Jacobi weights, fractional integration, and sharp Ulyanov inequalities

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