Variation and oscillation for semigroups associated with discrete Jacobi operators

Harmonic analysis operators related to symmetrized Jacobi expansions

Following a symmetrization procedure proposed recently by Nowak and Stempak, we consider the setting of symmetrized Jacobi expansions. In this framework we investigate mapping properties of several fundamental harmonic analysis operators, including Riesz transforms, Poisson semigroup maximal operator, Littlewood-Paley-Stein square functions and multipliers of Laplace and Laplace-Stieltjes transform type. Our paper delivers also some new results in the original setting of classical Jacobi expansions.

Maximal operators of exotic and non-exotic Laguerre and other semigroups associated with classical orthogonal expansions

Classical settings of discrete and continuous orthogonal expansions, like Laguerre, Bessel and Jacobi, are associated with second order differential operators playing the role of the Laplacian. These depend on certain parameters of type that are usually restricted to a half-line, or a product of half-lines if higher dimensions are considered. Following earlier research done by Hajmirzaahmad, we deal in this paper with Laplacians in the above-mentioned contexts with no restrictions on the type parameters and bring to attention naturally associated orthogonal systems that in fact involve the cla…

Variation operators associated with the semigroups generated by Schrödinger operators with inverse square potentials

By $\{T_t^a\}_{t>0}$ we denote the semigroup of operators generated by the Friedrichs extension of the Schrödinger operator with the inverse square potential $L_a=-Δ+\frac{a}{|x|^2}$ defined in the space of smooth functions with compact support in $\mathbb{R}^n\setminus\{0\}$. In this paper we establish weighted $L^p$-inequalities for the maximal, variation, oscillation and jump operators associated with $\{t^α\partial_t^αT_t^a\}_{t>0}$, where $α\geq 0$ and $\partial _t^α$ denotes the Weyl fractional derivative. The range of values $p$ that works is different when $a\geq 0$ and when $-\frac{(n…

Self-improving properties for abstract Poincaré type inequalities

We study self-improving properties in the scale of Lebesgue spaces of generalized Poincaré inequalities in the Euclidean space. We present an abstract setting where oscillations are given by certain operators (e.g., approximations of the identity, semigroups or mean value operators) that have off-diagonal decay in some range. Our results provide a unified theory that is applicable to the classical Poincaré inequalities and furthermore it includes oscillations defined in terms of semigroups associated with second order elliptic operators as those in the Kato conjecture. In this latter situation…

Heat kernels of the discrete Laguerre operators

For the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the corresponding norms. On the one hand, this helps us to answer basic questions (recurrence, stochastic completeness) regarding the associated Markovian semigroup. On the other hand, we prove the analogs of the Cwiekel-Lieb-Rosenblum and the Bargmann estimates for perturbations of the Laguerre operators, as well as the optimal Hardy inequality.

根系相关正交展开的Littlewood-Paley理论

Littlewood-Paley theory for orthogonal expansions associated with root systems

本文引入紧致Heckman-Opdam拉普拉斯算子的非对称热与Poisson半群，证明Littlewood-Paley $g$-函数及Riesz变换、位势的$L^p$有界性，并统一了Jacobi展开等特例。

关于极大算子 $L^p$ 范数的研究

On $L^p$ norms of maximal operators

本文确定了多种极大算子的精确 $L^p$ 范数为 $p'$，涵盖热、Poisson 等半群及 Hardy–Littlewood 算子，并证明高维情形范数严格大于 $p'$。