%% LyX 1.6.7 created this file. For more info, see http://www.lyx.org/. %% Do not edit unless you really know what you are doing. \documentclass[english]{beamer} \usepackage{amssymb} \usepackage{esint} %\makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. \pdfpageheight\paperheight \pdfpagewidth\paperwidth %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands. % this default might be overridden by plain title style \newcommand\makebeamertitle{\frame{\maketitle}}% \AtBeginDocument{ \let\origtableofcontents=\tableofcontents \def\tableofcontents{\@ifnextchar[{\origtableofcontents}{\gobbletableofcontents}} \def\gobbletableofcontents#1{\origtableofcontents} } \makeatletter \long\def\lyxframe#1{\@lyxframe#1\@lyxframestop}% \def\@lyxframe{\@ifnextchar<{\@@lyxframe}{\@@lyxframe<*>}}% \def\@@lyxframe<#1>{\@ifnextchar[{\@@@lyxframe<#1>}{\@@@lyxframe<#1>[]}} \def\@@@lyxframe<#1>[{\@ifnextchar<{\@@@@@lyxframe<#1>[}{\@@@@lyxframe<#1>[<*>][}} \def\@@@@@lyxframe<#1>[#2]{\@ifnextchar[{\@@@@lyxframe<#1>[#2]}{\@@@@lyxframe<#1>[#2][]}} \long\def\@@@@lyxframe<#1>[#2][#3]#4\@lyxframestop#5\lyxframeend{% \frame<#1>[#2][#3]{\frametitle{#4}#5}} \makeatother \def\lyxframeend{} % In case there is a superfluous frame end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. \usepackage{beamerthemeshadow} \makeatother \begin{document} \title{Complex powers of the Laplacian on Affine Nested Fractals as Calder\'on-Zygmund operators} \author{Marius Ionescu} \institute{Based on joint work with Luke Rogers} \date{AMS Fall Meeting, Syracuse} \makebeamertitle \lyxframeend{}\lyxframe{[<+-| alert@+>]Affine Nested Fractals} \begin{definitions}%{} \begin{enumerate} \item Let $\{F_{1},F_{2},\dots,F_{N}\}$ be an iterated function system of contractive similarities in $\mathbb{R}^{n}$. We let $K$ be the \emph{invariant} set of the i.f.s.\[ K=\bigcup_{i}F_{i}(K).\] \item $K$ is called postcritically finite (pcf) if $K$ is connected, and there exists a finite set $V_{0}\subseteq K$ called the \emph{boundary}, such that\[ F_{w}K\bigcap F_{w^{\prime}}K\subseteq F_{w}V_{0}\bigcap F_{w^{\prime}}V_{0}\] for $w\ne w^{\prime}$ with $\vert w\vert=\vert w^{\prime}\vert$. \item $K$ is called \emph{affine nested} if, in addition, it has a large symmetry group. \end{enumerate} \end{definitions}%{} \lyxframeend{}\lyxframe{[<+-| alert@+>]Energy and Laplacian} \begin{definitions}%{} \begin{enumerate} \item We assume the existence of a \emph{self similar measure} $\mu$ on $K$\[ \mu(A)=\sum_{i}\mu_{i}\mu(F_{i}^{-1}A).\] \item We also assume the existence of a \emph{self-similar energy} $\mathcal{E}$ on $K$\[ \mathcal{E}(u)=\sum_{i}r_{i}\mathcal{E}(u\circ F_{i}).\] \item The \emph{Laplacian} is defined using the weak formulation: $\Delta u=f$ if\[ \mathcal{E}(u,v)=-\int fvd\mu\] for all $v\in\operatorname{dom}_{0}\mathcal{E}$. \end{enumerate} \end{definitions}%{} \lyxframeend{}\lyxframe{[<+-| alert@+>]Fractal Blow-ups and Fractafolds} \begin{definitions}%{} \begin{enumerate} \item Let $w\in\{1,\dots,N\}^{\infty}$ be an infinite word. Then one defines a \emph{fractal blow-up} $K_{\infty}$ via\[ K_{\infty}=\bigcup_{m=1}^{\infty}F_{w_{1}}^{-1}\dots F_{w_{m}}^{-1}K.\] \item A \emph{fractafold} based on $K$ is a set $X$ such that every point in $X$ has a neighborhood that is similar to a neighborhood of a point in $K$. \end{enumerate} \end{definitions}%{} \lyxframeend{}\lyxframe{[<+-| alert@+>]Set up} \begin{itemize} \item Let $K$ be an affine nested fractal such that $V_{0}$ contains $3$ or more points. \item Let $X$ denote either a fractal blow-up without boundary based on $K$ or $K$ itself \item Then one can extend the definition of the energy $\mathcal{E}$ and the measure $\mu$ to $X$. \item We define the Laplacian $\Delta$ on $X$ via the weak formulation. \item If $X$ is non-compact then $\Delta$ has pure point spectrum. \item The \emph{effective resistance metric} $R(x,y)$ on $X$ is defined via\[ R(x,y)^{-1}=\min\{\mathcal{E}(u)\,:\, u(x)=0\mbox{ and }u(x)=1\}.\] \end{itemize} \lyxframeend{}\lyxframe{[<+-| alert@+>]Heat Kernels} \begin{itemize} \item The heat operator $e^{t\Delta}$ is given by integration with respect to a positive heat kernel $h_{t}(x,y)$\[ e^{t\Delta}u=\int_{X}h_{t}(x,y)u(y)d\mu(y).