Log-Sobolev inequalities on graphs with positive curvature Текст научной статьи по специальности «Математика»

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DOI: https://doi.Org/10.15688/mpcm.jvolsu.2017.3.8

UDC 517 LBC 22.161

LOG-SOBOLEV INEQUALITIES ON GRAPHS WITH POSITIVE CURVATURE1

Yong Lin

Professor, Department of Mathematics, Renmin University of China linyong01@ruc.edu.cn

59 Zhongguancun Street, Haidian District Beijing, 100872, P.R. China

Shuang Liu

Student, Department of Mathematics, Renmin University of China cherrybu@ruc.edu.cn

59 Zhongguancun Street, Haidian District Beijing, 100872, P.R. China

Hongye Song

Lecturer, Department of Mathematics, Renmin University of China,

59 Zhongguancun Street, Haidian District Beijing, 100872, P.R. China

songhongye@bisu.edu.cn

Beijing International Studies University,

1 Dingfuzhuang Nanli, Chaoyang District, Beijing, China

o Abstract. Based on a global estimate of the heat kernel, some important

, inequalities such as Poincaré inequality and log-Sobolev inequality, furthermore a

M tight logarithm Sobolev inequality are presented on graphs, just under a positive

J curvature condition CDE'(n,K) with some K > 0. As consequences, we obtain

, exponential integrability of integrable Lipschitz functions and moment bounds at

d the same assumption on graphs.

Zj

c

^ Key words: Log-Sobolev inequality, Laplacian, CDE'(n,K).

1. Notations and main results

Let G = (V,E) be a symmetric weighted, connected and locally finite graph. The weight function denote by uxy, we assume uxy = uyx (symmetry), and m(x) :=J2y~x < < x, for any x e V (locally finite). Moreover we assume

^min = inf Wxy > 0.

y ~ X

x,y £ v

Given a positive measure - : V ^ R+ on graph, and assume that

m(x)

Vu := max —-— < x, xev u(x)

n -(x)

:= max - < x.

x,y £ V y

We denote by VR the space of real functions on V, by ip(V, -) = {f e VR : : l>2xev —(x)lf (x)lP < (}, 1 < P < x, the space of integrable functions on V with respect to the measure -. For p = x, let £^(V, -) = {f e VR : sup^eV |/(x)| < x} be the set of bounded functions. If for any f,g e 12(V., -), let the inner product as if, 9) = J2 xev -(x)f (x)g(x), then the space of £2(V, -) is a Hilbert space. For every function f e £P(V, u), 1 < p < x, we can define the norm as follows

\xev J

-(x)lf (x)lpj , 1 < p< X.

We denote by C0(V) C £P(V) the dense subset of functions f : V ^ R with finite support. For any graph, it associated with a Dirichlet form, see [7],

Q(d) :D(Q) x D(Q) ^ R 1 2

(f,9) ^ шху(f(у) - f (х))(9(у) - g(x)),

x~y

where the form domain D(Q) is defined as the completion of C0(V) under the norm || ■ ||q given by

11/IIq = 11/ll*W) + 1E^(f (y) - f (x))2,Vf e C0(V),

x,y

see Keller and Lenz [9]. For the Dirichlet form Q(D), its infinitesimal generator A is called the discrete Laplacian A on G by, for any x e V,

Af (x) = -1- E ^(f (V) - f (*)),

^ ' y^x

it is a bounded operator because of the assumption of Du. And the associated semigroup Pt : £2(V, -) ^ £2(V, -), for any x e V,

Ptf (x) = ^2 -(y)p(t ,x,y)f (y),

yev

V

where p(t,x,y) is the fundamental solution of the heat equation (heat kernel), we refer from [9].

Now we introduce the notion of the CDE' inequality on graphs from [8]. First we need to recall the definition of two bilinear forms associated to the --Laplacian. Definition 1.1. The gradient form r and the iterated gradient form r2 are respectively defined by, for any f,g e VR,

2r(f,g)(x) = -L £ Uxy (f (y) - f (X))(g(y) - g(X)),

and

2r2(f, g) = Ar(f, g) - r(f, Ag) - r(Af, g). We write r(f) = r(f, f), and ^(f) = ^(f, f).

