# Lorentz norm of a monotone non-increasing function

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.

• Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures. – José Carlos Santos Aug 7 '18 at 9:12
• Sorry, my bad! I'll remember that for next time – user582283 Aug 7 '18 at 9:15