\] \item The heat kernel satisfies the following sub-Gaussian estimates\[ h_{t}(x,y)\lesssim t^{-\beta}\mbox{exp}\left(-c\left(\frac{R(x,y)^{d+1}}{t}\right)^{\gamma}\right).\] \end{itemize} \lyxframeend{}\lyxframe{Heat Kernels are Integrable} \begin{theorem}%{} If $y\in X$ we have that \[ \int_{X}e^{-c\bigl(\frac{R(x,y)^{d+1}}{t}\bigr)^{\gamma}}d\mu(x)\lesssim t^{\frac{d}{d+1}}\] for all $t>0$. \end{theorem}%{} \lyxframeend{}\lyxframe{Calder\'on-Zygmund operators on fractals} \begin{definition}%{} \global\long\def\supp{\operatorname{supp}} \global\long\def\D{\mathcal{D}} An operator $T$ bounded on $L^{2}(\mu)$ is called \emph{a Calderón-Zygmund operator} if $T$ is given by integration with respect to a kernel \[ Tu(x)=\int_{X}K(x,y)u(y)d\mu(y)\] such that $K(x,y)$ is a function off the diagonal which satisfies \[ \vert K(x,y)\vert\lesssim R(x,y)^{-d}\] and\[ \vert K(x,y)-K(x,\overline{y})\vert\lesssim\eta\left(\frac{R(y,\overline{y})}{R(x,\overline{y})}\right)R(x,y)^{-d},\] for some Dini modulus of continuity $\eta$ and some $c>1$. \end{definition}%{} \lyxframeend{}\lyxframe{[<+-| alert@+>]Singular Integral operators on fractals} \begin{definition}%{} \begin{itemize} \item We say that $K(x,y)$ is \emph{a standard kernel}. \item We say that the operator $T$ is a \emph{singular integral operator} if the kernel $K(x,y)$ is singular at $x=y$. \end{itemize} \end{definition}%{} \lyxframeend{}\lyxframe{Main theorem} \begin{theorem}%{} Suppose that $X$ is an infinite blow-up of a PCF fractal $F$, and assume that $K(x,y)$ is a smooth function off the diagonal and satisfies the estimates\[ \vert K(x,y)\vert\lesssim R(x,y)^{-d}\] and\[ \vert\Delta_{2}K(x,y)\vert\lesssim R(x,y)^{-2d-1}.\] Then $K(x,y)$ satisfies also\[ \vert K(x,y)-K(x,\overline{y})\vert\lesssim\left(\frac{R(y,\overline{y})}{R(x,\overline{y})}\right)^{1/2}R(x,y)^{-d},\] for all $x,y,\overline{y}\in X$ such that $R(x,y)\ge cR(y,\overline{y})$, for some $c>1$. \end{theorem}%{} \lyxframeend{}\lyxframe{[<+-| alert@+>]Imaginary powers of the Laplacian} \begin{definitions}%{} Let $\alpha\in\mathbb{R}$ and let $\mathcal{D}$ be the set of finite linear combinations of eigenfunctions of $\Delta$. \begin{enumerate} \item We define\[ (-\Delta)^{i\alpha}\varphi=\lambda^{i\alpha}\varphi\] for $\varphi$ an eigenfunction with eigenvalue $\lambda$. \item Then, for $u\in\D$, \[ (-\Delta)^{i\alpha}u=C_{\alpha}\Delta\bigl(\int_{0}^{\infty}e^{t\Delta}ut^{-i\alpha}dt\bigr),\] where $C_{\alpha}>0$ is a constant. \item The kernel of $(-\Delta)^{i\alpha}$ is\[ K_{i\alpha}(x,y)=C_{\alpha}\int_{0}^{\infty}\Delta_{1}h_{t}(x,y)t^{-i\alpha}dt.\] \end{enumerate} \end{definitions}%{} \lyxframeend{}\lyxframe{Properties} \begin{theorem}%{} For any $\alpha\in\mathbb{R}$, the kernel $K_{i\alpha}(x,y)$ is smooth off the diagonal and satisfies the following estimates\[ \vert K_{i\alpha}(x,y)\vert\lesssim R(x,y)^{-d}\] and\[ \vert\Delta_{2}K_{i\alpha}(x,y)\vert\lesssim R(x,y)^{-2d-1}.\] \end{theorem}%{} \lyxframeend{}\lyxframe{$L^{p}$-boundedness} \begin{corollary}%{} The operators $(-\Delta)^{i\alpha}$ are Calderón-Zygmund operators. Therefore $(-\Delta)^{i\alpha}$ extends to a bounded operator on $L^{p}(\mu)$ for all $1
]Operators of Laplace transform type} \begin{definition}%{} \begin{itemize} \item A function $p:[0,\infty)\to\mathbb{R}$ is said to be of \emph{Laplace transform type} if\[ p(\lambda)=\lambda\int_{0}^{\infty}m(t)e^{-t\lambda}dt,\] where $m$ is uniformly bounded. \item For such a $p$ we can define an operator\[ p(-\Delta)u=(-\Delta)\int_{0}^{\infty}m(t)e^{t\Delta}udt\] with a kernel \[ K_{p}(x,y)=\int_{0}^{\infty}(-\Delta_{1})h_{t}(x,y)m(t)dt.\] \end{itemize} \end{definition}%{} \lyxframeend{}\lyxframe{Laplace transform operators are Calder\'on-Zygmund operators} \begin{theorem}%{} The kernel $K_{p}$ is smooth off the diagonal and it satisfies the estimates\[ \vert K_{p}(x,y)\vert\lesssim R(x,y)^{-d}\] and\[ \vert\Delta_{2}K_{p}(x,y)\vert\lesssim R(x,y)^{-2d-1}.\] Therefore $p(-\Delta)$ is a Calderón-Zygmund operator, and it is bounded on $L^{q}(\mu)$, $1