Definition 1.2. We say that a graph G satisfies the exponential curvature dimension inequality CDE(x,n,K) if for any positive function f : V ^ R+ such that Af (x) < 0, we have

W)(x) = W)(x) - r(f, ^fi) (x) > -(Af)(x)2 + Kr(f)(x).

f ) n

We say that CDE(n, K) is satisfied if CDE(x, n, K) is satisfied for all x e V.

We say that a graph G satisfies the CDE'(x,n,K), if for any positive function f : : V ^ R+, we have

T2(f )(x) > -f (x)2 (Alog f) (x)2 + KT(f )(x). n

We say that CIDE'(n, K) is satisfied if CDE'(x, n, K) holds for all x e V.

We write fv fd^ = J2xev V-(x)f (x). Now we introduce Poincaré inequality and log-Sobolev inequality on graphs. For all integrable functions f, let

/r \2

p2,

V(1 ) = JV 1~{JV fdЦ)

be the variance of the function, and for any positive integrable function f such that

fv f \ loS f Ин <

E(f ) = í f log fdv - Í fdvlog Í fdv,

Jv Jv Jv

be the entropy.

A graph G = (V., E) is said to satisfy a Poincaré inequality with constant С > 0, for any f e D(Q), if

v(f) < с í T(f)dvL, (P(C))

Jv

a log-Sobolev inequality with constants С > 0,D > 0, for any f e D(Q), if

E(f2) < 2C Í T(f)dp + D f f(LS(C,D))

Jv Jv

There are a useful fact we will use later, that is, we just examine whether the log-Sobolev inequality holds for any positive value function, since

E(f) = E(\f |2) < 2C f r(|/+ d\ \f< 2C f r(f)d^ + d\ fd^,

Jv Jv Jv Jv

since \\a\ - \b\ \ < \a - b\ for any a,b e R, then r(\f \) < r(f). When D = 0, the logarithmic Sobolev inequality will be called tight, denoted by LS(C).

The main results we derived in this paper are log-Sobolev inequality and the tight log-Sobolev inequality with appropriate constants on graphs, just under a condition with a positive curvature.

Theorem 1.1. Let G = (V,E) be a locally finite, connected graph satisfying CDE'(n,K) for any K > 0, then for any t0 > 0, such that the graph satisfies log-Sobolev inequality LS(C, D), with

C =2t0, D = 2 log M,

where M = M(n, K) = 1-

( l-e-2Tt0

Theorem 1.2. Under the same conditions of the above theorem, then the graph satisfies a tight log-Sobolev inequality LS(C") with

„„ 3 (( nKC' N / nKC' \ nKC' nKC' \ ^

= K{\1 + —) log{1 +- ~log —) + ,

where C = 1536D^DMn.

We also have two applications of the above theorem, including exponential integrability and moment bounds, as follows.

Theorem 1.3. Let G = (V,E) be a locally finite, connected graph satisfying CDE'(n,K) for any K > 0, then there exist a constant 0 < C = C"(n,K) < x such that, if f is 1-Lipschitz and J fd^ < <x, we have

f esf d^ < es ¡vfd^+Cs2, (1.1)

v

and

1 a2(fvfdri2

e a2 f du< , = e 2(l-2Ca2) ^ QO,

lv VT—2CÔ*

for every a2 < .

Theorem 1.4. Under the same conditions of the above theorem, then there exist a constant 0 < G = C''(n, K) < x such that for every p > 2 and every f e (P(V, |), we have

11/ IU <11/12 + 2C(p - 2) r(f )§P .

If f is Lipschitz, then

< II/112 + 2C(p - 2) II/II

Lip '

P

2. log-Sobolev inequality

In this section we focus on log-Sobolev inequality and tight log-Sobolev inequality with the assumption of positive curvature.

Proof of Theorem 1.1 We divide the proof into several steps.

