On a class of sublinear operators in variable exponent Morrey-type spaces

Abstract

For a class of sublinear operators, we find conditions on the variable exponent Morrey-type space $L^{p(\cdot ),q,\omega (\cdot ,\cdot )}\left({\mathbb{R}}^n\right)$ ensuring the boundedness in this space. A priori assumptions on this class are that the operators are bounded in $L^{p(\cdot )}\left({\mathbb{R}}^n\right)$ and satisfy some size condition. This class includes in particular the maximal operator, singular operators with the standard kernel, and the Hardy operators. We also prove embedding of variable exponent Morrey-type spaces into weighted $L^{p(\cdot )}$-spaces.

Keywords:

• Sublinear operators
• Morrey-type spaces
• variable exponent

AMS Subject Classifications:

• 46E30
• 42B25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of H. Rafeiro was supported by a Start-up Grant from United Arab Emirates University, Al Ain, United Arab Emirates [Grant $N^{\underset{\_}{o}}$ G00002994]. The research of S. Samko was supported by (a) Russian Foundation for Basic Research [grant $N^{\underset{\_}{o}}$ 19-01-00223] and (b) TUBITAK and Russian Foundation for Basic Research [grant $N^{\underset{\_}{o}}$ 20-51-46003].