On embeddings of weighted tensor product Hilbert spaces

We study embeddings between tensor products of weighted reproducing kernel Hilbert spaces. The setting is based on a sequence of weights γj>0γj>0 and sequences 1+γjk1+γjk and 1+lγj1+lγj of reproducing kernels kk such that H(1+γjk)=H(1+lγj)H(1+γjk)=H(1+lγj), in particular. We derive necessary and sufficient conditions for the norms on ⨂j=1sH(1+γjk) and ⨂j=1sH(1+lγj) to be equivalent uniformly in ss. Furthermore, we study relaxed versions of uniform equivalence by modifying the sequence of weights, e.g., by constant factors, and by analyzing embeddings of the respective spaces. Likewise, we analyze the limiting case s=∞s=∞.