Liggett-Stroock form of Nash-type inequalities

Mu-Fa Chen1

Department of Mathematics, Beijing Normal University, Beijing 100875, The People's Republic of China (mfchen@ns.bnu.edu.cn)

Abstract

In the recent years, a great effort has been made in developing a new ergodic theory for Markov processes. It concerns mainly the study on several different inequalities. Some of them are very classical but some of them are rather new. The talk introduces the inequalities and discuss their comparison. Some recent results are also reported.

1  Notations

Let (E, E, p) be a measure space with s-finite non-negative measure p. Denote by Lp(p) the real Lp-space with norm || ·||p. Consider a symmetric form D on L2(p) with domain D (D). For instance, one particular form we are interested is as follows:
D(f) = 1
2
 ó
õ

E×E 
 J(dx, dy)[f(y)-f(x)]2+ ó
õ

E 
 K(dx)f(x)2,
D (D) = {f Î L2(p): D(f) < ¥},
(1)
where J is a non-negative symmetric measure having no change on the diagonals {(x, x): x Î E} and K(dx) is a non-negative measure. As usual, D (f, g): = [D(f+g)-D(f-g)]/4.

2  The Liggett-Stroock form of Nash-type inequalities

The main inequality we are interested is the following:

||f||2 £ C D(f)1/p V(f)1/q,        f Î L2(p),
(2)
where ||f|| = ||f||2, C = C(p) is a constant and 1/p+ 1/q = 1 with 1 £ p < ¥. Now, only the functional V ³ 0 has to be specified. The easier case is that p = 1 and hence 1/q = 0, then there is nothing to do about V and (2) is then reduced to the Poincaré inequality (1890):
||f||2 £ C D(f),        f Î L2(p).
(3)

Thus, we may assume in what follows that 1 < p < ¥. In view of the invariance of (2), it is natural to make the assumption that V(c f) = c2 V(f) for all constant c. However, if we take V(f) = D(f) or ||f||2, (2) is again reduced to (3).

Next, take V(f) = ||f||r2. We may also assume that r Î [1, 2) since the case that r = 2 is treated above and the case of r Î (2, ¥) can be reduced to the one of r Î [1, 2) by duality. When r = 1, it is called the Nash inequality (1958):

||f||2 £ C D(f)1/p ||f||12/q,        f Î L2(p).
(4)
By setting p = 1+2/n (n > 0), one gets the more familiar form of the Nash inequality ||f||2+4/n £ C D(f) ||f||14/n, f Î L2(p). It is interesting that for all r Î [1, 2), the inequality (2) with V(f) = ||f||r2 is equivalent to (4) and hence we need only to consider (4).

Of course, there are many other choices of V: V(f) = supx |f(x)|2, supx ¹ 0| [(f(x)-f(0))/( r(x, 0))] |2, supx ¹ y| [(f(y)-f(x))/( r(x, y))] |2, where r is a distance in E . The last one was used by Liggett (1991) and we may call the corresponding inequality the Liggett inequality.

3  Alternative form of (2)

From now on, we restrict ourselves to the case that p is a probability measure and the form (D, D (D)) satisfies D(1) = 0. Then, the right-hand side of (2) becomes zero for constant function f º 1. Thus, it is necessary to make a change of the left-hand side of (2). For this, one simply uses the variation of f: j(f) = p(f2)-p(f)2 instead of ||f||2, where p(f) = òf dp. Then we obtain the alternative form of (2) as follows.

j(f) £ C D(f)1/p V(f)1/q,        f Î L2(p),
(5)

4  The second class of inequalities

Keeping the right-hand side of (3) but making a change of the left-hand side, one gets the following inequality

ì
í
î 		ó
õ 		|f|2p/(p-1)U æ
è 		f2/||f||2 ö
ø 		dp ü
ý
þ 		(p-1)/p

 
 £ C D(f),        f Î L2(p).
(6)
When U = 1, it is just the Sobolev inequality (1936):
||f||2p/(p-1) £ C D(f),        f Î L2(p).
Since 2p/(p-1) ³ 2, the inequality is stronger than the Poincaré one except p = 2. When U = log and p = ¥, (6) is the logarithmic Sobolev inequality (L. Gross, 1975):
ó
õ 		|f|2log æ
è 		f2/||f||2 ö
ø 		dp £ C D(f),        f Î L2(p).
(7)
The advantage of the last inequality is that it is a poweful tool in the study of infinite dimensional analysis but not the Sobolev one.

5  The relation of the above inequalities

There is a simple comparison between the above inequalities:

Nash ineq.ÞLogS ineq.ÞPoincaré ineq.ÞLiggett ineq.

6  The isoperimetric and Cheeger's constants

To present some sufficient conditions for the inequalities of the form given by (1) with K(dx) = 0, we need to introduce some analogue of the isoperimetric or the Cheeger's constants. For Poincaré ineq.:

k(1/2) =
inf
p(A) Î (0, 1) 
 J(1/2)(A×Ac)
p(A)Ùp(Ac)
For Nash ineq.:

k(1/2) =
inf
p(A) Î (0, 1) 
 J(1/2)(A×Ac)
[p(A)Ùp(Ac)](n-1)/n
 ,    n = 2(q-1)
For LogS ineq.:
k(1/2) =
inf
0 < p(A) << 1 
 J(1/2)(A×Ac)
p(A)   æ
Ö
log[e+ p(A)-1]
 
.

Here only J(1/2) has not defined yet. Note that the original kernel J can be very unbounded. To avoid this, choose a symmetric founction r(x, y) so that J(1)(dx, E)/p(dx) £ 1, p-a.e., where J( a)I{r(x, y) > 0}(dx, dy) = [J(dx, dy)/( r(x, y) a )], a Î [0, 1].

7  Main theorem

Having the constants at hand, it is a simple matter to state our main result.

Theorem 1 For the form given by (1) with K(dx) = 0, if k(1/2) > 0, then the correspongding inequality in (5) or (7) holds.

We remark that even though the above condition is in general not necessary but it can be sharp qualitatively.

Footnotes:

1 Research supported in part by NSFC (No. 19631060), Math. Tian Yuan Found., Qiu Shi Sci. & Tech. Found., RFDP and MCME.

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On 29 Apr 1999, 00:08.