Jorge J. Betancor, Juan C. Fariña, Lourdes Rodriguez-Mesa, Alejandro Sanabria-Garcia
In this paper we investigate Lp-boundedness properties for the higher order Riesz transforms associated with Laguerre operators. Also we prove that the k-th Riesz transform is a principal value singular integral operator (modulus a constant times of the function when k is even). To establish our results we exploit a new identity connecting Riesz transforms in the Hermite and Laguerre settings.
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We show that for $δ$ below certain critical indices there are functions whose Jacobi or Laguerre expansions have almost everywhere divergent Cesaro and Riesz means of order $δ$.
In this paper using the well known asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials. We develop a transference method to obtain the $L^p$-continuity of the Gaussian-Riesz transform and the $L^p$-continuity of the Laguerre-Riesz transform from the $L^p$-continuity of the Jacobi-Riesz transform, in dimension one.
In this paper we establish $L^p$-boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions.
Riesz transforms, Hardy spaces and Campanato spaces associated with Laguerre expansions
本文研究与拉盖尔展开相关的Riesz变换，证明其为Calderón-Zygmund算子以解决公开问题，建立对应Hardy空间与Campanato空间理论，并证明该变换在这些空间上有界，完善了其有界性刻画。
The aim of this paper is to prove Kolmogorov-Riesz type theorems via Bessel and Laguerre translations, and Pego-type theorems by the corresponding transformations.
Hardy spaces and Campanato spaces associated with Laguerre expansions and higher order Riesz transforms
本文针对拉盖尔微分算子\mathcal{L}_ν，建立了全范围\\\\(p \in (0,1]\\\\)的相关哈代空间理论，证明了对应BMO型空间是其对偶空间，并验证了高阶里斯变换在勒贝格空间等上的有界性。
In this paper we prove weighted mixed norm estimates for Riesz transforms associated to Hermite and special Hermite operators. The estimates are shown to be equivalent to vectorvalued esimates for a sequence of operators defined in terms of Laguerre functions of different type.
In this paper we will study the boundedness of Riesz Potentials, Bessel potentials and Fractional Derivatives on Gaussian Besov-Lipschitz spaces $B_{p,q}^α(γ_d)$. Also these results can be extended to the case of Laguerre or Jacobi expansions and even further to the general framework of diffusions semigroups.
In this paper we show weighted estimates in mixed norm spaces for the Riesz transform associated with the harmonic oscillator in spherical coordinates. In order to prove the result we need a weighted inequality for a vector-valued extension of the Riesz transform related to the Laguerre expansions which is of independent interest. The main tools to obtain such extension are a weighted inequality for the Riesz transform independent of the order of the involved Laguerre functions and an appropriate adaptation of Rubio de Francia's extrapolation theorem.
Weighted norm inequalities of higher-order Riesz transforms associated with Laguerre expansions
该研究针对与拉盖尔展开相关的高阶里斯变换，通过分析微分算子L_ν及偏导数δ_{ν_j}，探究其加权估计，补全了高阶里斯变换在全范围参数下有界性的描述。
Weighted norm inequalities of some singular integrals associated with Laguerre expansions
本文对拉盖尔算子$\mathcal L_ν$相关的奇异积分（含极大函数、里斯变换和平方函数）进行加权估计研究，完成了里斯变换在$ν\in (-1,\infty)^n$全范围的刻画，显著改进了已有文献结果。
In a previous paper the boundedness properties of Riesz Potentials, Bessel potentials and Fractional Derivatives were studied in detail on Gaussian Besov-Lipschitz spaces $B_{p,q}^α(γ_d)$. In this paper we will continue our study proving the boundedness of those operators on Gaussian Triebel-Lizorkin spaces $F_{p,q}^α(γ_d)$. Also these results can be extended to the case of Laguerre or Jacobi expansions and even further to the general framework of diffusions semigroups.
