Adam Sikora

Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators

Alessio Martini, Adam Sikora

We study the Grushin operators acting on $R_{x^{'}}^{d_{1}} \times R_{x "}^{d_{2}}$ and defined by the formula \[ L=-\sum_{\jone=1}^{d_1}\partial_{x'_\jone}^2 - (\sum_{\jone=1}^{d_1}|x'_\…

Small time asymptotics of diffusion processes

A. F. M. ter Elst, Derek W. Robinson, Adam Sikora

We establish the short-time asymptotic behaviour of the Markovian semigroups associated with strongly local Dirichlet forms under very general hypotheses. Our results apply to a wi…

math.FA2023

Hardy spaces meet harmonic weights revisited

Marcin Preisner, Adam Sikora

We investigate Hardy spaces $H_{L}^{1} (X)$ corresponding to self-adjoint operators $L$ . Our main aim is to obtain a description of $H_{L}^{1} (X)$ in terms of atomic decompositions similar…

Comparison of the classical BMO with the BMO spaces associated with operators and applications

Donggao Deng, Xuan Thinh Duong, Adam Sikora +1

Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMO_L space associated with L which was introduced recently by Duong…

math.FA2024

Harmonic functions for Bessel operators

MichaÅ Dymowski, Marcin Preisner, Adam Sikora

We verify the continuity of the Riesz transform from the operator related Hardy space to $L^{1}$ - Lebesgue space of integrable functions. For the standard Euclidean Laplace operator…

math.AP2019

Riesz transforms on a class of non-doubling manifolds II

Andrew Hassell, Daniel Nix, Adam Sikora

We consider a class of manifolds $M$ obtained by taking the connected sum of a finite number of $N$ -dimensional Riemannian manifolds of the form $(\mathbb{R}^{n_i}, Î´) \…

math.AP2016

Gaussian heat kernel estimates: from functions to forms

Thierry Coulhon, Baptiste Devyver, Adam Sikora

On a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there…

math.FA2010

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Xuan Thinh Duong, Adam Sikora, Lixin Yan

Let $L$ be a non-negative self adjoint operator acting on $L^{2} (X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{- t L}$ whose kerne…

math.AP2014

GruÅ¡in operators, Riesz transforms and nilpotent Lie groups

Derek W Robinson, Adam Sikora

We establish that the Riesz transforms of all orders corresponding to the GruÅ¡in operator $H_{N} = - \nabla_{x}^{2} - ∣ x ∣^{2 N} \nabla_{y}^{2}$ , and the first-order operators $(\nabla_{x},x^…

Riesz meets Sobolev

Thierry Coulhon, Adam Sikora

We show that the $L^{p}$ boundedness, $p > 2$ , of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to…

math.AP2016

Spectral multipliers via resolvent type estimates on non-homogeneous metric measure spaces

Peng Chen, Adam Sikora, Lixin Yan

We describe a simple but surprisingly effective technique of obtaining spectral multiplier results for abstract operators which satisfy the finite propagation speed property for th…

Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means

Peng Chen, El Maati Ouhabaz, Adam Sikora +1

We consider abstract non-negative self-adjoint operators on $L^{2} (X)$ which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we…

math.AP2008

Multivariable spectral multipliers and quasielliptic operators

Adam Sikora

We study multivariable spectral multipliers $F (L_{1}, L_{2})$ acting on Cartesian product of ambient spaces of two self-adjoint operators $L_{1}$ and $L_{2}$ . We prove that if $F$ satisfies…

math.CA2002

Bochner-Riesz summability for analytic functions on the m-complex unit sphere and for cylindrically symmetric functions on R^{n-1} times R

Adam Sikora, Terence Tao

We prove that spectral projections of Laplace-Beltrami operator on the m-complex unit sphere E_{Delta_{S^{2m-1}}}([0,R)) are uniformly bounded as an operator from H^p(S^{2m-1}) to…

Markov uniqueness of degenerate elliptic operators

Derek W. Robinson, Adam Sikora

Let $I^c ◯$ be an open subset of $\Ri^d$ and $H_{I^} c ◯ = - \sum_{i, j = 1}^{d} \partial_{i} c_{ij} \partial_{j}$ a second-order partial differential operator on $L_{2} (I^c ◯)$ with domain $C_{c}^{\infty} (I^c ◯)$ …

math.AP2014

Bochner-Riesz profile of anharmonic oscillator $L = - \frac{d ^{2}}{d x ^{2}} + ∣ x ∣$

