On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators

Aysegul Ç Alabalik,Stefan Samko,Alexandre Almeida

doi:10.1007/s43037-019-00049-7

Abstract

We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of $$C_0^\infty ({\mathbb {R}^n})$$ in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy–Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators.

Highlights

Morrey spaces play an important role in the study of local behaviour and regularity properties of solutions to PDE, including heat equations and NavierStokes equations
In this paper we are interested in studying the behavior of those classical operators in certain subspaces of Morrey spaces
The boundedness of classical operators in vanishing Morrey spaces at the origin was already studied in some papers, including the case of generalized parameters, see [25, 29, 32, 33]

Summary

Introduction

Morrey spaces play an important role in the study of local behaviour and regularity properties of solutions to PDE, including heat equations and NavierStokes equations. Many classical operators from Harmonic Analysis such as maximal operators, singular operators, potential operators and Hardy operators, are known to be bounded in Morrey spaces. In this paper we are interested in studying the behavior of those classical operators in certain subspaces of Morrey spaces. The boundedness of classical operators in vanishing Morrey spaces at the origin was already studied in some papers, including the case of generalized parameters, see [25, 29, 32, 33]. Up to authors’ knowledge, the boundedness of classical operators in the vanishing Morrey spaces V∞Lp,λ(Rn) and V (∗)Lp,λ(Rn) was not touched so far, apart some results in [5, Theorem 3.8], [6, Corollary 4.3] where it was observed that convolution operators with integrable kernels are bounded in those subspaces.

Preliminaries We use the following notation

Additional results

Methods