# An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family

Authors: Maxim J. Goldberg; Seonja Kim
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

LetX be a topological space equipped with a complete positive σ-finite measure and T a subset of the reals with 0 as an accumulation point. Let atx,y be a nonnegative measurable function on X×X which integrates to 1 in each variable. For a function f∈L2X and t∈T, define Atfx≡∫atx,yfydy. We assume that Atf converges to f in L2, as t⟶0 in T. For example, At is a diffusion semigroup (with T=0,∞). For W a finite measure space and w∈W, select real-valued hw∈L2X, defined everywhere, with hwL2X≤1. Define the distance D by Dx,y≡hwx−hwyL2W. Our main result is an equivalence between the smoothness of an L2X function f (as measured by an L2-Lipschitz condition involving at·,· and the distance D) and the rate of convergence of Atf to f.


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## Body

## 1. Introduction

One of the questions that arise in harmonic analysis is the connection between the smoothness of a given function and the rate of approximation by members of a specified family of functions. An important example is the relationship between the smoothness of a function and the speed of convergence of its diffused version to itself, in the limit as time goes to zero. As mentioned in the Introduction of [1], for the Euclidean setting and the heat kernel, see for example [2, 3].In a more general setting, for a diffusion semigroupTtft≥0 on a topological space X with a positive σ-finite measure given, for t>0, by an integral kernel operator: Ttfx≡∫Xρtx,yfydy, Coifman and Leeb in [1, 4] introduce a family of multiscale diffusion distances and establish quantitative results about the equivalence of a bounded function f being Lipschitz and the rate of convergence of Ttf to f, as t→0+. The respective authors of [5–7] consider different aspects of the connection between the smoothness of a function and the rate of convergence of its diffused versions to itself.As mentioned in, for instance, the Introductions of [5–7], the interest in diffusion semigroups is natural since they play an important role in analysis, both theoretical and applied. Diffusion semigroups include the heat semigroup and, more generally, as discussed in, e.g., [8], arise from considering large classes of elliptic second-order (partial) differential operators on domains in Euclidean space or on manifolds.For examples of theoretical results involving diffusion semigroups, the interested reader may refer to Chavel [9], Cowling [10], Stein [8], Sturm [11], and Wu [12]. Some applications of diffusion semigroups to dimensionality reduction, embedding, clustering, data representation, manifold parametrization, and multiscale analysis of complex structures can be found in, e.g., [13–22]. Various definitions and procedures for efficient computation of natural diffusion distances can be found in, e.g., [1, 4, 23, 24].In the present work, we consider a more general family than a diffusion semigroup. ForT a subset of the reals having 0 as an accumulation point, for t∈T, let atx,y be a nonnegative measurable function on X×X which integrates to 1 in each variable. For a function f∈L2X and t∈T, define Atfx=∫Xatx,yfydy. We assume that for every f∈L2X, Atf−fL2⟶0, as t⟶0,t∈T. No assumption is made that the family At is symmetric or is a semigroup nor is anything assumed about T other than that T has 0 as an accumulation point.For a finite measure spaceW, selecting hw∈L2X for every w∈W, we define a distance between points x,y∈X by Dx,y≜hwx−hwyL2W. We next introduce an L2 version of being Lipschitz (relative to At) using this distance D. Our main result is that a function f∈L2X is L2-Lipschitz if and only if we have an estimate of the rate of convergence of f,f−Atf to 0, namely, 0≤f,f−Atf≤cgDt, where gDt=2∫Whw,hw−Athwdw⟶0, as t⟶0,t∈T.Our paper is organized as follows. Following a notation and assumptions section (Section2), we state the main definitions, provide some examples, and establish our results in Section 3. The paper ends with the Conclusions and Acknowledgments sections.


## 2. Notation and Assumptions

LetX be a topological space equipped with a complete positive σ-finite measure. The measure on X will be denoted by dx and dy. W is a finite measure space, with measure denoted by dw. We assume all spaces involved are such that Fubini’s theorem holds on any product of these spaces; e.g., the spaces are σ-finite. All functions are assumed to be real-valued and measurable on the respective spaces; in particular, functions of several variables are assumed to be measurable on the appropriate product spaces.T will denote a subset of the reals, with 0 as an accumulation point. From now on, t⟶0 will mean t⟶0,t∈T. For every t∈T, let atx,y be a nonnegative measurable function on X×X with the property that ∫Xatx,ydy=∫Xatx,ydx=1. For t∈T and a function f∈L2X, define At:L2⟶L2 by Atfx=∫Xatx,yfydy. We assume that for every f∈L2X, Atf−fL2⟶0, as t⟶0.No assumption is made that the familyAt is symmetric or is a semigroup nor is anything assumed about T other than that T has 0 as an accumulation point.Note thatAt is indeed bounded on L2 with norms not exceeding one, since(1)g,Atf≤∬atx,yfxgydxdy≤∬atx,yf2xdxdy1/2∬atx,yg2ydxdy1/2=∫atx,ydyf2xdx1/2∫atx,ydxg2ydy1/2=fL2gL2.In particular,f,f−Atf≥0.We will define a familyΔ of symmetric distances on X×X satisfying the triangle inequality with the following properties for every D∈Δ:(i)
gDt≡∬atx,yD2x,ydxdy<∞ for every t∈T(ii)
gDt⟶0, as t⟶0


