Cones of monotone functions generated by a generalized fractional maximal function

TL;DR

This paper introduces new cones of monotone functions generated by a generalized fractional maximal function, characterizes their properties, and establishes criteria for embedding these spaces into rearrangement invariant spaces, including identifying the optimal RIS.

Contribution

It defines and analyzes cones of monotone functions generated by generalized fractional maximal functions and provides embedding criteria into rearrangement invariant spaces.

Findings

Characterization of cones of monotone functions generated by $M_\

Embedding criteria for these cones into RIS spaces

Identification of the optimal rearrangement invariant space for embedding

Abstract

In this paper, we consider the generalized fractional maximal function and use it to introduce the space of generalized fractional maximal functions and the various cones of monotone functions generated by generalized fractional maximal functions $M_\Phi f$. We introduced three function classes. We give equivalent descriptions of such cones when the function $\Phi$ belongs to some function classes. The conditions for their mutual covering are given. Then, these cones are used to construct a criterion for embedding the space of generalized fractional maximal functions into the rearrangement invariant spaces (RIS). The optimal RIS for such embedding is also described.