Suppose we have a product space $(X_1\times X_2,d\mu=d\mu_1\times d\mu_2)$, each with finite measure and $p>1$.
Is there a possibility that an inequality of this form holds on the product space?
$$\|f\|_{L^pL^p}\leq  C_1\|f\|_{L^1L^p} + C_2\|f\|_{L^pL^1}$$
where $\|f\|_{L^pL^q}=\big(\int_X\big(\int_Y |f|^pd\mu_2\big)^{q/p}d\mu_1 \big)^{1/q}$.