Spectral estimates of Cauchy's transform inL 2(?) (original) (raw)
Journal of Interpolation and Approximation in Scientific Computing, 2012
In this article, we introduce a new class of analytic functions of the unit disc D namely the Exponential Cauchy Transforms K e defined by f (z) = ∫ T exp [K (xz)] dµ(x) where K (z) = (1 − z) −1 is classical Cauchy kernel and µ(x) is a complex Borel measures and x belongs to the unit circle T. We use Laguerre polynomials to explore the coefficients of the Taylor expansions of the kernel and Peron's formula to study the asymptotic behavior of the Taylor coefficients. Finally we investigate relationships between our new class K e , the classical Cauchy space K and the Hardy spaces H p .
Propagation of analytic and Gevrey singularities for operators with non- involutive characteristics
Journal of Mathematics of Kyoto University, 1993
Propagation of analytic and Gevrey singularities for operators with non-involutive characteristics By Massimo CICOGNANI and Luisa ZANGHIRATI I. Introduction In this paper we cosider a class of analytic operators with multiple non-involutive characteristics and study the propagation of analytic and Gevrey singularities. To state precisely our result, we begin by recalling that f E C-(X), X an open set in Rn, is said to be of Gevrey class G', 1<d <00, at x0E X if there exists a neighborhood V of xo, VC X, such that SUVIDal(X)1 C l a l + l a ! d X E for a constant C independent on a E Z + n. We denote by G d (X) the space of all f E C-(X) which are of class G d at every xoE X , and write Go d (X) for G d (X)n co-(x). ci(x) is the space of analytic functions in X. For d >1, the spaces of d-ultradistributions G (d) '(X), Go (d) "(X) are the dual spaces of G d (X) and Go d (X) respectively and G (d) "(X) can be identified with the space of all elements of Go (d) "(X) with compact support. We recall also that the space D '(X) of all distributions in X can be identified with a subspace of Go (d Y (X) for a ll d > 1. If l< d then we have Go (d Y(X)OE Go (d i) "(X). The d-wave front set WFd(f) of f E Go (d) "(X), d >1, is defined as follows: for a fixed (xo, eo)E T*(X)\0, we say that (xo, eo)qWFd(f) if there exist Go d (X) with 0(x)=1 in a neighborhood of xo, and positive constants C, E such that the Fourier transform (çbf)^ of Of satisfies: (1.2) 1 (sbf)"()1 < cexp(-Elel("fl for all e in a conic neighborhood of eo. We also define the 1-wave front set (analytic wave front set) WFI(f) of f EGo (d r(X), d >1: let us consider a sequence {q5,1c Go d (X) , q5,(x)=1 in a neighborhood of xo, such that there exists a constant C and for every 6 >0 a consotant Cc satisfying Communicated by Prof. N. Iwasaki,
Analytic Continuation of Cauchy and Exponential Transforms
International Society for Analysis, Applications and Computation, 2001
We review some recent results concerning analytic continuation properties of the Cauchy transform of a domain in the complex plane, of a corresponding exponential transform and of the resolvent of a hyponormal operator associated with the domain. The main result states the equivalence between the mentioned analytic continuations. As a corollary we obtain apriori regularity of boundaries admitting analytic continuation of the Cauchy transform.
On families of Cauchy transforms and BMOA
Journal of the Australian Mathematical Society, 2000
In this paper we prove a number of results on Cauchy transforms of generalized type given by Borel measures supported on the class of analytic functions mapping the unit disc into the unit disk. We also give a BMOA characterization using these families.
Cauchy Transforms and Univalent Functions
Fields Institute Communications, 2013
We use a formula of Pommerenke relating the primitives of functions which are the Cauchy transforms of measures on the unit circle to their behavior in the space of functions of bounded mean oscillation. This is a linear process and it has some smoothness. Further, there is a non-linear map from the Cauchy transforms into the normalized univalent functions. We show that for the subspace H 1 of Cauchy transforms the univalent functions so obtained have quasi-conformal extensions to all of the plane.
On the asymptotic expansion of Tian-Yau-Zelditch
2011
We give a purely complex geometric proof of the existence of the Bergman kernel expansion. Our method actually provides a sharper estimate, and in the case that the metrics are real analytic, we prove that the remainder decays faster than any polynomial.