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Adam Sikora

Publications and source records attributed to Adam Sikora.

At least 37 records · Page 2Linked to original sources
arXiv2015

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

The unit sphere $\mathbb{S}$ in $\mathbb{C}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $\Box_b$. We prove a Hörmander spectral multiplier theorem for $\Box_b$ with critical index $n-1/2$, that is, half the topological dimension of $\mathbb{S}$. Our proof is mainly based on representation theory and on a detailed analysis of the spaces of differential forms on $\mathbb{S}$.

math.AP↗
arXiv2014

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

We investigate spectral multipliers, Bochner-Riesz means and convergence of eigenfunction expansion corresponding to the Schrödinger operator with anharmonic potential ${\mathcal L}=-\frac{d^2}{dx^2}+|x|$. We show that the Bochner-Riesz profile of the operator ${\mathcal L}$ completely coincides with such profile of the harmonic oscillator ${\mathcal H}=-\frac{d^2}{dx^2}+x^2$. It is especially surprising because the Bochner-Riesz profile for the one-dimensional standard Laplace operator is known to be essentially different and the case of operators ${\mathcal H}$ and ${\mathcal L}$ resembles more the profile of multidimensional Laplace operators. Another surprising element of the main obtained result is the fact that the proof is not based on restriction type estimates and instead entirely new perspective have to be developed to obtain the critical exponent for Bochner-Riesz means convergence.

math.AP↗
arXiv2014

Grušin operators, Riesz transforms and nilpotent Lie groups

We establish that the Riesz transforms of all orders corresponding to the Grušin operator $H_N=-\nabla_{x}^2-|x|^{2N}\,\nabla_{y}^2$, and the first-order operators $(\nabla_{x},x^ν\,\nabla_{y})$ where $x\in \Ri^n$, $y\in\Ri^m$, $N\in\Ni_+$, and $ν\in\{1,\ldots,n\}^N$, are bounded on $L_p(\Ri^{n+m})$ for all $p\in\langle1,\infty\rangle$ and are also weak-type $(1,1)$. Moreover, the transforms of order less than or equal to $N+1$ corresponding to $H_N$ and the operators $(\nabla_{x}, |x|^N\nabla_{y})$ are bounded on $L_p(\Ri^{n+m})$ for all $p\in\langle1,\infty\rangle$. But all transforms of order $N+2$ are bounded if and only if $p\in\langle1,n\rangle$. The proofs are based on the observation that the $(\nabla_{x},x^ν\,\nabla_{y})$ generate a finite-dimensional nilpotent Lie algebra, the corresponding connected, simply connected, nilpotent Lie group is isometrically represented on the spaces $L_p(\Ri^{n+m})$ and $H_N$ is the corresponding sublaplacian

math.AP↗
arXiv2013

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

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 global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo-Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber-Krahn inequalities or localized Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. A crucial role is played by the finite propagation speed property for the associated wave equation, and our main result holds for an abstract semigroup of operators satisfying the Davies-Gaffney estimates.

math.AP↗
arXiv2013

The limitations of the Poincar{é} inequality

We examine the validity of the Poincaré inequality for 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 $a_1H_δ\geq H\geq a_2H_δ$ for some $a_1,a_2>0$ where $H_δ$ is a generalized Grušin operator, \[ H_δ=-\nabla_{x_1}\,|x_1|^{(2δ_1,2δ_1')}\,\nabla_{x_1}-|x_1|^{(2δ_2,2δ_2')}\,\nabla_{x_2}^2 \;. \] Here $x_1\in\Ri^n$, $x_2\in\Ri^m$, $δ_1,δ_1'\in[0,1\rangle$, $δ_2,δ_2'\geq0$ and $|x_1|^{(2δ,2δ')}=|x_1|^{2δ}$ if $|x_1|\leq 1$ and $|x_1|^{(2δ,2δ')}=|x_1|^{2δ'}$ if $|x_1|\geq 1$. \smallskip We prove that the Poincaré inequality, formulated in terms of the Riemannian geometry corresponding to $H$, is valid if $n\geq 2$, or if $n=1$ and $δ_1\veeδ_1'\in[0,1/2\rangle$ but it fails if $n=1$ and $δ_1\veeδ_1'\in[1/2,1\rangle$. The failure is caused by the leading term. If $δ_1\in[1/2, 1\rangle$ it is an effect of the local degeneracy $|x_1|^{2δ_1}$ but if $δ_1\in[0, 1/2\rangle$ and $δ_1'\in [1/2,1\rangle$ it is an effect of the growth at infinity of $|x_1|^{2δ_1'}$. If $n=1$ and $δ_1\in[1/2, 1\rangle$ then the semigroup $S$ generated by the Friedrichs' extension of $H$ is not ergodic. The subspaces $x_1\geq 0$ and $x_1\leq 0$ are $S$-invariant and the Poincaré inequality is valid on each of these subspaces. If, however, $n=1$, $δ_1\in[0, 1/2\rangle$ and $δ_1'\in [1/2,1\rangle$ then the semigroup $S$ is ergodic but the Poincaré inequality is only valid locally. \smallskip Finally we discuss the implication of these results for the kernel of the semigroup $S$.

