A fundamental condition for harmonic analysis in anisotropic generalized Orlicz spaces

Abstract

Anisotropic generalized Orlicz spaces have been investigated in many recent papers, but the basic assumptions are not as well understood as in the isotropic case. We study the greatest convex minorant of anisotropic $\Phi$-functions and prove the equivalence of two widely used conditions in the theory of generalized Orlicz spaces, usually called (A1) and (M). This provides a more natural and easily verifiable condition for use in the theory of anisotropic generalized Orlicz spaces for results such as Jensen's inequality which we obtain as a corollary.

arXiv commentAn earlier version of this paper was titled "Anisotropic generalized Orlicz spaces"

References in corpus (12)

• Maximal regularity for local minimizers of non-autonomous functionals
• A pocket guide to nonlinear differential equations in the Musielak--Orlicz spaces
• Lipschitz bounds and nonautonomous integrals
• Gossez's approximation theorems in the Musielak-Orlicz-Sobolev spaces
• Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space
• The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth
• Double phase image restoration
• Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak-Orlicz spaces in the class of renormalized solutions
• Regularity theory for non-autonomous partial differential equations without Uhlenbeck structure
• Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon
• Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon
• Parabolic equations in Musielak -- Orlicz spaces with discontinuous in time $N$-function

Cited by in corpus (4)

• Regularity theory for non-autonomous problems with a priori assumptions
• A revised condition for harmonic analysis in generalized Orlicz spaces on unbounded domains
• Bounded variation spaces with generalized Orlicz growth related to image denoising
• Monotone approximation of differentiable convex functions with applications to general minimization problems