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Adam Sikora

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

    Tue, 02/21/2012 - 15:01 — SomNambul
    Authors: Adam Sikora, Lixin Yan, Peng Chen, El Maati Ouhabaz
    Subjects: Analysis of PDEs
    Abstract

    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 introduce a restriction type condition which in
    the case of the standard Laplace operator is equivalent to $(p,2)$ restriction
    estimate of Stein and Tomas. Next we show that in the considered abstract
    setting our restriction type condition implies sharp spectral multipliers and
    endpoint estimates for the Bochner-Riesz summability.

  2. Restriction and spectral multiplier theorems on asymptotically conic manifolds.

    Mon, 12/20/2010 - 16:01 — SomNambul
    Authors: Adam Sikora, Colin Guillarmou, Andrew Hassell
    Subjects: Analysis of PDEs
    Abstract

    The classical Stein-Tomas restriction theorem is equivalent to the statement
    that the spectral measure $dE(\lambda)$ of the square root of the Laplacian on
    $\RR^n$ is bounded from $L^p(\RR^n)$ to $L^{p'}(\RR^n)$ for $1 \leq p \leq
    2(n+1)/(n+3)$, where $p'$ is the conjugate exponent to $p$, with operator norm
    scaling as $\lambda^{n(1/p - 1/p') - 1}$.

  3. Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers.

    Wed, 03/10/2010 - 16:01 — SomNambul
    Authors: Adam Sikora, Xuan Thinh Duong, Lixin Yan
    Subjects: Functional Analysis
    Abstract

    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 $e^{-tL}$ whose kernels $p_t(x,y)$ have Gaussian upper bounds but
    possess no regularity in variables $x$ and $y$. In this article, we study
    weighted $L^p$-norm inequalities for spectral multipliers of $L$. We show sharp
    weighted H\"ormander-type spectral multiplier theorems follow from Gaussian
    heat kernel bounds and appropriate $L^2$ estimates of the kernels of the
    spectral multipliers.

  4. Markov uniqueness of degenerate elliptic operators.

    Thu, 12/24/2009 - 16:01 — SomNambul
    Authors: Derek W. Robinson, Adam Sikora
    Subjects: Analysis of PDEs
    Abstract

    Let $\Omega$ be an open subset of $\Ri^d$ and
    $H_\Omega=-\sum^d_{i,j=1}\partial_i c_{ij} \partial_j$ a second-order partial
    differential operator on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where
    the coefficients $c_{ij}\in W^{1,\infty}(\Omega)$ are real symmetric and
    $C=(c_{ij})$ is a strictly positive-definite matrix over $\Omega$.

    In particular, $H_\Omega$ is locally strongly elliptic.

  5. Riesz meets Sobolev.

    Fri, 11/13/2009 - 16:01 — SomNambul
    Authors: Adam Sikora, Thierry Coulhon
    Subjects: Analysis of PDEs
    Abstract

    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 a certain form of Sobolev inequality. We also
    characterize in such terms the heat kernel gradient upper estimate on manifolds
    with polynomial growth.

  6. Degenerate elliptic operators in one dimension.

    Fri, 09/04/2009 - 12:10 — SomNambul
    Authors: Derek W. Robinson, Adam Sikora
    Subjects: Analysis of PDEs
    Abstract

    Let $H$ be the symmetric second-order differential operator on $L_2(\Ri)$
    with domain $C_c^\infty(\Ri)$ and action $H\varphi=-(c \varphi')'$ where $ c\in
    W^{1,2}_{\rm loc}(\Ri)$ is a real function which is strictly positive on
    $\Ri\backslash\{0\}$ but with $c(0)=0$. We give a complete characterization of
    the self-adjoint extensions and the submarkovian extensions of $H$.

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