In Terence Tao's notes on harmonic analysis (http://www.math.ucla.edu/~tao/247a.1.06f/notes1.pdf) problem 6.2 on page 18 asks a (what I suspect to be easy) question:
if $f: \mathbf{R}^{+} \rightarrow \mathbf{R}^{+}$ is a monotone non-increasing function, show that $$\lVert f\rVert_{L^{p,q}(X,\mu)} = \lVert f(t)t^{1/p}\rVert_{L^{q}(\mathbf{R}^{+},\frac{dt}{t})} $$
Following his cue, I'm trying to prove this first for a smoother function, but I'm having a hard time applying the hypothesis and working with the distribution function.