Don't understand a $L^\infty$ bound argument involving measure of set

Question

I'm trying to understand the proof of Proposition 2.2, part 2 of this paper. this is where I am stuck.

For any $k > 0$, we have $$k^{\frac{2(N+1)}{N}}|\{|u|^m > k\}| \leq c_1k^{\frac{2(m-1)}{mN}+1}(D)^{\frac{N+2}{N}}.$$ This is supposed to imply: $$\lVert u \rVert_{L^{m+2/N, \infty}(\Omega \times (0,T))} \leq C_1(D)^{\frac{N+2}{mN+2}}$$ but I have no idea how.

I am also unsure of what exactly the norm on the LHS is. the paper does not say.

Answer 1

If $p\gt 1$, we define the $L^{p,\infty}$ semi-norm by $$\lVert f\rVert_{p,\infty}^p:=\sup_{t\gt 0}t^p\lambda\{s, |f(s)|\gt t\}$$ (this is equivalent to a norm, namely, $\sup_{A,\lambda(A)\in (0,\infty)}\mu(A)^{1/p-1}\int_A|f|\mathrm d\lambda$).

If we define $x:=k^{1/m}$ and if we use the inequality, we obtain $$x^{2m\frac{N+1}N}\lambda\{|u|\lt x\}\leqslant c_1(D)^{\frac{N+2}N}x^{2\frac{m-1}N+m},$$ which gives the wanted estimate.