Riesz transforms for the Dunkl harmonic oscillator

We propose an approach to the theory of Riesz transforms in a framework emerging from certain reflection symmetries in Euclidean spaces. Relying on Rösler’s construction of multivariable generalized Hermite functions associated with a finite reflection group on \({\mathbb R^d}\), we define and investigate a system of Riesz transforms related to the Dunkl harmonic oscillator. In the case isomorphic with the group \({\mathbb{Z}^d_2}\) it is proved that the Riesz transforms are Calderón–Zygmund operators in the sense of the associated space of homogeneous type, thus their mapping properties follow from the general theory.     

Higher Riesz transforms and imaginary powers associated to the harmonic oscillator

Summary We consider higher order Riesz transforms for the multi-dimensional Hermite function expansions. The Riesz transforms occur to be Calderón--Zygmund operators hence their mapping properties follow by using results from a general theory. Then we investigate higher order conjugate Poisson integrals showing that at the boundary they approach appropriate Riesz transforms of a given function. Finally, we consider imaginary powers of the harmonic oscillator by using tools developed for studying Riesz transforms.     

Limiting weak-type behaviors for Riesz transforms and maximal operators in Bessel setting

We establish the limiting weak type behaviors of Riesz transforms associated to the Bessel operators on IR+. which are closely related to the best constants of the weak type (1,1) estimates for such operators. Meanwhile, the corresponding results for Hardy-Littlewood maximal operator and fractional maximal operator in Bessel setting are also obtained.     

Higher-order Riesz operators for the Ornstein–Uhlenbeck Semigroup

We prove that the second-order Riesz transforms associated to the Ornstein–Uhlenbeck semigroup are weak type (1,1) with respect to the Gaussian measure in finite dimension. We also show that they are given by a principal value integral plus a constant multiple of the identity. For the Riesz transforms of order three or higher, we present a counterexample showing that the weak type (1,1) estimate fails.     

Variation Inequalities Related to Schr?dinger Operators on Morrey Spaces

The authors establish the boundedness of the variation operators associated with the heat semigroup, Riesz transforms and commutators generated by the Riesz transforms and BMO-type functions in the Schr?dinger setting on the Morrey spaces.     

Riesz–Jacobi Transforms as Principal Value Integrals

We establish an integral representation for the Riesz transforms naturally associated with classical Jacobi expansions. We prove that the Riesz–Jacobi transforms of odd orders express as principal value integrals against kernels having non-integrable singularities on the diagonal. On the other hand, we show that the Riesz–Jacobi transforms of even orders are not singular operators. In fact they are given as usual integrals against integrable kernels plus or minus, depending on the order, the identity operator. Our analysis indicates that similar results, existing in the literature and corresponding to several other settings related to classical discrete and continuous orthogonal expansions, should be reinvestigated so as to be refined and in some cases also corrected.     

Riesz transforms on a solvable Lie group of polynomial growth

We study Riesz transforms associated with a sublaplacian H on a solvable Lie group G, where G has polynomial volume growth. It is known that the standard second order Riesz transforms corresponding to H are generally unbounded in L^p(G). In this paper, we establish boundedness in L^p for modified second order Riesz transforms, which are defined using derivatives on a nilpotent group G_N associated with G. Our method utilizes a new algebraic approach which associates a distinguished choice of Cartan subalgebra with the sublaplacian H. We also obtain estimates for higher derivatives of the heat kernel of H, and give a new proof (without the use of homogenization theory) of the boundedness of first order Riesz transforms. Our results can be generalized to an arbitrary (possibly non-solvable) Lie group of polynomial growth.     

相伴于非散度型椭圆算子的Riesz变换的L~p有界性

本文利用小波方法在一般阶的非散度椭圆算子的系数BMO模非常小的情形下,证明了广义 Riesz变换的 L~p(2≤p<+∞)有界性。     

Classes of weights related to Schrödinger operators

In this work we obtain boundedness on weighted Lebesgue spaces on Rd of the semi-group maximal function, Riesz transforms, fractional integrals and g-function associated to the Schrödinger operator −Δ+V, where V satisfies a reverse Hölder inequality with exponent greater than d/2. We consider new classes of weights that locally behave as Muckenhoupt's weights and actually include them. The notion of locality is defined by means of the critical radius function of the potential V given in Shen (1995) [8].     

