The Equivalent Norms in Spaces with a Weight Having Singularity at the Domain Boundary
NRU MPEI, Pavel V. Zubkov · Vestnik MEI · 2017
Earlier, it was proven that every periodic function that is quadratically summable with a power weight in a half-band is uniquely represented as an orthogonal sum of analytic and coanalytic components. Therefore, it is natural to regard the coanalytic component as a definite characteristic of the non-analyticity of the function. The aim of this article is to prove the equivalence near the boundary of the considered norm of weight space and the norm that is convenient for formulating and investigating the coanalytic problem in spaces with a weight having a singularity at the domain boundary. This fact, as well as the known — for the equivalent norm — exact inequalities between the norms of functions and their traces, and the estimated solutions of boundary value problems for degenerate elliptic equations in bounded and unbounded domains open the possibility to investigate the problem about minimizing the coanalytic deviation when the specified boundary function is extended to inside the domain in the corresponding weight spaces.