Bessel potential spaces with variable exponent

Abstract

Abstract. We show that a variable exponent Bessel potential space coincides with the variable exponent Sobolev space if the Hardy-Littlewood maximal operator is bounded on the underlying variable exponent Lebesgue space. Moreover, we study the Hölder type quasi-continuity of Bessel potentials of the first order.

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

“…are the Bessel potential spaces (fractional Sobolev spaces) of variable integrability which were introduced in [1] and in [12]. The integrability p has to belong to C log (R n ) with 1 < p − ≤ p + < ∞ and s ≥ 0 (Theorem 4.5 in [9]).…”

“…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.…”

Hölder Quasicontinuity in Variable Exponent Sobolev Spaces

“…For a different point of view, which is related to [15], in the variable exponent case, we refer to [19]. Bounds for maximal functions in variable exponent spaces have been obtained in [20][21][22][23][24][25][26][27].…”