Jacobi weights, fractional integration, and sharp Ulyanov inequalities

TL;DR

This paper establishes sharp inequalities relating fractional integrals, moduli of smoothness, and K-functionals for weighted Lp spaces with Jacobi weights, advancing the understanding of fractional integration in weighted function spaces.

Contribution

It introduces new sharp Ulyanov-type inequalities for fractional integrals in weighted Lp spaces with Jacobi weights, connecting fractional integration with smoothness measures.

Findings

Proves Hardy--Littlewood type inequalities for fractional integrals with Jacobi weights.

Derives sharp (L_p, L_q) Ulyanov-type inequalities for moduli of smoothness.

Establishes connections between fractional integrals and K-functionals in weighted spaces.

Abstract

We consider functions L_p-integrable with Jacobi weights on [-1,1] and prove Hardy--Littlewood type inequalities for fractional integrals. As applications, we obtain the sharp (L_p, L_q) Ulyanov-type inequalities for the Ditzian--Totik moduli of smoothness and the K-functionals of fractional order.