On approximation by rational functions in Musielak–Orlicz spaces

In this chapter we study modular spaces and Musielak–Orlicz spaces which provide the framework for a variety of different function spaces, including classical (weighted) Lebesgue and Orlicz spaces and variable exponent Lebesgue spaces. Although our aim mainly is to study the latter, it is important to see the connections between all of these spaces. Many of the results in this chapter can be found in a similar form in [295], but we include them to make this exposition self-contained. Research in the field of Musielak–Orlicz functions is still active and we refer to [67] for newer results and references.KeywordsModular SpaceFunction SpaceDual SpaceSimple FunctionOrlicz SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We show that several definitions of algebras of continuous Fourier multipliers on variable Lebesgue spaces over the real line are equivalent under some natural assumptions on variable exponents. Some of our results are new even in the case of standard Lebesgue spaces and give answers on two questions about algebras of continuous Fourier multipliers on Lebesgue spaces over the real line posed by Mascarenhas, Santos and Seidel.

We prove fractional Leibniz rules and related commutator estimates in the settings of weighted and variable Lebesgue spaces. Our main tools are uniform weighted estimates for sequences of square-function-type operators and a bilinear extrapolation theorem. We also give applications of the extrapolation theorem to the boundedness on variable Lebesgue spaces of certain bilinear multiplier operators and singular integrals.

Let ( 𝕏 , d , μ ) {(\mathbb{X},d,\mu)} be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and let X ⁢ ( 𝕏 ) {X(\mathbb{X})} be a ball quasi-Banach function space on 𝕏 {\mathbb{X}} . In this article, the authors introduce the weak Hardy space W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} associated with X ⁢ ( 𝕏 ) {X(\mathbb{X})} via the Lusin area function. Then the authors characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the molecule, the grand maximal function, and the Littlewood–Paley g-function and g λ * {g^{*}_{\lambda}} -function. Moreover, all these results have a wide generality. Particularly, the results of this article are also new even when they are applied, respectively, to weighted Lebesgue spaces, Orlicz spaces, and variable Lebesgue spaces, which actually are new even on RD-spaces (that is, spaces of homogeneous type with additional reverse doubling condition). The main novelties of this article exist in that the authors take full advantage of the geometrical properties of 𝕏 {\mathbb{X}} expressed by both the dyadic cubes and the exponential decay of the approximations of the identity to overcome the difficulties caused by the deficiencies of both the explicit expression of the quasi-norm of X ⁢ ( 𝕏 ) {X(\mathbb{X})} and the reverse doubling condition of μ, and that the authors use the tent space on 𝕏 × ℤ {\mathbb{X}\times\mathbb{Z}} to characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the Littlewood–Paley g λ * {g^{*}_{\lambda}} -function, where the range of λ might be best possible in some cases.

Maximal operator on variable Lebesgue spaces with radial exponent

A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces

We prove the Stechkin inequality for Fourier multipliers on variable Lebesgue spaces under some natural assumptions on variable exponents.

In this paper, the continuous wavelet transform is studied in variable $$L^{p(\cdot )}$$ spaces. Parseval relation and Reconstruction formula for the continuous wavelet transform in variable $$L^{p(\cdot )} $$ spaces are derived. The characterization of variable Besov spaces associated with the continuous wavelet transform is also presented.

Characterizations for boundedness of fractional maximal function commutators in variable Lebesgue spaces on stratified groups

We study totally bounded sets in variable Lebesgue spaces. The full characterization of this kind of sets is given for the case of variable Lebesgue space on metric measure spaces. Furthermore, the sufficient conditions for compactness are shown without assuming log-Holder continuity of the exponent.

The present paper deals with the Riemann boundary value problem for analytic functions in the framework of the new function spaces introduced by the first two authors, the so-called grand variable exponent Lebesgue spaces which unify two non-standard type function spaces: variable exponent Lebesgue spaces and grand Lebesgue spaces.

We prove that the space of pointwise multipliers between two distinct Musielak–Orlicz spaces is another Musielak–Orlicz space and the function defining it is given by an appropriately generalized Legendre transform. In particular, we obtain characterization of pointwise multipliers between Nakano spaces. We also discuss factorization problem for Musielak–Orlicz spaces and exhibit some differences between Orlicz and Musielak–Orlicz cases.

In this paper a two-weight criterion for multidimensional Hardy type operator and its dual operator acting from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces is proved. As application we prove the boundedness of multidimensional geometric mean operator in the weighted Musielak-Orlicz spaces. In particular, from obtained results implies the boundedness of multidimensional Hardy operator and its dual operator acting from usual weighted Lebesgue spaces into weighted variable Lebesgue spaces. In this paper we establish integral-type neces- sary and sufficient condition on weights, which provides the boundedness of the multidimensional Hardy type operator from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces.

In Hudzik and Landes, the convexity coefficient of Musielak–Orlicz function spaces over a non‐atomic measure space equipped with the Luxemburg norm is computed whenever the Musielak–Orlicz functions are strictly convex see [6]. In this paper, we extend this result to the case of Musielak–Orlicz spaces equipped with the Orlicz norm. Also, a characterization of uniformly convex Musielak–Orlicz function spaces as well as k‐uniformly convex Musielak–Orlicz spaces equipped with the Orlicz norm is given.

In this paper, the authors present some new characterizations of the Musielak–Orlicz–Sobolev spaces with even smoothness order via ball averages and their derivatives on the radius. Consequently, as special examples of the Musielak–Orlicz–Sobolev spaces studied in this paper, the corresponding characterizations for some weighted Sobolev spaces, Orlicz–Sobolev spaces and variable Sobolev spaces are also obtained. Since these characterizations depend only on ball averages and their derivatives on the radius, they provide some possible ways to introduce the corresponding function spaces on any metric measure space.