Assume that $f(t)$ belongs to $W^{s_1,2}(0,T)$ and $h(x,t)$ belongs to $W^{s_2,2}(0,T;H)$ for some $s_1,s_2<\frac12$ where $H$ is a Hilbert space. It is known that for any $s<s_1+s_2-\frac12$, $s<s_1,s_2$ that the pointwise multiplication of functions is a continuous bilinear map $W^{s_1,2}(\mathbb{R})\times W^{s_2,2}(\mathbb{R}) \rightarrow W^{s,2}(\mathbb{R})$. Does there exist such a result for product of Banach/Hilbert space-valued functions?
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+1 interesting question. Have you seen this paper https://arxiv.org/abs/1512.07379 ? They have a proof of the known result you mention. Does the proof carry over to your case? – JackT Jun 22 '23 at 00:07
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Yes, I have seen that paper. But I believe they too don't answer the question of products of functions in $W^{s_1,p_1}(0,T;H_1)$ and $W^{s_2,p_2}(0,T;H_2)$, where $H_1$ and $H_2$ are infinite dimensional spaces (say Hilbert). – math Jun 22 '23 at 14:43