# Conditions for the embeddig of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$

Let $$I$$ be a compact interval of $$\mathbb{R}$$ and $$U$$ be a bounded subset of $$\mathbb{R}^2$$.

If $$f \in L^\infty(I, W^{1,2}(U))$$, what (non-trivial) condition ($$L^p$$-estimate on $$f$$ or decay-like estimate on the $$W^{1,2}$$-norm of $$f$$) is sufficient to obtain that $$f \in L^\infty(I \times \Omega)$$?

• Can you give some idea of what sort of $L^p$ estimates you have in mind? – Nate Eldredge Dec 11 '18 at 23:36
• @NateEldredge Honestly I don't have a precise idea. – Rene Dec 12 '18 at 1:19
• Well, doesn't $W^{1,2}(U)$ contains functions like $\ln(x^2+y^2)$, which should be in every $L^p$ but unbounded? So I don't see how any simple sort of $L^p$ bound is going to help at all. – Nate Eldredge Dec 12 '18 at 1:34
• @NateEldredge That's true. Maybe some decay-like estimate for some $L^p$ norms or the $H^1$ norm could help? – Rene Dec 12 '18 at 1:43
• What is a decay-like condition on a norm? – Hannes Dec 12 '18 at 10:37