Deducing that $f$ is in $L^p$

Question

Suppose $f : \mathbb{R} \to \mathbb{R}$ is integrable and there is positive $K$ and $0<c<1$ such that $$\int_{B} \left\vert \:f(x)\right\vert \:\mathrm{d}x \leq Km(B)^{c}$$ for every Borel measurable subset $B$ of $\mathbb{R}$. Here $m$ is Lebesgue measure.

Prove that $f \in L^{p}(\mathbb{R})$ for some $p>1$.

If $f \in L^{p}$, then $K$, $c$ are given by $\left\vert \left\vert\,f\,\right\vert\right\vert_p$ and $1-\frac{1}{p}$ by the Holder's inequality. This is the converse problem.

Answer 1

Let $p:=\frac 1{1-c}$. We have $\sup_t t^p\lambda\{|f|\geqslant t\}\lt\infty$. Indeed, fix $t$, take $B:=\{|f|\geqslant t\}$ and use the relationship $t\lambda(B)\leqslant \int_B |f(x)|\mathrm d\lambda(x)$.

We then conclude that $f\in\mathbb L^q$ for any $1\leqslant q\lt p$ using the formula $$\int |h|^q\mathrm d\lambda=q\int_0^\infty t^{q-1}\lambda\{|h|\geqslant t\}\mathrm dt.$$

This is linked with the $\mathbb L^{p,\infty}$ (weak $\mathbb L^p$ spaces), defined by $f\in\mathbb L^{p,\infty}$ if $\sup_t t^p\lambda\{|f|\geqslant t\}\lt\infty$. For $p\gt 1$, we can endow them with the norm $$\lVert f\rVert_{p,\infty}:=\sup_{\lambda(A)\gt 0}\lambda(A)^{1-\frac 1p}\int_A|f|\mathrm d\lambda.$$ We have $\mathbb L^{p,\infty}\subset\mathbb L^q$ for $1\leqslant q\lt p$.

Answer 2

Suppose $c = \frac{1}{q}$. Then for all indicator functions $\chi_B$ we have $$\left| \int_\mathbb{R} \! f \chi_B \, dx\right| \le \int_B \! |f| \, dx \le Km(B)^{1/q} = K\|\chi_B\|_q.$$

Now for some simple function $\chi = \sum_{j} c_j \chi_j$ and $B_j = \mbox{supp }\chi_j$ we deduce the same estimate: $$\left| \int_\mathbb{R} \! f \chi \, dx\right| \le \sum_j |c_j|\int_{B_j} \! |f| \, dx\le K\sum_j (|c_j|^q m(B_j))^{1/q}$$ $$ = \sum_j K\|c_j\chi_j\|_q \le K\|\chi\|_q.$$ For the last inequality we used that the $\chi_j$ have disjoint supports and applied Jensen's inequality to the concave function $x \to x^{1/q}$. By density, $g \to \int_{\mathbb{R}} \! fg \, dx$ extends to a continuous linear functional on $L^q(\mathbb{R})$. By the Riesz Representation Theorem, $f \in L^p(\mathbb{R})$ for $p^{-1} + q^{-1} = 1$.