Step 1. For any function 0 < f e D(Q) and every t > 0,

i f log f2d- < 2t i r(f)d— + / f2 log(Ptf)2d-. (2.1)

Jv Jv Jv

This is a relationship between entropy and energy along the semigroup from [C] without diffusion property. We consider a functional s) = fv fj2 logPsfd—, for any s > 0, and 0 < f e D(Q), where fk = nkf, and nk is a nondecreasing sequence of finitely supported functions {nksuch that

lim nk = 1.

k—^^o

Then, since uxy = uyx, and by the Green's formula,

ik

Jv and

r i k

v Psf Jv (psf

s )= /fc-FTT" dV = - ,Psf )dvL,

^,psn* = 2^ £ цP§)- P§)) vm - ps!(*» 1 '

V. (f2(v)+ f2(x) (x) ^^(У)]<

У fk(x) РШ PSKxr) -

X,yEV

- 2 E^xy {fke(y) + fi2(x) - 2 fk (x) fk (y))

x,y€Y

= r( fk) d-

Jv

So that $'( s) > - fv r( fk)d— for every s > 0. Integrating from 0 to t yields

22

/ f2k log f2dv - 2t / T( fk)dv + f2k log(Ptf )2dv.

v v v

We have

v

r(ik)dv = 1 E шху(fk(У) - fk(x))2 -

2 E -xy(h (V) - h(x)2

x,y£V

- 2 E -xy (fk(y) - f(y))2 + 2J2 -xy (f(y) - f(x))2 +

x,y€V x,y€V

1

+ 2 E -xy(f(x) - fk(x))2 = YJ^(x)f(x)2(y\k(x) - 1)2 + r(f)dv -

x,yev xev v

- i T(f)dv + Dv E V(x)f(x)2(m(x) - 1)2,

" v x^v

let k ^ x,

£ l(x) f (x)2(nk(x) - 1)2 ^ 0,

xev

and log f, logPtf E 11(V., |), so we derive what we desire.

Step 2. If for some t0 > 0, there exists a constant M > 0 such that for all 0 <fe D(Q),

\\PtJ 11« <m 11/H2,

then a logarithmic Sobolev inequality LS(C, D) holds, with

C = 2t0, D = 2logM. When ||/\\2 = 1, since for any x E V, p0f(x)l < \\PtJ\\«, we have from (2.1),

/ f2 log f2di < 2t i T(f)d| + 2log(\\Ptof \\«) < 2t i T(f)d| + 2logM.

v v v

If \\/\\2 = 1, we can apply f = jj^ to the above inequality, which immediately leads to the above result.

Step 3. We obtain LS(C,D) on graph just under the assumption of the positive curvature.

We first introduce the global estimate of heat kernel. If G = (V, E) be a locally finite, connected graph satisfying CDE'(n,K) with K > 0, then the measure | is finite (see [6]). We may then assume | is probability measure, then limt^«p(t,x, ■) = 1 in the case of probability measure. Under the same condition, for any x,y ev, t > 0,

p(t,x, y) < 7-w. (2.2)

(1 - e- 2ft)

(see [8, Proposition 7.5]).

Then, by Holder inequality and combining with (2.2), we obtain for some t0 > 0,

\\PtJ\\« <M\\/\2,

where M = -12K . we together with step 2 immediately end the proof.

1-e 3-4°

Proof of Theorem 1.2 From [5], we can divide the proof of the tight log-Sobolev inequality at the assumption of curvature into several steps.

Step 1. A logarithmic Sobolev inequality LS(C,D) together with Poincare inequality P(C') implies a tight logarithmic Sobolev inequality LS(C + C(^ + 1)).

We first introduce Rothaus's Lemma in the discrete condition from [10]. If f E VR such that fv f2 log(1 + f2)di < x, then for every a E R,

E((f + a)2) <E(f2) + 2i f2di.

v

Applied to f = f — fv fdi with a = fv fdi of the above inequality, yields

E(f2) < E(f2) + 2 i f2di,

v

by the logarithmic Sobolev inequality LS(C,D) applied to /, and combining to the above inequality we get,

E(f2) < 2C i T(f)d* +(D + 2) i f2d^, Jv Jv

and since fv f2d| = fv f2d| — (fvfd|2 for the probability measure | (the finiteness of measure is true when the graph satisfies CDE'(n,K) with K > 0, see [8]), it remains to use the Poincare inequality P(C'), the conclusion is therefore established.

Before the next step, we first show two theorems from [8] and [2]. It will be useful to prove the graph also satisfy a Poincare inequality when the curvature of graph is bounded by a positive number.