math.AP↗
arXiv2013

Analysis of degenerate elliptic operators of Grušin type

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 $a_1H_δ\geq H\geq a_2H_δ$ for some $a_1,a_2>0$ where \[ H_δ=-{\nabla}_{x_1}\cdot(c_{δ_1, δ'_1}(x_1)\,\nabla_{x_1})-c_{δ_2, δ'_2}(x_1)\,\nabla_{x_2}^2 \;. \] Here $x_1\in\Ri^n$, $x_2\in\Ri^m$ and $c_{δ_i, δ'_i}$ are positive measurable functions such that $c_{δ_i, δ'_i}(x)$ behaves like $|x|^{δ_i}$ as $x\to0$ and $|x|^{δ_i'}$ as $x\to\infty$ with $δ_1,δ_1'\in[0,1\rangle$ and $δ_2,δ_2'\geq0$. Our principal results state that the submarkovian semigroup $S_t=e^{-tH}$ is conservative and its kernel $K_t$ satisfies bounds \[ 0\leq K_t(x\,;y)\leq a\,(|B(x\,;t^{1/2})|\,|B(y\,;t^{1/2})|)^{-1/2} \] where $|B(x\,;r)|$ denotes the volume of the ball $B(x\,;r)$ centred at $x$ with radius $r$ measured with respect to the Riemannian distance associated with $H$. The proofs depend on detailed subelliptic estimations on $H$, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive, nor continuous.

math.AP↗
arXiv2013

Boundedness of maximal functions on non-doubling manifolds with ends

Let $M$ be a manifold with ends constructed in \cite{GS} and $Δ$ be the Laplace-Beltrami operator on $M$. In this note, we show the weak type $(1,1)$ and $L^p$ boundedness of the Hardy-Littlewood maximal function and of the maximal function associated with the heat semigroup $\M_Δf(x)=\sup_{t> 0} |\exp (-tΔ)f(x)| $ on $L^p(M)$ for $1 < p \le \infty$. The significance of these results comes from the fact that $M$ does not satisfies the doubling condition.

math.AP↗
arXiv2012

Sharp spectral multipliers for operators satisfying generalized Gaussian estimates

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 $p_t(x,y)$ satisfy generalized $m$-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein-Tomas type estimates. These results are applicable to spectral multipliers for large classes of operators including $m$-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals.

math.AP↗
arXiv2012

Resolvent at low energy III: the spectral measure

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 $M$ in such a way that $g$ becomes a scattering metric on $M$. Let $Δ$ be the positive Laplacian associated to $g$, and $P = Δ+ V$, where $V$ is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure $dE(λ) = (λ/πi) \big(R(λ+i0) - R(λ- i0) \big)$ of $P_+^{1/2}$, where $R(λ) = (P - λ^2)^{-1}$, as $λ\to 0$, in a manner similar to that done previously by the second author and Vasy, and by the first two authors. The main result is that the spectral measure has a simple, `conormal-Legendrian' singularity structure on a space which is obtained from $M^2 \times [0, λ_0)$ by blowing up a certain number of boundary faces. We use this to deduce results about the asymptotics of the wave solution operators $\cos(t \sqrt{P_+})$ and $\sin(t \sqrt{P_+})/\sqrt{P_+}$, and the Schrödinger propagator $e^{itP}$, as $t \to \infty$. In particular, we prove the analogue of Price's law for odd-dimensional asymptotically conic manifolds. This result on the spectral measure has been used in a follow-up work by the authors (arXiv:1012.3780) to prove sharp restriction and spectral multiplier theorems on asymptotically conic manifolds.

math.AP↗
arXiv2012

Restriction and spectral multiplier theorems on asymptotically conic manifolds

The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $dE(λ)$ 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 $λ^{n(1/p - 1/p') - 1}$. We prove a geometric generalization in which the Laplacian on $\RR^n$ is replaced by the Laplacian, plus suitable potential, on a nontrapping asymptotically conic manifold, which is the first time such a result has been proven in the variable coefficient setting. It is closely related to, but stronger than, Sogge's discrete $L^2$ restriction theorem, which is an $O(λ^{n(1/p - 1/p') - 1})$ estimate on the $L^p \to L^{p'}$ operator norm of the spectral projection for a spectral window of fixed length. From this, we deduce spectral multiplier estimates for these operators, including Bochner-Riesz summability results, which are sharp for $p$ in the range above.

