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 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).

Weak type estimates for maximal operators with a cylindric distance function

2006 ◽  
Vol 253 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Sunggeum Hong ◽  
Paul Taylor ◽  
Chan Woo Yang
Keyword(s):  
Distance Function ◽  
Weak Type ◽  
Maximal Operators ◽  
Weak Type Estimates

scholarly journals Weak type estimates for the Hardy-Littlewood maximal functions

1974 ◽  
Vol 49 (3) ◽  
pp. 217-223 ◽  
Author(s):  
Luis Caffarelli ◽  
Calixto Calderón
Keyword(s):  
Weak Type ◽  
Maximal Functions ◽  
Weak Type Estimates

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.

Sign in / Sign up

Export Citation Format

Share Document