Characterization of Riesz and Bessel potentials on variable Lebesgue spaces

By Alexandre Almeida, S. Samko. 2006

71 citations recorded by the source corpus.

Riesz and Bessel potential spaces are studied within the framework of the Lebesgue spaces with variable exponent. It is shown that the spaces of these potentials can be characterized in terms of convergence of hypersingular integrals, if one assumes that the exponent satisfies natural regularity conditions. As a consequence of this characterization, we describe a relation between the spaces of Riesz or Bessel potentials and the variable Sobolev spaces.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a59780cf5bba1d06e00f93282f9a01638aa4f2e9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. Ciqiang Zhuo
4. Yoshihiro Sawano
5. Jan Vyb'iral
6. H. Kempka
7. P. Hasto
8. S. Samko

In Mathematical Physics

1. Dachun Yang
2. Wen Yuan
3. Ciqiang Zhuo
4. Yoshihiro Sawano
5. Jan Vyb'iral
6. H. Kempka
7. P. Hasto
8. Jingshi Xu

Citations in the corpus

• Spaces of Variable Smoothness and Integrability: Characterizations by Local Means and Ball Means of Differences (2011, background): “If the smoothness s ∈ R is a constant and p ∈ P log(Rn) with p− > 1, then F s p(·),2(Rn) = Lsp(·)(R) are the variable Bessel potential spaces from [2] and [23] with its special cases F 0 p(·),2(Rn) = Lp(·)(R) and F p(·),2(Rn) = W p(·)(Rn) for k ∈ N0, see [16].”
• 2-Microlocal Besov and Triebel-Lizorkin Spaces of Variable Integrability (2009, background): “For example we have F s p(·),2(R ) = Ls,p(·)(Rn), where Ls,p(·)(Rn) are the Bessel potential spaces (fractional Sobolev spaces) of variable integrability which were introduced in [1] and in [12].”
• SOBOLEV AND JAWERTH EMBEDDINGS FOR SPACES WITH VARIABLE SMOOTHNESS AND INTEGRABILITY (2009, methodology): “[32] Samko, S.: Convolution and potential type operators in Lp(x)(Rn).”
• Sobolev's inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space Lp(⋅)(logL)q(⋅) (2008, unknown): “8 below) is given in Section 2, which is an extension of Almeida-Samko [3], Diening [6], Futamura-Mizuta [10], Futamura-Mizuta-Shimomura [11, 12], Harjulehto-HästöPere [18], Kokilashvili-Samko [21], Mizuta-Shimomura [27] and Samko-Vakulov [31].”
• Hardy inequality in variable exponent Lebesgue spaces (2007, background): “The boundedness of the operator Bfi in the space Lp(¢)(Rn) was earlier shown in [2], [ 1 ] without the condition (4.13) on the variation of 1 p(x) , but under the assumption that besides the decay condition, p satisfies everywhere the local log-condition.”
• An atomic decomposition of variable Besov and Triebel-Lizorkin spaces (2009, background): “This result was obtained firstly by Almeida and Samko in [3] and again by Gurka, Harjulehto and Nekvinda in [12].”
• Function spaces of variable smoothness and integrability (2007, methodology): “Alrea dy somewhat earlier Almeida & Samko [4] and Gurka, Harjulehto & Nekvinda [26] had extende d variable integrability Sobolev spaces to Bessel potential spaces Wα,p(·) for constant but non-integer α.”
• VARIABLE BESOV AND TRIEBEL-LIZORKIN SPACES (2008, background): “Remark 3. Almeida and Samko in [ 2 ] and, independently, Gurka, Harjulehto and Nekvinda in [12] have introduced Bessel potential spaces with variable exponents.”
• Atomic, molecular and wavelet decomposition of 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability (2010, background): “So it was natural, that several extensions in various more advanced function spaces as Bessel potential spaces H p(·) and Besov and Triebel-Lizorkin spaces F s p(·),q(R), B p(·),q(R) were made ([3], [18] and [43]).”
• Besov spaces with variable smoothness and integrability (2010, background): “[6] A. Almeida and S. Samko: Embeddings of variable HajÃlasz–Sobolev spaces into Hölder spaces of variable order, J. Math.”
• Triebel-Lizorkin-Type Spaces with Variable Exponents (2015, background): “A. Almeida and S. Samko, Characterization of Riesz and Bessel potentials on variable Lebesgue spaces, J. Funct.”
• Bessel potential spaces with variable exponent (2007, background): “Remark 5.8. One of the referees pointed out to us a recent preprint [ 1 ] where the spaces of Bessel potentials over the spaces Lp(¢)(Rn) were investigated.”
• FUNCTION SPACES WITH VARIABLE EXPONENTS – AN INTRODUCTION – (2014, methodology): “Samko [4] introduced the spaces of Riesz and Bessel potentials with densities in Lebesgue spaces with variable exponents.”
• Sobolev inequalities with variable exponent attaining the values $1$ and $n$ (2008, unknown): “, in [1], [2], [4], [5], [8], [10], [11], [12], [13], [15], [16], [20], [24], [28] in the variable exponent setting.”