Generalizations of Herz-Morrey spaces and boundedness of the Calderón-Zygmund operators

In this paper, we use the grand variable Herz-Morrey spaces, and our main objective is to prove the boundedness of multilinear Calderón-Zygmund operators on the product of grand variable Herz-Morrey spaces. We first define the Lebesgue spaces with variable exponents and some basic lemmas including Hölder's inequality for Lebesgue spaces. To prove the boundedness of multilinear Calderón-Zygmund operators on grand variable Herz-Morrey spaces, we use the well known Hölder's inequality and Minkonwski's inequality. We split the summation of grand variable Herz-Morrey spaces to find the $L^{p(\cdot )}({\mathbb{R}}^n)$ estimate of the characteristic function under some conditions. After finding estimate of each term, we conclude that multilinear Calderón-Zygmund operators are bounded on grand variable Herz-Morrey spaces. Our results generalize some results on variable Herz spaces and grand variable Herz spaces.