Peng Chen, Waldemar Hebisch, Adam Sikora

We investigate spectral multipliers, Bochner-Riesz means and convergence of eigenfunction expansion corresponding to the SchrÃ¶dinger operator with anharmonic potential ${\mathcal…

Degenerate elliptic operators in one dimension

Derek W. Robinson, Adam Sikora

Let $H$ be the symmetric second-order differential operator on $L_2(\Ri)$ with domain $C_c^\infty(\Ri)$ and action $H I ¨ = - (c I ¨ ^{'})^{'}$ where $ c\in W^{1,2}_{\rm loc}(\Ri)$ is a real fu…

math.AP2024

Vertical Maximal Functions on Manifolds with Ends

Himani Sharma, Adam Sikora

We consider the setting of manifolds with ends which are obtained by compact perturbation (gluing) of ends of the form $R^{n_{i}} \times M_{i}$ . We investigate family…

math.MG2017

Gradient estimates for heat kernels and harmonic functions

Thierry Coulhon, Renjin Jiang, Pekka Koskela +1

Let $(X, d, I^¼)$ be a doubling metric measure space endowed with a Dirichlet form $\E$ deriving from a "carrÃ© du champ". Assume that $(X,d,Î¼,\E)$ supports a scale-invariant $L^{2}$ -P…

Restriction and spectral multiplier theorems on asymptotically conic manifolds

Colin Guillarmou, Andrew Hassell, Adam Sikora

The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $d E (I^»)$ of the square root of the Laplacian on $\RR^n$ is bounded from $L^p(…

math.AP2020

Probabilistic approach to quantum separation effect for Feynman-Kac semigroup

Adam Sikora, Jacek Zienkiewicz

Quantum tunnelling phenomenon allows a particle in SchrÃ¶dinger mechanics tunnels through a barrier that it classically could not overcome. Even the infinite potentials do not alwa…

Sharp spectral multipliers for operators satisfying generalized Gaussian estimates

Adam Sikora, Lixin Yan, Xiaohua Yao

Let $L$ be a non-negative self adjoint operator acting on $L^{2} (X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{- t L}$ whose kerne…

math.AP2018

Riesz transforms on a class of non-doubling manifolds

Andrew Hassell, Adam Sikora

We consider a class of manifolds $M$ obtained by taking the connected sum of a finite number of $N$ -dimensional Riemannian manifolds of the form $(\mathbb{R}^{n_i}, Î´) \…

math.AP2010

$L_{1}$ -uniqueness of degenerate elliptic operators

Derek W. Robinson, Adam Sikora

Let $I^c ◯$ be an open subset of $\Ri^d$ with $0 \in I^c ◯$ . Further let $H_{I^} c ◯ = - \sum_{i, j = 1}^{d} \partial_{i} c_{ij} \partial_{j}$ be a second-order partial differential operator with domain $…

Gaussian heat kernel upper bounds via PhragmÃ©n-LindelÃ¶f theorem

Thierry Coulhon, Adam Sikora

We prove that in presence of $L^{2}$ Gaussian estimates, so-called Davies-Gaffney estimates, on-diagonal upper bounds imply precise off-diagonal Gaussian upper bounds for the kernels…

Degenerate elliptic operators: capacity, flux and separation

Derek W. Robinson, Adam Sikora

Let $S = {S_{t}}_{t \geq 0}$ be the semigroup generated on $L_2(\Ri^d)$ by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients.…

Second-order operators with degenerate coefficients

A. F. M. ter Elst, Derek W. Robinson, Adam Sikora +1

We consider properties of second-order operators $H = - \sum_{i, j = 1}^{d} \partial_{i} c_{ij} \partial_{j}$ on $\Ri^d$ with bounded real symmetric measurable coefficients. We assume…

math.AP2019

Lie group approach to Grushin operators

Jacek DziubaÅski, Adam Sikora

We consider a finite system ${X_{1}, X_{2}, \dots, X_{n}}$ of complete vector fields acting on smooth manifolds $M$ equipped with a smooth positive measure. We assume that the system…

math.AP2024

Regularity and Separation for $p$ -Laplace operators

Daniel Hauer, Adam Sikora

We analyze $p$ -Laplace operators with degenerate elliptic coefficients. This investigation includes GruÅ¡in type $p$ -Laplace operators. We describe a \emph{separation phenomenon} i…

