## 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

Zbl 0818.42009
Full Text:

### 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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