Inequality of strong $L^p$ and weak $L^p$ norm on a finite set with counting measure

Question

If $X$ is a counting finite set with counting measure. Let $f : X \to \mathbb C$ be a complex valued function. For any $ 0 < p < \infty$, show that $$ ||f||_{ L^p } \le C_p (\log (1 + |X| ))^{1/p} ||f||_{L^{p,\infty}} $$ for some constant $C_p$ depending only on $p$. Where $||f||_{ L^p }$ is the strong $L^p$ norm and $||f||_{L^{p,\infty}}$ is the weak $L^p$ norm.

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The hint says normalise $||f||_{ L^{p , \infty}} = 1$, and then manually dispose of the regions of $X$ where $f$ is too large or too small.

But I have no idea about it. Any help is appreciated.

Answer 1

Do the normalization. Remember that $\|f\|_{L^{p,\infty}}^p=p\int_0^\infty \lambda_f(t)t^{p-1}dt$. We have that $\#\{|f(x)|\geq t\} \leq t^{-p}$ but just inputting that isn't good enough so let's throw away some values. Note that the trivial bound $|X|$ is lower that $t^{-p}$ over some small values of $t$, and that if $t>1$, $\#\{|f(x)|\geq t\} <1$ but that value has to be an integer so it is zero. Now break up that integral based on which values of $t$ each bound is relevant for and you should get your answer.

Answer 2

Denote $X:=\{x_1,\dots,x_n\}$. First assume that the number $|f(x_j)|$ are pairwise distinct. Then $$|f(x_j)|^p(N-j+1)=|f(x_j)|^p\cdot \mu\{x, |f(x)|\geqslant |f(x_j)|\}\leqslant C_p\lVert f\rVert_{p,\infty}^p.$$ Then $\lVert f\rVert^p_p\leqslant C_p\lVert f\rVert_{p,\infty}^p\sum_{j=1}^N\frac 1{N-j+1}$.

For the general case, we approximate $f$ by below by functions $f_k$ for which the numbers $|f_k(x_j)|$ are pairwise distinct.