scholarly journals Heat semigroup and singular PDEs

2016 ◽  
Vol 270 (9) ◽  
pp. 3344-3452 ◽  
Author(s):  
I. Bailleul ◽  
F. Bernicot
Keyword(s):  
Heat Semigroup ◽  
Singular Pdes

scholarly journals Heat kernels of the discrete Laguerre operators

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
Aleksey Kostenko
Keyword(s):  
Hardy Inequality ◽  
Jacobi Polynomials ◽  
Heat Kernels ◽  
The Other ◽  
Heat Semigroup ◽  
Stochastic Completeness ◽  
Markovian Semigroup ◽  
Other Hand ◽  
The One

AbstractFor 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.

Boundedness of Differential Transforms for Heat Semigroups Generated by Schrödinger Operators

2020 ◽  
pp. 1-34
Author(s):  
Zhang Chao ◽  
José L. Torrea
Keyword(s):  
Maximal Operator ◽  
Singular Integrals ◽  
Heat Semigroup ◽  
Real Sequence ◽  
Differential Transform ◽  
Intermediate Size ◽  
Heat Semigroups ◽  
Image Position ◽  
Reverse Hölder ◽  
Littlewood Maximal Operator

Abstract In this paper we analyze the convergence of the following type of series $$\begin{eqnarray}T_{N}^{{\mathcal{L}}}f(x)=\mathop{\sum }_{j=N_{1}}^{N_{2}}v_{j}\big(e^{-a_{j+1}{\mathcal{L}}}f(x)-e^{-a_{j}{\mathcal{L}}}f(x)\big),\quad x\in \mathbb{R}^{n},\end{eqnarray}$$ where ${\{e^{-t{\mathcal{L}}}\}}_{t>0}$ is the heat semigroup of the operator ${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V$ with $\unicode[STIX]{x1D6E5}$ being the classical laplacian, the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q}$ with $q>n/2$ and $n\geqslant 3$ , $N=(N_{1},N_{2})\in \mathbb{Z}^{2}$ with $N_{1}<N_{2}$ , ${\{v_{j}\}}_{j\in \mathbb{Z}}$ is a bounded real sequences, and ${\{a_{j}\}}_{j\in \mathbb{Z}}$ is an increasing real sequence. Our analysis will consist in the boundedness, in $L^{p}(\mathbb{R}^{n})$ and in $BMO(\mathbb{R}^{n})$ , of the operators $T_{N}^{{\mathcal{L}}}$ and its maximal operator $T^{\ast }f(x)=\sup _{N}T_{N}^{{\mathcal{L}}}f(x)$ . It is also shown that the local size of the maximal differential transform operators (with $V=0$ ) is the same with the order of a singular integral for functions $f$ having local support. Moreover, if ${\{v_{j}\}}_{j\in \mathbb{Z}}\in \ell ^{p}(\mathbb{Z})$ , we get an intermediate size between the local size of singular integrals and Hardy–Littlewood maximal operator.

scholarly journals Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

2020 ◽  
Vol 59 (3) ◽  
Author(s):  
Patricia Alonso-Ruiz ◽  
Fabrice Baudoin ◽  
Li Chen ◽  
Luke Rogers ◽  
Nageswari Shanmugalingam ◽  
...  
Keyword(s):  
Heat Kernel ◽  
Heat Semigroup ◽  
Dirichlet Spaces ◽  
Kernel Estimates ◽  
Besov Class ◽  
Heat Kernel Estimates ◽  
Bv Functions

scholarly journals On the summability of divergent power series satisfying singular PDEs

2019 ◽  
Vol 357 (3) ◽  
pp. 258-262 ◽  
Author(s):  
Hua Chen ◽  
Zhuangchu Luo ◽  
Changgui Zhang
Keyword(s):  
Power Series ◽  
Divergent Power Series ◽  
Singular Pdes

scholarly journals A note on bounded variation and heat semigroup on Riemannian manifolds

2007 ◽  
Vol 76 (1) ◽  
pp. 155-160 ◽  
Author(s):  
A. Carbonaro ◽  
G. Mauceri
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Heat Kernel ◽  
Ricci Curvature ◽  
Bounded Variation ◽  
Riemannian Manifolds ◽  
Heat Semigroup ◽  
Functions Of Bounded Variation ◽  
Fixed Radius ◽  
Lower Bound Estimate

In a recent paper Miranda Jr., Pallara, Paronetto and Preunkert have shown that the classical De Giorgi's heat kernel characterisation of functions of bounded variation on Euclidean space extends to Riemannian manifolds with Ricci curvature bounded from below and which satisfy a uniform lower bound estimate on the volume of geodesic balls of fixed radius. We give a shorter proof of the same result assuming only the lower bound on the Ricci curvature.

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