unfortunately, the operator DTα does not satisfy the Leibniz rule for derivative of a product.The operatorDTα given by Definition 11 can be extended to the maximal domain φ∈L2ℚpn:ξpαFφξ∈L2ℚpn; thus, this operator is not bounded, and also, its spectrum is pjj∈ℤ∪0.Theorem 13.
Let−n<α<0, 1<q−≤q+≤n/α, and q∈W0∞ℚpn. Then, the Taibleson operator
(71)DTα:Lq·ℚpn⟶Lq#·ℚpn,is well-defined and bounded, where 1/q#=1/qx+α/n. If q∈W0Ωpn, then DTα:Lq·Ωpn⟶Lq#·Ωpn is bounded.Proof.
Note that, under the hypothesis of the theorem, we haveDTαfx=I−αfx. The result now follows from Theorem 6 and Theorem 10.In what follows, we are interested to study the following semilinear Cauchy problem (α<0):
(72)∂ux,t∂t+DTαux,t=fux,t,ux,0=u0x,wheref:ℝ⟶ℝ is a Lipschitz function.Lemma 14.
Let−n<α<0, 1<q−≤q+<n/α, q∈W0Ωpn, where Ωpn<∞. Then, the operator Tux,t≔−DTαux,t+fux,t is Lipschitz, i.e.,
(73)Tux,t−Tvx,tLq·Ωpn≤Cux,t−vx,tLq·Ωpn.Proof.
From the embeddingLq#·Ωpn↪Lq·Ωpn and Theorem 10, we see that
(74)Tux,t−Tvx,tLq·Ωpn≤Tux,t−Tvx,tLq#·Ωpn+fux,t−fvx,tLq·Ωpn≤DTαu−vx,tq#·+C1ux,t−vx,tLq·Ωpn≤Cux,t−vx,tLq·Ωpn,which ends the proof.Theorem 15.
Let−n<α<0, 1<q−≤q+<n/∣α∣, q∈W0Ωpn, where Ωpn<∞. Then, the Cauchy problem (72) has a unique solution in Lq·ℚpn.Proof.
This follows immediately from the Cauchy-Lipschitz-Picard theorem.



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