Research Article | Open Access

Emin V. Guliyev, Yagub Y. Mammadov, "Some Embeddings into the Morrey and Modified Morrey Spaces Associated with the Dunkl Operator", *Abstract and Applied Analysis*, vol. 2010, Article ID 291345, 10 pages, 2010. https://doi.org/10.1155/2010/291345

# Some Embeddings into the Morrey and Modified Morrey Spaces Associated with the Dunkl Operator

**Academic Editor:**Ağacik Zafer

#### Abstract

We consider the generalized shift operator, associated with the Dunkl operator , . We study some embeddings into the Morrey space (-Morrey space) , and modified Morrey space (modified -Morrey space) associated with the Dunkl operator on . As applications we get boundedness of the fractional maximal operator , , associated with the Dunkl operator (fractional -maximal operator) from the spaces to for and from the spaces to for .

#### 1. Introduction

On the real line, the Dunkl operators are differential-difference operators introduced in 1989 by Dunkl [1] and are denoted by , where is a real parameter . These operators are associated with the reflection group on . Rösler in [2] shows that the Dunkl kernel verifies a product formula. This allows us to define the Dunkl translations , .

In the theory of partial differential equations, together with weighted spaces, Morrey spaces play an important role. Morrey spaces were introduced by Morrey in 1938 in connection with certain problems in elliptic partial differential equations and calculus of variations (see [3]). Later, Morrey spaces found important applications to Navier-Stokes [4, 5] and Schrödinger [6–8] equations, elliptic problems with discontinuous coefficients [9, 10], and potential theory [11–13]. An exposition of the Morrey spaces can be found in the book [14].

In the present work, we study some embeddings into the -Morrey and modified -Morrey spaces. As applications we give boundedness of the fractional -maximal operator in the -Morrey and modified -Morrey spaces.

The paper is organized as follows. In Section 2, we present some definitions and auxiliary results. In Section 3, we give some embeddings into the -Morrey and modified -Morrey spaces. In Section 4, we prove the boundedness of the fractional -maximal operator from the spaces to for and from the spaces to for .

#### 2. Preliminaries

On the real line, we consider the first-order differential-difference operator defined by

which is called the Dunkl operator. For , the Dunkl kernel on was introduced by Dunkl in [1] (see also [15–17]) and is given by

where is the normalized Bessel function of the first kind and order [18], defined by

The Dunkl kernel is the unique analytic solution on of the initial problem for the Dunkl operator (see [1]).

Let be the weighted Lebesgue measure on given by

For every , we denote by the spaces of complex-valued functions , measurable on such that

For we denote by the weak spaces defined as the set of locally integrable functions , with the finite norm

Note that

For all , we put

where

and is the Bessel kernel given by

where and .

In the sequel we consider the signed measure , on , given by

For and being a continuous function on , the Dunkl translation operator is given by

Using the change of variable , we have also (see [2])

where and .

Proposition 2.1 (see Soltani [16]). *For all the operator extends to , and we have for ,
*

Let , and , where . For (the space of locally integrable functions on ), we consider

Theorem 2.2 (see [19]). *
() If , then for every **
where is independent of .**
(2) If , , then and **
where is independent of .*

Corollary 2.3. *If , then
**
for a.e. .*

#### 3. Some Embeddings into the -Morrey and Modified -Morrey Spaces

*Definition 3.1 (see [20]). *Let , , and , . We denote by Morrey space ( -Morrey space) and by the modified Morrey space ( modified -Morrey space), associated with the Dunkl operator as the set of locally integrable functions , with the finite norms
respectively.

If or , then , where is the set of all functions equivalent to on .

Note that

*Definition 3.2 (see [19]). *Let , . We denote by the weak -Morrey space and by the modified weak -Morrey space as the set of locally integrable functions , with finite norms
respectively.

We note that

Lemma 3.3 (see [20]). *Let . Then
*

Lemma 3.4. *Let , . Then
*

*Proof. *Let . Then by (3.3) we have

Let . Then

Therefore, and the embedding is valid.

Thus .

From Lemmas 3.3 and 3.4 for , we have

Lemma 3.5. *Let . Then
*

*Proof. *The embedding is a consequence of Hölder's inequality and Proposition 2.1. Indeed,

On the -Morrey spaces, the following embedding is valid.

Lemma 3.6 (see [20]). *Let and . Then for **
where .*

On the modified -Morrey spaces, the following embedding is valid.

Lemma 3.7. *Let and . Then for *

*Proof. *Let , , and . By the Hölder's inequality, we have

Note that

Therefore, and

#### 4. Some Applications

In this section, using the results of Section 3, we get the boundedness of the fractional -maximal operator in the -Morrey and modified -Morrey spaces.

