It is well known that $H^s(R^d)$ satisfies the product estimate $ \Vert uv \Vert \leq \Vert u \Vert \Vert v \Vert $ for any $ u , v \in H^s(R^d) $ as long as $ d/2 < s $(Equivalently, it is an algebra). I am now interested in multi-variable (I heard them being called 'non-isotropic') Sobolev Spaces $H^{s,r}(R^2)$ with $ \Vert u \Vert_{H^{s,r}(R^2)} = \Vert (1+{\xi}^2)^{s/2} (1+ {\eta}^2)^{r/2} F(u)(\xi,\eta) \Vert_{L^2(R^2)} $. Does anybody know if (and under what conditions) there is a similar product estimate for the aforementioned spaces ? To be more specific, for which $(s,r)$ does $ \Vert uv \Vert_{H^{s,r}} \leq \Vert u \Vert_{H^{s,r}} \Vert v \Vert_{H^{s,r}} $ hold for all $ u,v \in H^{s,r} $ ? Additionally, are there any good sources regarding those types of Sobolev-spaces ?
2026-04-12 13:31:00.1776000660
Product estimate in multi-variable (non-isotropic) Sobolev Spaces $H^{s,r}(R^2)$
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