Fractional Operators in <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 10.3455 13.4804" width="10.3455pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg>-</span>adic Variable Exponent Lebesgue Spaces and Application to <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M2" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 10.3455 13.4804" width="10.3455pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><use xlink:href="#g113-113"/></g></svg>-</span>adic Derivative

Chacón-Cortés, Leonardo Fabio; Rafeiro, Humberto

doi:https://doi.org/10.1155/2021/3096701

Journal of Function Spaces

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Function Spaces, Hardy-Type Inequalities, and their Applications

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Research Article | Open Access

Volume 2021 | Article ID 3096701 | https://doi.org/10.1155/2021/3096701

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

Leonardo Fabio Chacón-Cortés¹and Humberto Rafeiro²

Academic Editor: Douadi Drihem

Received03 Jun 2021

Accepted07 Aug 2021

Published18 Sept 2021

Abstract

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

1. Introduction

The field of -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 -adic setting it is not possible to define the derivative in the classical sense.

Variable exponent Lebesgue spaces generalize the notion of -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 -adic Lebesgue spaces with a variable exponent. The corresponding result for classical -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 -adic analysis and variable exponent Lebesgue spaces on the -adic numbers, necessary for the development of this work. In Section 3, the boundedness of the fractional maximal operatoris studied in the framework of variable exponent -adic Lebesgue spaces. The boundedness of the special case , 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 to , where is the Sobolev limiting exponent; see (31) for the corresponding definition. The boundedness of the fractional integral operatoris obtained from the boundedness of the fractional maximal operator and Welland’s pointwise inequality tailored for the -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 -adic Lebesgue spaces with variable exponent, which is the analogue of the derivative in the spatial variable (), and study the nonhomogeneous Cauchy problem (72) associated with this operator.

The notation denotes the existence of a constant for which , means that and .

2. Preliminaries

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

2.1. The Field of -adic Numbers

By we denote a prime number. The field is given as the completion of with respect to the -adic norm , given by where are integers coprime with . The integer is denoted as the -adic order of . This norm can be extended to as and satisfies the so-called strong triangular inequality with equality when . If , it follows that . The set is a complete ultrametric space and, as a topological space, is homeomorphic to a Cantor-like subset of the real line. A -adic number has a unique series expansion, viz., where and . For , we denote by the ball of radius with center at and by the corresponding sphere. We denote and note that

Note that , where is the one-dimensional ball. In there exists the additive Haar measure (by we denote the Haar measure of the set ), since the field is a locally compact commutative group with respect to addition. Normalizing the measure by we get a unique measure. From here onwards, we use the normalized Haar measure; thus, for any .

Note that the collection of all disjoint balls of the same radius forms a partition of , since inequality (6) implies that any two balls in with the same radius are either identical or disjoint.

2.2. Some Function Spaces

A complex-valued function defined on is called locally constant if for any , there exists an integer such that

A function is called a Schwartz-Bruhat function (or a test function) if it is locally constant with the compact support. The -vector space of Schwartz-Bruhat functions is denoted by .

A measurable function belongs to the Lebesgue space , , when where if the limit exists.

We now introduce the notion of -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 . By we denote the set of all variable exponent satisfying , where and

For by , we denote the space of measurable functions such that 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., where and are conjugate exponents, viz., .

For , we say that when there is a positive constant , for which

for all and any . We say that when there is a positive constant , for which

for any .

The class is defined as . The importance of the class stems from the fact that it is a sufficient condition for the boundedness of the maximal operator in : if , then see Theorem 5.2 of [26] for the corresponding proof.

In the case where is a bounded subset of , we have the following: if , then, see Theorem 5.1 of [26].

Lemma 1. Let be a -Lipschitz function, for some Then, .

Proof. We give the proof only for the case since the other case is immediate. Since is a continuous function, for any ball , there exists a maximum (respectively, minimum) point (respectively, ). From the Lipschitzianity of , 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 , where is a bounded subset of . Then, there exists an extension function 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 . Let us define as with where comes from equation (21). Since is an increasing and concave function for and approaches zero with , then from Theorem 2 of [29], we have that . In order to prove that , it suffices to check for . Since is an increasing function and taking and as in the proof of Lemma 1, we see that which ends the first step.
Second step: we show that there exists an extension function . Since is a bounded set, let us take such that . We define two Urysohn functions, and , as follows: The functions and are -Lipschitz with , due to the fact that , see Prop. 2.1.1 of [30]. Defining the exponent as, we see that since the class is closed under addition and multiplication, and because in the exterior of the ball .

