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

Singular spherical maximal operators on a class of degenerate two-step nilpotent Lie groups

“…The proof of (i) and (ii) of Theorem 2.1 will be given in Sections 4 and 5 below, respectively. As pointed out in the introduction, p > d/(d − 1) is necessary in the above theorem and it can be seen by testing M on the function f given by f (y, v) = |y| 1−d (log |y|) −1 χ(y, v) with a suitable cutoff function χ, which equals to 1 if |(y, v)| ≤ 1 2 and supp χ ⊆ B(0, 1). Let µ be the induced Lebesgue measure on the sphere S d−1 .…”

“…which are uniform in t ∈ [1,2]. The proof of (4.15), (4.16), (4.17) and (4.18) will be given in Sections 4.1.2 and 4.1.3 below, which are obtained by using the cancellation of the kernels of K j,k and s d ds K j,k s to show almost orthogonality properties for these operators and certain estimates for oscillatory integrals to establish the decay estimates.…”

“…J T = −J). The most prominent examples are the Heisenberg groups H n which arise when d = 2n and J = 1 2 J 1 with…”

“…In [7], Müller and Seeger independently proved this optimal range for p by using Fourier integral operators in a study which concerns more general surfaces Σ with a nonvanishing rotational curvature in "non-degenerate" two-step nilpotent groups including all H-type groups. This topic has attracted a lot of attention in the last decades, and has been a very active research topic in harmonic analysis; see, for example [1,3,4,7,6,9,10,12,13] and the references therein for further details.…”

“…We remark that p > d/(d − 1) is necessary in Theorem 1.1 and it can be seen by testing M on the function f given by f (y, v) = |y| 1−d (log |y|) −1 χ(y, v) with a suitable nonnegative cutoff function χ, which equals to 1 if |(y, v)| ≤ 1 2 and supp χ ⊆ B(0, 1) (see [15, Chapter XI] and Section 2 below). However, it is not clear for us to establish L p boundedness for range d/(d − 1) < p ≤ (d − 1)/(d − 2) for d ≥ 4.…”