## 3. Main Definitions and Results

We start by describing the familyΔ of symmetric distances on X×X.Definition 1.
Select a finite measure spaceW.  For eachw∈W, selecthw∈L2X, defined everywhere, withhwL2X≤1. Note that somehwmay be chosen to beidentically0. Then, the distanceD∈Δis given by(2)Dx,y≡hwx−hwyL2W=∫Whwx−hwy2dw1/2.Clearly,D is symmetric and satisfies the triangle inequality (the latter fact follows from the triangle inequality for L2W).Before looking at some examples of such distances, we define ourL2-Lipschitz condition.Definition 2.
ForD∈Δ, we say thatf∈L2XisL2-Lipschitz (relative toAt) if(3)∬atx,yfx−fy2dxdy≤c∬atx,yD2x,ydxdy,for every t∈T.Now let us consider some examples of distancesD∈Δ. For the first one, let X be a bounded subset of ℝn having (some) finite measure dx. Let W=1,2,⋯,n, with dw indicating unit masses assigned at each point of W. For k=1,⋯,n, let hkx=cxi, for x=x1,⋯,xn∈X, where c is a suitable constant to ensure that ∥hk∥L2≤1. Then, D1x,y≡∫Whwx−hwy2dw1/2=c2∑k=1nxk−yk21/2, a multiple of the Euclidean distance on X.For our second example, letX be a finite measure space with measure dx. Let W=0,1×X, with dw=sα−1dsdx, where α>0. For w=s,u∈W, let hwx=asx,u. Clearly, hwL2=∫asx,udx=1. Then,(4)D22x,y≡∫X∫01asx,u−asy,u2sα−1dsdu=∫01asx,·−asy,·L22sα−1ds,an analog of the distance considered by Coifman and Leeb in [1, 4] for a semigroup. Note that asx,u and asy,u are normalized in L2 with respect to u as well.To gain some understanding of this distanceD2 (although we will use the case of ℝn with Lebesgue measure, not a finite measure space), let us calculate the distance D2=∫01asx,·−asy,·L22sα−1ds1/2 for the basic case when X=ℝn with dx Lebesgue measure and At is the heat flow semigroup. (While the derivation right after the statement of Proposition 2.6 in [5] by Coifman and Goldberg has a calculation of this distance, we present a more detailed computation here.)We easily see that∫Xasx,u−asy,u2du=2−2e−x−y2/16s, where ∣x−y∣ is the Euclidean distance between the points x and y. Thus, D22x,y=2∫011−e−x−y2/16ssα−1ds.If∣x−y∣≥1, e−x−y2/16s is bounded away from 1 for 0<s<1, so D22x,y~∫01sα−1ds=c.If∣x−y∣<1, write(5)∫011−e−x−y2/16ssα−1ds=∫0x−y21−e−x−y2/16ssα−1ds+∫x−y211−e−x−y2/16ssα−1ds.For the first summand, observe that∫0x−y21−e−x−y2/16ssα−1ds~∫0x−y2sα−1ds=cx−y2α. For the second summand, an easy calculation shows that, for α≠1,(6)∫x−y211−e−x−y2/16ssα−1ds~∫x−y21x−y2ssα−1ds=c1α−1x−y2−x−y2α~x−y2α,0<α<1.x−y2,α>1.Combining with the estimate for the first summand, and with the case∣x−y∣≥1, we obtain that for 0<α<1,(7)D2x,y~x−yα,∣x−y∣<1,1,∣x−y∣≥1,while for α>1,(8)D2x,y~x−y,x−y<1.1,x−y≥1.Our third example is a variation of our second example above. As in the second example, letX be a finite measure space with measure dx and let W=0,1×X, with dw=sα−1dsdx, where α>0. For w=s,u∈W, let hwx=asx,u/as·,uL2. Clearly, hwL2=1.Then, our new distance is given by(9)D32x,y=∫X∫01asx,uas·,uL2−asy,uas·,uL22sα−1dsdu.