Boundedness of some operators on grand generalized Morrey spaces over non-homogeneous spaces

Abstract

<abstract><p>The aim of this paper is to obtain the boundedness of some operator on grand generalized Morrey space $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $ over non-homogeneous spaces, where $ G\subset $ $ \mathbb{R}^{n} $ is a bounded domain. Under assumption that functions $ \varphi $ and $ \phi $ satisfy certain conditions, the authors prove that the Hardy-Littlewood maximal operator, fractional integral operators and $ \theta $-type Calderón-Zygmund operators are bounded on the non-homogeneous grand generalized Morrey space $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $. Moreover, the boundedness of commutator $ [b, T^{G}_{\theta}] $ which is generated by $ \theta $-type Calderón-Zygmund operator $ T_{\theta} $ and $ b\in\mathrm{RBMO}(\mu) $ on spaces $ \mathcal{L}^{p), \varphi, \phi}_{\mu}(G) $ is also established.</p></abstract>

On some differential properties of functions in generalized grand Sobolev-Morrey spaces

“…After a vast amount of research about grand Lebesgue, small Lebesgue, grand Lebesgue-Morrey, grand grand Lebesgue-Morrey, grand grand Sobolev-Morrey, small small Sobolev-Morrey, grand grand Nikolskii Morrey and generalized grand Lebesgue-Morrey spaces has been introduced and studied by many mathematicians (see, e.g. [2,3], [5]- [14]) etc.…”