Some Extensions of E. Stein’s Work on Littlewood–Paley Theory Applied to Symmetric Diffusion Semigroups

Abstract

Discrete diffusion semigroups have proven to be highly effective tools for machine learning and data analysis due to the interplay between diffusion processes (of inferences), and their naturally associated geometries. Inspired by the harmonic analysis machinery that E. Stein developed for symmetric diffusion semigroups acting on $L_p$ spaces, we show that a correspondence between the rate of diffusion approximation and a Besov-type version of smoothness exists in the general continuous case, even without a local kernel representation. Specifically, let $\left\{ A_t\right\} _{t\ge 0}$ be a symmetric diffusion semigroup on $L_p(X)$, for X a complete positive $\sigma$-finite measure space. We first establish that for $1<p<\infty$ the semigroup $\left\{ A_t\right\} _{t\ge 0}$ acting on $L_p(X)$ is strongly continuous and holomorphic. We then show that for $f\in L_p(X)$ and $0<\alpha <1$, the following are equivalent:

• (1) $\left\| \left( A_{t}-I\right) f\right\| _{L_p(X)}\le ct^\alpha$,   for $0 < t\le 1$;

• (2) $|\langle f,h \rangle |\le c\int ^1_0 s^\alpha \left\| A_{s}h\right\| _{L_{p'}(X)}\frac{ds}{s}$,   for all $h\in L_{p'}(X)$; and

• (3) $\Vert {\mathcal {A}}A_tf\Vert _{L_p(X)} \le c t^{\alpha -1}$,  for $0 < t\le 1$, where ${\mathcal {A}}$ is the infinitesimal generator.

We also present some extensions, including results for the $p=\infty$ case.

Appendix

In this section, we give some examples of symmetric diffusion semigroups that are not given by kernels. As before, X is a complete positive $\sigma$-finite measure space.

To begin our construction, for $1\le p\le \infty$, let $P:L_p(X)\rightarrow L_p(X)$ be a bounded operator with the following properties:

• (1) $\left\| P f\right\| _{L_p(X)} \le \left\| f\right\| _{L_p(X)}$, for $1\le p\le \infty$;

• (2) $P 1 = 1$;

• (3) $P f\ge 0$ if $f\ge 0$;

• (4) P is a self-adjoint operator on $L_2(X)$; and

• (5) $P^2 = P$.

Let $Q=I-P$. Note that $Q^2=Q$ and that $QP=PQ=0$. Now fix a non-negative real number a. Finally, define the uniformly continuous semigroup $\left\{ A_t:L_p(X)\rightarrow L_p(X), 1\le p \le \infty \right\} _{t\ge 0}$ by

$$\begin{aligned} A_t=e^{-atQ}. \end{aligned}$$

Using that $Q^2=Q$, we easily see that

$$\begin{aligned} \begin{aligned} A_t&=e^{-atQ} = I+\left[ (-at)+\dfrac{(-at)^2}{2!}+\dots \right] Q\\&=I+\left( e^{-at}-1\right) Q = P + e^{-at}Q =\left( 1- e^{-at}\right) P+e^{-at}I. \end{aligned} \end{aligned}$$

Using the assumption that $P 1 = 1$, we immediately see that $A_t 1=\left( 1- e^{-at}\right) P1+e^{-at}1=1$. Also, since $1- e^{-at}\ge 0$, and $f\ge 0$ implies $P f\ge 0$ by assumption, we have that $A_t f = \left( 1- e^{-at}\right) Pf+e^{-at}f \ge 0$ for $f\ge 0$. Since P is assumed to be a self-adjoint operator on $L_2(X)$, $A_t$ is self-adjoint as well. Clearly, since $Q=I-P$ is a bounded operator on every $L_p(X)$, $1\le p\le \infty$, $A_t=e^{-atQ}$ is a uniformly continuous, hence strongly continuous, semigroup on every $L_p(X)$. Finally, since $1- e^{-at}\ge 0$ and $\left\| P f\right\| _{L_p(X)} \le \left\| f\right\| _{L_p(X)}$ by assumption,

$$\begin{aligned} \begin{aligned} \left\| A_t f\right\| _{L_p(X)}&\le \left( 1- e^{-at}\right) \left\| Pf\right\| _{L_p(X)} + e^{-at}\left\| f\right\| _{L_p(X)}\\&\le \left( 1- e^{-at}\right) \left\| f\right\| _{L_p(X)} + e^{-at}\left\| f\right\| _{L_p(X)}=\left\| f\right\| _{L_p(X)}. \end{aligned} \end{aligned}$$

Hence, $A_t$ satisfies all the requirements to be a symmetric diffusion semigroup.

We will now consider some examples of an operator P as above so that the associated symmetric diffusion semigroup $A_t$ does not have a kernel representation.

Example 1

For $X={\mathbf {R}}^n$, let $Pf(x)=\frac{1}{2}(f(x)+f(-x))= Ef(x)$, the even part of f. It is immediate to verify that the five required properties for P hold and that $A_t f = Pf + e^{-at}\left( I-P\right) f=Ef+e^{-at}Of$, where Of is the odd part of f. $A_t$ does not have a kernel representation. This is the example mentioned in the Introduction.

Example 2

Let $X={\mathbf {R}}^2$. For a function f defined on ${\mathbf {R}}^2$ and $x\in {\mathbf {R}}^2$, define $Pf(x)=\frac{1}{2\pi }\int _0^{2\pi }f(|x|,\theta )\,d\theta$, where $f(|x|,\theta )$ is the function f with polar coordinate arguments. So Pf(x) is the average of the values of f on the circle of radius |x|, center at the origin. Note that $P1=1$ and that $P f\ge 0$ if $f\ge 0$. It is easily seen that $P^2=P$, P is self-adjoint on $L_2({\mathbf {R}}^2)$, and that (by Jensen’s inequality) $\left\| P f\right\| _{L_p({\mathbf {R}}^2)} \le \left\| f\right\| _{L_p({\mathbf {R}}^2)}$ (the case $p=\infty$ being obvious). Note that $A_t=\left( 1- e^{-at}\right) P+e^{-at}I$ does not have a kernel representation: P integrates over one-dimensional curves in two-dimensional space, while $If=f$ is point evaluation.

Example 3

The previous example can be extended to other changes of variables in ${\mathbf {R}}^n$, with Pf defined by the normalized integral of the product of f and the change of variables Jacobian, over k of the new variables.