For we define the fractional maximal functions

In the case , we denote by . Note that .

Lemma 4.1. *Let , and . Then and the following equality
**
is valid.*

*Proof. *

Taking in Lemma 4.1 and using Lemma 3.3, we get for the following result.

Corollary 4.2. *Let . Then
*

Lemma 4.3. *Let , and . Then and the following equality
**
is valid.*

Corollary 4.4. *Let and . Then the operator is bounded from to for . Moreover,
*

Corollary 4.5. * . Then the operator is bounded from to for . Moreover,
*

#### Acknowledgment

The authors express their thank to the referees for careful reading, helpful comments and suggestions on the manuscript of this paper.

#### References

- C. F. Dunkl, “Integral kernels with reflection group invariance,”
*Canadian Journal of Mathematics*, vol. 43, no. 6, pp. 1213–1227, 1991. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Rösler, “Bessel-type signed hypergroups on $\mathbb{R}$,” in
*Probability Measures on Groups and Related Structures, XI (Oberwolfach, 1994)*, H. Heyer and A. Mukherjea, Eds., pp. 292–304, World Scientific, River Edge, NJ, USA, 1995. View at: Google Scholar | Zentralblatt MATH | MathSciNet - C. B. Morrey Jr., “On the solutions of quasi-linear elliptic partial differential equations,”
*Transactions of the American Mathematical Society*, vol. 43, no. 1, pp. 126–166, 1938. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. L. Mazzucato, “Besov-Morrey spaces: function space theory and applications to non-linear PDE,”
*Transactions of the American Mathematical Society*, vol. 355, no. 4, pp. 1297–1364, 2003. View at: Publisher Site | Google Scholar | MathSciNet - M. E. Taylor, “Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations,”
*Communications in Partial Differential Equations*, vol. 17, no. 9-10, pp. 1407–1456, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Pérez, “Two weighted norm inequalities for Riesz potentials and uniform ${L}_{p}$-weighted Sobolev inequalities,”
*Indiana University Mathematics Journal*, vol. 39, no. 1, pp. 31–44, 1990. View at: Publisher Site | Google Scholar | MathSciNet - A. Ruiz and L. Vega, “Unique continuation for Schrödinger operators with potential in Morrey spaces,”
*Publicacions Matemàtiques*, vol. 35, no. 1, pp. 291–298, 1991. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Z. Shen, “The periodic Schrödinger operators with potentials in the Morrey class,”
*Journal of Functional Analysis*, vol. 193, no. 2, pp. 314–345, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Caffarelli, “Elliptic second order equations,”
*Rendiconti del Seminario Matematico e Fisico di Milano*, vol. 58, pp. 253–284, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Di Fazio, D. K. Palagachev, and M. A. Ragusa, “Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients,”
*Journal of Functional Analysis*, vol. 166, no. 2, pp. 179–196, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. R. Adams, “A note on Riesz potentials,”
*Duke Mathematical Journal*, vol. 42, no. 4, pp. 765–778, 1975. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. R. Adams, “Choquet integrals in potential theory,”
*Publicacions Matemàtiques*, vol. 42, no. 1, pp. 3–66, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet - F. Chiarenza and M. Frasca, “Morrey spaces and Hardy-Littlewood maximal function,”
*Rendiconti di Matematica e delle sue Applicazioni*, vol. 7, no. 3-4, pp. 273–279, 1987. View at: Google Scholar | Zentralblatt MATH | MathSciNet - A. Kufner, O. John, and S. Fučík,
*Function Spaces. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis*, Noordhoff International, Leyden, The Netherlands; Academia, Prague, Czech Republic, 1977. View at: MathSciNet - M. Sifi and F. Soltani, “Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 92–106, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Soltani, “${L}_{p}$-Fourier multipliers for the Dunkl operator on the real line,”
*Journal of Functional Analysis*, vol. 209, no. 1, pp. 16–35, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Trimèche, “Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators,”
*Integral Transforms and Special Functions*, vol. 13, no. 1, pp. 17–38, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. N. Watson,
*A Treatise on the Theory of Bessel Functions*, Cambridge University Press, Cambridge, UK, 1966. - V. S. Guliyev and Y. Y. Mammadov, “On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line,”
*Journal of Mathematical Analysis and Applications*, vol. 353, no. 1, pp. 449–459, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Y. Mammadov, “Some embeddings into the Morrey spaces associated with the Dunkl operator on R,” in
*Proceedings of NASA, Embedding Theorems. Harmonic Analysis*, pp. 258–269, 2007. View at: Google Scholar

#### Copyright

Copyright © 2010 Emin V. Guliyev and Yagub Y. Mammadov. 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.