math.AP↗
arXiv2012

Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators

We study the Grushin operators acting on $\R^{d_1}_{x'}\times \R^{d_2}_{x"}$ and defined by the formula \[ L=-\sum_{\jone=1}^{d_1}\partial_{x'_\jone}^2 - (\sum_{\jone=1}^{d_1}|x'_\jone|^2) \sum_{\jtwo=1}^{d_2}\partial_{x"_\jtwo}^2. \] We obtain weighted Plancherel estimates for the considered operators. As a consequence we prove $L^p$ spectral multiplier results and Bochner-Riesz summability for the Grushin operators. These multiplier results are sharp if $d_1 \ge d_2$. We discuss also an interesting phenomenon for weighted Plancherel estimates for $d_1 <d_2$. The described spectral multiplier theorem is the analogue of the result for the sublaplacian on the Heisenberg group obtained by D. Müller and E.M. Stein and by W. Hebisch.

math.AP↗
arXiv2012

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

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. We also observe that this restriction estimate holds for operators satisfying dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrödinger operators with inverse square potentials on $\RR^n$, the harmonic oscillator, elliptic operators on compact manifolds and Schrödinger operators on asymptotically conic manifolds.

math.AP↗
arXiv2010

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

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 Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate $L^2$ estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials on complete Riemannian manifolds.

math.FA↗
arXiv2010

$L_1$-uniqueness of degenerate elliptic operators

Let $Ω$ be an open subset of $\Ri^d$ with $0\in Ω$. Further let $H_Ω=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j$ be a second-order partial differential operator with domain $C_c^\infty(Ω)$ where the coefficients $c_{ij}\in W^{1,\infty}_{\rm loc}(\barΩ)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $0 0$ where $μ(s)=\int^s_0dt\,c(t)^{-1/2}$ then we establish that $H_Ω$ is $L_1$-unique, i.e.\ it has a unique $L_1$-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e.\ it has a unique $L_2$-extension which generates a submarkovian semigroup. Moreover these uniqueness conditions are equivalent with the capacity of the boundary of $Ω$, measured with respect to $H_Ω$, being zero. We also demonstrate that the capacity depends on two gross features, the Hausdorff dimension of subsets $A$ of the boundary the set and the order of degeneracy of $H_Ω$ at $A$.

math.AP↗
arXiv2009

Markov uniqueness of degenerate elliptic operators

Let $Ω$ be an open subset of $\Ri^d$ and $H_Ω=-\sum^d_{i,j=1}\partial_i c_{ij} \partial_j$ a second-order partial differential operator on $L_2(Ω)$ with domain $C_c^\infty(Ω)$ where the coefficients $c_{ij}\in W^{1,\infty}(Ω)$ are real symmetric and $C=(c_{ij})$ is a strictly positive-definite matrix over $Ω$. In particular, $H_Ω$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_Ω$, i.e. the self-adjoint extensions which generate submarkovian semigroups. Our main result establishes that $H_Ω$ is Markov unique, i.e. it has a unique submarkovian extension, if and only if $\capp_Ω(\partialΩ)=0$ where $\capp_Ω(\partialΩ)$ is the capacity of the boundary of $Ω$ measured with respect to $H_Ω$. The second main result establishes that Markov uniqueness of $H_Ω$ is equivalent to the semigroup generated by the Friedrichs extension of $H_Ω$ being conservative.

math.AP↗
arXiv2009

Riesz meets Sobolev

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.

math.AP↗
arXiv2009

Degenerate elliptic operators in one dimension

Let $H$ be the symmetric second-order differential operator on $L_2(\Ri)$ with domain $C_c^\infty(\Ri)$ and action $Hφ=-(c φ')'$ 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 $H$. In particular if $ν=ν_+\veeν_-$ where $ν_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $ν\not\in L_2(0,1)$ and a unique submarkovian extension if and only if $ν\not\in L_\infty(0,1)$. In both cases the corresponding semigroup leaves $L_2(0,\infty)$ and $L_2(-\infty,0)$ invariant. In addition we prove that for a general non-negative $ c\in W^{1,\infty}_{\rm loc}(\Ri)$ the corresponding operator $H$ has a unique submarkovian extension.

math.AP↗