We develop a technique of proving standard estimates in the setting of Laguerre function expansions of convolution type, which works for all admissible type multi-indices $α$ in this context. This generalizes a simpler method existing in the literature, but being valid for a restricted range of $α$. As an application, we prove that several fundamental operators in harmonic analysis of the Laguerre expansions, including maximal operators related to the heat and Poisson semigroups, Riesz transforms, Littlewood-Paley-Stein type square functions and multipliers of Laplace and Laplace-Stieltjes tra…
In this paper we give sufficient conditions on a measurable function $p:(0,\infty)^n\rightarrow [1,\infty)$ in order that harmonic analysis operators (maximal operators, Riesz transforms, Littlewood--Paley functions and multipliers) associated with $α$-Laguerre polynomial expansions are bounded on the variable Lebesgue space $L^{p(\cdot)} ((0,\infty)^n, μ_α)$, where $dμ_α(x)=2^n\prod_{j=1}^n \frac{x_j^{2α_j+1} e^{-x_j^2}}{Γ(α_j+1)} dx$, being $α=(α_1, \dots, α_n)\in [0,\infty)^n$ and $x=(x_1,\dots,x_n)\in (0,\infty)^n$.
We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calderón-Zygmund theory we prove that these operators are bounded on weighted $L^p$ spaces, $1 < p < \infty$, and from weighted $L^1$ to weighted weak $L^1$. …
Analogous of Riesz potentials and Riesz transforms are defined and studied for the Dunkl transform associated with a family of weighted functions that are invariant under a reflection group. The $L^p$ boundedness of these operators is established in certain cases.
In this paper, an analogous of Heisenberg inequality is established for Laguerre-Bessel transform. Also, a local uncertainty principle for this transform is investigate
W prove a dimension-free estimate for the $L^2(\mathbb{R}^d)$ norm of the maximal truncated Riesz transform in terms of the $L^2(\mathbb{R}^d)$ norm of the Riesz transform. Consequently, the vector of maximal truncated Riesz transforms has a dimension-free estimate on $L^2(\mathbb{R}^d)$. We also show that the maximal function of the vector of truncated Riesz transforms has a dimension-free estimate on all $L^p(\mathbb{R}^d)$ spaces, $1<p<\infty.$
In this paper we obtain the $L^p$-boundedness of Riesz transforms for Dunkl transform for all $1<p<\infty$.
The solution of conformable Laguerre differential equation using conformable Laplace transform
本文利用合形拉普拉斯变换求解合形拉盖尔及关联拉盖尔微分方程，验证解与幂级数解一致，还讨论了拉盖尔多项式性质并提出证明合形Rodriguez公式与生成函数。
In this paper we give a criterion to prove boundedness results for several operators from $H^1((0,\infty),γ_α)$ to $L^1((0,\infty),γ_α)$ and also from $L^\infty((0,\infty),γ_α)$ to $\BMO((0,\infty),γ_α)$, with respect to the probability measure $dγ_α(x)=\frac{2}{Γ(α+1)} x^{2α+1} e^{-x^2} dx$ on $(0,\infty)$ when ${α>-\frac12}$. We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood-Paley functions, multipliers of Laplace transform type, fractional integrals and variation operators in the Laguerre setting.
We present a new criterion for the weighted $L^p-L^q$ boundedness of multiplier operators for Laguerre and Hermite expansions that arise from a Laplace-Stieltjes transform. As a special case, we recover known results on weighted estimates for Laguerre and Hermite fractional integrals with a unified and simpler approach.
We define and investigate a system of Riesz transforms related to the Dunkl harmonic oscillator.
We estimate in Lp the maximal Riesz transform in terms of the Riesz transform itself for p greater than 1. In the limiting case p=1 the weak L1 inequality is shown to fail. Surprisingly, the weak L1 inequality for the maximal Beurling transform in terms of the Beurling transform does hold.
We define and study Riesz transforms and conjugate Poisson integrals associated with multi-dimensional Jacobi expansions.
In the present paper, we establish that Riesz transforms for Dunkl Hermite expansion as introduced in [4] are singular integral operators with Hörmander's type conditions and we show that are bounded on $L^p(\mathbb{R}^d; dμ_k) 1 < p < 1.
This paper provides a constructive proof of the weak factorizations of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of ${\rm BMO}(\mathbb{R}^n)$ (the dual of $H^1(\mathbb{R}^n)$) via commutators of the multilinear Riesz transforms.
