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On the weak-type (1,1) of the uncentered Hardy-Littlewood maximal operator associated with certain measures on the plane. (English) Zbl 1316.42021

Let \(\mu\) be a positive Borel measure on \(\mathbb{R}^2\), finite on compact sets and with \(\mu (B)>0\) for all Euclidean balls \(B\). For a function \(f\in L^1_{\mu}(\mathbb{R}^2)\), the uncentered Hardy-Littlewood maximal operator \(M_{\mu}\) is defined by \[ M_{\mu} f(x)=\sup_{x\in B}\frac{1}{\mu(B)}\int_B|f(y)| d\mu(y), \] where \(B\) are open balls.
Assume that \(\mu=\nu\times \lambda\), where \(\nu\) is a measure on the circle \(\mathcal {S}^1\) and \(\lambda\) is a measure on \(\mathbb{R}^+:=(0,+\infty)\). The measure \(\nu\) is doubling if \(\nu(2I)\leq C\nu(I)\) for some \(C>0\) and every interval \(I\subseteq \mathcal{S}^1\). \(\nu\) is somewhere doubling if there exists an interval \(I\subseteq \mathcal{S}^1\) such that \(\nu(2J)\leq C\nu(J)\) for some \(C>0\) whenever \(J\) and \(2J\) are contained in \(I\). The measure \(\lambda\) on \(\mathbb{R}^+\) is doubling away from the origin, that is, there exists a constant \(C>0\) such that for \(0< r\leq 10 a\), \[ \lambda([a, a+2r])\leq C\lambda([a+r/2, a+3r/2]). \]
The main results in this paper are as follows:
{Theorem 1}. Let \(\mu=\nu\times \lambda\) be a measure for which \(\nu\) is doubling and \(\lambda\) is doubling away from the origin. Then \(M_{\mu}\) is of weak type \((1,1)\).
Furthermore, the authors give the necessary and sufficient conditions for the operator \(M_{\mu}\) to be of weak type \((1,1)\).
{Theorem 2}. If either \(\nu\) or \(\lambda\) is somewhere doubling, then \(M_{\mu}\) is of weak type \((1,1)\) if and only if \(\nu\) is doubling and \(\lambda\) is doubling away from the origin.
They also use the upper and lower \(s\)-densities of Radon measures to provide a characterization of some measure \(\mu=\nu\times\lambda\). For a Radon measure \(\sigma\) defined on \(\mathbb{R}^2\), the upper and lower \(s\)-densities are defined by \[ \theta_*^s(\sigma,x)={\lim\inf}_{r\downarrow 0}\frac{\sigma(B(x,r))}{(2r)^s},\quad \theta^{*s}(\sigma,x)={\lim\sup}_{r\downarrow 0}\frac{\sigma(B(x,r))}{(2r)^s}, \] respectively, where \(B(x,r)\) is the closed ball centered at \(x\) with radius \(r\) and \(s>0\).
{Theorem 3}. Let \(\mu=\nu\times \lambda\) for Radon measures \(\nu\) and \(\lambda\). Suppose that either one of the following conditions holds:
(i) there is a set \(A\subset \mathcal{S}^1\), \(\nu(A)>0\), with \(0<\theta_*^s(\nu,x)\leq \theta^{*s}(\nu,x)<\infty\) for \(x\in A\);
(ii) there is a set \(A\subset \mathbb{R}^+\), \(\lambda(A)>0\), with \(0<\theta_*^s(\lambda,x)\leq \theta^{*s}(\lambda,x)<\infty\) for \(x\in A\);
then \(M_{\mu}\) is of weak type \((1,1)\) if and only if \(\nu\) is doubling and \(\lambda\) is doubling away from the origin.
At last, the authors’ results also generalize those of A. M. Vargas [Stud. Math. 110, No. 1, 9–17 (1994; Zbl 0818.42009)] when \(n=2\) by taking \(\nu\) equal to the Lebesgue measure on \(\mathcal{S}^1\).

MSC:

42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0818.42009
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References:

[1] Kaufman, R. and Wu, J. M., Two problems on doubling measures, Rev. Mat. Iberoam.11 (1995), 527-545. · Zbl 0862.28005 · doi:10.4171/RMI/183
[2] Mattila, P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, New York, 1995. · Zbl 0819.28004 · doi:10.1017/CBO9780511623813
[3] Sjögren, P., A remark on the maximal function for measures in \(\mathbb{R}^n\), Amer. J. Math.105 (1983), 1231-1233. · Zbl 0528.42007 · doi:10.2307/2374340
[4] Vargas, A. M., On the maximal function for rotation invariant measures in \(\mathbb{R}^n\), Studia Math.110 (1994), 9-17. · Zbl 0818.42009
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