Lebesgue Space Estimates for Spherical Maximal Functions on Heisenberg Groups

Abstract

We prove $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve currently known bounds on sparse domination for global maximal operators. We also consider lacunary variants and extensions to Métivier groups.

On the Korányi Spherical maximal function on Heisenberg groups

“…They are suitable modifications of the examples in [1,22,24] for the Euclidean case, which were in turn adapted from standard examples for spherical means and maximal functions. The first four counter-examples are similar to the corresponding ones in [23] for the maximal function associated to codimension two spheres on the Heisenberg groups. The fifth one is new and can be viewed as the replacement of a standard Knapp type example.…”

“…In light of the recent L p → L q estimates for the local maximal operator associated with codimension two spheres in the Heisenberg groups [2,23] (see the following remarks), it is natural to seek similar estimates for M . Such an estimate would also imply sparse bounds for the global maximal operator M. Our main theorem contains L p → L q estimates for M which are sharp up to endpoints.…”

“…Remarks 1.3. (i) Non-vanishing Rotational Curvature: A different maximal function on the Heisenberg group, considered in [2,3,[18][19][20]23], is associated to averages over codimension two spheres contained in a dilation invariant subspace of H n . In contrast, the averaging operator A t in (1.1) is associated to a surface of codimension one which respects the non-isotropic dilation structure on H n .…”

Lattice Points Close to the Heisenberg Spheres

“…A number of recent results have appeared investigating mapping properties of maximal averaging operators associated with the Heisenberg groups. See, for instance, [32] and the references therein.…”

On the Korányi spherical maximal function on Heisenberg groups

“…In light of the recent L p → L q estimates for the local maximal operator associated with codimension two spheres in the Heisenberg groups [2,26] (see Remark 1.3), it Fig. 2 The region R in Theorem 1.1, for n = 2 is natural to seek similar estimates for M. Our main theorem contains L p → L q estimates for M which are sharp up to endpoints (with the off-diagonal estimates being new) (Figs.…”