Introduction to Riesz Transforms

What is/are Riesz Transforms?

Riesz Transforms - We consider Riesz transforms of any order associated to an Ornstein–Uhlenbeck operator with covariance given by a real, symmetric and positive definite matrix, and with drift given by a real matrix whose eigenvalues have negative real parts. ^[1] We obtain boundedness criteria in terms of Muckenhoupt weights for the Hardy–Littlewood maximal operator and Riesz transforms in weighted grand Morrey spaces $$M^{p),q,\varphi}_w$$. ^[2] We study $L^p$ bounds for two kinds of Riesz transforms on $\mathbb{R}^d$ related to the harmonic oscillator. ^[3] Our results are based on Riesz transforms, minimax approach and Concentration–compactness Lemma. ^[4] In this paper, we prove that the Riesz transforms associated with $\mathscr{A}$ of order one or two are of weak type $(1,1)$, but that those of higher order are not. ^[5] We prove sharp power-weighted $L^p$, weak type and restricted weak type inequalities for the heat semigroup maximal operator and Riesz transforms associated with the Bessel operator $B_{\nu}$ in the exotic range of the parameter $-\infty < \nu < 1$. ^[6] Our applications include the spectral multipliers and the Riesz transforms associated to Schrodinger operators in various settings, ranging from the magnetic Schrodinger operators in Euclidean spaces to the Laguerre operators. ^[7] As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces. ^[8] In this survey we review this approach in three concrete examples: the Jacobian estimate by Coifman-Lions-Meyer-Semmes, the Coifman-Rochberg-Weiss commutator estimate for Riesz transforms, and a Kato-Ponce-Vega-type inequality. ^[9] As applications, we obtain the HL1$H^{1}_{\mathcal{L}}$-boundedness of Riesz transforms and the imaginary power related to L$\mathcal{L}$. ^[10] We also point out that Riesz transforms associated to H are bounded on some modulation spaces. ^[11] In this work, the boundedness of the spherical maximal function, the mapping properties of the fractional spherical maximal functions, the variation and oscillation inequalities of Riesz transforms on Herz spaces have been established. ^[12] In this sense the Hilbert or Riesz transforms or their tensor products are a representative testing class for Calderón-Zygmund or Journé operators. ^[13] We establish the limiting weak type behaviors of Riesz transforms associated to the Bessel operators on ℝ+; which are closely related to the best constants of the weak type (1; 1) estimates for such operators. ^[14] We study $L^p$ bounds for two kinds of Riesz transforms on $\mathbb{R}^d$ related to the harmonic oscillator. ^[15] Classical Riesz transforms are involved in the Schrödinger-type identity method as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace, and some generalized transforms. ^[16] This paper establishes the boundedness of the variation operators associated with Riesz transforms and commutators generated by the Riesz transforms and BMO-type functions in the Schrödinger setting on the weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class. ^[17] In this paper, the weighted grand Lebesgue spaces with mixed-norms are introduced and boundedness criteria in these spaces of strong maximal functions and Riesz transforms are presented. ^[18] The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. ^[19] By using the denseness of the set of smooth functions with compact support, we obtain the oscillation and variation inequalities of Riesz transforms on this Morrey type space. ^[20] As a consequence we conclude that the Hilbert and Riesz transforms are not compact multilinear transforms. ^[21] We introduce the iterated commutator for the Riesz transforms in the multi-parameter flag setting, and prove the upper bound of this commutator with respect to the symbol $b$ in the flag BMO space. ^[22] The main substance of this counterexample is a result of interest in its own right, that even a weak-type estimate for the Riesz transforms fails on the space $L^1(\mathcal{H}^{d-\beta}_\infty)$, $\beta \in [1,d)$. ^[23] It turns out that the whole class of quasi-monogenic signals has similar properties than the monogenic signal based on the Riesz transforms. ^[24] A key ingredient is the deduction of a new commutator estimate involving Riesz transforms. ^[25] Related to the Schrödinger operator $$L=-\Delta +V$$L=-Δ+V, the behaviour on $$L^p$$Lp of several first and second order Riesz transforms was studied by Shen (Ann Inst Fourier (Grenoble) 45(2):513–546, 1995). ^[26] The aim of this paper is to extend the study of Riesz transforms associated to Dunkl Ornstein-Uhlenbeck operator considered by A. ^[27] 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. ^[28]

riesz transforms associated

In this paper, we prove that the Riesz transforms associated with $\mathscr{A}$ of order one or two are of weak type $(1,1)$, but that those of higher order are not. ^[1] We prove sharp power-weighted $L^p$, weak type and restricted weak type inequalities for the heat semigroup maximal operator and Riesz transforms associated with the Bessel operator $B_{\nu}$ in the exotic range of the parameter $-\infty < \nu < 1$. ^[2] Our applications include the spectral multipliers and the Riesz transforms associated to Schrodinger operators in various settings, ranging from the magnetic Schrodinger operators in Euclidean spaces to the Laguerre operators. ^[3] We also point out that Riesz transforms associated to H are bounded on some modulation spaces. ^[4] We establish the limiting weak type behaviors of Riesz transforms associated to the Bessel operators on ℝ+; which are closely related to the best constants of the weak type (1; 1) estimates for such operators. ^[5] The aim of this paper is to extend the study of Riesz transforms associated to Dunkl Ornstein-Uhlenbeck operator considered by A. ^[6]