Spherical means on the Heisenberg group: Stability of a maximal function estimate

Abstract

Consider the surface measure µ on a sphere in a nonvertical hyperplane on the Heisenberg group H n , n ≥ 2, and the convolution f * µ. Form the associated maximal function M f = sup t>0 |f * µt| generated by the automorphic dilations. We use decoupling inequalities due to Wolff and Bourgain-Demeter to prove L p -boundedness of M in an optimal range.

Lebesgue space estimates for spherical maximal functions on Heisenberg groups

“…(ii) Under the smallness condition (1.4), we give an alternative proof of the result for maximal operators in [1] which relied on decoupling estimates to prove local L p → L p (2n−1)/p ′ regularity results for the averaging operators acting on compactly supported functions on H n , in the range 1 < p < 4n+2 2n+3 . Our approach is to use L p space-time estimates instead.…”

“…(8.2) below). If q > p the operator norm in such estimates will depend on I and it is no loss of generality to assume I = [1,2]. In [3] it is proved that M maps L p (H n ) to L q (H n ) provided that ( 1p , 1 q ) belongs to the interior of the triangle with corners (0, 0), ( 2n−1 2n , 2n−1 2n ), ( 3n+1 3n+7 , 6 3n+7 ).…”

“…Our approach is to use L p space-time estimates instead. However, the L p -Sobolev result in [1] is interesting in its own right, and is still needed for the L p (H n ) boundedness of the maximal operator in the range p > 2n 2n−1 if one does not impose any condition on Λ. The use of L 2 space-time estimates is implicit already in the work by Narayanan and Thangavelu [24] who use the group Fourier transform on the Heisenberg group to estimate a relevant square-function involving generalizations of spherical means.…”

Circular maximal functions on the Heisenberg group

“…We can observe that 1 2 (ξ + ξ ′ ) and 1 2 (η + η ′ ) coincide, which means that (η, η ′ ) satisfies (ξ, ξ ′ ) ∼ µ,ν (η, η ′ ) in (4.1) with (η, η ′ ) = (ξ, ξ ′ ). (4.4) which shows the existence of an elliptic conjugate of (ξ, ξ ′ ).…”

“…More recently in [1], the horizontal plane is replaced by the general hyperplane in the Heisenberg group H n . The purpose of this paper is to prove the L p (H 1 ) boundedness of the circular maximal average as Nevo and Thangavelu conjectured (1.4) with n = 1:…”