scholarly journals Riesz transforms of a general Ornstein–Uhlenbeck semigroup

2021 ◽  
Vol 60 (4) ◽  
Author(s):  
Valentina Casarino ◽  
Paolo Ciatti ◽  
Peter Sjögren
Keyword(s):  
Invariant Measure ◽  
Weak Type ◽  
Riesz Transform ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Riesz Transforms ◽  
Real Matrix

AbstractWe 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. In this general Gaussian context, we prove that a Riesz transform is of weak type (1, 1) with respect to the invariant measure if and only if its order is at most 2.

scholarly journals Estimates for Operators Related to the Sub-Laplacian with Drift in Heisenberg Groups

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Hong-Quan Li ◽  
Peter Sjögren
Keyword(s):  
Heisenberg Group ◽  
Weak Type ◽  
Riesz Transforms ◽  
Heisenberg Groups ◽  
Related Measure

AbstractIn the Heisenberg group of dimension $$2n+1$$ 2 n + 1 , we consider the sub-Laplacian with a drift in the horizontal coordinates. There is a related measure for which this operator is symmetric. The corresponding Riesz transforms are known to be $$L^p$$ L p bounded with respect to this measure. We prove that the Riesz transforms of order 1 are also of weak type (1, 1), and that this is false for order 3 and above. Further, we consider the related maximal Littlewood–Paley–Stein operators and prove the weak type (1, 1) for those of order 1 and disprove it for higher orders.

M-convolutions of products and ratios, statistical distributions and fractional calculus

Analysis ◽  
2016 ◽  
Vol 36 (1) ◽  
Author(s):  
Arakaparampil M. Mathai
Keyword(s):  
Fractional Calculus ◽  
Random Variables ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Positive Definite Matrices ◽  
The Real ◽  
Real Scalar ◽  
Scalar Variable ◽  
The Matrix

AbstractIt is shown that Mellin convolutions of products and ratios in the real scalar variable case can be considered as densities of products and ratios of two independently distributed real scalar positive random variables. It is also shown that these are also connected to Krätzel integrals and to the Krätzel transform in applied analysis, to reaction-rate probability integrals in astrophysics and to other related aspects when the random variables have gamma or generalized gamma densities, and to fractional calculus when one of the variables has a type-1 beta density and the other variable has an arbitrary density. Matrix-variate analogues are also discussed. In the matrix-variate case, the M-convolutions introduced by the author are shown to be directly connected to densities of products and ratios of statistically independently distributed positive definite matrix random variables in the real case and to Hermitian positive definite matrices in the complex domain. These M-convolutions reduce to Mellin convolutions in the scalar variable case.

scholarly journals On the dimensional weak-type (1,1) bound for Riesz transforms

2020 ◽  
pp. 2050072
Author(s):  
Daniel Spector ◽  
Cody B. Stockdale
Keyword(s):  
Linear Combination ◽  
Upper Bound ◽  
General Class ◽  
Absolute Constant ◽  
Weak Type ◽  
Riesz Transform ◽  
Type Inequality ◽  
Riesz Transforms ◽  
Finite Linear Combination ◽  
Finite Linear

Let [Formula: see text] denote the [Formula: see text] Riesz transform on [Formula: see text]. We prove that there exists an absolute constant [Formula: see text] such that [Formula: see text] for any [Formula: see text] and [Formula: see text], where the above supremum is taken over measures of the form [Formula: see text] for [Formula: see text], [Formula: see text], and [Formula: see text] with [Formula: see text]. This shows that to establish dimensional estimates for the weak-type [Formula: see text] inequality for the Riesz transforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calderón–Zygmund operators.

scholarly journals Harmonic analysis operators associated with multidimensional Bessel operators

2012 ◽  
Vol 142 (5) ◽  
pp. 945-974 ◽  
Author(s):  
Jorge J. Betancor ◽  
Alejandro J. Castro ◽  
Jezabel Curbelo
Keyword(s):  
Harmonic Analysis ◽  
Maximal Operator ◽  
Weak Type ◽  
Riesz Transforms ◽  
Heat Semigroup ◽  
Strong Type ◽  
Bessel Operators

We establish that the maximal operator and the Littlewood–Paley g-function associated with the heat semigroup defined by multidimensional Bessel operators are of weak type (1, 1). We also prove that Riesz transforms in the multidimensional Bessel setting are of strong type (p, p), for every 1 < p < ∞, and of weak type (1, 1).

scholarly journals Weak Type Estimates for Riesz-Laguerre Transforms

2007 ◽  
Vol 75 (3) ◽  
pp. 397-408 ◽  
Author(s):  
Emanuela Sasso
Keyword(s):  
Weak Type ◽  
Riesz Transforms ◽  
First Order ◽  
Weak Type Estimates

We prove that the first order Riesz transforms associated to the Laguerre semigroup are weak-type (1, 1). We also present a counterexample showing that for the Riesz transforms of order three or higher the weak type (1, 1) estimate fails.

scholarly journals Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces

2021 ◽  
Vol 11 (1) ◽  
pp. 72-95
Author(s):  
Xiao Zhang ◽  
Feng Liu ◽  
Huiyun Zhang
Keyword(s):  
Hilbert Transform ◽  
Besov Spaces ◽  
Singular Integrals ◽  
Riesz Transform ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Variation Operators

Abstract This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ Lq (S n−1) (q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. 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.

scholarly journals RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

2021 ◽  
pp. 1-37
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Riesz Transform ◽  
Wreath Products ◽  
Compact Quantum Group ◽  
Riesz Transforms ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Free Products ◽  
Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

scholarly journals A geometric statement of the Harnack inequality for a degenerate Kolmogorov equation with rough coefficients

2019 ◽  
Vol 21 (07) ◽  
pp. 1850057 ◽  
Author(s):  
Francesca Anceschi ◽  
Michela Eleuteri ◽  
Sergio Polidoro
Keyword(s):  
Maximum Principle ◽  
Harnack Inequality ◽  
Weak Solutions ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Strong Maximum Principle ◽  
Kolmogorov Equation ◽  
Measurable Coefficients ◽  
Rough Coefficients ◽  
Vector Formula

We consider weak solutions of second-order partial differential equations of Kolmogorov–Fokker–Planck-type with measurable coefficients in the form [Formula: see text] where [Formula: see text] is a symmetric uniformly positive definite matrix with bounded measurable coefficients; [Formula: see text] and the components of the vector [Formula: see text] are bounded and measurable functions. We give a geometric statement of the Harnack inequality recently proved by Golse et al. As a corollary, we obtain a strong maximum principle.

Sign in / Sign up

Export Citation Format

Share Document