The Riesz transform is a natural multi-dimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of a wavelet often has slow decay. One can nevertheless overcome this problem by requiring the original wavelet to have sufficient smoothness, decay, and vanishing moments. In this paper, we derive necess…
In this article we prove dimension free $L^p$-boundedness of Riesz transforms associated with a Bessel diferential operator. We obtain explicit estimates of the $L^p$-norms for the Bessel-Riesz transforms in terms of p, establishing a linear behaviour with respect to p. We use the Bellman function technique to prove a bilinear dimension free inequality involving Poisson semigroups defined through this Bessel operator.
We investigate Laplace type and Laplace-Stieltjes type multipliers in the $d$-dimensional setting of the Dunkl harmonic oscillator with the associated group of reflections isomorphic to $\mathbb{Z}_2^d$ and in the related context of Laguerre function expansions of convolution type. We use Calderón-Zygmund theory to prove that these multiplier operators are bounded on weighted $L^p$, $1<p<\infty$, and from $L^1$ to weak $L^1$.
For 0<n<d integers and r>2, we prove that an n-dimensional Ahlfors-David regular measure M in R^d is uniformly n-rectifiable if and only if the r-variation for the Riesz transform with respect to M is a bounded operator in L^2(M). This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the L^2(M) boundedness of the Riesz transform to the uniform rectifiability of M.
We study weighted $(L^p, L^q)$-boundedness properties of Riesz potentials and fractional maximal functions for the Dunkl transform. In particular, we obtain the weighted Hardy-Littlewood-Sobolev type inequality and weighted week $(L^1, L^q)$ estimate. We find a sharp constant in the weighted $L^p$-inequality, generalizing the results of W. Beckner and S. Samko.
In this paper we represent the $k$-th Riesz transform in the ultraspherical setting as a principal value integral operator for every $k\in \mathbb{N}$. We also measure the speed of convergence of the limit by proving $L^p$-boundedness properties for the oscillation and variation operators associated with the corresponding truncated operators.
We discuss weighted estimates for the squares of the Riesz transforms, R^{2}, on L^{2}(W) where W is a matrix A2 weight. We prove that if W is close to the Identity matrix Id, then the operator norm of R^{2} is close to its unweighted norm on L^{2} which is one. This is done by the use of the Bellman function technique.
This paper is the continuation of the study on discrete harmonic analysis related to Jacobi expansions initiated in [1]. Considering the operator $\mathcal{J}^{(α,β)}=J^{(α,β)}-I$, where $J^{(α,β)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator, we focus on the study of weighted inequalities for the Riesz transform associated with it.
The aim of this paper is to estimate the $L^2$-norms of vector-valued Riesz transforms $R_ν^s$ and the norms of Riesz operators on Cantor sets in $\R^d$, as well as to study the distribution of values of $R_ν^s$. Namely, we show that this distribution is "uniform" in the following sense. The values of $|R_ν^s|^2$ which are comparable with its average value are attended on a "big" portion of a Cantor set. We apply these results to give examples demonstrating the sharpness of our previous estimates for the set of points where Riesz transform is large, and for the corresponding Riesz capacities. …
Let $\mathcal G$ be a stratified Lie group and $\{\X_j\}_{1 \leq j \leq n}$ a basis for the left-invariant vector fields of degree one on $\mathcal G$. Let $Δ= \sum_{j = 1}^n \X_j^2 $ be the sub-Laplacian on $\mathcal G$ and the $j^{\mathrm{th}}$ Riesz transform on $\mathcal G$ is defined by $R_j:= \X_j (-Δ)^{-\frac{1}{2}}$, $1 \leq j \leq n$. In this paper we give a new version of the lower bound of the kernels of Riesz transform $R_j$ and then establish the Bloom-type two weight estimates as well as a number of endpoint characterisations for the commutators of the Riesz transforms and BMO fu…
In this paper, let $L=L_{0}+V$ be a Schrödinger type operator where $L_{0}$ is higher order elliptic operator with complex coefficients in divergence form and $V$ is signed measurable function, under the strongly subcritical assumption on $V$, the authors study the $L^{q}$ boundedness of Riesz transforms $\nabla^{m}L^{-1/2}$ for $q\leq 2$ and obtain a sharp result. Furthermore, the authors impose extra regularity assumptions on $V$ to obtain the $L^{q}$ boundedness of Riesz transforms $\nabla^{m}L^{-1/2}$ for $q>2$. As an application, the main results can be applied to the operator $L=(-Δ)^{m}…
We prove that the energy conditions in arXiv:1302.5093v7 are implied by the fractional A2 conditions and testing conditions for the vector of fractional Riesz transforms when one measure is supported on a line. Then we apply the main theorem in arXiv:1302.5093v7 to obtain a T1 theorem for the fractional Riesz transforms when one weight is supported on a line.
