Characterization of Riesz and Bessel potentials on variable Lebesgue spaces

Abstract

(Communicated by Vakhtang Kokilashvili)2000 Mathematics Subject Classification. 46E35, 26A33, 42B20, 47B38.Keywords and phrases. Lebesgue space with variable exponent, hypersingular integral, Riesz potential, Bessel potential.Abstract. 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.

Generalized maximal functions and related operators on weighted Musielak-Orlicz spaces

“…In fact, they seem to be the natural context in order to describe the behaviour of certain classes of fluids, called electrorheological fluids, which have the ability to significantly modify its mechanical properties when an electric field is applied (see for example [47]). Other applications that find in these spaces an adequate development framework for their theory are the processes of image restoration [6] and partial differential equations (see for instance [1] and [24]). …”

“…The boundedness of many operators in harmonic analysis that appear in connection with the study of regularity properties of the solutions of partial differential equations were widely considered in the variable context by different authors, see for instance [9], [11], [14], [15], [16], [30], [31], [32], [38], [39] and [40] for the HardyLittlewood maximal function M, [5], [21], [22] and [28] for the fractional maximal function M α , [18] and [33] for Calderón-Zygmund operators and their commutators, and [1], [10], [24] and [28] for potential type operators (see [13] for other classical operators).…”

2-Microlocal Besov and Triebel-Lizorkin Spaces of Variable Integrability

“…The variable exponent spaces have interesting applications in fluid dynamics, PDE's and image processing. In that connection, Sobolev spaces with variable exponent have been introduced and studied in detail in [1] and [12]. From the point of view of Harmonic analysis, the breakthrough for variable exponent spaces was achieved by Diening, when he showed in [7] that the HardyLittlewood maximal operator is bounded on L p(·) (Ω) for p satisfying some regularity condition inside a large ball B R and outside p is constant.…”

Fractional and Hypersingular Operators in Variable Exponent Spaces on Metric Measure Spaces

“…We refer to the books [49] and [47] for hypersingular integrals in general. Hypersingular operators in the variable exponent setting were firstly studied by the authors in [1], [2], [46] in related to the problem of inversion and characterization of the Riesz potentials in the Euclidean case, and in [3] there where studied some mapping properties of hypersingular integrals in the context of variable exponent Sobolev spaces on metric measure spaces.…”