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

The field of p-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. Evenmore sowhen in the p-adic setting it is not possible to define the derivative in the classical sense. Variable exponent Lebesgue spaces generalize the notion of q-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 the p-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. Section 2 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


Introduction
The field of p-adic numbers are an interesting and useful tool to study phenomena in physics, biology, and medicine, among other sciences; see, e.g., [1][2][3][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.
In this article, we are interested in the boundedness of the fractional integral and maximal fractional operator on the p-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. Section 2 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 is studied in the framework of variable exponent p-adic Lebesgue spaces. The boundedness of the special case M 0 , 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 L qð·Þ ðℚ n p Þ to L q # ð·Þ ðℚ n p Þ, where q # is the Sobolev limiting exponent; see (31) for the corresponding definition. The boundedness of the fractional integral operator 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 As already mentioned, we will use a reverse approach. Finally, in Section 4, 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 ∈ ℚ n p ), and study the nonhomogeneous Cauchy problem (72) associated with this operator.
The notation a ≲ b denotes the existence of a constant C for which a ≤ Cb, a≍b means that a ≲ b and b ≲ a.

Preliminaries
For an exposition on the p-adic analysis, see [25,28].
2.1. The Field of p-adic Numbers. By p we denote a prime number. The field ℚ p is given as the completion of ℚ with respect to the p-adic norm j·j p , given by where a, b are integers coprime with p. The integer γ ≔ ord ðxÞðordð0Þ ≔ +∞Þ is denoted as the p-adic order of x. This norm can be extended to ℚ n p as and satisfies the so-called strong triangular inequality with equality when kxk p ≠ kyk p . If ordðxÞ ≔ min 1≤i≤n ford ðx i Þg, it follows that kxk p = p −ordðxÞ . The set ðℚ n p , k·k p Þ is a complete ultrametric space and, as a topological space, ℚ p is homeomorphic to a Cantor-like subset of the real line. A p-adic number x ≠ 0 has a unique series expansion, viz., where x j ∈ f0, 1, 2, ⋯, p − 1g and x 0 ≠ 0. For γ ∈ ℤ, we denote by the ball of radius p γ with center at a = ða 1 , ⋯, a n Þ ∈ ℚ n p and by the corresponding sphere. We denote and note that for any a ∈ ℚ n p . Note that the collection of all disjoint balls of the same radius γ forms a partition of ℚ n p , since inequality (6) implies that any two balls in ℚ n p with the same radius are either identical or disjoint.

Some Function Spaces.
A complex-valued function φ defined on ℚ n p is called locally constant if for any x ∈ ℚ n p , there exists an integer lðxÞ ∈ ℤ such that where ð if the limit exists.

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We now introduce the notion of p-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 if q : ℚ n p ⟶ ½1,∞Þ. By Qðℚ n p Þ we denote the set of all variable exponent satisfying q + < ∞, where q + ≔ ess sup x∈ℚ n p qðxÞ and q − ≔ ess inf x∈ℚ n p qðxÞ: For q ∈ Qðℚ n p Þ by L qð·Þ ðℚ n p Þ, we denote the space of measurable functions f : where For the Lebesgue space with a variable exponent, we have The Hölder inequality is valid, up to a multiplicative constant, in the framework of Lebesgue spaces with variable exponent, viz., ð where q and q ′ are conjugate exponents, viz., 1 = 1/qðxÞ + 1/q′ðxÞ. For q ∈ Qðℚ n p Þ, we say that q ∈ W 0 ðℚ n p Þ when there is a positive constant C, for which for all γ ∈ ℤ and any x ∈ ℚ n p . We say that q ∈ W ∞ ðℚ n p Þ when there is a positive constant C, for which for any x, y ∈ ℚ n p .
The importance of the class W ∞ 0 ðℚ n p Þ stems from the fact that it is a sufficient condition for the boundedness of the maximal operator in L qð·Þ ðℚ n p Þ: if q ∈ W ∞ 0 ðℚ n p Þ, then see Theorem 5.2 of [26] for the corresponding proof.
In the case where Ω n p is a bounded subset of ℚ n p , we have the following: if q ∈ W 0 ðℚ n p Þ, then, see Theorem 5.1 of [26].
Proof. We give the proof only for the case γ ≤ 0 since the other case is immediate. Since q is a continuous function, for any ball B n γ ðxÞ ⊂ ℚ n p , there exists a maximum (respectively, minimum) point x ∈ B n γ ðxÞ (respectively, x ∈ B n γ ðxÞ). From the Lipschitzianity of q, we have 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.
Let q ∈ W 0 ðΩ n p Þ, where Ω n p is a bounded subset of ℚ n p . Then, there exists an extension functionq ∈ W ∞ 0 ðℚ n p Þ 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 functionq ∈ W 0 ðℚ n p Þ. Let us defineq as with where C comes from equation (21). Since ωðtÞ is an increasing and concave function for t ≥ 0 and approaches zero with t, then from Theorem 2 of [29], we have that jqðxÞ −qðyÞj ≤ ωðkx − yk p Þ. In order to prove thatq ∈ W 0 ðℚ n p Þ, 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 3

