Singular Integrals with Flag Kernels and Analysis on Quadratic CR Manifolds

Abstract

We study a class of operators on nilpotent Lie groups G given by convolution with flag kernels. These are special kinds of product-type distributions whose singularities are supported on an increasing subspace (0)We show that product kernels can be written as finite sums of flag kernels, that flag kernels can be characterized in terms of their Fourier transforms, and that flag kernels have good regularity, restriction, and composition properties.We then apply this theory to the study of the g b -complex on certain quadratic CR submanifolds of C n . We obtain L p regularity for certain derivatives of the relative fundamental solution of g b and for the corresponding Szego projections onto the null space of g b by showing that the distribution kernels of these operators are finite sums of flag kernels.

Besov and Triebel-Lizorkin Spaces Related to Singular Integrals with Flag Kernels

“…It was proved in [5,6] that flag kernels on R 2 are closely connected with product kernels on R 2 × R. We denote any element of R 2 × R by the 3-tuple x = (x 1 , x 2 , x 3 ), where (x 1 , x 2 ) ∈ R 2 and x 3 ∈ R. We endow R 2 with the following dilation that for any δ > 0 and x = (x 1 , x 2 ) ∈ R 2 , δx = (δx 1 , δ 2 x 2 ) and the norm that x = (x 2 1 + |x 2 |) 1/2 , which is equivalent to |x 1 | + |x 2 | 1/2 . Obviously, the homogeneous dimension of R 2 is 3.…”

“…In order to study the b -complex on certain quadratic CR submanifolds of C n , Nagel, Ricci, and Stein [6] introduced the notion of singular integrals with flag kernels on R n . Since the flag singular integral is a special case of product singular integrals, the boundedness of flag singular integrals on L p (R n ) with p ∈ (1, ∞) is a simple corollary of the boundedness of the corresponding product singular integrals.…”

“…Recently, Han and Lu in [3] developed a corresponding Hardy space theory for flag singular integrals on R n . Motivated by [3,6,7], letting s 1 , s 2 ∈ (−1, 1) and s = (s 1 , s 2 ), in this paper, we introduce the Besov spaceḂ s pq (R 2 ) with p, q ∈ [1, ∞] and the Triebel-Lizorkin spaceḞ s pq (R 2 ) with p ∈ (1, ∞) and q ∈ (1, ∞] associated to singular integrals with flag kernels. Some basic properties, including their dual spaces, some equivalent norm characterizations via Littlewood-Paley functions, lifting properties, and some embedding theorems on these spaces are given.…”

Product kernels adapted to curves in the space

“…This follows from a slight modification of the arguments in Lemma 2.2.3 in [12]. More explicitly, denote by ϕ each function ϕ J in the decomposition (3.6).…”

“…A precise definition of such kernels, which are called "product kernels", was introduced in terms of certain differential inequalities and suitable cancellation conditions. Several conditions on K, guaranteeing the L p boundedness of the operator T , have been introduced [4], and many applications of the product theory to the operators arising in certain boundary value problems have been studied [12], [11]. Moreover, the euclidean spaces R d j , j = 1, 2, have been replaced by appropriate nilpotent groups [9], [12], and by smooth manifolds with a geometry determined by a control distance [10].…”