The circular maximal operator on Heisenberg radial functions

Lebesgue space estimates for spherical maximal functions on Heisenberg groups

“…The original interest in [25] was in pointwise convergence and ergodic results and hence in L p -estimates for the maximal function A sharp result was proved by Müller and the second author [23] and, independently and by a different method, by Narayanan and Thangavelu [24]; namely for n ≥ 2, the L p boundedness of M holds if and only if p > 2n 2n−1 . It is conjectured that this statement holds true even when n = 1 but this problem is currently still open (see [4] for a recent positive result for M acting on L p (H 1 ) functions of the form x → f • (|x|, x 3 )).…”

$${\mathbf {L}}^{{\mathbf {p}}}$$-Sobolev Estimates for a Class of Integral Operators with Folding Canonical Relations

“…To check that the cone Σ x has one nonvanishing curvature everywhere one verifies that the plane curve 2 . In order to apply our main result one also needs to check that the projection π R (for the adjoint R * β ) has only fold singularities; this turns out to be the case when β = 0. π R is given by…”

“…However in this case sharp L p -Sobolev estimates and maximal function bounds for n ≥ 2 have been recently established in [1], using methods which are closely related to the current paper. For a more recent result on circular maximal functions on the Heisenberg group see [2] where the case of Heisenberg radial functions is considered. 4.3.2.…”

On the Korányi Spherical maximal function on Heisenberg groups

“…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 .…”