In this paper, we introduce a class of non-convolution-type Calderón-Zygmund operators, whose kernels are certain sums involving the products of the Daubechies wavelets and their convolutions. And we obtain the continuity on the Besov spaces B 0,q p (1 ≤ p, q ≤∞), which is mainly dependent on the properties of the Daubechies wavelets and Lemari's T1 theorem for Besov spaces.
The paper has two parts. We first briefly survey recent studies on the equivalence problem for real submanifolds in a complex space under the action of biholomorphic transformations. We will mainly focus on some of the recent studies of Bishop surfaces, which, in particular, includes the work of the authors. In the second part of the paper, we apply the general theory developed by the authors to explicitly classify an algebraic family of Bishop surfaces with a vanishing Bishop invariant. More precisely, we let M be a real submanifold of C 2 defined by an equation of the form w = zz + 2Re(z s + az s+1 ) with s≥ 3 and a a complex parameter. We will prove in the second part of the paper that for s≥ 4 two such surfaces are holomorphically equivalent if and only if the parameter differs by a certain rotation. When s = 3, we show that surfaces of this type with two different real parameters are not holomorphically equivalent.
HUANG XiaoJun 1, & YIN WanKe 2 1 Department of Mathematics, Rutgers University, New Brunswick, NJ 08902, USA
This paper focuses on the study of the boundedness of convolution-type Calderón-Zygmund operators on some endpoint Triebel-Lizorkin spaces. Applying wavelets, molecular decomposition and interpolation theory, the author establishes the boundedness on certain endpoint Triebel-Lizorkin spaces F˙10 ,q(2 < q ≤∞) under a very weak pointwise regularity condition.
