Local Integrability of a Maximal Function

In summary, local integrability for a maximal function refers to the property of a function where the integral over any compact set is finite, allowing us to study the behavior of a function on a small scale. It is important because it allows us to define and analyze functions that may not be integrable on a global scale. The Hardy-Littlewood maximal function is a specific example of a maximal function that generates the local integrability condition. Some examples of locally integrable maximal functions include the Hardy-Littlewood maximal function, the Hilbert transform, and the Riesz transform. Local integrability is also related to the dominated convergence theorem, which is often used to prove the local integrability of functions by allowing the interchange of integration and limit under certain
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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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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)##.
 
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