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ELECTRONIC COMMUNICATIONS in PROBABILITY

POINCARÉ INEQUALITY AND THE

L

CONVERGENCE OF

SEMI-GROUPS.

PATRICK CATTIAUX

Institut de Mathématiques de Toulouse. UMR 5219. Université Paul Sabatier,

118 route de Narbonne, F-31062 Toulouse cedex 09. email: cattiaux@math.univ-toulouse.fr ARNAUD GUILLIN

Laboratoire de Mathématiques. UMR 6620, Université Blaise Pascal, avenue des Landais, F-63177 Aubière.

email: guillin@math.univ-bpclermont.fr CYRIL ROBERTO

Laboratoire d’Analyse et Mathématiques Appliquées. UMR 8050, Universités de Paris Est Marne la Vallée et de Paris 12-Val-de-Marne

Boulevard Descartes, Cité Descartes, Champs sur Marne F-77454 Marne la Vallée Cedex 2

email: cyril.roberto@univ-mlv.fr

SubmittedMarch 3, 2010, accepted in final formJune 8, 2010 AMS 2000 Subject classification: 26D10, 39B62, 47D07, 60G10, 60J60

Keywords: Poincaré inequality, rate of convergence

Abstract

We prove that for symmetric Markov processes of diffusion type admitting a “carré du champ”, the Poincaré inequality is equivalent to the exponential convergence of the associated semi-group in one (resp. all) Lp(µ)spaces for 1<p<+_∞. We also give the optimal rate of convergence. Part of these results extends to the stationary, not necessarily symmetric situation.

1 Introduction and main results.

Let X_t be a general Markov process taking values in some Polish space E. We denote by L its infinitesimal generator. We assume that the extended domain of L contains a nice core _D of uniformly continuous functions which is an algebra. Functions in_Dwill be called “smooth”, and constant functions are assumed to be in_D. Next, we define the “carré du champ” operator

Γ(f,g) =1

2 L(f g)−f L g−g L f

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For a given probability measure µ on E, the associated Dirichlet form can thus be defined for smooth f’s as

E(f,f):= ₋

f L f dµ=

Γ(f,f)dµ.

In the sequel we assume thatµis an invariant measure. Thus L generates a µstationary semi-groupPt, which is a contraction semi-group on allLp(µ)for 1≤p≤+∞. We shall also look at cases where the followingL2_{ergodic theorem holds (this is automatic in the symmetric case): for} all f _∈L2_(µ)_,

lim

t→+∞kPtf −

f dµ_kL2₍_µ₎= 0 .

For the above, seee.g.[3].

Here and in the sequel, for anyp∈[1,_∞),_kf kLp₍_µ₎or shortlyk f k_p, stands for theLp(µ)-norm of f with respect toµ:_k f _kp_p:=R_|f_|pdµ.

In the sequel we shall assume thatΓcomes from a derivation, i.e.

Γ(f g,h) = fΓ(g,h) +gΓ(f,h),

i.e. (in the terminology of[1]) thatX.is a diffusion. We also recall the chain rule: if ϕis aC2 function,

L(ϕ(f)) =ϕ′(f)L f +ϕ′′(f) Γ(f,f).

It is well known that the following two statements are equivalent

(H-Poinc). µsatisfies a Poincaré inequality, i.e. there exists a constantCP such that for all smooth f,

Varµ(f):=

f2dµ₋

f dµ

≤ C_P

Γ(f,f)dµ. (H-2-1). There exists a constantλ2such that

Var_µ(P_tf)_≤ e−2λ2t_Var

µ(f). If one of these assumptions is satisfied we haveλ₂=1/C_P.

The symmetric situation is peculiar due to the following well known reinforcement of the previous equivalence

Lemma 1.1. Assume that µ is symmetric. If there exists β >0 such that for all f ∈ C, C be-ing an everywhere dense subset ofL2(µ)the following property holds Varµ(Ptf)≤ cf e−2βt, then Varµ(Ptf)≤e−2βtVarµ(f)for all f ∈L2(µ), i.e. the Poincaré inequality holds with CP≤1/β. There are several proofs of this lemma (using the spectral resolution or some convexity tools), see e.g.[8]lemma 2.6. In particular any exponential decay of the Variance

(H-2) Varµ(Ptf)≤ K2e−2βtVarµ(f), implies (H-2-1) withβ=λ₂, i.e. we may takeK₂=1.

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A similar statement is false in the non-symmetric case. Examples are known where (H-2) holds withK₂>1 but where the Poincaré inequality does not hold (see the kinetic Ornstein-Uhlenbeck process studied in[6](also see[2]section 6)).