Analysis of degenerate elliptic operators of Grushin type

Derek W. Robinson, Adam Sikora

We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_{2} (R^{n} \times R^{m})$ . We assume the coefficients are real symmetric and $a_1H_Î´\ge…

math.AP2007

Riesz transforms in one dimension

Andrew Hassell, Adam Sikora

We study the boundedness on $L^{p}$ of the Riesz transform $\nabla L^{- 1/2}$ , where $L$ is one of several operators defined on $R$ or $R_{+}$ , endowed with the measure $r^{d - 1} d r$ ,…

math.AP2003

Riesz transform, Gaussian bounds and the method of wave equation

Adam Sikora

For an abstract self-adjoint operator $L$ and a local operator $A$ we study the boundedness of the Riesz transform $A L^{- I^\pm}$ on $L^{p}$ for some $I^\pm > 0$ . A very simple proof of the o…

math.AP2025

On the Interplay Between Hodge Projections and Bounded Harmonic Functions on Manifolds with Ends

Dangyang He, Adam Sikora

We investigate the $L^{p}$ -boundedness of the Hodge projection in the setting of manifolds with ends. We examine its relationship to the Riesz transform and the space of bounded harm…

Dirichlet forms and degenerate elliptic operators

A. F. M. ter Elst, Derek W. Robinson, Adam Sikora +1

It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $\Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form ir…

A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces

Salahaddine Boutayeb, Thierry Coulhon, Adam Sikora

On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of glob…

Analysis of degenerate elliptic operators of GruÅ¡in type

Derek W. Robinson, Adam Sikora

We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficients are real symmetric and $a_1H_Î´\geq H\geq…

math.AP2015

Spectral multipliers, Bochner-Riesz means and uniform Sobolev inequalities for elliptic operators

Adam Sikora, Lixin Yan, Xiaohua Yao

This paper comprises two parts. In the first, we study $L^{p}$ to $L^{q}$ bounds for spectral multipliers and Bochner-Riesz means with negative index in the general setting of abstract…

math.AP2019

Sharp spectral multipliers without semigroup framework and application to random walks

Peng Chen, El Maati Ouhabaz, Adam Sikora +1

In this paper we prove spectral multiplier theorems for abstract self-adjoint operators on spaces of homogeneous type. We have two main objectives. The first one is to work outside…

Resolvent at low energy III: the spectral measure

Colin Guillarmou, Andrew Hassell, Adam Sikora

Let $M^{\circ}$ be a complete noncompact manifold and $g$ an asymptotically conic Riemaniann metric on $M^{\circ}$ , in the sense that $M^{\circ}$ compactifies to a manifold with boundary…

math.AP2021

Vertical and horizontal Square Functions on a Class of Non-Doubling Manifolds

Julian Bailey, Adam Sikora

We consider a class of non-doubling manifolds $M$ that are the connected sum of a finite number of $N$ -dimensional manifolds of the form $\mathbb{R}^{n_{i}} \times \mathc…

Flows and invariance for elliptic operators

A. F. M. ter Elst, Derek W. Robinson, Adam Sikora

Let $S$ be the submarkovian semigroup on $L_{2} (R^{d})$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with $W^{1, \infty}$ coefficients $c_{kl}…

math.FA2021

Hardy spaces meet harmonic weights

Marcin Preisner, Adam Sikora, Lixin Yan

We investigate the Hardy space $H_{L}^{1}$ associated with a self-adjoint operator $L$ defined in a general setting in [S. Hofmann, et. al., Hardy spaces associated to non-negative sel…

math.AP2015

Spectral multipliers for the Kohn Laplacian on forms on the sphere in $C^{n}$

Valentina Casarino, Michael G. Cowling, Alessio Martini +1

The unit sphere $S$ in $C^{n}$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $□_{b}$ . We prove a HÃ¶rmander spectral multipli…

math.AP2018

Bounds on the maximal Bochner-Riesz means for elliptic operators

Peng Chen, Sanghyuk Lee, Adam Sikora +1

We investigate $L^{p}$ boundedness of the maximal Bochner-Riesz means for self-adjoint operators of elliptic type. Assuming the finite speed of propagation for the associated wave op…

Derek W. Robinson, Adam Sikora

We examine the validity of the PoincarÃ© inequality for degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficien…

Ellipticity and Ergodicity

Derek W. Robinson, Adam Sikora

Let $S = {S_{t}}_{t \geq 0}$ be the submarkovian semigroup on $L_2(\Ri^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous c…

Boundedness of maximal functions on non-doubling manifolds with ends

Xuan Thinh Duong, Ji Li, Adam Sikora

Let $M$ be a manifold with ends constructed in \cite{GS} and $I^$ be the Laplace-Beltrami operator on $M$ . In this note, we show the weak type $(1, 1)$ and $L^{p}$ boundedness of the…