Grand Morrey Type Spaces

Abstract

The so called grand spaces nowadays are one of the main objects in the theory of function spaces. Grand Lebesgue spaces were introduced by T. Iwaniec and C. Sbordone in the case of sets $\Omega$ with finite measure $|\Omega|<\infty$, and by the authors in the case $|\Omega|=\infty$. The latter is based on introduction of the notion of grandizer. The idea of "grandization" was also applied in the context of Morrey spaces. In this paper we develop the idea of grandization to more general Morrey spaces $L^{p,q,w}(\mathbb{R}^n)$, known as Morrey type spaces. We introduce grand Morrey type spaces, which include mixed and partial grand versions of such spaces. The mixed grand space is defined by the norm $$ \sup_{\varepsilon,\delta} \varphi(\varepsilon,\delta)\sup_{x\in E} \left(\int\limits_{0}^{\infty}{w(r)^{q-\delta}}b(r)^{\frac{\delta}{q}} \left(\,\int\limits_{|x-y|<r}\big|f(y)\big|^{p-\varepsilon} a(y)^{\frac{\varepsilon}{p}}\,dy\right)^{\frac{q-\delta}{p-\varepsilon}} \frac{dr}{r}\right)^{\frac{1}{q-\varepsilon}} $$ with the use of two grandizers $a$ and $b$. In the case of grand spaces, partial with respect to the exponent $q$, we study the boundedness of some integral operators. The class of these operators contains, in particular, multidimensional versions of Hardy type and Hilbert operators.