3. Main Results

3.1. Boundedness in

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

3.1.1. Fractional Maximal Operator

The classical result regarding boundedness of the fractional maximal operator says that if and , then is bounded (this follows at once from inequality (3) and the boundedness of the operator , cf. [25]). For further goals, we need estimate (32).

Lemma 3. Let , be an exponent function such that , and we define the Sobolev limiting exponent by Then,

Proof. From and Hölder’s inequality (20), we have Taking the supremum over all , we establish the desired inequality.

Theorem 4. Let and be an exponent function such that . Suppose, moreover, that , where is the Sobolev limiting exponent (31). Then, the operator is bounded.

Proof. Taking , the estimate (32), the relation (19), and the boundedness of the maximal operator in , we have The general case follows from homogeneity considerations.

3.1.2. Fractional Potential Operator

The well-known Sobolev theorem states that the fractional potential operator (2), sometimes introduced with a normalizing factor, is bounded from to where 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 the -adic setting; see [31] for the Euclidean counterpart.

Lemma 5. Let , , and . Then,

Proof. Let , then On the other hand, Taking the previous estimates into consideration, we have The inequality (35) is obtained taking into account (38) with given by since

Theorem 6. Let and . Then, the fractional potential operator, is bounded, where is the Sobolev limiting exponent (31).

Proof. From the definition of the variable exponent Lebesgue norm and homogeneity, it suffices to show that when , see Lemma 3.3 of [26] for more details.
From (35) and Hölder’s inequality, we obtain where and are conjugate exponents and will be chosen appropriately. To estimate , from (17), we have Defining by , we have, by Lemma 3, that the operator is bounded, where . From (43) and (18), it follows that which end the estimate for .
The estimate for follows, mutatis mutandis, as taking into account that defining as , we get that and are indeed conjugate exponents and is the right exponent for the boundedness of ; the details are left to the reader.

3.2. Boundedness in

The fractional integral operator can be defined for an open set in the following way:

We are interested in proving the boundedness for the operator , where is defined by (31). We begin with two lemmas, which are important on their own.

Although we need Lemma 7 for bounded set , we give the lemma for general measurable sets which include, as a particular case, .

Lemma 7. Let . Then, where

Proof. We split the proof into three cases: (1) , (2) and , and (3) and .

Case 1. . Since , we see that thus, . Integrating the last inequality over , we obtain from which it follows, using the definition of variable exponent norm and (49), that

Case 2. and . In this case, we have (because ). Since , then taking limit when , we get from which we obtain consequently, . Integrating the last inequality over , we have from which, by the definition of variable exponent norm, we get

Case 3. and . By the ultrametricity and the condition on , we have . Then, from which, by the definition of variable exponent norm, we get which completes the proof.

Lemma 8. Let . Then, where is defined in (47).

Proof. We first prove the theorem for the case . When , by the ultrametricity condition, we have ; thus, From the pointwise estimate (58), the boundedness of the maximal operator (see (23)), and (46), we see that which proves the estimate for . The case for general follows from the case and the identity .

Lemma 9. Let with and , where is a bounded open subset of . Then, where is the Sobolev limiting exponent defined by (31) and the constant does not depend on and .

Proof. Assuming for , we have where the estimate in the second inequality follows taking in (36) (we are indeed allowed to take in that estimate).

To estimate the integral over the exterior of the ball, using the extension exponent (30), Hölder’s inequality, and the estimate (57), we have

where, in the last inequality, we use the fact that for . From the previous estimates, when , we have

Choosing and replacing it in (63), we obtain (60).

Theorem 10. Let with and be a bounded open set in . Then, the operator is bounded, where is defined by (31).

Proof. Let us take . From (60), (18), and the boundedness of the maximal operator (24), we have The result now follows from homogeneity of the norm.

4. Applications to -adic Derivative

When we work with functions of a -adic variable, , 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 -adic numbers that plays the role of the derivative is the Taibleson operator (not local operator).