Our aim in this note will be be to investigate theLp_{situation. For} _f _∈Lp_(µ)_define N_p(f):=_kf ₋

f dµ_k_p .

Consider the operators defined on Lp _by _T_t₍_f_{) =} _P_t₍_f ₋_µ(_f₎₎_{, where}_µ(_f_{) =} R _{f d}_µ_. _T_t _{is a} bounded operator inLp_{with operator norm denoted by}_k_T_t_k_p_{. For all}_t_≥_0,_k_T_t _k_p_≤_{2, and for} p=2 we havekTt k2≤1. According to the Riesz-Thorin interpolation theorem, using the pairs

(1, 2)or(2,+_∞)depending whether 1≤p≤2 orp≥2, we thus have

kTtkp≤21−rp with rp=2 min

₁

p, 1− 1 p

. (1.2)

The same interpolation argument (with the pairs(1,p),(p,+_∞)and possiblyp=2) yields Theorem 1.3. The following statements are equivalent.

(1) (H-2) is satisfied.

(2) There exist some1<p<+_∞and positive constantsλ_p and K_psuch that for all f _∈Lp_(µ) and all t>0,

N_p(P_tf)_≤ K_pe−λpt_N p(f).

(3) for all1<p<+_∞, there exist positive constantsλ_pand Kp such that for all f ∈Lp(µ)and all t>0,

N_p(P_tf)_≤ K_pe−λpt_N p(f). We shall denote by (H-p) the property (2) for a given p.

Furthermore, ifµis symmetric, these statements are also equivalent to (H-Poinc). In addition if (H-2) is satisfied, we have

Kp≤21−rpK2 and λp≥rpλ2. (1.4) Note that one cannot directly use the Riesz-Thorin theorem on the closed subspace of functions with zero mean, so the prefactor 21−rp _{cannot be (at least immediately) avoided.}

Our goal will be to obtain the best possible constants λ_p and Kp, assuming that the Poincaré inequality holds (i.e. K2=1).

There are two interests in such a result. In the first case it is obviously the rate of convergence at infinity. In this case, it is important to get the largest possible λ_p, despite the (reasonable) value of Kp. In the second case it is the opposite: to obtain the result with Kp =1 so that the inequality becomes an equality at time t = 0. In this case, one may use the result in order to obtain (for example) isoperimetric controls. The ideal situation is, of course, the one where we can reach these goals simultaneously (as forp=2). Unfortunately we did not succeed in obtaining this ideal result. Nevertheless, we will improve on the result obtained via interpolation in both directions. More precisely, we will prove the two results described below.

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Theorem 1.5. If (H-Poinc) is satisfied, then for all p_≥2, (H-p) holds with K_p=1and

λ_p_≥ 1

82k CP

, if p_≤2kfor some k_≥2.

Note that forp=2 (k=1) we recover a worse constant than the knownλ₂=1/C_P. Theorem 1.6. If (H-Poinc) is satisfied, then for all p≥1, (H-p) holds with

λ_p_≥4(p−1)

p2_C P

=4λ₂1

1₋1 p

and an appropriate K_p. For p≥4one may choose Kp =2

2−4

p _{and for}₂< _p ≤₄_{one may choose K}

p = p2/(p−2)2. For 1_≤p<2we have K_p_≤22p−1_K

p/(p−1).

Note that the constantλpobtained in Theorem 1.6 satisfies

λ_p=2 max

₁

p, 1− 1 p

r_pλ₂,

and so improves upon the one obtained by interpolation (see (1.4)) by a factor almost equal to 2 for largep′s. Actually we believe thatλp=4λ21_p

1₋1 p

is the best constant for allp’s, but we did not succeed in giving a rigorous proof of this fact.

Once it was seen that the Poincaré inequality implies the same exponential decay for bothP_t and its adjointP_t∗, the case 1<p<2 is obtained via a simple duality argument, so that we shall only consider the casep_≥2 in the sequel.

Note also that forp=2kfork_≥1, our proof givesKp=21−2/pand even a better one for largep. There is something surprising in the previous theorem, namely the explosion ofKp whenp goes to 2. Actually the proof will furnish a “better” result for small t’s, namely

Theorem 1.7. If (H-Poinc) is satisfied, then for all2≤p≤4, (H-p) holds with

λ_p= 1

C_P

₁

p + 1 2

and Kp=2 2−4

p_. In this situation it also holds

Np(Ptf)≤

1+4(p−1)t

pC_P

1/p

e−4λ21_p

1−1p

t Np(f).