Lemma 2.1. If a graph be a locally finite, connected, and satisfy CDE'(n,K) with K > 0, then the diameter of the natural distance on the graph is finite, moreover the upper bound quantitative estimation is

6 D„n D < 2ш' ц

K

We refer a lower bound estimate of eigenvalues from [2]. Lemma 2.2. Let a finite graph Q satisfy CDE(n, p) with p > 0, then

1

Ai >

64(5n + 1)DUD2'

Since CDE'(n,K) implies CDE(n,K), see [3], so we have the same result under the condition of CDE'(n, K) with K > 0. From Lemma 2.1, we find the graph is finite when it satisfies CDE'(n,K) with K > 0. So we can use the above lemma when the curvature is bounded by a positive number.

Step 2. If a graph be a locally finite, connected, and satisfies CDE'(n,K) with K > 0, then the graph satisfies a spectral inequality P(C') with

С' = С'(n, К) = 1536DDnn(5nr+ 1 < ж.

К

For any f G D(Q), from spectral theory on graph G, let Ài be the z-th eigenvalue, and li!-1 (N is the number of vertices in the subgraph Q) be an orthonormal basis of eigenfunctions, i.e.

A^ = Ài^i,

and

(Фг, Фj) = E (Х)Ф j(X) = bij,

then we can write f = YIîLq1 аФг.

xev

rv .1 К .

г=0

We obtain

N-1 N-1

-A f = аФг =E а\Фг

а

г=0 i=0

since

N-1 N-1 N-1

E аФг • E аЛгФг > Л1 f

=0 =0 =0

therefore

/ Г(f)dp = - fAfdp > AW f2dp. iv Jv Jv

Let C' > 1, we may apply f = f — fvfdp to the above equality, we obtain the Poincare inequality,

v(f) <C i T(f)dp.

v

Combining Lemma 2.1 and Lemma 2.2, we obtain the spectral inequality. Step 3. Combining Theorem 1.1, step 1 and step 2, we obtain with the condition CDE'(n, 0), the tight log-Sobolev inequality LS(C'') holds with

C'' = C''(n, K) = 2to + C' (1 + logM), where M = -2»Kt and C' = 1536D^DUn. Minimizing the right-hand side of

the above equality with respect to t0 > 0, we have

3 / nKC' \

'o = 2K log{1 + —)

so we can get the result from simple computation.

3. Its applications

3.1. Exponential integrability

In this section we prove every integrable Lipschitz function is exponentially integrable if the Poincaré inequality and the tight log-Sobolev inequality holds respectively, moreover

at the assumption of positive curvature on graphs. For a given Lipschitz function f e D(Q),

i

we denote its Lipschitz norm by ||/||Lip = \\r(f)\\<^. A function is said to be 1-Lipschitz if

ll/lk < 1.

Proposition 3.1. Suppose that the graph satisfies the Poincare inequality with a constant C. Then, if f is 1-Lipschitz and fv fdp < <x, we have

e A f dp < oo,

v

for any A <

Proof. For any n G Z+. let tyn(t) = (—n) Vt An. It is easy to know tyn(t) satisfies, for any ti, t2 G R

№n(ti) — 4n(^)| < |ii — t2|.

Considering fn(x) = 4n° f(x), which converges to f(x) when n ^ x, for every x gV, we have r(fn, fn) < T(fn, f) < T(f, f) < 1. Therefore, using Fatou's lemma, we may restrict ourselves to the case where is bounded by replacing to n.

Then we consider the function g = e 2, r( f) < 1. Since a important discrete estima-

tion from Proposition 6.7 in [8], then

J r(e ¥ )dp < J eÀ J r( f) dp < y J e A/dp, (3.1)

setting $(X) = fv eÀf dp., applied the the Poincaré inequality P(C') to g, we obtain

C'A2 .o.À,

Л2 Г , A2

Ф(Л)(1 -^f-) < ф2^^.

If 1 - ^A2 > 0, then we have

^(A) < ^2(A)(i — CA-)-1,

and it remains to iterate the procedure replacing A by | to get the result.

Remark 3.2. We have, of course, a similar result to the general Lipschitz function f with T(f) < c2 for changing f into f.

Proof of Theorem 1.3. As before, we may restrict ourselves to bounded 1-Lipschitz functions. Consider the function (p(s) = fvesfdu, observe that

p'( s) = fesfdi,

v

while

E(esf ) = s / fesf d| — p(s)log p(s) = sp'(s) — p(s)log p(s).

v

On the other hand, since T(f) < 1, and (3.1),

r sf s2

J T(e-)d| < -p(S).