(For a related example, see Section 4 of [23] and the very last example in Section 2 of [5].)Let us specialize to the case of a symmetric diffusion semigroup with the following additional requirement:asz,z is constant over z∈X (but varies with s). Let as·,· denote the value of asz,z for every z∈X. Under these assumptions, using the semigroup property, we easily obtain(10)D32x,y=∫012−2a2sx,ya2s·,·sα−1ds.In the very special subcase ofX=ℝn equipped with Lebesgue measure and At the heat flow semigroup,(11)2−2a2sx,ya2s·,·=2−2e−x−y2/8s,and we thus obtain the same estimates for D3x,y as for D2x,y above.We now return to the general development. The following simple result is the key tautology to prove our Theorem6.Proposition 3.
Forf∈L2X, ∬atx,yfx−fy2dxdy=2f,f−Atf.Proof.
Using Fubini’s theorem and the assumption thatatx,y integrates to 1 in each variable, we see that(12)∬atx,yfx−fy2dxdy=∬atx,yf2x+f2y−2fxfy2dxdy=∬atx,yf2xdxdy+∬atx,yf2ydxdy−2∫fx∫atx,yfydydx=2fL2X2−2f,Atf=2f,f−2f,Atf=2f,f−Atf.ForD∈Δ, letting gDt≡∬atx,yD2x,ydxdy, we obtain the following result.Proposition 4.
gDt=2∫Whw,hw−Athwdw.Proof.
Using the definition ofDx,y, Fubini’s theorem, and Proposition 3, we observe that(13)∬atx,yD2x,ydxdy=∫W∬atx,yhwx−hwy2dxdydw=2∫Whw,hw−Athwdw.Corollary 5.
gDt<∞for everyt∈TandgDt⟶0ast⟶0.Proof.
Sincehw,hw−Athw≤2 and ∫Wdw<∞, the result that gDt<∞, for every t∈T, follows from Proposition 4. From one of our initial assumptions that for every f∈L2X, Atf−fL2⟶0, as t⟶0, we obtain that hw,hw−Athw⟶0, as t⟶0, for every w. Hence, gDt⟶0, as t⟶0, by the dominated convergence theorem.Recalling Definition2, we can now prove the following theorem, which is of interest only due to Corollary 5.Theorem 6.
ForD∈Δandf∈L2X, fisL2-Lipschitz if and only iff,f−Atf≤cgDt, for everyt∈T.Proof.
First, suppose thatf∈L2X is L2-Lipschitz. Then, by Proposition 3, we have(14)f,f−Atf=12∬atx,yfx−fy2dxdy≤c∬atx,yD2x,ydxdy=cgDt,for t∈T.
Conversely, supposef,f−Atf≤cgDt, for every t∈T. Then, by Proposition 3 again,(15)∬atx,yfx−fy2dxdy=2f,f−Atf≤cgDt=c∬atx,yD2x,ydxdy.
Thus,f is L2-Lipschitz.It is easy to see that1/2f−AtfL22≤f,f−Atf≤fL2f−AtfL2, so Theorem 6 establishes an equivalence between f being L2-Lipschitz and having an estimate of the speed of convergence of f−AtfL2 to 0, as t→0.Note that ifAt is a symmetric semigroup (and T=0,∞), then(16)f,f−Atf=f,f−f,Atf=f,f−At/2f,At/2f=fL22−At/2fL22=fL2−At/2fL2fL2+At/2fL2,so fL2fL2−At/2fL2≤f,f−Atf≤2fL2fL2−At/2fL2. Hence, if At is a symmetric semigroup, f,f−Atf~fL2−At/2fL2.


## 4. Conclusions

ForX a topological space equipped with a complete positive σ-finite measure, W a finite measure space, and selecting everywhere-defined real-valued hw∈L2X for every w∈W with hwL2X≤1, we have defined a distance D by Dx,y≜hwx−hwyL2W.ForT a subset of the reals having 0 as an accumulation point and for t∈T, letting atx,y be a nonnegative measurable function on X×X which integrates to 1 in each variable, we have considered bounded operators At on L2X given by Atfx=∫Xatx,yfydy. Assuming that for every f∈L2X, Atf−fL2⟶0, as t⟶0,t∈T, we have shown that ∬atx,yfx−fy2dxdy≤c∬atx,yD2x,ydxdy, for every t∈T if and only if 0≤f,f−Atf≤cgDt, where gDt=2∫Whw,hw−Athwdw⟶0, as t⟶0,t∈T.



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