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 $\|f\|\_{L^pL^q}=\big(\int\_X\big(\int\_Y |f|^pd\mu\_2\big)^{q/p}d\mu\_1 \big)^{1/q}$.



Mixed norm inequality
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. 