First, we need the Fourier transform. We set for . The map is an additive character on , i.e., a continuous map from into the unit circle satisfying , .

Given and , we set . The Fourier transform of is defined as where is the Haar measure on normalized by the condition .

The Fourier transform is a linear isomorphism from onto itself satisfying . We will also use the notation and for the Fourier transform of .

Definition 11. The Taibleson pseudodifferential is given by operator

This operator can be expressed in the following way:

where is the -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 satisfying

Remark 12. We note that the constant functions ; furthermore, . Unfortunately, the operator does not satisfy the Leibniz rule for derivative of a product.

The operator given by Definition 11 can be extended to the maximal domain ; thus, this operator is not bounded, and also, its spectrum is .

Theorem 13. Let , , and . Then, the Taibleson operator is well-defined and bounded, where . If , then is bounded.

Proof. Note that, under the hypothesis of the theorem, we have . The result now follows from Theorem 6 and Theorem 10.

In what follows, we are interested to study the following semilinear Cauchy problem ():

where is a Lipschitz function.

Lemma 14. Let , , , where . Then, the operator is Lipschitz, i.e.,

Proof. From the embedding and Theorem 10, we see that which ends the proof.

Theorem 15. Let , , , where . Then, the Cauchy problem (72) has a unique solution in .

Proof. This follows immediately from the Cauchy-Lipschitz-Picard theorem.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to this work. And all authors typed, read, and approved the final manuscript.

Acknowledgments

L.F. Chacón-Cortés was partially supported by the research project with ID-PPTA: 00009123, of the Faculty of Sciences of the Pontificia Universidad Javeriana, Bogotá, Colombia. H. Rafeiro was supported by a Research Start-Up Grant of the United Arab Emirates University via Grant No. G00002994.

