Gossez's approximation theorems in Musielak–Orlicz–Sobolev spaces (original) (raw)
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Density of smooth functions in Musielak–Orlicz spaces
Banach Journal of Mathematical Analysis
We provide necessary and sufficient conditions for the space of smooth functions with compact supports C ∞ C (Ω) to be dense in Musielak-Orlicz spaces L Φ (Ω) where Ω is an open subset of R d. In particular we prove that if Φ satisfies condition ∆2, the closure of C ∞ C (Ω) ∩ L Φ (Ω) is equal to L Φ (Ω) if and only if the measure of singular points of Φ is equal to zero. This extends the earlier density theorems proved under the assumption of local integrability of Φ, which implies that the measure of the singular points of Φ is zero. As a corollary we obtain analogous results for Musielak-Orlicz spaces generated by double phase functional and we recover the well known result for variable exponent Lebesgue spaces.
An approximation theorem in Musielak-Orlicz-Sobolev spaces
In this paper we prove the uniform boundedness of the operators of convolution in the Musielak-Orlicz spaces, and the density of D (R^n) in the Musielak-Orlicz-Sobolev spaces by assuming a condition of Log-Hölder type of continuity.
Some approximation results in Musielak-Orlicz spaces
arXiv (Cornell University), 2017
We give sufficient conditions for the continuity in norm of the translation operator in the Musielak-Orlicz LM spaces. An application to the convergence in norm of approximate identities is given, whereby we prove density results of the smooth functions in LM , in both modular and norm topologies. These density results are then applied to obtain basic topological properties.
A P\'olya-Szeg\"o principle for general fractional Orlicz-Sobolev spaces
arXiv (Cornell University), 2019
In this article we prove modular and norm Pólya-Szegö inequalities in general fractional Orlicz-Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of theses spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary we prove a Rayleigh-Faber-Krahn type inequality for Dirichlet eigenvalues under nonlocal nonstandard growth operators.
Approximative Characteristics of Modular Orlicz Spaces
Journal of Mathematical Sciences, 2019
We obtain the exact values of the best approximations, basic widths and Kolmogorov widths for some sets of images of multipliers in the modular Orlicz spaces l M. We give a description of the space S M,N of all multipliers from the space l M to l N .
2019
In weighted Orlicz type spaces S p, μ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is in a certain sense the best. Some applications of the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduli of smoothness are given. Equivalence between moduli of smoothness and certain Peetre K-functionals is shown in the spaces S p, μ .
Modular convergence in HHH-Orlicz spaces of Banach valued functions
arXiv (Cornell University), 2022
In this article we develop the theory of H-Orlicz space generated by generalised Young function. Modular convergence of H-Orlicz space for the case of vector-valued functions and norm convergence in H θ (X, µ) where X is any Banach space are discussed. Relationships of modular convergence and norm convergence of H-Orlicz spaces are discussed.
Some Remarks About the Density of Smooth Functions in Weighted Sobolev Spaces
1994
Introduction and statement of main resultsIn this note we deal with the problem of the density of smooth functions in weightedSobolev spaces (for general results and references on this topic see, for instance, [14], [10],[2], and the bibliography therein). In order to introduce some definitions, let us fix abounded openset\Omega` IRn, a real number p ? 1, and a function