The latter result suggests that one can expect a “reasonable” Kp for p close to 2 and λp = 4λ₂1

1−1_p. The same duality argument allows us to consider the cases p ≤ 2. One can also remark that the final bound yields, fortunately,N_p(P_tf)/N_p(f)_≤1 for all t_≥0 contrary to the previous one obtained in Theorem 1.6.

The casep=1 is extensively studied in[4]and the Poincaré inequality is no longer sufficient in general to obtain an exponential decay inL1_(µ)_{. Replacing}Lp _{norms by Orlicz norms (weaker} than anyN_pforp>1) is possible provided one reinforces the Poincaré inequality into aF-Sobolev inequality (see[4]Theorem 3.1 and also[5]) as it is well known in the caseF=log for the Orlicz spaceL_logL_.

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We could not find the statement of our results in the literature, perhaps because the interpolation argument appeared to be immediate. However recall that in [7], F.Y. Wang used the equivalent Beckner type formulation of Poincaré inequality to give a partial answer to the problem i.e., a Poincaré inequality with constantC_P is equivalent to the following: for any 1<p_≤2 and for any non-negative f,

(P_tf)p_d_µ₋

f dµ

≤e−

4(p−1)t p CP

(f)p_d_µ₋

f dµ

One has to take care with the constants since a factor 2 may or may not appear in the definition of Γ, depending on authors and of papers by the same authors. This result cannot be used to study the decay to the mean inLp_{norm, but it is of particular interest when studying densities of} probability.

Note that the decay rate we obtain in Theorem 1.6 is the same as the one in Wang’s result.

Acknowledgements

We warmly thank M. Ledoux who asked us about the exponential convergence inLp_(p₆₌_{2). Our} debt is actually bigger: Michel preciously saved a copy of one of our main arguments, that we lost in our perfectly disordered office.

The third author also would like to warmly thank Fabio Martinelli and the University of Rome 3 where part of this work was done.

We sincerely thank Susan Antoniadis for her careful reading.

Last but not least we warmly thank an anonymous referee for the careful reading of the manuscript, and many suggestions. In particular, by pointing out the possibility of a direct interpolation, he obliged us to substantially improve the first draft of the paper.

2 Poincaré inequalities and

L

_spaces.

This section is dedicated to the proof of our results.

2.1 Proof of Theorem 1.5.

A natural idea is to study the time derivative ofN_p(P_tf), namely d

d tN p

p(Ptf) =p

si g n(Ptf −µ(f))|Ptf −µ(f)|p−1L Ptf dµ. It follows that the following staments are equivalent:

There exists a constantC(p)such that for all f, N_pp(P_tf)_≤e−

pt C(p)_Np

p(f). (2.1)

There exists a constantC(p)such that for all f _{∈ D}withµ(f) =0, N_pp(f) _{≤ −}C(p)

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In order to compare all the inequalities (2.2) to the Poincaré inequality (i.e. p=2) one is tempted to make the change of function f _7→si g n(f)_|f_|2/p _(or _f _7→_{si g n}₍_f₎_|_f_|p/2_{) and to use the chain} rule. Unfortunately, firstϕ(u) =u2/pis notC2_{, and secondly}_µ(_{si g n}₍_f₎_|_f_|2/p₎

=0 (the same for p/2 for the second argument).

However, forp≥2, one can integrate by parts in (2.2) which thus becomes N_pp(f)_≤ C(p) (p₋1)

|f_|p−2Γ(f,f)dµ = C(p)4(p−1)

Γ(_|f_|p/2,_|f_|p/2)dµ. (2.3) Thus, it remains to show that the Poincaré inequality implies (2.3) for allp≥2. This will be done in three steps. First we will show that (2.3) holds forp=4. Next we shall show that it holds for all 2_≤p≤4. Finally we shall show that if (2.3) holds for pthen it holds for 2p. This will complete the proof by an induction argument. In this procedure, starting withp=2, only the last two steps are necessary. Anyway, we believe that the details for 2p =4 will help the reader to follow the scheme of proof for the general cases.

