Estimates of Lp norms for sums of positive functions

Ilgiz Kayumov

doi:10.2478/v10062-012-0026-9

Abstract

We present new inequalities of \(L_p\) norms for sums of positive functions. These inequalities are useful for investigation of convergence of simple partial fractions in \(L_p(\mathbb{R})\).

Highlights

In the paper [2] we proved the following theorem
The proof of Theorem 1 is based on the following fact
It turns out that there exists a nontrivial generalization of this result for arbitrary positive functions from arbitrary measurable space

Summary

Introduction

In [1] the problem to find necessary and sufficient conditions for convergence of the series g∞ in Lp(R) was posed. Kayumov converges in Lp(R) and all zk lie in the angle |z| ≤ C|y| with a fixed C, for all ε > 0 the following condition holds: In the paper [2] we proved the following theorem. If g∞(t) converges in Lp(R), the sequence |yn| is increasing and |zk| ≤ C|yk|, the condition (3) holds. The proof of Theorem 1 is based on the following fact.

Objectives

Results

Conclusion