We provide a study of the Riesz transforms on stratified nilpotent Lie groups, and obtain a certain version of the pointwise lower bound of the Riesz transform kernel. Then we establish the characterisation of the BMO space on stratified nilpotent Lie groups via the boundedness of the commutator of the Riesz transforms and the BMO function. This extends the well-known Coifman, Rochberg, Weiss theorem from Euclidean space to the setting of stratified nilpotent Lie groups. In particular, these results apply to the well-known example of the Heisenberg group. As an application, we also study the c…
Riesz-Jacobi transforms as principal value integrals
本文为经典Jacobi展开的Riesz变换建立积分表示，证明奇数阶变换为含不可积奇异性的主值积分，偶数阶为非奇异算子，并指出相关文献结果需重新审视与修正。
Given a domain D in R^d with mild geometric measure theoretic assumptions on its boundary, we show that boundedness of the principal value Riesz tranforms (witn kernel of homogeneity -(d-1)) on Hölder spaces of order alpha on the boundary of D is equivalent to D being a Lyapunov domain of order alpha (i.e., the boundary of D is an hypersurface of class 1+alpha). Another equivalent condition involving Riesz transforms on D is discussed. We also prove that on Lyapunov domains of order alpha the higher order Riesz transforms associated with an odd polynomial are bounded on the Hölder space of ord…
$L^p$ norm of truncated Riesz transform and an improved dimension-free $L^p$ estimate for maximal Riesz transform
该研究利用径向Fourier乘子分部积分法，证明极大截断Riesz变换的$L^p$范数界无维数依赖，给出具体估计式，同时得到截断Riesz变换的范数收缩性与精确范数等式。
Riesz transform and commutators in the Dunkl setting
该研究在 Dunkl 框架下刻画了 Dunkl Riesz 变换核的最优逐点大小与正则性估计，建立了仅用欧氏度量的核下界，证明了交换子的上下界分别对应 Dunkl 与欧氏度量的 BMO 空间，并讨论了紧性与两类 VMO 性质。
The linearization problem for nonlinear second-order ODEs to the Laguerre form by means of generalized Sundman transformations (S-transformations) is considered, which has been investigated by Duarte et al. earlier. A characterization of these S-linearizable equations in terms of first integral and procedure for construction of linearizing S-transformations has been given recently by Muriel and Romero. Here we give a new characterization of S-linearizable equations in terms of the coefficients of ODE and one auxiliary function. This new criterion is used to obtain the general solutions for the…
We study the problem of an appropriate choice of derivatives associated with discrete Fourier-Bessel expansions. We introduce a new so-called essential measure Fourier-Bessel setting, where the relevant derivative is simply the ordinary derivative. Then we investigate Riesz transforms and Sobolev spaces in this context. Our main results are $L^p$-boundedness of the Riesz transforms (even in a multi-dimensional situation) and an isomorphism between the Sobolev and Fourier-Bessel potential spaces. Moreover, throughout the paper we collect various comments concerning two other closely related Fou…
Let $M_1$, $\cdots$, $M_\ell$ be complete, connected and non-collapsed manifolds of the same dimension, where $2\le \ell\in\mathbb{N}$, and suppose that each $M_i$ satisfies a doubling condition and a Gaussian upper bound for the heat kernel. If each manifold $M_i$ has volume growth either bigger than two or equal to two, then we show that the Riesz transform $\nabla Ł^{-1/2}$ is bounded on $L^p(M)$ for each $1<p<2$ on the gluing manifold $M=M_1\#M_2\#\cdots \# M_\ell$.