The casep=4. We proceed with the proof forp=4,i.e.we prove that the Poincaré inequality implies (2.3) forp=4. Assume thatµ(f) =0. First, applying the Poincaré inequality tof2we get

f4dµ_≤

f2dµ

+4CP

f2Γ(f,f)dµ, so that it remains to prove that

f2dµ

≤C

f2Γ(f,f)dµ,

for some constantC. For anyu>0 letϕ =ϕu:R7→R be the 2-Lipschitz function defined by

ϕ(s) =0 if_|s_{| ≤}u,ϕ(s) =sif_|s_{| ≥}2uand linear in between. Applying Poincaré inequality toϕ(f)

yields

(ϕ(f))2_d_µ_≤

ϕ(f)dµ

+4CP

{|f|≥u}

Γ(f,f)dµ. But

(ϕ(f))2_d_µ_≥

{|f|≥2u}

f2dµ_≥

f2dµ₋4u2, and sinceµ(f) =0,

ϕ(f)dµ

≤

|ϕ(f)₋f|dµ=

{|f|≤u}

|f|dµ+

{u≤|f|≤2u}

|ϕ(f)₋f|dµ

≤u µ(_{|f_{| ≤}u_}) +µ(_{u_{≤ |}f_{| ≤}2u_})

≤u (2.4)

where we have used that_|ϕ(s)₋s_{| ≤}ufor anys_∈[₋2u,₋u]_∪[u, 2u]. Summarizing, it follows that

f2dµ_≤5u2+4CP

{|f|≥u}

Γ(f,f)dµ_≤5u2+ 4

u2CP

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Optimizing inu2_{finally yields}

f2dµ

≤90CP

f2Γ(f,f)dµ, and in turn

N₄4(f)_≤94C_P

f2Γ(f,f)dµ.

The constant 94 is clearly not optimal. Note that replacing the constant 2 by(1+ρ),ρ >0, in the definition ofϕwould yield a worse constant.

The case2_≤p_≤4. In the previous part we used the ordinary form of the Poincaré inequality. Actually if 1_≤a_≤2 we have

|f|2_d_µ₋

|f|a_d_µ

2/a

≤Varµ(f)≤CP

Γ(f,f)dµ. (2.5) Hence if 2_≤p≤4, applying (2.5) with_|f|p/2_{, and choosing}_a₌₄_/_p_{we have}

|f_|pdµ_≤

f2dµ

p/2

+CP(p2/4)

|f_|p−2Γ(f,f)dµ. (2.6) Now recall that, in the previous step, we have shown

f2dµ_≤5u2₊₄_C P

{|f|≥u}

Γ(f,f)dµ, hence

f2dµ_≤5u2+ 4

up−2 CP

|f_|p−2Γ(f,f)dµ. Again, we may optimize inu, and obtain

|f|p_d_µ_≤

p2 4 +2

2+₂p₅p₂−1

|f|p−2Γ(f,f)dµ, (2.7) so that (2.3) holds for 2≤p≤4, with

C(p)_≤

4 +2

2+p₂₅p₂−1 p−1 CP.

From p to2p. Now assume that (2.3) holds for some p_≥2 and that the Poincaré inequality also holds with constantC_P. First we apply Poincaré inequality to the function_|f_|p _{to get}

|f|2p_d_µ_≤

|f|p_d_µ

+CPp2

|f|2p−2_Γ(_f_,_f₎_d_µ_. Our aim is to bound the quantityR_|f|p_d_µ_{using (2.3). On the one hand we have}

|ϕ(f)_|pdµ_≥

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On the other hand, using that_|a+b_|p_≤₂p−1₍_|_a_|p₊_|_b_|p₎_{and (2.4), it follows from (2.3) applied} toϕ(f)₋µ(ϕ(f))that (recall thatϕis 2-Lipschitz)

|ϕ(f)_|p_d_µ_≤₂p−1_|_µ(ϕ(_f₎₎_|p₊₂p−1

|ϕ(f)₋µ(ϕ(f))_|p_d_µ

≤2p−1up+2p−1(p−1)C(p)

{|f|≥u}

|ϕ(f)₋µ(ϕ(f))_|p−2_Γ(ϕ(_f₎_,_ϕ(_f₎₎_d_µ

≤2p−1up+2p+1(p−1)C(p)

{|f|≥u}

|ϕ(f)₋µ(ϕ(f))_|p−2Γ(f,f)dµ. Now, again by (2.4) we have

{|f|≥u}

|ϕ(f)₋µ(ϕ(f))_|p−2Γ(f,f)dµ_≤

{|f|≥u}

||f|+u|p−2Γ(f,f)dµ

= 1

{|f|≥u}

up_|f_|p−2

1+ u

|f|

p−2

Γ(f,f)dµ

≤2

p−2 up

|f|2p−2Γ(f,f)dµ. Summarizing we end up with

|f_|pdµ_≤2p+2p−1up+2

2p−1

up (p−1)C(p)

|f_|2p−2Γ(f,f)dµ. Since 2p₊₂p−1₌₃_·₂p−1_{, optimizing over}_u_{leads to}

|f_|pdµ

≤3_·23p(p₋1)C(p)

|f_|2p−2Γ(f,f)dµ. In turn,

|f_|2pdµ_≤3_·23p(p₋1)C(p) +p2C_P

|f_|2p−2Γ(f,f)dµ. This is equivalent to say that

C(2p)(2p₋1)_≤3_·23p(p₋1)C(p) +p2C_P. (2.8) Note that the previous computation applied to p = 2 (recall that C(2) = C_P) leads to a worse constant than 94. From (2.8) we easily deduce that

