Vol 53, No 6 (2018)

The paper considers Cauchy problem in the Gevre type multianisotropic spaces. Necessary and sufficient conditions for unique solvability of this problem are obtained and the properties of operators (polynomials) that are hyperbolic with a specified weight are investigated.

In this paper, we show that the norm of the Bergman projection on L^p,q-spaces in the upper half-plane is comparable to csc(π/q). Then we extend this result to a more general class of domains, known as the homogeneous Siegel domains of type II.

In this paper, we prove that for any ε ∈ (0, 1) there exists ameasurable set E ∈ [0, 1) with measure |E| > 1 − ε such that for any function f ∈ L¹[0, 1), it is possible to construct a function \(\tilde f \in {L^1}[0,1]\) coinciding with f on E and satisfying \(\int_0^1 {|\tilde f(x) - f(x)|dx < \varepsilon } \), such that both the Fourier series and the greedy algorithm of \(\tilde f\) with respect to a bounded Vilenkin system are almost everywhere convergent on [0, 1).

In this paper, we study certain classes of analytic functions which satisfy a subordination condition and are associated with the crescent-shaped regions. We first give certain integral representations for the functions belonging to these classes and also present a relevant example. Making use of some known lemmas, we derive sufficient conditions for the functions to be in these classes. Some results on coefficient estimates are also obtained.

The problem of the sine representation for the support function of centrally symmetric convex bodies is studied. We describe a subclass of centrally symmetric convex bodies which is dense in the class of centrally symmetric convex bodies. Also, we obtain an inversion formula for the sine-transform.