scholarly journals Calderón–Zygmund Operators Associated to Ultraspherical Expansions

2007 ◽  
Vol 59 (6) ◽  
pp. 1223-1244 ◽  
Author(s):  
Dariusz Buraczewski ◽  
Teresa Martinez ◽  
José L. Torrea
Keyword(s):  
Differential Operator ◽  
Higher Order ◽  
Riesz Transforms ◽  
Homogeneous Type ◽  
Type Space ◽  
Ultraspherical Expansions ◽  
Homogeneous Type Space

AbstractWe define the higher order Riesz transforms and the Littlewood–Paley g-function associated to the differential operator Lλf(θ) = –f′′(θ)–2λ cot θ f′(θ) + λ2f(θ). We prove that these operators are Calderón–Zygmund operators in the homogeneous type space ((0, π), (sin t)2λdt). Consequently, Lp weighted, H1 – L1 and L∞ – BMO inequalities are obtained.

scholarly journals ON LITTLEWOOD-PALEY FUNCTIONS ASSOCIATED WITH BESSEL OPERATORS

2009 ◽  
Vol 51 (1) ◽  
pp. 55-70 ◽  
Author(s):  
J. J. BETANCOR ◽  
J. C. FARIÑA ◽  
A. SANABRIA
Keyword(s):  
Higher Order ◽  
Doubling Measure ◽  
Homogeneous Type ◽  
Type Space ◽  
Bessel Operators ◽  
Homogeneous Type Space

AbstractIn this paper, we study Lp-boundedness properties for higher order Littlewood-Paley g-functions in the Bessel setting. We use the Calderón-Zygmund theory in a homogeneous-type space (in the sense of Coifman and Weiss) ((0, ∞), d, γα), where d represents the usual metric on (0, ∞) and γα denotes the doubling measure on (0, ∞) with respect to d defined by dγα(x) = x2α+1dx, with α > −1/2.

scholarly journals A symmetrized conjugacy scheme for orthogonal expansions

2013 ◽  
Vol 143 (2) ◽  
pp. 427-443 ◽  
Author(s):  
Adam Nowak ◽  
Krzysztof Stempak
Keyword(s):  
Differential Operator ◽  
Higher Order ◽  
Riesz Transforms ◽  
Orthogonal System ◽  
Order Differential Operator ◽  
Orthogonal Expansions ◽  
Partial Derivatives ◽  
Poisson Integrals ◽  
Conjugate Poisson Integrals ◽  
Classical Shape

We establish a symmetrization procedure in the context of general orthogonal expansions associated with a second-order differential operator L, a Laplacian. Combining with a unified conjugacy scheme from an earlier paper by Nowak and Stempak permits, using a suitable embedding, a differential-difference Laplacian $\mathbb{L}$ to be associated with the initially given orthogonal system of eigenfunctions of L, so that the resulting extended conjugacy scheme has the natural classical shape. This means, in particular, that the related partial derivatives decomposing $\mathbb{L}$ are skew-symmetric in an appropriate L2 space and they commute with Riesz transforms and conjugate Poisson integrals. The results also shed new light on the issue of defining higher-order Riesz transforms for general orthogonal expansions.

Hardy spaces on ZN

2002 ◽  
Vol 132 (1) ◽  
pp. 25-43 ◽  
Author(s):  
Santiago Boza ◽  
María J. Carro
Keyword(s):  
Maximal Function ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Classical Case ◽  
Positive Answer ◽  
Riesz Transforms ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Definition Of ◽  
Open Question

The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.In this paper, we give a positive answer to this question.

scholarly journals The Attractors of the Common Differential Operator Are Determined by Hyperbolic and Lacunary Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-19
Author(s):  
Wolf Bayer
Keyword(s):  
Differential Operator ◽  
Taylor Series ◽  
Analytic Functions ◽  
Chaotic Behavior ◽  
Higher Order ◽  
Limit Behavior ◽  
Hyperbolic Functions ◽  
Limit Sets ◽  
The Common

For analytic functions, we investigate the limit behavior of the sequence of their derivatives by means of Taylor series, the attractors are characterized by -limit sets. We describe four different classes of functions, with empty, finite, countable, and uncountable attractors. The paper reveals that Erdelyiéshyperbolic functions of higher orderandlacunary functionsplay an important role for orderly or chaotic behavior. Examples are given for the sake of confirmation.

scholarly journals Riesz Transforms Associated with Higher-Order Schrödinger Type Operators

2017 ◽  
Vol 49 (3) ◽  
pp. 381-410
Author(s):  
Qingquan Deng ◽  
Yong Ding ◽  
Xiaohua Yao
Keyword(s):  
Higher Order ◽  
Riesz Transforms ◽  
Schrödinger Type Operators

scholarly journals Higher order Riesz transforms associated with Bessel operators

2008 ◽  
Vol 46 (2) ◽  
pp. 219-250 ◽  
Author(s):  
Jorge J. Betancor ◽  
Juan C. Fariña ◽  
Teresa Martinez ◽  
Lourdes Rodríguez-Mesa
Keyword(s):  
Higher Order ◽  
Riesz Transforms ◽  
Bessel Operators
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