Spline wavelets and Riemann–Liouville operators in Besov type spaces

With help of spline wavelet systems and corresponding to them decomposition theorems, conditions are found for the fulfillment of inequalities relating the norms of images and pre-images of Riemann–Liouville operators $I_\alpha$ of natural and fractional orders $\alpha >0$ in weighted Besov type spaces $B_{pq}{}^s$ on the real axis and semi-axis. Here $0 < p,q < \infty$ and $-\infty < s < \infty$ are summation and smoothness parameters, respectively. With some restrictions on weights, it is possible to generalize the obtained results to multi-dimensional case.
To solve the problem:
(1) special systems of spline wavelets of natural orders are constructed;
(2) decompositions of elements of the spaces $B_{pq}{}^s$ on $R^n$ with Muckenhoupt weights of local type are presented in terms of such systems, and an isometric isomorphism of $B_{pq}{}^s$ with the corresponding sequential spaces is established.
The proofs also use fractional-order spline wavelets developed by T. Blue and M. Unser. Decompositions corresponding to them are used in the work to extract one-sided estimates.
As an application of the main results, we study the behavior of sequences of characteristic (approximation and entropy) numbers of Riemann–Liouville operators. From the inequalities for $I_\alpha$ with $0 < \alpha < 1$ we also derive boundedness conditions for the Hilbert transform on subclasses in $B_{pq}{}^s$.