Let $G$ be the Lie group given by the semidirect product of $R^2$ and $R^+$ endowed with the Riemannian symmetric space structure. Let $X_0, X_1, X_2$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $Δ=-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_iΔ^{-1/2}$ and $S_i=Δ^{-1/2}X_i$, for $i=0,1,2$. We prove that the operators $R_i$, but not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show that the second order Riesz transforms $T_{ij}=X_iΔ^{-1}X_j$ are bounded from $H^1$ …
The fractional Riesz transform and their commutator in Dunkl setting
本文研究 Dunkl 设置下分数阶 Riesz 变换的有界性，并给出其交换子在中心 BMO 与 Dunkl BMO 空间有界性的充要条件，进而用 VMO 空间刻画交换子的紧性。
In the work of S. Petermichl, S. Treil and A. Volberg it was explicitly constructed that the Riesz transforms in any dimension $n \geq 2$ can be obtained as an average of dyadic Haar shifts provided that an integral is nonzero. It was shown in the paper that when $n=2$, the integral is indeed nonzero (negative) but for $n \geq 3$ the nonzero property remains unsolved. In this paper we show that the integral is nonzero (negative) for $n=3$. The novelty in our proof is the delicate decompositions of the integral for which we can either find their closed forms or prove an upper bound.
Let $L=-Δ+V$ be a Schrödinger operator acting on $L^2(\mathbb R^n)$, $n\ge1$, where $V\not\equiv 0$ is a nonnegative locally integrable function on $\mathbb R^n$. In this article, we will introduce weighted Hardy spaces $H^p_L(w)$ associated with $L$ by means of the area integral function and study their atomic decomposition theory. We also show that the Riesz transform $\nabla L^{-1/2}$ associated with $L$ is bounded from our new space $H^p_L(w)$ to the classical weighted Hardy space $H^p(w)$ when $\frac{n}{n+1}<p<1$ and$w\in A_1\cap RH_{(2/p)'}$.
Let $\mathcal L=-Δ+V$ be a Schrödinger operator, where $Δ$ is the Laplacian on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_q$ for $q\geq d$. The Riesz transform associated with the operator $\mathcal L=-Δ+V$ is denoted by $\mathcal R=\nabla{(-Δ+V)}^{-1/2}$ and the dual Riesz transform is denoted by $\mathcal R^{\ast}=(-Δ+V)^{-1/2}\nabla$. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class $RH_q$ for $q\geq d$. Then we will establish the boundedness pro…
Endpoint estimates for higher order Gaussian Riesz transforms
本文证明高阶高斯Riesz变换在原子Hardy空间到L1空间的有界性，并推广Bruno的一阶结果，适用于任意阶数和维数。
Maximal Riesz transform in terms of Riesz transform on quantum tori and Euclidean space
本文在量子环面和量子欧几里得空间上建立了最大Riesz变换的$L_p$有界性，并证明当$2\leq p<\infty$时范数常数与维数无关。
Dimension-free $L^p$ estimates for higher order maximal Riesz transforms in terms of the Riesz transforms
本文证明了高阶极大Riesz变换向量可以用对应的Riesz变换表示的无维数$L^p(\mathbb{R}^d)$估计，并给出了常数对$p$的显式依赖。
Let $M$ be a complete non-compact Riemannian manifold. In this paper, we derive sufficient conditions on metric perturbation for stability of $L^p$-boundedness of the Riesz transform, $p\in (2,\infty)$. We also provide counter-examples regarding in-stability for $L^p$-boundedness of Riesz transform.
We study the validity of the $L^p$ inequality for the Riesz transform when $p>2$ and of its reverse inequality when $p<2$ on complete Riemannian manifolds under the doubling property and some Poincaré inequalities.
In this paper, we introduce a discrete Riesz transforms associated with the non-symmetric trigonometric Heckman-Opdam polynomials of type $A_1$. We prove that they can be extended to a bounded operators on $\ell^p(\mathbb{Z})$, $1<p<\infty$.
We prove that directional wavelet projections and Riesz transforms are related by interpolatory estimates. The exponents of interpolation depend on the Hölder estimates of the wavelet system. This paper complements and continues previous work on Haar projections.
On a complete non-compact Riemannian manifold $M$, we prove that a so-called quasi Riesz transform is always $L^p$ bounded for $1<p\leq 2$. If $M$ satisfies the doubling volume property and the sub-Gaussian heat kernel estimate, we prove that the quasi Riesz transform is also of weak type $(1,1)$.
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