C(2k)_≤2k_·23·2k from which the expected result of Theorem 1.5 follows.

2.2 Proof of Theorem 1.6

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Lemma 2.9. Assume that (H-Poinc) holds, and that (H-p) holds for some p_≥1withλ_p=θ_p/p C_P and K_p.

Then, provided pθp>2p−1, (H-2p) holds with

λ_2p=4(2p₋1)/(2p)2C_P and K_2p2p=1+ 2p−1

pθ_p₋(2p−1)K

2p p .

Lemma 2.10. Assume that (H-Poinc) holds. Then for all2_≤p≤4, (H-p) holds with

λ_p=4(p₋1)/p2C_P and Kp p=

(p₋2)2. In this situation we also have

N_p(P_tf)_≤

1+4(p−1)t

pCP

1/p

e−

4(p−1)t p2CP _N

p(f).

Proof of Lemma 2.9.

Without loss of generality we assume thatR f dµ= 0 and denote byU_p(t):=Np

p(Ptf). Recall that, since 2p_≥2,

U_2p′ (t) = 2p

si g n(Ptf)|Ptf|2p−1L Ptf dµ

= ₋2p(2p−1)

(Ptf)2p−2Γ(Ptf,Ptf)dµ

= ₋4(2p−1)

Γ(_|Ptf|p,|Ptf|p)dµ. In addition the Poincaré inequality, applied to_|Ptf|p, yields

U2p(t)≤U2p(t) +CP

Γ(_|Ptf|p,|Ptf|p)dµ. Putting these inequalities together we thus have

U_2p′ (t)_{≤ −}4(2p−1)

2p CP

U_2p(t) + 4(2p−1)

2p CP

U_p2(t). (2.11) We may thus apply Gronwall’s lemma and obtain

U_2p(t)_≤e−

4(2p−1)t

2p CP

U_k(0) +4(2p−1)

2p CP

Z t

0 e

4(2p−1)s

2p CP _U2 p(s)ds

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Applying (H-p) we obtain U2p(t) ≤ e−

4(2p−1)t

2p CP

U2p(0) +

4(2p₋1)

2p C_P

Z t

0 e

4(2p−1)s

2p CP

K_ppe− pθp s

p CP _U p(0)

≤ e−

4(2p−1)t

2p CP

U_2p(0) +4(2p−1)

2p CP

Z t

0 e−

2(pθp−(2p−1))s p CP _K2p

p U 2 p(0)ds

≤ e−

4(2p−1)t

2p CP

U_2p(0) + 2p−1

pθ_p₋(2p−1)K

2p p U

2 p(0)

≤ e−

4(2p−1)t

2p CP _U_2p₍₀₎

1+ 2p−1

pθ_p₋(2p−1)K

2p p

sinceU_p2(0)_≤U2p(0)thanks to the Cauchy-Schwarz inequality. Proof of Lemma 2.10.

If 2_≤p_≤4, writep=2qwith 1_≤q_≤2. We have U_p′(t) =₋4(p−1)

Γ(_|P_tf_|q,_|P_tf_|q)dµ.

As in the second step of the proof of theorem 1.5, we may use the following consequence of the Poincaré inequality,

N_2q2q(_|Ptf|)−N 2q

2 (Ptf)≤CP

Γ(_|Ptf|q,|Ptf|q)dµ. (2.12) As in the previous proof we thus obtain

U_p′(t)_{≤ −}4(p−1)

p C_P Up(t) +

4(p₋1)

p C_P U q 2(t).

Using Gronwall’s lemma and applying the Poincaré inequality forU2, we thus have Up(t) ≤ e−

4(p−1)t p CP

Up(0) +

4(p₋1)

p C_P

Z t

0 e

4(p−1)s p CP _e−

sp CP_Uq

2(0)ds

≤ e−

4(p−1)t p CP

Up(0) +

4(p₋1)

p CP

Z t

0 e−

s(p−2)2

p CP _Uq 2(0)ds

≤ e−

4(p−1)t p CP _U

p(0)

1+ 4(p−1) (p₋2)2

yielding the first part of the lemma. The second one immediately follows from the second inequal-ity above.

