Strongly singular integral operators associated to different quasi-norms on the Heisenberg group

Abstract

Abstract. In this article we study the behavior of strongly singular integrals associated to three different, albeit equivalent, quasi-norms on Heisenberg groups; these quasi-norms give rise to phase functions whose mixed Hessians may or may not drop rank along suitable varieties. In the particular case of the Koranyi norm we improve on the arguments in [7] and obtain sharp L 2 estimates for the associated operators.

L 2 and Hp boundedness of strongly singular operators and oscillating operators on Heisenberg groups

“…(i) a 2 < C β and α ≤ (n + 1 2 )β, (ii) a 2 = C β and α ≤ (n + 1 4 )β, (iii) a 2 > C β and α ≤ (n + 1 3 )β. In [10] Laghi-Lyall reduced the boundedness problem for operators on the Heisenberg group to that for the local operators and used a version of Hömander's L 2 -boundedness theorem on the Heisenberg group. However, as we shall show, we may view the operators on the Heisenberg group as operators on Euclidean space R 2n+1 .…”

“…For p > 1, L p boundedness can be obtained by interpolation between the L 2 boundedness estimates and some L 1 boundedness estimates for dyadic-piece operator. We refer to Laghi-Lyall [10,Theorem 5] for the case a 2 < C β except the endpoint. Using the interpolation technique, we shall get the L p boundedness in the case a 2 ≥ C β .…”

“…The determinant is calculated in Laghi-Lyall [10]. However we give a somewhat simpler computation by considering the matrix L associated naturally with the matrix H (see below), which will also be useful in Lemma 4.6 and the proof of Proposition 3.2 and Proposition 3.3.…”

“…Reviwer's comment: As I said, this is a very well written article. One minor typo that I noticed was two incidences where the Laghi-Lyall [10] was simply referred to as Laghi [10](page 3 line 6 and page 8 line 16), this should be changed.…”

Strongly singular convolution operators on the Heisenberg-type groups

“…对应地, 我们也可以在一些流形上研究类似的强奇异积分算子. 事实上, 在文献 [8,9] 中, 一些强奇异 卷积算子已经在 Heisenberg 群上研究过. 本文的目的是将文献 [8] 中的主要结果通过更简单、有效、 易懂的技巧推广至一般 Heisenberg 型群的情形.…”