Pointwise Estimates for Block-Radial Functions of Sobolev Classes

Abstract

The paper gives sharp pointwise estimates for functions belonging tȯ H s, p (R N ) with radial symmetry in m blocks of variables, for m < sp < N . The estimates are formulated in terms of multiradial monomials. The form of the monomials depends on the structure of the group of block-radial symmetries and the distances of the given point to the hyperplanes in R N that contain the singular orbits of the group. For some exceptional set of parameters the logarithmic factor is needed. Weak continuity related to the estimates is also considered.

Some properties of block-radial functions and Schrödinger type operators with block-radial potentials

“…The same tool was used to prove the Strauss inequality for block radial functions cf. [26]. In [4] we estimates from below and from above entropy numbers of compact Sobolev embeddings of block-radial Besov spaces R γ B s p,q (R d ) and fractional Sobolev spaces R γ H s p (R d ), 1 < p < ∞.…”