Restrict everything to a ball in $n$ dimensions, let $x$ represent the first $n-1$ variables, and $t$ the $n-$th variable. It is obvious by Holder's Inequality that $$ \int\limits_t\left(\int\limits_x|f|^p\right)^{\frac1p}\leq\left(\int\limits_t\int\limits_x|f|^p\right)^{\frac1p}|\{f\neq0\}|^{1-\frac1p} $$ I am interested in whether the following inequality holds

$$ \int\limits_t\left(\int\limits_x|f|^p\right)^{\frac1p}\leq C\left(\int\limits_t\int\limits_x|f|^{p-\epsilon}\right)^{\frac1{p-\epsilon}} $$ for $\epsilon>0$ and $f$ such that $\Vert f\Vert_{L^p(x)}$ and $\Vert f\Vert_{L^p(x,t)}$ are small ($<1$ in particular). Any results related to this question are welcome.