Singular integral operators with PC symbols on the spaces with general weights (original) (raw)

Matematicas Puras Study of Some P~Operties of Analytic Functions (*)

2016

Reminder.- The field C is an cl qel.rc aver the field R. The elements 1,i farm a basi s of Cover R. An analytic function f, may be regarded as a map: (x,y)~u(x,y) + iv(x,y) = f(x,y), from ~2. intothealge!:.ra C, such that, Thi s formula remains meaningful when is a di stri bution • We are going to denote by K a field equal to R orto C;!:.y A a commutative algebra over K (K- algebra) with unit element, so that K may be identified with a subalgebra of A; by 0, the space of the k v vcl ued C"'funtion,s with compact support defined i R",equipped with the usual topology (Ref. 1, pag 67); and by 0 ' the topological dual space of 0, call ed the space of di stri bution s • Let us fix a base (ax)>.EL of A. Let us denote pr ~ the map r;. z x a:x---? z tA from A onto K. Let L ~ be the space of all K-linear maps T of 0 into A, such thar pr:>,oTEO for every),€L. (*) Thesis work submitted to the Faculty of the Groduate School of the

On Certain Classes of Analytic Functions Defined by a Linear Operator

2011

In this paper, we define and study two new classes of analytic functions which generalize a number of classes studied earlier. Sharp coefficient bound and some inclusion results are discussed. We also observe that these classes are preserved under the Bernadi integral transform. 2000 mathematics subject classification: 30C45 • 30C50

Some Applications on Subclasses of Analytic Functions Involving Linear Operator

2019

The object of this paper is to give some applications of differential subordination concept on subclasses of univalent functions for some convolution operators which are defined on the space of univalent meromorphic functions in the punctured open unit disc. Moreover, we derive some sandwich theorems, we study some geometric properties like coefficient bounds, distortion theorem, and radii of starlikeness and convexity for these classes of functions. Extreme points and integral operator are also investigated.

On analytic families of operators

Israel Journal of Mathematics, 1969

The classical Riesz-Thorin interpolation theorem [6] was extended by Hirschman [2] and Stein [5] to analytic families of operators. We recall the notions: Let F(z), z = x+iy, be analytic in 0< Re z< 1 and continuous in 0 =< Re z _< 1. F(z) is said to be of admissible growth iff Sup log iF(x + iY) I < Ae~Iyl where a < 7z. O<_x~l The significance of this notion is in the following lemma due to Hirschman [2]: LEMMA. lf F(z) is of admissible growth and ifloglF(it) l ~ ao(t), log IF(i+ it) I <= a~(t) then log I F(0) ]_<f_% Po(0, t)ao(t)dt + f 2~ P~(O, t)a~(t)dt where P~(O, t) are the values of the Poisson kernel Jor the strip, on Rez = 0, Rez = 1. We next define analytic families of linear operators: Let (M,/~) (N, v) be two measure spaces. Let {~} be a family of linear operators indexed by z, 0 ~ Re z ~ 1 so that for each z, Tz is a mapping of simple functions on M to measurable functions on N. {T~} is called an analytic family iff for any measurable set E of M of finite measure, for almost every y 6 N, the function qSr(z) = T~(X~)(y) is analytic in 0 < Re z < 1, continuous in 0 ~ Re z __< 1. The analytic family is of admissible growth iff for almost every y ~ N, ~by(z) is of admissible growth. We finally recall the notion of L(p, q) spaces. An exposition of these spaces can be found in Hunt [3]. Let f be a complex valued measurable function defined on a ~-finite measure space (M,/~). # is assumed to be non-negative. We assume that f is finite valued a.e., and denoting Ey = {x/If(x)] > Y}, 2r(y) = /~(Ey), we assume also that for some y > 0, 2;(y)< oo. We define f*(t) = Inf{y > Oily(y) < t}.

Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators

Progress in Mathematics, 1997

Boundedness of the Cauchy singular integral In this chapter our leading actor, the Cauchy singular integral operator S, enters the scene. A very deep theorem, which should actually be Theorem 1 of this book, says that S is bounded on LP(r, w) (1 < p < 00) if and only ifr is a Carleson curve and w is a Muckenhoupt weight in Ap(r). The proof of this theorem is difficult and goes beyond the scope of this book. We nevertheless decided to write down a proof, but this proof will only be given in Chapter 5. The purpose of this chapter is to provide the reader with some facts and results that should suffice to understand Chapters 6 to 10 without browsing in Chapter 5. This chapter contains a proof of the necessity portion of the afore-mentioned theorem, i.e. we show that if S is bounded on LP (r, w) then w E Ap (r). We also give an elementary proof of the sufficiency part for certain classes of oscillating curves ro and oscillating weights woo These curves ro and weights Wo playa fundamental role in the spectral theory of singular integral and Toeplitz operators, because all possible phenomena, or, to be more precise, all possible indicator sets are already attained by these curves and weights: given any Carleson curve r and any weight wE Ap(r), there exist ro and Wo such that Nt(r,p, w) = Nt(ro,p, wo). We conclude the chapter by some technical constructions which imply that in connection with boundedness questions everything can be reduced to the case of unbounded simple arcs. 4.1 The Cauchy singular integral Let r be a simple locally rectifiable curve and equip r with length measure IdTI. Recall that local rectifiability means that r n {z E C : Izl < R} is rectifiable for all R > o. Bounded curves of infinite length, such as are not locally rectifiable in our sense. Thus, a locally rectifiable curve is rectifiable if and only if it is bounded.