Spectral Multipliers of Self-Adjoint Operators on Besov and Triebel–Lizorkin Spaces Associated to Operators

Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ that satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Hörmander-type spectral multiplier theorem for $L$ on the Besov and Triebel–Lizorkin spaces associated to $L$. Our work not only recovers the boundedness of the spectral multipliers on $L^p$ spaces and Hardy spaces associated to $L$ but also is the 1st one that proves the boundedness of a general spectral multiplier theorem on Besov and Triebel–Lizorkin spaces.

Remarks on L p -limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems

This paper comprises two parts. We first investigate an {L^{p}} -type of limiting absorption principle for Schrödinger operators {H=-\Delta+V} on {\mathbb{R}^{n}} ( {n\geq 3} ), i.e., we prove the ϵ-uniform {L^{{2(n+1)}/({n+3})}} – {L^{{2(n+1)}/({n-1})}} -estimates of the resolvent {(H-\lambda\pm i\epsilon)^{-1}} for all {\lambda>0} under the assumptions that the potential V belongs to some integrable spaces and a spectral condition of H at zero is satisfied. As applications, we establish a sharp Hörmander-type spectral multiplier theorem associated with Schrödinger operators H and deduce {L^{p}} -bounds of the corresponding Bochner–Riesz operators. Next, we consider the fractional Schrödinger operator {H=(-\Delta)^{\alpha}+V} ( {0<2\alpha<n} ) and prove a uniform Hardy–Littlewood–Sobolev inequality for {(-\Delta)^{\alpha}} , which generalizes the corresponding result of Kenig–Ruiz–Sogge [20].

OBSERVATIONS ON GAUSSIAN UPPER BOUNDS FOR NEUMANN HEAT KERNELS

Given a domain ${\rm\Omega}$ of a complete Riemannian manifold ${\mathcal{M}}$, define ${\mathcal{A}}$ to be the Laplacian with Neumann boundary condition on ${\rm\Omega}$. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound $$\begin{eqnarray}h(t,x,y)\leq \frac{C}{[V_{{\rm\Omega}}(x,\sqrt{t})V_{{\rm\Omega}}(y,\sqrt{t})]^{1/2}}\biggl(1+\frac{d^{2}(x,y)}{4t}\biggr)^{{\it\delta}}e^{-d^{2}(x,y)/4t}\quad \text{for}~t>0,~x,y\in {\rm\Omega}.\end{eqnarray}$$ Here $d$ is the geodesic distance on ${\mathcal{M}}$, $V_{{\rm\Omega}}(x,r)$ is the Riemannian volume of $B(x,r)\cap {\rm\Omega}$, where $B(x,r)$ is the geodesic ball of centre $x$ and radius $r$, and ${\it\delta}$ is a constant related to the doubling property of ${\rm\Omega}$. As a consequence we obtain analyticity of the semigroup $e^{-t{\mathcal{A}}}$ on $L^{p}({\rm\Omega})$ for all $p\in [1,\infty )$ as well as a spectral multiplier result.