Proof of Theorem 1.6 and of Theorem 1.7.

The two previous lemmata yieldλ₄=3/4C_P andK4=

It follows that Tt is bounded in L4 with an operator norm smaller than or equal to 2e−3t/4CP. Interpolating betweenL2_andL4_{yields for 2}_≤_p_≤_4,

λp= 1 CP ₁ p+ 1 2

and Kp=2

(2−4

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Of course this value ofλ_p is smaller, for 2< p <4, than the expected 4(p₋1)/p2_C

P, but it is already better than the one obtained via direct interpolation.Kpis exactly(2

1−2

p)2_. It is immediate that for 2≤p≤4,θ_p=1+p

2satisfiespθp>2p−1, so that we may apply Lemma 2.9 for 2≤p≤4. Now ifθ_p=4(p−1)/pandp≥2 it is immediate thatpθ_p_≥2p−1. Hence we have obtained that for all p_≥4 we may chooseλ_p=4(p₋1)/p2_C

P. It is not difficult, although tedious, to check that ifK_p_≤22−4p _then_K

2p≤22−

2p _{(one has to deal} first with the case 2≤p≤4 withpθ_p=p+p2

2, and then with the casep≥4).

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[8] L. Wu. Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition.Ann. Prob., 34(5):1960–1989, 2006. MR2271488

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In order to compare all the inequalities (2.2) to the Poincaré inequality (i.e. p=2) one is tempted to make the change of function f _7→si g n(f)_|f_|2/p _(or _f _7→_{si g n(f}₎_|_f_|p/2_{) and to use the chain}

rule. Unfortunately, firstϕ(u) =u2/p_{is not}_C2_{, and secondly}_{µ(si g n(}_f₎_|_f_|2/p₎

=0 (the same for p/2 for the second argument).

However, forp≥2, one can integrate by parts in (2.2) which thus becomes N_pp(f)_≤ C(p) (p₋1)

|f_|p−2Γ(f,f)dµ = C(p)4(p−1) p2

Γ(_|f_|p/2,_|f_|p/2)dµ. (2.3) Thus, it remains to show that the Poincaré inequality implies (2.3) for allp≥2. This will be done in three steps. First we will show that (2.3) holds forp=4. Next we shall show that it holds for all 2_≤p≤4. Finally we shall show that if (2.3) holds for pthen it holds for 2p. This will complete the proof by an induction argument. In this procedure, starting withp=2, only the last two steps are necessary. Anyway, we believe that the details for 2p =4 will help the reader to follow the scheme of proof for the general cases.

The casep=4. We proceed with the proof forp=4,i.e.we prove that the Poincaré inequality implies (2.3) forp=4. Assume thatµ(f) =0. First, applying the Poincaré inequality tof2we get

f4dµ_≤

f2dµ

+4CP

f2Γ(f,f)dµ, so that it remains to prove that

f2dµ

≤C

f2Γ(f,f)dµ,

for some constantC. For anyu>0 letϕ =ϕu:R7→R be the 2-Lipschitz function defined by ϕ(s) =0 if_|s_{| ≤}u,ϕ(s) =sif_|s_{| ≥}2uand linear in between. Applying Poincaré inequality toϕ(f) yields

(ϕ(f))2_dµ_≤ Z

ϕ(f)dµ

+4CP

{|f|≥u}

Γ(f,f)dµ. But

(ϕ(f))2_dµ_≥ Z

{|f|≥2u}

f2dµ_≥

f2dµ₋4u2, and sinceµ(f) =0,

ϕ(f)dµ

≤

|ϕ(f)₋f|dµ=

{|f|≤u}

|f|dµ+

{u≤|f|≤2u}

|ϕ(f)₋f|dµ

≤u µ(_{|f_{| ≤}u_}) +µ(_{u_{≤ |}f_{| ≤}2u_})

≤u (2.4)

where we have used that_|ϕ(s)₋s_{| ≤}ufor anys_∈[₋2u,₋u]_∪[u, 2u]. Summarizing, it follows that

f2dµ_≤5u2+4CP

{|f|≥u}

Γ(f,f)dµ_≤5u2+ 4 u2CP

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Optimizing inu2_{finally yields} Z

f2dµ

2 ≤90CP

f2Γ(f,f)dµ, and in turn

N₄4(f)_≤94C_P

f2Γ(f,f)dµ.

