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Grand Net Spaces and Applications to Integral Operators

Durvudkhan Suragan, Muhammad Asad Zaighum

Abstract

This paper introduces the concept of grand net spaces, a new framework that provides a unified setting for studying various function spaces. Building on the seminal works of [8] and [15], we define grand net spaces and establish their key properties, including embedding results, norm equivalences, and interpolation theorems. We prove that these spaces coincide with grand Lorentz spaces under certain conditions and derive boundedness criteria for integral operators acting on grand net spaces. The latter extends the Nursultanov-Tikhonov theorem established in [16].

Grand Net Spaces and Applications to Integral Operators

Abstract

This paper introduces the concept of grand net spaces, a new framework that provides a unified setting for studying various function spaces. Building on the seminal works of [8] and [15], we define grand net spaces and establish their key properties, including embedding results, norm equivalences, and interpolation theorems. We prove that these spaces coincide with grand Lorentz spaces under certain conditions and derive boundedness criteria for integral operators acting on grand net spaces. The latter extends the Nursultanov-Tikhonov theorem established in [16].

Paper Structure

This paper contains 6 sections, 9 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Grand Net Spaces
  3. Equivalence of Norms
  4. Real Interpolation results
  5. Integral Operators on Grand Net spaces
  6. Acknowledgement

Key Result

Theorem 2.1

Let $|\Omega|=1$, $s\in(1,\infty)$ and $0<p<\infty$, then for $\theta>0$ we have

Theorems & Definitions (20)

  • Theorem 2.1
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.1
  • Remark 2.2
  • ...and 10 more