Maximal and Calderón–Zygmund operators on the local variable Morrey–Lorentz spaces and some applications

Abstract

In this paper, we give the definition of local variable Morrey–Lorentz spaces ${\mathcal{M}}_{p(\cdot ),q(\cdot ),\lambda }^{loc}({\mathbb{R}}^n)$ which are a new class of functions. Also, we prove the boundedness of the Hardy–Littlewood maximal operator M and Calderón–Zygmund operators T on these spaces. Finally, we apply these results to the Bochner–Riesz operator $B_r^{\delta }$, identity approximation $A_{\epsilon }$ and the Marcinkiewicz operator ${\mu }_{\mathrm{\Omega }}$ on the spaces ${\mathcal{M}}_{p(\cdot ),q(\cdot ),\lambda }^{loc}({\mathbb{R}}^n)$.

Keywords:

• Local variable Morrey–Lorentz space
• Hardy–Littlewood maximal function
• Calderón–Zygmund operators

AMS Subject Classifications:

• Primary 42B25
• 42B35
• Secondary 47G10

Disclosure statement

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

Additional information

Funding

The research of Kucukaslan was supported by the grant from The Scientific and Technological Research Council of Turkey [TUBITAK Grant-1059B191600675]. The research of Guliyev was partially supported by the grant from Elmin Inkişaf Fondu [Agreement No. EIF-BGM-4-RFTF-1/2017-21/01/1-M-08]. The research of Guliyev and Serbetci was partially supported by the grant from Cooperation Program 2532 TUBITAK-RFBR (Russian Foundation for Basic Research) with Agreement Number No. 119N455.