POTW Local Integrability of a Maximal Function

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Let $f$ be a measurable function supported on some ball $B = B(x,\rho)\subset \mathbb{R}^n$. Show that if $f \cdot \log(2 + |f|)$ is integrable over $B$, then the same is true for the Hardy-Littlewood maximal function $Mf : y \mapsto \sup_{0 < r < \infty}|B(y,r)|^{-1} \int_{B(y,r)} |f(z)|\, dz$.

 

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Spoiler

Note $$\int_B Mf\, dx = \int_{B\cap (Mf < 1)} Mf\, dx + \int_{B\cap (Mf \ge 1)} Mf\, dx \le |B| + \int_0^\infty |B\cap (Mf \ge \max\{1,\lambda\})|\, d\lambda$$ by layer-cake representation. By the weak type (1,1)-estimate of the maximal function, the last integral is controlled by $$|(Mf \ge 1)| + \int_1^\infty \frac{C}{\lambda}\int_{|f| > \lambda/2} |f(x)|\, dx\, d\lambda$$ where $C$ is a constant. By Fubini's theorem the latter expression may be rewritten $$|(Mf \ge 1)| + C\int_{\mathbf{R}^n}\int_1^{2|f|}\, |f|\, \frac{d\lambda}{\lambda}\, dx = |(Mf \ge 1)| + C\int_B |f|\log(2|f|)\, dx$$which, in turn, is dominated by $$|(Mf \ge 1)| + 2C\int_B |f|\log(2 + |f|)\, dx < \infty$$Hence, $Mf\in L^1(B)$.