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.

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 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.

Convergence of Marcinkiewicz Means of Integrable Functions with Respect to Two-Dimensional Vilenkin Systems

2004 ◽  
Vol 11 (3) ◽  
pp. 467-478
Author(s):  
György Gát
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Integrable Function ◽  
Two Dimensional ◽  
Vilenkin Groups ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Marcinkiewicz Means

Abstract We prove that the maximal operator of the Marcinkiewicz mean of integrable two-variable functions is of weak type (1, 1) on bounded two-dimensional Vilenkin groups. Moreover, for any integrable function 𝑓 the Marcinkiewicz mean σ 𝑛𝑓 converges to 𝑓 almost everywhere.

scholarly journals Generalized Heisenberg Groups and Shtern's Question

2004 ◽  
Vol 11 (4) ◽  
pp. 775-782
Author(s):  
M. Megrelishvili
Keyword(s):  
Heisenberg Group ◽  
Hilbert Spaces ◽  
Normed Space ◽  
Unitary Representations ◽  
Heisenberg Groups ◽  
Minimal Subgroups ◽  
Weakly Continuous

Abstract Let 𝐻(𝑋) := (ℝ × 𝑋) ⋋ 𝑋* be the generalized Heisenberg group induced by a normed space 𝑋. We prove that 𝑋 and 𝑋* are relatively minimal subgroups of 𝐻(𝑋). We show that the group 𝐺 := 𝐻(𝐿4[0, 1]) is reflexively representable but weakly continuous unitary representations of 𝐺 in Hilbert spaces do not separate points of 𝐺. This answers the question of A. Shtern.

Hermite Conjugate Functions and Rearrangement Invariant Spaces

1973 ◽  
Vol 16 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Kenneth F. Andersen
Keyword(s):  
Weak Type ◽  
Conjugate Function ◽  
Poisson Integral ◽  
Conjugate Functions ◽  
Rearrangement Invariant Spaces ◽  
Image Position ◽  
Invariant Spaces

The Hermite conjugate Poisson integral of a given f ∊ L1(μ), dμ(y)= exp(—y2) dy, was defined by Muckenhoupt [5, p. 247] aswhereIf the Hermite conjugate function operator T is defined by (Tf) a.e., then one of the main results of [5] is that T is of weak-type (1, 1) and strongtype (p,p) for all p>l.

scholarly journals Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space

2020 ◽  
pp. 1-27
Author(s):  
Hong-Quan Li ◽  
Peter Sjögren
Keyword(s):  
Euclidean Space ◽  
Exponential Growth ◽  
Euclidean Distance ◽  
Weak Type ◽  
Riesz Transforms ◽  
Poisson Semigroups

Abstract Let $v \ne 0$ be a vector in ${\mathbb {R}}^n$ . Consider the Laplacian on ${\mathbb {R}}^n$ with drift $\Delta _{v} = \Delta + 2v\cdot \nabla $ and the measure $d\mu (x) = e^{2 \langle v, x \rangle } dx$ , with respect to which $\Delta _{v}$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood–Paley–Stein functions associated with the heat and the Poisson semigroups.

scholarly journals A note on weighted maximal inequalities

1997 ◽  
Vol 40 (1) ◽  
pp. 193-205
Author(s):  
Qinsheng Lai
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Strong Type ◽  
Maximal Inequalities ◽  
Littlewood Maximal Operator

In this paper, we obtain some characterizations for the weighted weak type (1, q) inequality to hold for the Hardy-Littlewood maximal operator in the case 0<q<1; prove that there is no nontrivial weight satisfying one-weight weak type (p, q) inequalities when 0<p≠q< ∞, and discuss the equivalence between the weak type (p, q) inequality and the strong type (p, q) inequality when p≠q.

scholarly journals Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups

2020 ◽  
Vol 10 (03) ◽  
pp. 2050016
Author(s):  
Michael Ruzhansky ◽  
Bolys Sabitbek ◽  
Durvudkhan Suragan
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Hardy Inequality ◽  
Euclidean Distance ◽  
Sobolev Inequalities ◽  
Sharp Constant ◽  
Heisenberg Groups ◽  
Hardy Inequalities ◽  
Angle Function ◽  
Distance To The Boundary

In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: [Formula: see text] which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.

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