The constant 94 is clearly not optimal. Note that replacing the constant 2 by(1+ρ),ρ >0, in the definition ofϕwould yield a worse constant.

The case2_≤p_≤4. In the previous part we used the ordinary form of the Poincaré inequality. Actually if 1_≤a_≤2 we have

|f|2_dµ₋ Z

|f|a_dµ

2/a

≤Varµ(f)≤CP

Γ(f,f)dµ. (2.5) Hence if 2_≤p≤4, applying (2.5) with_|f|p/2_{, and choosing}_a₌_4/p_{we have}

|f_|pdµ_≤

f2dµ

p/2

+CP(p2/4)

|f_|p−2Γ(f,f)dµ. (2.6) Now recall that, in the previous step, we have shown

f2dµ_≤5u2₊₄_C

{|f|≥u}

Γ(f,f)dµ, hence

f2dµ_≤5u2+ 4 up−2 CP

|f_|p−2_Γ(_f_,_f₎_dµ_.

Again, we may optimize inu, and obtain

|f|p_dµ_≤

p2 4 +2

2+₂p₅p₂−1

|f|p−2_Γ(_f_,_f₎_dµ_, _(2.7)

so that (2.3) holds for 2≤p≤4, with C(p)_≤

4 +2

2+p₂₅p₂−1

p−1 CP.

From p to2p. Now assume that (2.3) holds for some p_≥2 and that the Poincaré inequality also holds with constantC_P. First we apply Poincaré inequality to the function_|f_|p _{to get}

|f|2p_dµ_≤

|f|p_dµ

+CPp2

|f|2p−2_Γ(f_,_f₎_dµ_.

Our aim is to bound the quantityR_|f|p_dµ_{using (2.3). On the one hand we have}

|ϕ(f)_|pdµ_≥

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On the other hand, using that_|a+b_|p_≤₂p−1₍_|_a_|p₊_|_b_|p₎_{and (2.4), it follows from (2.3) applied} toϕ(f)₋µ(ϕ(f))that (recall thatϕis 2-Lipschitz)

|ϕ(f)_|p_dµ_≤₂p−1_|_µ(ϕ(f₎₎_|p₊₂p−1 Z

|ϕ(f)₋µ(ϕ(f))_|p_dµ

≤2p−1up+2p−1(p₋1)C(p)

{|f|≥u}

|ϕ(f)₋µ(ϕ(f))_|p−2_Γ(ϕ(f_),_ϕ(_f_))dµ ≤2p−1up+2p+1(p₋1)C(p)

{|f|≥u}

|ϕ(f)₋µ(ϕ(f))_|p−2_Γ(f_,_f_)dµ.

Now, again by (2.4) we have

{|f|≥u}

|ϕ(f)₋µ(ϕ(f))_|p−2_Γ(_f_,_f_)dµ_≤ Z

{|f|≥u}

||f|+u|p−2_Γ(f_,_f_)dµ

= 1 up

{|f|≥u}

up_|f_|p−2

1+ u

|f|

p−2

Γ(f,f)dµ

≤2

p−2

|f|2p−2_Γ(_f_,_f_)dµ.

Summarizing we end up with

|f_|pdµ_≤2p+2p−1up+2

2p−1

up (p−1)C(p)

|f_|2p−2Γ(f,f)dµ. Since 2p₊₂p−1₌₃_·₂p−1_{, optimizing over}_u_{leads to}

Z |f_|pdµ

≤3_·23p(p₋1)C(p)

|f_|2p−2Γ(f,f)dµ. In turn,

|f_|2p_dµ_≤

3_·23p_(p₋_{1)C(p) +}_p2_C

|f_|2p−2_Γ(_f_,_f_)dµ.

This is equivalent to say that

C(2p)(2p₋1)_≤3_·23p(p₋1)C(p) +p2C_P. (2.8) Note that the previous computation applied to p = 2 (recall that C(2) = C_P) leads to a worse constant than 94. From (2.8) we easily deduce that

C(2k)_≤2k_·23·2k from which the expected result of Theorem 1.5 follows.

2.2 Proof of Theorem 1.6

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Lemma 2.9. Assume that (H-Poinc) holds, and that (H-p) holds for some p_≥1withλ_p=θ_p/p C_P and K_p.

Then, provided pθp>2p−1, (H-2p) holds with

λ₂_p=4(2p₋1)/(2p)2C_P and K₂2_pp=1+ 2p−1 pθ_p₋(2p₋1)K

2p p .

Lemma 2.10. Assume that (H-Poinc) holds. Then for all2_≤p≤4, (H-p) holds with λ_p=4(p₋1)/p2C_P and Kp

p= p2 (p₋2)2.

