Boundedness of strong maximal functions with respect to non-doubling measures

Abstract

Abstract The main purpose of this paper is to establish a boundedness result for strong maximal functions with respect to certain non-doubling measures in R n $\mathbb{R}^{n}$ . More precisely, let d μ ( x 1 , … , x n ) = d μ 1 ( x 1 ) ⋯ d μ n ( x n ) $d\mu(x_{1}, \ldots, x_{n})=d\mu_{1}(x_{1})\cdots d\mu_{n}(x_{n})$ be a product measure which is not necessarily doubling in R n $\mathbb{R}^{n}$ (only assuming d μ i $d\mu_{i}$ is doubling on R $\mathbb{R}$ for i = 2 , … , n $i=2, \ldots, n$ ), and let ω be a nonnegative and locally integral function such that ω i ( ⋅ ) = ω ( x 1 , … , x i − 1 , ⋅ , x i + 1 , … , x n ) $\omega _{i}(\cdot)=\omega(x_{1}, \ldots, x_{i-1}, \cdot, x_{i+1}, \ldots, x_{n})$ is in A ∞ 1 ( d μ i ) $A_{\infty}^{1}(d\mu_{i})$ uniformly in x 1 , … , x i − 1 , x i + 1 , … , x n $x_{1}, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{n}$ for each i = 1 , … , n − 1 $i=1, \ldots, n-1$ , let d ν = ω d μ $d\nu=\omega \,d\mu$ , ν ( E ) = ∫ E ω ( y ) d μ ( y ) $\nu(E)=\int_{E} \omega(y)\,d\mu(y)$ , and M ω d μ n $M_{\omega \,d\mu}^{n}$ be the strong maximal function defined by M ω d μ n f ( x ) = sup x ∈ R ∈ R 1 ν ( R ) ∫ R | f ( y ) | ω ( y ) d μ ( y ) , $$M_{\omega \,d\mu}^{n} f(x)=\sup_{x\in R\in\mathcal{R}} \frac{1}{\nu (R)} \int_{R} \bigl\vert f(y) \bigr\vert \omega(y)\,d\mu(y), $$ where R $\mathcal{R}$ is the collection of rectangles with sides parallel to the coordinate axes in R n $\mathbb{R}^{n}$ . Then we show that M ω d μ n $M_{\omega \,d\mu}^{n} $ is bounded on L ω d μ p ( R n ) $L^{p}_{\omega \,d\mu}(\mathbb{R}^{n})$ for 1 < p < ∞ $1< p<\infty$ . This extends an earlier result of Fefferman (Am. J. Math. 103:33-40, 1981) who established the L p $L^{p}$ boundedness when d μ = d x $d\mu=dx$ is the Lebesgue measure on R n $\mathbb{R}^{n}$ and d ν = ω d μ $d\nu=\omega \,d\mu$ is doubling with respect to rectangles in R n $\mathbb{R}^{n}$ , ω satisfies a uniform A ∞ 1 $A^{1}_{\infty}$ condition in each of the variables except one. Moreover, we also establish some boundedness result for the Cordoba maximal functions (Córdoba A. in Harmonic Analysis in Euclidean Spaces, pp. 29-50, 1978) associated with the Córdoba-Zygmund dilation in R 3 $\mathbb{R}^{3}$ with respect to some non-doubling measures. This generalizes the result of Fefferman-Pipher (Am. J. Math. 119:337-369, 1997).

Keywords

• strong maximal functions
• non-doubling measures
• A ∞ $A^{\infty}$ weights
• reverse Hölder inequality
• geometric covering lemmas
• Córdoba’s maximal functions