Centered Hardy-Littlewood maximal function on product manifolds

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.