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which ends the first step. Second step: we show that there exists an extension functionq ∈ W ∞ 0 ðℚ n p Þ. Since Ω n p is a bounded set, let us take γ > 0 such that Ω n p ⊂ B n γ ð0Þ. We define two Urysohn functions, u 0 and u ∞ , as follows: The functions u 0 and u ∞ are L-Lipschitz with L ≤ 1/p, due to the fact that dðΩ n p , ℚ n p \ B n γ+1 ð0ÞÞ ≥ p, see Prop. 2.1.1 of [30]. Defining the exponentq as, we see thatq ∈ W 0 ðℚ n p Þ since the class W 0 ðℚ n p Þ is closed under addition and multiplication, andq ∈ W ∞ ðℚ n p Þ becauseqðxÞ ≡ q + in the exterior of the ball B n γ+1 ð0Þ.

Boundedness in ℚ n p .
In this section, we study the boundedness of the operators in the case of ℚ n p .

Fractional Maximal
Operator. The classical result regarding boundedness of the fractional maximal operator says that if 1 < q < n/α and 1/s = 1/q − α/n, then M α : L q ðℚ n p Þ ⟶ L s ðℚ n p Þ is bounded (this follows at once from inequality (3) and the boundedness of the operator I α : L q ðℚ n p Þ ⟶ L s ðℚ n p Þ, cf. [25]). For further goals, we need estimate (32).
Proof. Taking k f k L qð·Þ ðℚ n p Þ = 1, the estimate (32), the relation (19), and the boundedness of the maximal operator M in L q # ð·Þ ðℚ n p Þ, we have The general case follows from homogeneity considerations.

Fractional Potential
Operator. The well-known Sobolev theorem states that the fractional potential operator (2), sometimes introduced with a normalizing factor, is bounded from L q ðℚ n p Þ to L q # ðℚ n p Þ where 1/q # = 1/q − α/n is the socalled 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 the p-adic setting; see [31] for the Euclidean counterpart.

Lemma 5.
Let 0 < α < n, 0 < ε < max ðα, n − αÞ, and f ∈ L 1 loc ðℚ n p Þ. Then, Journal of Function Spaces On the other hand, Taking the previous estimates into consideration, we have The inequality (35) is obtained taking into account (38) with γ given by since x ≤ dxe < x + 1: Then, the fractional potential operator, 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 when k f k L qð·Þ ðℚ n p Þ ≤ 1, see Lemma 3.3 of [26] for more details.
The estimate for I 2 follows, mutatis mutandis, as I 1 taking into account that defining q 2 as 1/qðxÞ − 2/½q # ðxÞq 2 ðxÞ = ðα + εÞ/n, we get that q 1 and q 2 are indeed conjugate exponents and q 2 is the right exponent for the boundedness of M α+ε ; the details are left to the reader.

Boundedness in Ω n
p . The fractional integral operator I α can be defined for an open set Ω n p ⊂ ℚ n p in the following way: We are interested in proving the L qð·Þ ðΩ n p Þ ⟶ L q # ð·Þ ðΩ n p Þ boundedness for the operator I α , where q # is defined by (31). We begin with two lemmas, which are important on their own.
Although we need Lemma 7 for bounded set Ω n p , we give the lemma for general measurable sets Ω n p which include, as a particular case, Ω n p = ℚ n p .