For which parameters $0 < p,q,s \leq \infty$ do we have a bound $$ \left\| \left( \sum_{n = 1}^\infty |f_n|^s \right)^{1/s} \right\|_{L^{p,q}(X)} \lesssim_{p,q,s} \left( \sum_{n=1}^\infty \| f_n \|_{L^{p,q}(X)}^s \right)^{1/s}, $$ whenever $\{ f_n \}$ is a family of functions in $L^{p,q}(X)$. For instance, the equation is true when $p = q = s$, in which case $\| f \|_{L^{p,p}(X)} \sim_p \| f \|_{L^p(X)}$ and one can exploit the equation $$ \left\| \left( \sum_{n = 1}^\infty |f_n|^p \right)^{1/p} \right\|_{L^p(X)} = \left( \sum_{n=1}^\infty \| f_n \|_{L^p(X)}^p \right)^{1/p}. $$ For what other parameters is it true?
For a function $f: X \to \mathbf{R}$, the quantity $\| f \|_{L^{p,q}(X)}$ is equal to $$ \| f \|_{L^{p,q}(X)} = \left( \sum_{k = -\infty}^\infty 2^{kq} W_k^{q/p} \right)^{1/q}$$ where $W_k$ is the measure of the set of points $x \in X$ where $2^k \leq |f(x)| < 2^{k+1}$. Equivalently, up to a multiplicative constant, this quantity is proportional to $$ \left( \int_{-\infty}^\infty f^*(s) s^{1/p-1} ds \right) $$ where $f^*$ is the decreasing rearrangement of $f$.