Embedding constants and embedding functions for Sobolev-like spaces on the unit sphere (Q600749)

Let \(m\in \mathbb N\) and let \(S\) be the unit sphere of \(\mathbb R^n\), \(n\geq 2\). The author denotes by \(X_2^m(S)\) the Hilbert space given by the inner product \[ (\phi,\psi)=\frac{1}{\sigma_{n-1}^2} \bigg(\int_S \phi(\theta)\,d\theta \bigg) \bigg(\int_S \psi(\theta)\,d\theta \bigg) + \int_S \mathcal D^m \phi(\theta)\mathcal D^m \psi(\theta)\,d\theta, \] where \(\mathcal D^m\) is the \(m\)-power of the Laplace-Beltrami operator. An extension of the Sobolev-like spaces \(X_2^r(S)\) to a fractional smoothness \(r\) is presented. This is done by using another description of \(X^m_2(S)\) in terms of decompositions into spherical harmonics which allows to replace \(m\) by any positive real number \(r\). The author finds, for \(r\geq (n-1)/4\), the precise value of the embedding constant from \(X^r_2(S)\) into \(C(S)\) as the sum of an explicit convergent series. He also manages to obtain a function for which the embedding norm of \(X_2^m\subset C(S)\) is achieved using the weak solution of a nonhomogeneous polyharmonic equation on \(S\).