Approximation in generalized Morrey spaces

Alexandre Almeida,Stefan Samko

doi:10.1515/gmj-2018-0023

Alexandre Almeida, Stefan Samko

Open Access

https://doi.org/10.1515/gmj-2018-0023

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Abstract

Abstract In this paper we study the approximation of functions from generalized Morrey spaces by nice functions. We introduce a new subspace whose elements can be approximated by infinitely differentiable compactly supported functions. This provides, in particular, an explicit description of the closure of the set of such functions in generalized Morrey spaces.

Highlights

Morrey spaces play an important role in applications to regularity properties of solutions to PDE including heat equations and Navier-Stokes equations, see [31, 32] and references therein for further details
B(x,r) λ−n f (t·) = t Lp,λ(Rn) p f, Lp,λ(Rn) t > 0, which implies a modication of the scaling factor in comparison with Lp-spaces. This property reveals the homogeneous nature of the spaces Lp,λ(Rn) and it is very useful in the study of partial dierential equations
The main goal of this paper is to extend the approximation scheme developed in [3] for classical Morrey spaces to the case of generalized Morrey spaces

Summary

Introduction

Morrey spaces play an important role in applications to regularity properties of solutions to PDE including heat equations and Navier-Stokes equations, see [31, 32] and references therein for further details. In [3] the authors have introduced a new (closed) subspace of Lp,λ(Rn), strictly smaller than Zorko class, and have shown that all elements in this new class, denoted by V0(,∗∞) Lp,λ(Rn), can be approximated by C0∞-functions in Morrey norm It was obtained in [3] an explicit description of the closure of C0∞(Rn) in Lp,λ(Rn). The main goal of this paper is to extend the approximation scheme developed in [3] for classical Morrey spaces to the case of generalized Morrey spaces The latter are dened by replacing the power rλ in (1.1) by a more general positive function φ(r) (cf (2.1), (2.2)), mainly satisfying monotonicity type conditions. The set of such Morrey functions provides an explicit description of the closure of C0∞(Rn) in the generalized Morrey spaces Lp,φ(Rn)

Preliminaries on generalized Morrey spaces

New and known Morrey subspaces

Convolution in Morrey spaces

Strict embeddings between Morrey subspaces

Approximation in subspaces of generalized Morrey spaces