Applying L S( C) to p( ), then we have

p — p log p < C 2p,

integrating the above inequality. To this end, let F(s) = Ss log p(s) (with F(0) = (/)), so that

spL — log p

F'(s) = -^ < C,

2

it follows,

F(s) < f fdi +Cs, v

which amounts to (1.1) immediately. Integrating (1.1) in the with respect to the measure

_ s2

e z^ds on R. Since

J etx-^dt = e

by Fubini's Theorem, we have

e~f du < , = e 2(1-2Cl2) < w,

Jv V1 — 2C a2

for every a2 < . Therefore the second claim holds and the proof ends.

3.2. Moment bounds on graphs

In this section, we obtain the moment bounds on graphs with positive curvature. Proof of Theorem 1.4. As before, we first consider positive bounded functions. Consider the functional 4(p) = ||/||2, p > 2, for a function 0 < f G V, |) the derivative of 4(p) given by,

4'(p) = ^4(p)i-fE(fp) > 0,

applying LS (C) to fi, combining the above equation, we obtain

4'(p) < ^Mp)i-t / T(f t

2 v

We need estimate the term fv T(f ?)dp for the symmetric property and Holder inequality, since

£ E ^ (f(x) i — f(y) i )2 =

1 — sgn( f(x) — f(y))

xEV у~ x

f(x) < f(у)

E "xV(f(x)§ - f(y)§)2--2-

xEV у ~ x

E £ «yx(m§ - m§)21 - ^ - и*»

xEV у ~ x

£ E ^y(f(*)§ - f(y)§)2

xEV у ~ x

f (x) > f (у)

so we have

i Г(f §)dp = £ £ ^y(f(x)§ - f(y)§)2 <

V

xE V у ~ x

f( x) > f( у)

< E E Uxy (If (*) §-1(f(x) - f(y)j)

xEV у~ x

f( x) > f( у)

2

2

J2 f(*)P-2 J2 Шxy(f(x) - f(y))2 <

xEV у~ x

f( x) > f( у)

2

< т E f(*)p-2r(!)(*) <

p ¡(*Y "r(!)(*)

xEV

1_ 2 2 < p2 (e p(*)fp(*)) P (e p(*)r(f)§ (*))P

\xEV J \xEV J

(/v r( § 4

^4(p) §-1 r(f) § dp

So substitute to the above inequality, we have

2

4'(p) < 2C^jv T(f),

by integration with respect to , and monotonicity, we can end the proof if is bounded.

As before, in the general case then follows from consideration of fn, since we know r(fn) < r(f), furthermore, \\fn\\P ^ \\f\\P with 1 <p<m from fn ^ f and |fnl < Ul for any n E Z+. We complete what we desire.

REMARK

1 Supported by the National Natural Science Foundation of China (Grant No. 11671401).

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ЛОГАРИФМИЧЕСКИЕ НЕРАВЕНСТВА СОБОЛЕВА НА ГРАФАХ ПОЛОЖИТЕЛЬНОЙ КРИВИЗНЫ

Йон Лин

Профессор, Факультет математики, Renmin University of China linyong01@ruc.edu.cn

59 Zhongguancun Street, Haidian District Beijing, 100872, P.R. China

Шуан Лиу

Студент, Факультет математики, Renmin University of China cherrybu@ruc.edu.cn

59 Zhongguancun Street, Haidian District Beijing, 100872, P.R. China

Хонье Сон

Лектор, факультет математики, Renmin University of China,

59 Zhongguancun Street, Haidian District Beijing, 100872, P.R. China

songhongye@bisu.edu.cn

Beijing International Studies University,

1 Dingfuzhuang Nanli, Chaoyang District, Beijing, China

Аннотация. В работе представлены некоторые важные неравенства на графах, такие как неравенство Пуанкаре и логарифмическое неравенство Соболева, а также плотное логарифмическое неравенство Соболева, полученные на основе глобальной оценки ядра уравнения теплопроводности при наложении только условия положительности кривизны CDE'(п,К) с некоторым К > 0. В качестве следствий мы получили экспоненциальную интегрируемость интегрируемых липшицевых функций и границ момента на графах при том же предположении.

Ключевые слова: логарифмическое неравенство Соболева, лапласиан, CDE'(п, К).