Best constants in the weak-type estimates for uncentered maximal operators (Q2902688)

Let \(\mu\) be a Borel measure on \(\mathbb{R}^N\). The uncentered maximal function of a function \(f\) defined on \(\mathbb{R}^N\) with respect to \(\mu\) is given by the expression NEWLINE\[NEWLINE({\mathcal M}_{\mu} f)(x)=\sup_{x\in B} \frac{1}{\mu(B)} \int_{B} |f| \;d\mu,NEWLINE\]NEWLINE where the supremum is taken over all closed balls \(B\) containing the point \(x\).NEWLINENEWLINEThe main result of the paper consists of establishing the following weak-type inequality NEWLINE\[NEWLINE\|{\mathcal M}_{\mu} f \|_{L^{q,\infty} (A, \mu)} \leq C_p \|f\|_{L^p(\mathbb{R}^N,\mu)} \mu(A)^{\frac{1}{q}-\frac{1}{p}},NEWLINE\]NEWLINE provided that \(A\) is any Borel subset of \(\mathbb{R}^N\) and \(1\leq p <\infty\), \(q\in(0,p]\).NEWLINENEWLINEMoreover, it is shown that the constants NEWLINE\[NEWLINEC_p=\displaystyle{\frac{(p-1)(2^{p/p-1}-1)}{p} ((p-1)(2^{p/p-1}-2))^{-1/p}}NEWLINE\]NEWLINE for \(1<p<\infty\), and \(C_1=2\), are the best possible in the previous estimate in the case that \(\mu\) is equal to the Lebesgue measure.NEWLINENEWLINEA similar result is obtained when the weak norms of \(f\) and \({\mathcal M}_{\mu} f\) are compared, in that case, with some other constant \(c_p\) involved. The results are an extension, to the weak case, of some sharp \(L^p\) estimates for \({\mathcal M}_{\mu}\) obtained by \textit{L. Grafakos} and \textit{S. Montgomery-Smith} [Bull. Lond. Math. Soc. 29, No. 1, 60--64 (1997; Zbl 0865.42020)] in the case \(1<p<\infty\), also for \(\mu\) equal to the Lebesgue measure.NEWLINENEWLINEAs an application, some related optimal weak-type bounds are obtained for the strong maximal operator where balls are replaced by rectangles with sides parallel to the axes and \(\mu\) is a product measure on \(\mathbb{R}^N\) whose factors are Borel measures on \(\mathbb{R}\).