# Functional Analysis – On the potential bound of \$ L ^ 2 (0, T; L ^ 1 ( partial U)) \$ norm by \$ L ^ 1 (0, T; L ^ 1 ( partial U)) \$

Let $$U$$ to be an open, bounded and connected subset of $$mathbb R ^ 3$$ with a $$C ^ 2-$$border $$U partial$$. If we know that $$f$$ is delimited in $$L ^ 1 (0, T; L ^ 1 (partial U))$$is there an inequality I could miss for the moment that provides an estimate for: $$int_0 ^ T big ( int _ { partial U} f big) ^ 2$$?

I would like to link this standard using the $$L ^ 1-$$estimate from the top but impossible to do it directly because $$L ^ 1$$ is not injected into $$L ^ 2$$. However, there may be an interpolation-type inequality that could be useful.

Any help is very appreciated. Thanks in advance