References

V. A. Avetisov, A. K. Bikulov, and V. A. Osipov, “p-adic models of ultrametric diffusion in the conformational dynamics of macromolecules,” Proceedings of the Steklov Institute of Mathematics, vol. 245, no. 2, pp. 48–57, 2004.
View at: Google Scholar
V. A. Avetisov, A. K. Bikulov, and V. A. Osipov, “p-adic description of characteristic relaxation in complex systems,” Journal of Physics A: Mathematical and General, vol. 35, no. 15, pp. 4239–4246, 2003.
View at: Google Scholar
B. Dragovich, A. Y. Khrennikov, S. V. Kozyrev, and I. V. Volovich, “On p-adic mathematical physics,” p-Adic Numbers, Ultrametric Analysis, and Applications, vol. 1, no. 1, pp. 1–17, 2009.
View at: Publisher Site | Google Scholar
R. Rammal, G. Toulouse, and M. A. Virasoro, “Ultrametricity for physicists,” Review of Modern Physics, vol. 58, no. 3, pp. 765–788, 1986.
View at: Google Scholar
W. Orlicz, “Über konjugierte exponentenfolgen,” Studia Mathematica, vol. 3, no. 1, pp. 200–211, 1931.
View at: Publisher Site | Google Scholar
O. Kováčik and Rákosník, “On spaces L^q(x) and W^k,q(x),” Czechoslovak Mathematical Journal, vol. 41, no. 4, pp. 592–618, 1991.
View at: Google Scholar
S. N. Antontsev and J. F. Rodrigues, “On stationary thermo-rheological viscous flows,” Annali Dell'universita' Di Ferrara, vol. 52, no. 1, pp. 19–36, 2006.
View at: Publisher Site | Google Scholar
E. Acerbi and G. Mingione, “Resultats de regularite pour les fluides electrorheologiques : le cas stationnaire,” Comptes Rendus Mathematique, vol. 334, no. 9, pp. 817–822, 2002.
View at: Publisher Site | Google Scholar
E. Acerbi and G. Mingione, “Regularity results for stationary electro-rheological fluids,” Archive for Rational Mechanics and Analysis, vol. 164, no. 3, pp. 213–259, 2002.
View at: Publisher Site | Google Scholar
M. Ružička, “Electrorheological fluids, modeling and mathematical theory,” in Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000.
View at: Google Scholar
M. Ružička, “Modeling, mathematical and numerical analysis of electrorheological fluids,” Applications of Mathematics, vol. 49, no. 6, pp. 565–609, 2004.
View at: Google Scholar
P. Harjulehto, P. Hästö, Ú. V. Lê, and M. Nuortio, “Overview of differential equations with non-standard growth,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 72, no. 12, pp. 4551–4574, 2010.
View at: Publisher Site | Google Scholar
G. Mingione, “Regularity of minima: an invitation to the dark side of the calculus of variations,” Applications of Mathematics, vol. 51, no. 4, pp. 355–426, 2006.
View at: Publisher Site | Google Scholar
R. Aboulaich, S. Boujena, and E. El Guarmah, “Sur un modèle non-linéaire pour le débruitage de l'image,” in Comptes Rendus Mathématique, vol. 345, no. 8, pp. 425–429, Académie des Sciences, Paris, 2007.
View at: Google Scholar
R. Aboulaich, D. Meskine, and A. Souissi, “New diffusion models in image processing,” Computers & Mathematics with Applications, vol. 56, no. 4, pp. 874–882, 2008.
View at: Publisher Site | Google Scholar
P. Blomgren, T. Chan, P. Mulet, and C. K. Wong, “Total variation image restoration, numerical methods and extensions,” in Proceedings of the 1997 IEEE International Conference on Image Processing, vol. 3, pp. 384–387, Santa Barbara, CA, USA, 1997.
View at: Google Scholar
E. M. Bollt, R. Chartrand, S. Esedoḡlu, P. Schultz, and K. R. Vixie, “Graduated adaptive image denoising: local compromise between total variation and isotropic diffusion,” Advances in Computational Mathematics, vol. 31, no. 1-3, pp. 61–85, 2009.
View at: Publisher Site | Google Scholar
Y. Chen, W. Guo, Q. Zeng, and Y. Liu, “A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images,” Inverse Problems and Imaging, vol. 2, no. 2, pp. 205–224, 2008.
View at: Publisher Site | Google Scholar
Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406, 2006.
View at: Publisher Site | Google Scholar
T. Wunderli, “On time flows of minimizers of general convex functionals of linear growth with variable exponent in BV space and stability of pseudo-solutions,” Journal of Mathematical Analysis and Applications, vol. 364, no. 2, pp. 591–598, 2010.
View at: Google Scholar
N. M. Chuong and H. D. Dung, “Maximal functions and weighted norm inequalities on local fields,” Applied and Computational Harmonic Analysis, vol. 29, no. 3, pp. 272–286, 2010.
View at: Publisher Site | Google Scholar
D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Birkhäuser, 2013.
L. Diening, P. Harjulehto, P. Hästö, and M. Ružička, “Lebesgue and Sobolev spaces with variable exponents,” in Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2011.
View at: Google Scholar
H. Rafeiro and E. Rojas, Espacios de Lebesgue con exponente variable: un espacio de Banach de funciones medibles (Spanish), IVIC–Instituto Venezolano de Investigaciones Científicas, 2014.
M. H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, 1975.
L. F. Chacón-Cortés and H. Rafeiro, “Variable exponent Lebesgue spaces and Hardy-Littlewood maximal function on p-adic numbers,” p-Adic Numbers, Ultrametric Analysis, and Applications, vol. 12, no. 2, pp. 90–111, 2020.
View at: Publisher Site | Google Scholar
C. Capone, D. Cruz-Uribe, and A. Fiorenza, “The fractional maximal operator and fractional integrals on variable L^p spaces,” Revista Matemática Iberoamericana, vol. 23, no. 3, pp. 743–770, 2007.
View at: Publisher Site | Google Scholar
V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-adic Analysis and Mathematical Physics, World Scientific, 1994.
E. J. McShane, “Extension of range of functions,” Bulletin American Mathematical Society, vol. 40, no. 12, pp. 837–843, 1934.
View at: Publisher Site | Google Scholar
S. Cobzas, R. Miculescu, and A. Nicolae, Lipschitz Functions, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2011.
G. V. Welland, “Weighted norm inequalities for fractional integrals,” Proceedings of the American Mathematical Society, vol. 51, pp. 143–148, 1975.
View at: Google Scholar

Copyright

Copyright © 2021 Leonardo Fabio Chacón-Cortés and Humberto Rafeiro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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