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