Discrete hilbert transforms and rearrangement invariant sequence spaces

A pair of rearrangement inequalities are obtained for a discrete analogue of the Hilbert transform which lead to necessary and sufficient conditions for certain discrete analogues of the Hilbert transform to be bouonded as linear operators between rearrangement invariant sequence spaces. In particular, if X is a rearrangement invariant space with indices α and β, then 0<β≤α<1 is both necessary and sufficient for these transforms to be bounded from X into itself, which generalizes a well known result of M. Riesz. Applications are made to discerete Hilbert transforms in higher dimensions, in particular, the discrete Riesz transforms are bounded from X into itself if and only if 0<β≤α<1.     

Characterization of Riesz bases of wavelets generated from multiresolution analysis

We investigate Riesz bases of wavelets generated from multiresolution analysis. This investigation leads us to a study of refinement equations with masks being exponentially decaying sequences. In order to study such refinement equations we introduce the cascade operator and the transition operator. It turns out that the transition operator associated with an exponentially decaying mask is a compact operator on a certain Banach space of sequences. With the help of the spectral theory of the compact operator we are able to characterize the convergence of the cascade algorithm associated with an exponentially decaying mask in terms of the spectrum of the corresponding transition operator. As an application of this study we establish the main result of this paper which gives a complete characterization of all possible Riesz bases of compactly supported wavelets generated from multiresolution analysis. Several interesting examples are provided to illustrate the general theory.     

Higher order Riesz transforms,fractional derivatives,and Sobolev spaces for Laguerre expansions

We study different Sobolev spaces associated with multidimensional Laguerre expansions. To do this we establish an analogue of P.A. Meyer's multiplier theorem, prove some transference results between higher order Riesz–Hermite and Riesz–Laguerre transforms, and introduce fractional derivatives and integrals corresponding to the Laguerre setting. Hypercontractivity of the Laguerre semigroups and Calderón's reproduction formula are also discussed.     

Weighted Grand Lebesgue Space with a Mixed Norm and Integral Operators

Doklady Mathematics - Weighted grand Lebesgue spaces with mixed norms are introduced, and criteria for the boundedness of strong maximal functions and Riesz transforms in these spaces are given.     

Riesz Transforms Characterizations of Hardy Spaces $$H^1$$ for the Rational Dunkl Setting and Multidimensional Bessel Operators

We characterize the Hardy space \(H^1\) in the rational Dunkl setting associated with the reflection group \(\mathbb {Z}_2^n\) by means of special Riesz transforms. As a corollary we obtain Riesz transforms characterization of \(H^1\) for product of Bessel operators in \((0,\infty )^n\).     

保不交算子值域的一些性质

研究了保不交算子值域的性质,建立了保不交算子值域为Riesz子空间的一个刻画;又讨论了主理想和主带在保不交算子作用后的象的性质,一些相关结果也得以讨论.     

Characterising Pairs of Refinable Splines

A complete characterisation is given, in terms of Fourier transforms, of pairs of refinable univariate spline functions, with knots at the integers, whose integer translates form a Riesz basis.     

Operators Associated with the Ornstein-Uhlenbeck Semigroup

Some of the arguments and techniques developed by the authorsin a previous paper are applied to the study of the boundednessof certain operators associated with the Ornstein–Uhlenbecksemigroup. In particular, a simple proof is given of the weaktype 1 with respect to the Gaussian measure of the Riesz transformsof order 1 and the Littlewood–Paley g-function which thenis extended to show the same property for the Riesz transformsof order 2. For Riesz transforms of higher order boundednessis shown in appropriate spaces close to L¹.     

On second-order periodic elliptic operators in divergence form

We consider second-order, strongly elliptic, operators with complex coefficients in divergence form on . We assume that the coefficients are all periodic with a common period. If the coefficients are continuous we derive Gaussian bounds, with the correct small and large time asymptotic behaviour, on the heat kernel and all its H?lder derivatives. Moreover, we show that the first-order Riesz transforms are bounded on the -spaces with . Secondly if the coefficients are H?lder continuous we prove that the first-order derivatives of the kernel satisfy good Gaussian bounds. Then we establish that the second-order derivatives exist and satisfy good bounds if, and only if, the coefficients are divergence-free or if, and only if, the second-order Riesz transforms are bounded. Finally if the third-order derivatives exist with good bounds then the coefficients must be constant. Received in final form: 28 February 2000 / Published online: 17 May 2001     

由schr(o)dinger算子定义的Riesz变换的Lp估计

本文主要讨论了当非负位势V(x)属于某逆Holder类时,由一致椭圆算子L=-div(A(x)(△))+V(x)所定义的Riesz变换在Lp空间的有界性.     

Variation Operators for Semigroups and Riesz Transforms on BMO in the Schrödinger Setting

In this paper we prove that the variation operators of the heat semigroup and the truncations of Riesz transforms associated to the Schrödinger operator are bounded on a suitable BMO type space.