Some properties on multi-radial functions

Abstract: W.Strauss proved pointwise estimates of radial functions belonging to the Sobolev classes. We consider the counterpart of  the estimates for  multi-radial functions. More precisely  We are interested in the  sharp pointwise estimates for functions in $\dot H^{s,p}({\mathbb R}^N)$ with radial symmetry in $m$ blocks of variables, for $m<sp<N$. The estimates take form of multi-radial monomials dependent on the dimensions of the blocks and the regions of order between the block radii (the simplest of which, for a subset of parameters is the Strauss radial estimate by $|x|^{\frac{sp-N}{p}}$), or, for an exceptional set of parameters, they become multi-radial monomials with a logarithmic factor.