In this situation we also have

N_p(P_tf)_≤

1+4(p−1)t pCP

1/p e−

4(p−1)t p2CP _N

p(f).

Proof of Lemma 2.9.

Without loss of generality we assume thatR f dµ= 0 and denote byU_p(t):=Np

p(Ptf). Recall that, since 2p_≥2,

U₂′_p(t) = 2p

si g n(P_tf)_|Ptf|2p−1L Ptf dµ = ₋2p(2p₋1)

(P_tf)2p−2_Γ(P

tf,Ptf)dµ = ₋4(2p−1)

Γ(_|Ptf|p,|Ptf|p)dµ. In addition the Poincaré inequality, applied to_|Ptf|p, yields

U2p(t)≤U2p(t) +CP

Γ(_|Ptf|p,|Ptf|p)dµ. Putting these inequalities together we thus have

U₂′_p(t)_{≤ −}4(2p−1) 2p CP

U₂_p(t) + 4(2p−1) 2p CP

U_p2(t). (2.11)

We may thus apply Gronwall’s lemma and obtain

U₂_p(t)_≤e−

4(2p−1)t

2p CP

U_k(0) +4(2p−1) 2p CP

Z t

4(2p−1)s

2p CP _U2 p(s)ds

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Applying (H-p) we obtain U2p(t) ≤ e−

4(2p−1)t

2p CP

U2p(0) +

4(2p₋1) 2p C_P

Z t

4(2p−1)s

2p CP

K_ppe− pθp s

p CP _U p(0)

≤ e−

4(2p−1)t

2p CP

U₂_p(0) +4(2p−1) 2p CP

Z t

e−

2(pθp−(2p−1))s p CP _K2p

p U

p(0)ds

≤ e−

4(2p−1)t

2p CP

U₂_p(0) + 2p−1 pθ_p₋(2p₋1)K

2p p U

p(0)

≤ e−

4(2p−1)t

2p CP _U₂ p(0)

1+ 2p−1 pθ_p₋(2p₋1)K

2p p

sinceU_p2(0)_≤U2p(0)thanks to the Cauchy-Schwarz inequality. Proof of Lemma 2.10.

If 2_≤p_≤4, writep=2qwith 1_≤q_≤2. We have U_p′(t) =₋4(p−1)

Γ(_|P_tf_|q,_|P_tf_|q)dµ.

As in the second step of the proof of theorem 1.5, we may use the following consequence of the Poincaré inequality,

N₂2_qq(_|Ptf|)−N

2 (Ptf)≤CP

Γ(_|Ptf|q,|Ptf|q)dµ. (2.12) As in the previous proof we thus obtain

U_p′(t)_{≤ −}4(p−1)

p C_P Up(t) +

4(p₋1) p C_P U

2(t).

Using Gronwall’s lemma and applying the Poincaré inequality forU2, we thus have

Up(t) ≤ e−

4(p−1)t p CP

Up(0) +

4(p₋1) p C_P

Z t

4(p−1)s p CP _e−

sp CP_Uq

2(0)ds

≤ e−

4(p−1)t p CP

Up(0) +

4(p₋1) p CP

Z t

e− s(p−2)2

p CP _Uq

2(0)ds

≤ e−

4(p−1)t p CP _U

p(0)

1+ 4(p−1) (p₋2)2

yielding the first part of the lemma. The second one immediately follows from the second inequal-ity above.

Proof of Theorem 1.6 and of Theorem 1.7.

The two previous lemmata yieldλ₄=3/4C_P andK4= p

It follows that Tt is bounded in L4 with an operator norm smaller than or equal to 2e−3t/4CP. Interpolating betweenL2_andL4_{yields for 2}_≤_p_≤_4,

λp= 1 CP ₁ p+ 1 2

and Kp=2

(2−4

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Of course this value ofλ_p is smaller, for 2< p <4, than the expected 4(p₋1)/p2_C

P, but it is already better than the one obtained via direct interpolation.Kpis exactly(2

1−2

p)2_. It is immediate that for 2≤p≤4,θ_p=1+p

2satisfiespθp>2p−1, so that we may apply Lemma

2.9 for 2≤p≤4. Now ifθ_p=4(p₋1)/pandp≥2 it is immediate thatpθ_p_≥2p−1. Hence we have obtained that for all p_≥4 we may chooseλ_p=4(p₋1)/p2_C

P. It is not difficult, although tedious, to check that ifK_p_≤22−4p _then_K

2p≤2