arXiv subjects
Publications and source records attributed to Adam Sikora.
Let $S$ be the submarkovian semigroup on $L_2({\bf R}^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with $W^{1,\infty}$ coefficients $c_{kl}$. Further let $Ω$ be an open subset of ${\bf R}^d$. Under mild conditions we prove that $S$ leaves $L_2(Ω)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $\sum_{l=1}^d c_{kl} \partial_l$ for all $k$.
Let $S=\{S_t\}_{t\geq0}$ be the submarkovian semigroup on $L_2(\Ri^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients $c_{ij}$. Further let $Ω$ be an open subset of $\Ri^d$. Under the assumption that $C_c^\infty(\Ri^d)$ is a core for $H$ we prove that $S$ leaves $L_2(Ω)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $Y_i=\sum^d_{j=1}c_{ij}\partial_j$.
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 Hörmander type differentiability condition then the operator $F(L_1,L_2)$ is of Calderón-Zygmund type. We apply obtained results to the analysis of quasielliptic operators acting on product of some fractal spaces. The existence and surprising properties of quasielliptic operators have been recently observed in works of Bockelman, Drenning and Strichartz. This paper demonstrates that Riesz type operators corresponding to quasielliptic operators are continuous on $L^p$ spaces.
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 $r^{d-1} dr$, $d > 1$, where $dr$ is Lebesgue measure. For integer $d$, this mimics the measure on Euclidean $d$-dimensional space, and in this case our setup is equivalent to looking at the Laplacian acting on radial functions on Euclidean space or variations of Euclidean space such as the exterior of a sphere (with either Dirichlet or Neumann boundary conditions), or the connected sum of two copies of $\R^d$. In this way we illuminate some recent results on the Riesz transform on asymptotically Euclidean manifolds. We are however interested in all real values of $d > 1$, and another goal of our analysis is to study the range of boundedness as a function of $d$; it is particularly interesting to see the behaviour as $d$ crosses 2. For example, in one of our cases which models radial functions on the connected sum of two copies of $\R^d$, the upper threshold for $L^p$ boundedness is $p=d$ for $d \ge 2$ and $p=d/(d-1)$ for $d < 2$. Only in the case $d=2$ is the Riesz transform actually bounded on $L^p$ when $p$ is equal to the upper threshold. We also study the Riesz transform when we have an inverse square potential, or a delta function potential; these cases provide a simple model for recent results of the first author and Guillarmou. Finally we look at the Hodge projector in a slightly more general setup.
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 and Yan. We discuss applications of the new BMO_L spaces in the theory of singular integration such as BMO_L estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space BMO_L might coincide with or might be essentially different from the classical BMO space.
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 of analytic families of operators on metric measure spaces.
We analyze degenerate, second-order, elliptic operators $H$ in divergence form on $L_2({\bf R}^{n}\times{\bf R}^{m})$. 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 \[ H_δ=-\nabla_{x_1} c_{δ_1, δ'_1}(x_1) \nabla_{x_1}-c_{δ_2, δ'_2}(x_1) \nabla_{x_2}^2 . \] Here $x_1\in{\bf R}^n$, $x_2\in{\bf R}^m$ 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 $δ_1,δ_1'\in[0,1>$ 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.
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 irregardless of the order of degeneracy. The behaviour of the corresponding evolution semigroup $S_t$ can be described in terms of a function $(A,B) \mapsto d(A ;B)\in[0,\infty]$ over pairs of measurable subsets of $\Ri^d$. Then \[ |(ϕ_A,S_tϕ_B)|\leq e^{-d(A;B)^2(4t)^{-1}}\|ϕ_A\|_2\|ϕ_B\|_2 \] for all $t>0$ and all $ϕ_A\in L_2(A)$, $ϕ_B\in L_2(B)$. Moreover $S_tL_2(A)\subseteq L_2(A)$ for all $t>0$ if and only if $d(A ;A^c)=\infty$ where $A^c$ denotes the complement of $A$.
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 wide class of strongly elliptic, subelliptic and degenerate elliptic operators. In the degenerate case the asymptotics incorporate possible non-ergodicity.
Let $S=\{S_t\}_{t\geq0}$ be the semigroup generated on $L_2(\Ri^d)$ by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients. Further let $Ω$ be an open subset of $\Ri^d$ with Lipschitz continuous boundary $\partialΩ$. We prove that $S$ leaves $L_2(Ω)$ invariant if, and only if, the capacity of the boundary with respect to $H$ is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result.
We consider properties of second-order operators $H = -\sum^d_{i,j=1} \partial_i \, c_{ij} \, \partial_j$ on $\Ri^d$ with bounded real symmetric measurable coefficients. We assume that $C = (c_{ij}) \geq 0$ almost everywhere, but allow for the possibility that $C$ is singular. We associate with $H$ a canonical self-adjoint viscosity operator $H_0$ and examine properties of the viscosity semigroup $S^{(0)}$ generated by $H_0$. The semigroup extends to a positive contraction semigroup on the $L_p$-spaces with $p \in [1,\infty]$. We establish that it conserves probability, satisfies $L_2$~off-diagonal bounds and that the wave equation associated with $H_0$ has finite speed of propagation. Nevertheless $S^{(0)}$ is not always strictly positive because separation of the system can occur even for subelliptic operators. This demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor Hölder continuous. In particular one can construct examples for which both upper and lower Gaussian bounds fail even with coefficients in $C^{2-\varepsilon}(\Ri^d)$ with $\varepsilon > 0$.
For an abstract self-adjoint operator $L$ and a local operator $A$ we study the boundedness of the Riesz transform $AL^{-α}$ on $L^p$ for some $α>0$. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on $L^p$ for all $p\in (1,2]$ for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on $L^p$ of the Laplace-Beltrami operator on Riemannian manifolds for $p>2$ .
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 L^p(S^{2m-1}) for all p --> (1,infinity). We also show that the Bochner-Riesz conjecture is true when restricted to cylindrically symmetric functions on R^{n-1} times R.