Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators

In this paper we study weighted mixed norm estimates for Riesz transforms associated to Dunkl harmonic oscillators. The idea is to show that the required inequalities are equivalent to certain vector valued inequalities for operator defined in terms of Laguerre expansions. In certain cases the main result can be deduced from the corresponding result for Hermite Riesz transforms.Comment: 25 page

Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups

This thesis is divided into three parts. In the first part we study mixed norm estimates for Riesz transforms associated with various differential operators. First we prove the mixed norm estimates for the Riesz transforms associated with Dunkl harmonic oscillator by means of vector valued inequalities for sequences of operators defined in terms of Laguerre function expansions. In certain cases, the result can be deduced from the corresponding result for Hermite Riesz transforms, for which we give a simple and an independent proof. The mixed norm estimates for Riesz transforms associated with other operators, namely the sub-Laplacian on Heisenberg group, special Hermite operator on C^d and Laplace-Beltrami operator on the group SU(2) are obtained using their L^pestimates and by making use of a lemma of Herz and Riviere along with an idea of Rubio de Francia. Applying these results to functions expanded in terms of spherical harmonics, we deduce certain vector valued inequalities for sequences of operators defined in terms of radial parts of the corresponding operators. In the second part, we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian ∆_κ on weighted L^P-spaces. In the general case, for the chaotic behavior of the Dunkl-heat semigroup on weighted L^p-spaces, we only have partial results, but in the case of the heat semigroup generated by the standard Laplacian, a complete picture of the chaotic behavior is obtained on the spaces L^p ( R^d,〖 (φ_iρ (x ))〗^2 dx) where φ_iρ the Euclidean spherical function is. The behavior is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar. In the last part, we study mixed norm estimates for the Cesáro means associated with Dunkl-Hermite expansions on〖 R〗^d. These expansions arise when one considers the Dunkl-Hermite operator (or Dunkl harmonic oscillator)〖 H〗_κ:=-Δ_κ+|x|^2. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Cesáro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters involved and apply a version of Three Lines Lemma

Mixed norm estimates for the Riesz transforms on SU (2)

In this paper we prove mixed norm estimates for Riesz transforms on the group SU(2). From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced

Mixed norm estimates for the Ces\`aro means associated with Dunkl--Hermite expansions

Our main goal in this article is to study mixed norm estimates for the Ces\`{a}ro means associated with Dunkl--Hermite expansions on $\mathbb{R}^d$. These expansions arise when one consider the Dunkl--Hermite operator (or Dunkl harmonic oscillator) $H_{\kappa}:=-\Delta_{\kappa}+|x|^2$, where $\Delta_{\kappa}$ stands for the Dunkl--Laplacian. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Ces\`{a}ro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters involved and apply a version of three lines lemma.Comment: 24 pages. Revised version following referee's comments. To appear in Transactions of the American Mathematical Societ

Mixed norm estimates for the Riesz transforms on SU (2)

In this paper we prove mixed norm estimates for Riesz transforms on the group SU(2). From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced

ON THE CHAOTIC BEHAVIOR OF THE DUNKL HEAT SEMIGROUP ON WEIGHTED L-p SPACES

In this paper we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian on weighted L-p spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces L-p(R-n, (phi(i rho)(x))(2)dx) where phi(i rho) is the Euclidean spherical function. The behavior is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar

ON THE CHAOTIC BEHAVIOR OF THE DUNKL HEAT SEMIGROUP ON WEIGHTED L-p SPACES

In this paper we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian on weighted L-p spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces L-p(R-n, (phi(i rho)(x))(2)dx) where phi(i rho) is the Euclidean spherical function. The behavior is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar

Mixed norm estimates for the Riesz transforms on the Heisenberg group

In this paper we prove weighted mixed norm estimates for Riesz transforms on the Heisenberg group and Riesz transforms associated to the special Hermite operator. From these results vector-valued inequalities for sequences of Riesz transforms associated to generalised Grushin operators and Laguerre operators are deduced