Riemann-Liouville integrals in Besov spaces
Elena P. Ushakova
TL;DR
This work addresses the problem of characterizing when Riemann–Liouville operators $I_{c}^{\boldsymbol{\alpha}}$ act boundedly between weighted Besov spaces $B_{pq}^{s,w}(\mathbb{R})$, with weights from ${\mathscr{A}}_\infty^{\rm loc}$ and related classes. The authors develop a spline-wavelet decomposition framework, establishing a precise, operator-norm criterion in terms of functionals $\mathfrak{C}_{\pm}^{\boldsymbol{\alpha}}(\kappa^*)$ (and their local/shifted analogues) that equivalently govern boundedness and sharp constants. They show that modifying the mother wavelet improves the constants and obtain refined results under additional assumptions on the weights, such as homogeneous anti-derivatives or average-type conditions, which yield simpler integral forms. The results advance the theory of weighted Besov spaces and fractional integral inequalities, with potential applications to weighted function-space embeddings and analysis of singular integral operators. Throughout, the analysis leverages Battle–Lemarié spline wavelets and detailed weighted norm controls, culminating in comprehensive criteria for norm inequalities with explicit weight-sensitive constants.
Abstract
Criteria for the fulfillment of inequalities in weighted smoothness function spaces of Besov type with Riemann-Liouville operators of natural orders on the real axis and semi-axes are found. The obtained estimates are refined under additional conditions on weights.