Mixed norm inequality

Question

Suppose we have a product space $(X_1\times X_2,\mu_1\otimes\mu_2)$, with finite measures $\mu_1,\mu_2$ 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 $\|f\|_{L^pL^q}=\big(\int_X\big(\int_Y |f|^pd\mu_2\big)^{q/p}d\mu_1 \big)^{1/q}$.

Answer 1

The answer is no. E.g., suppose that $X_1=X_2=[0,1]$, $\mu_1=\mu_2=$ Lebesgue measure, $f(x_1,x_2)=g(x_1)g(x_2)$, $g=1_{[0,u]}$, $u\in(0,1)$. Then your proposed inequality becomes $$u^{2/p}\le(C_1+C_2)u^{1+1/p},$$ which fails to hold for any given real $p>1$, $C_1$, $C_2$ if $u$ is small enough.