Calder\'on-Zygmund operators in the Bessel setting for all possible type indices

In this paper we adapt the technique developed in [17] to show that many harmonic analysis operators in the Bessel setting, including maximal operators, Littlewood-Paley-Stein type square functions, multipliers of Laplace or Laplace-Stieltjes transform type and Riesz transforms are, or can be viewed as, Calder\'on-Zygmund operators for all possible values of type parameter $\lambda$ in this context. This extends the results obtained recently in [7], which are valid only for a restricted range of $\lambda$.Comment: 12 page

On fundamental harmonic analysis operators in certain Dunkl and Bessel settings

We consider several harmonic analysis operators in the multi-dimensional context of the Dunkl Laplacian with the underlying group of reflections isomorphic to $\mathbb{Z}_2^n$ (also negative values of the multiplicity function are admitted). Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes transform type. Using the general Calder\'on-Zygmund theory we prove that these objects are bounded in weighted $L^p$ spaces, $1<p<\infty$, and from $L^1$ into weak $L^{1}$.Comment: 26 pages. arXiv admin note: text overlap with arXiv:1011.3615 by other author

Sharp estimates of the spherical heat kernel

We prove sharp two-sided global estimates for the heat kernel associated with a Euclidean sphere of arbitrary dimension. This solves a long-standing open problem.Comment: 9 pages, to appear in J. Math. Pures Appl. (9

Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex

We prove genuinely sharp two-sided global estimates for heat kernels on all compact rank-one symmetric spaces. This generalizes the authors' recent result obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar heat kernel bounds are shown in the context of classical Jacobi expansions, on a ball and on a simplex. These results are more precise than the qualitatively sharp Gaussian estimates proved recently by several authors.Comment: 16 page

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calder\'on-Zygmund theory we prove that these operators are bounded on weighted $L^p$ spaces, $1 < p < \infty$, and from weighted $L^1$ to weighted weak $L^1$. We also obtain similar results for analogous set of operators in the closely related multi-dimensional Laguerre-symmetrized framework. The latter emerges from a symmetrization procedure proposed recently by the first two authors. As a by-product of the main developments we get some new results in the multi-dimensional Laguerre function setting of convolution type

Riesz-Jacobi transforms as principal value integrals

We establish an integral representation for the Riesz transforms naturally associated with classical Jacobi expansions. We prove that the Riesz-Jacobi transforms of odd orders express as principal value integrals against kernels having non-integrable singularities on the diagonal. On the other hand, we show that the Riesz-Jacobi transforms of even orders are not singular operators. In fact they are given as usual integrals against integrable kernels plus or minus, depending on the order, the identity operator. Our analysis indicates that similar results, existing in the literature and corresponding to several other settings related to classical discrete and continuous orthogonal expansions, should be reinvestigated so as to be refined and in some cases also corrected.Comment: 30 page

Analysis in the multi-dimensional ball

We study the heat semigroup maximal operator associated with a well-known orthonormal system in the d-dimensional ball. The corresponding heat kernel is shown to satisfy Gaussian bounds. As a consequence, we can prove weighted $L^p$ estimates, as well as some weighted inequalities in mixed norm spaces, for this maximal operator.Comment: 18 page

Some remarks on oscillation inequalities

In this paper we establish uniform oscillation estimates on $L^p(X)$ with $p\in(1,\infty)$ for the polynomial ergodic averages. This result contributes to a certain problem about uniform oscillation bounds for ergodic averages formulated by Rosenblatt and Wierdl in the early 1990's. We also give a slightly different proof of the uniform oscillation inequality of Jones, Kaufman, Rosenblatt and Wierdl for bounded martingales. Finally, we show that oscillations, in contrast to jump inequalities, cannot be seen as an endpoint for $r$-variation inequalities.Comment: 23 pages, no figures, accepted for publication in the Ergodic Theory & Dynamical System