Boyd indices for quasi-normed function spaces with some bounds

Waqas Nazeer,Qaisar Mehmood,Shin Min Kang,Abdul Rauf Nizami

doi:10.1186/s13660-015-0754-9

Abstract

We calculate the Boyd indices for quasi-normed rearrangement invariant function spaces with some bounds. An application to Lorentz type spaces is also given.

Highlights

Let Lloc be the space of all locally integrable functions f on Rn and M+ be the cone of all locally integrable functions g ≥ on (, ) with the Lebesgue measure.Let f ∗ be the decreasing rearrangement of f given by f ∗(t) = inf λ > : μf (λ) ≤ t, t >, and μf be the distribution function of f defined by μf (λ) = x ∈ Rn : f (x) > λ n,| · |n denoting Lebesgue n-measure
We consider rearrangement invariant quasi-normed spaces E → L ( ) such that f E = ρE(f ∗) < ∞, where ρE is a quasi-norm rearrangement invariant defined on M+
The main goal of this paper is to provide formulas for the Boyd indices with some bounds of rearrangement invariant quasi-normed spaces and to apply these results to the case of Lorentz type spaces

Summary

Introduction

There is an equivalent quasi-norm ρp ≈ ρE that satisfies the triangle inequality ρpp(g + We apply this inequality for functions g ∈ M+ with some kind of monotonicity. The function hE is sub-multiplicative, increasing, hE( ) = , hE(u)hE( /u) ≥ and Consider the gamma spaces E = q(w), < q ≤ ∞, wpositive weight, that is, a positive function from M+, with a quasi-norm f q(w) := ρE(f ∗), ρE(g) := ρw,q(

Objectives

Results

Conclusion