Centered Hardy-Littlewood maximal function on product manifolds

Shiliang Zhao

doi:10.1515/anona-2021-0233

Abstract

Abstract Let X be the direct product of Xi where Xi is smooth manifold for 1 ≤ i ≤ k. As is known, if every Xi satisfies the doubling volume condition, then the centered Hardy-Littlewood maximal function M on X is weak (1,1) bounded. In this paper, we consider the product manifold X where at least one Xi does not satisfy the doubling volume condition. To be precise, we first investigate the mapping properties of M when X 1 has exponential volume growth and X 2 satisfies the doubling condition. Next, we consider the product space of two weighted hyperbolic spaces X 1 = (ℍ n+1, d, yα −n−1 dydx) and X 2 = (ℍ n+1, d, yβ −n−1 dydx) which both have exponential volume growth. The mapping properties of M are discussed for every α , β ≠ n 2 \alpha,\beta \ne {n \over 2} . Furthermore, let X = X 1 × X 2 × … Xk where Xi = (ℍ ni +1, yαi − ni −1 dydx) for 1 ≤ i ≤ k. Under the condition α i > n i 2 {\alpha_i} > {{{n_i}} \over 2} , we also obtained the mapping properties of M.

Highlights

Let S be a smooth manifold and L a second order di erential operator
The maximal function plays an important role in harmonic analysis and is closely related to the theory of singular integral operators, square functions([23])
The main purpose of this paper is to study that the mapping properties of the centered maximal functions on the product manifolds will to what extend be in uenced by that on submanifolds

Summary

Introduction

Let S be a smooth manifold and L a second order di erential operator. Denote by B(x, r) the ball centered at x with radius r > and by V(x, r) the volume of B(x, r) with respect to μ. The centered Hardy-Littlewood maximal function is de ned by: Mf (x) = sup r> V(x, r). The maximal function plays an important role in harmonic analysis and is closely related to the theory of singular integral operators, square functions([23]). If the manifold S supports the Besicovitch covering lemma, one can prove M is weak (1,1) bounded. The Besicovitch covering lemma is not easy to verify. It does not hold on some common manifolds.

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Volume Estimates

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And we have

Since x sup

Note that n

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