arXiv subjects

Michael Frazier

Publications and source records attributed to Michael Frazier.

3 recordsLinked to original sources

arXiv2019

Littlewood-Paley Theory for Matrix-Weighted Function Spaces

We define the vector-valued, matrix-weighted function spaces $\dot{F}^{\alpha q}_p(W)$ (homogeneous) and $F^{\alpha q}_p(W)$ (inhomogeneous) on $\mathbb{R}^n$, for $\alpha \in \mathbb{R}$, $0<p<\infty$, $0<q \leq \infty$, with the matrix weight $W$ belonging to the $A_p$ class. For $1<p<\infty$, we show that $L^p(W) = \dot{F}^{0 2}_p(W)$, and, for $k \in \mathbb{N}$, that $F^{k 2}_p(W)$ coincides with the matrix-weighted Sobolev space $L^p_k(W)$, thereby obtaining Littlewood-Paley characterizations of $L^p(W)$ and $L^p_k (W)$. We show that a vector-valued function belongs to $\dot{F}^{\alpha q}_p(W)$ if and only if its wavelet or $\varphi$-transform coefficients belong to an associated sequence space $\dot{f}^{\alpha q}_p(W)$. We also characterize these spaces in terms of reducing operators associated to $W$.

math.CA↗

arXiv2015

Positive solutions to Schr\"odinger's equation and the exponential integrability of the balayage

Let $\Omega \subset \mathbb{R}^n$, for $n \geq 2$, be a bounded $C^2$ domain. Let $q \in L^1_{loc} (\Omega)$ with $q \geq 0$. We give necessary conditions and matching sufficient conditions, which differ only in the constants involved, for the existence of very weak solutions to the boundary value problem $(-\triangle -q) u =0, \, \, u\ge 0 \, \, \text{on} \, \, \Omega, \, u=1 \, \text{on} \, \, \partial \Omega$, and the related nonlinear problem with quadratic growth in the gradient, $-\triangle u = |\nabla u|^2 + q \, \text{on} \, \Omega, \, u=0 \, \, \text{on} \, \, \partial \Omega$. We also obtain precise pointwise estimates of solutions up to the boundary. A crucial role is played by a new "boundary condition" on $q$ which is expressed in terms of the exponential integrability on $\partial \Omega$ of the balayage of the measure $\delta q \, dx$, where $\delta (x) = \text{dist} (x, \partial \Omega)$. This condition is sharp, and appears in such a context for the first time. It holds, for example, if $\delta q \, dx$ is a Carleson measure in $\Omega$, or if its balayage is in $BMO(\partial \Omega)$, with sufficiently small norm. This solves an open problem posed in the literature.

math.AP↗

arXiv2014

Global estimates for kernels of Neumann series and Green's functions

We obtain global pointwise estimates for kernels of the resolvents $(I-T)^{-1}$ of integral operators \[Tf(x) = \int_{\Omega} K(x, y) f(y) d \omega(y)\] on $L^2(\Omega, \omega)$ under the assumptions that $||T||_{L^2(\omega) \rightarrow L^2 (\omega)} <1$ and $d(x,y)=1/K(x,y)$ is a quasi-metric. Let $K_1=K$ and $K_j(x,y) = \int_{\Omega} K_{j-1} (x,z) K(z,y) \, d \omega (z)$ for $j \geq 1$. Then $$ K(x,y) e^{c K_2 (x,y)/K(x,y)} \leq \sum_{j=1}^{\infty} K_j(x,y) \leq K(x,y) e^{C K_2 (x,y)/K(x,y)}, $$ for some constants $c,C>0$. Our estimates yield matching bilateral bounds for Green's functions of the fractional Schr\"{o}dinger operators $(-\triangle)^{\alpha/2}-q$ with arbitrary nonnegative potentials $q$ on $\mathbb{R}^n$ for $0<\alpha<n$, or on a bounded non-tangentially accessible domain $\Omega$ for $0<\alpha \le 2$. In probabilistic language, these results can be reformulated as explicit bilateral bounds for the conditional gauge associated with Brownian motion or $\alpha$-stable L\'evy processes.

math.AP↗