Boundedness Analysis of the Fractional Maximal Operator in Grand Herz Space on the Hyperplane

Abstract

The primary purpose of this work was to prove the boundedness of the Fractional Maximal Operator in Grand Herz Spaces on the Hyperplane. Here, We defined Grand Herz Space in a continuous Case. For Simplicity, We divided our Problem into two theorems by taking two subsets of Hyperplane( ) as ( ) and its complement . We proved the boundedness of the Fractional Maximal Operator in Grand Herz Space on these two subsets of Hyperplane. We also defined the continuous Case of Grand Herz Space. We proved some results to use in our proof. We represented other terms this paper uses, i.e. the Hyperplane and Fractional Maximal operator. Our proof method relied on one of the corollaries we gave in this paper. We proved the condition to apply that corollary, and then by referring to this, we confirmed both of our theorems. This paper is helpful in Harmonic analysis and delivers ways to analyse the solutions of partial differential equations. The Problem of our discussion provides methods to study the properties of very complex functions obtained from different problems from Physics, Engineering and other branches of science. Solutions of nonlinear Partial Differential equations often resulted in such functions which required deep analysis. Our work helps check the boundedness of such types of functions.