Potential operators associated with Jacobi and Fourier-Bessel expansions

We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier-Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those $1 \le p,q \le \infty$, for which the potential operators are of strong type $(p,q)$, of weak type $(p,q)$ and of restricted weak type $(p,q)$. These results may be thought of as analogues of the celebrated Hardy-Littlewood-Sobolev fractional integration theorem in the Jacobi and Fourier-Bessel settings. As an ingredient of our line of reasoning, we also obtain sharp estimates of the Poisson kernel related to Fourier-Bessel expansions.Comment: 28 pages, 4 figures; v2 (some comments on Bessel potentials added

Sharp estimates for potential operators associated with Laguerre and Dunkl-Laguerre expansions

We study potential operators associated with Laguerre function expansions of convolution and Hermite types, and with Dunkl-Laguerre expansions. We prove qualitatively sharp estimates of the corresponding potential kernels. Then we characterize those $1 \le p,q \le \infty$, for which the potential operators are $L^p-L^q$ bounded. These results are sharp analogues of the classical Hardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and Dunkl-Laguerre settings.Comment: 25 pages, 2 figure

Sharp estimates of the potential kernel for the harmonic oscillator with applications

We prove qualitatively sharp estimates of the potential kernel for the harmonic oscillator. These bounds are then used to show that the $L^p-L^q$ estimates of the associated potential operator obtained recently by Bongioanni and Torrea are in fact sharp.Comment: 10 pages, 1 figure; v2 (corrections in Section 3 concerning Theorem 3.1 and its proof and Figure 1

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

Miejsce miejskich obszarów funkcjonalnych w procesie rozwoju regionalnego

Problematyka miejskich obszarów funkcjonalnych nie jest nowym przedmiotem badań naukowych ani na świecie, ani w Polsce. Od kilkudziesięciu lat prowadzone są badania nad obszarami wyróżniającymi się występowaniem relacji przestrzennych i społeczno-gospodarczych między miastem głównym (ośrodkiem rdzeniowym) a jednostkami położonymi w jego najbliższym otoczeniu (strefa peryferyjna). Artykuł zawiera uszczegółowienie terminów stosowanych w delimitacji tych terenów, a także systematyzację teorii rozwoju regionalnego, która stanowi podstawę do określenia roli miejskich obszarów funkcjonalnych, w tym ośrodka rdzeniowego i strefy peryferyjnej, w tych koncepcjach. Głównym celem pracy jest określenie zakładanego w koncepcjach teoretycznych znaczenia miejskich obszarów funkcjonalnych w procesie rozwoju regionalnego

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