Heat flow on 1-forms under lower Ricci bounds.
Functional inequalities, spectral theory, and heat kernel
This chapter is based on the author’s work [Bra20], from which large parts are taken over verbatim.
In this final chapter, we fix again an RCD(𝐾 ,∞) space(M,d,𝔪) according to Section 0.1,𝐾 ∈R. Occasionally, we will assume the more restrictive RCD∗(𝐾 , 𝑁)or RCD(𝐾 , 𝑁)condition,𝑁∈ [1,∞). Required details on these notions which are not yet contained in Section 0.1 or Section 1.2 are summarized in Section 4.2 below.
We retain the notations from Chapter 3 except the more standard ones for the (ex-tended) Cheeger energy domains𝑊1,2(M):=Fand S2(M):=Fe, recall Remark 3.2.4.
Readers who want to read this chapter as independently of Chapter 3 as possible are recommended to familiarize themselves with Subsection 3.2.4, Subsection 3.2.5 and Subsection 3.2.8 for notational purposes, with the calculus rules from Theorem 3.3.3, Theorem 3.4.3 and Theorem 3.5.5, and with the heat flow on1-forms(H𝑡)𝑡≥0introduced in Subsection 3.6.4.
Theorem 4.1.1. For every𝜔∈L2(𝑇∗M)and every𝑡≥0,
|H𝑡𝜔| ≤e−𝐾 𝑡P𝑡|𝜔| 𝔪-a.e.
Motivation and possible further extensions Our motivation to study the heat flow (H𝑡)𝑡≥0in more detail than in [Gig18] comes from different directions. First, we provide a further contribution to the large diversity of works generalizing important “classical smooth” statements to nonsmooth spaces. Second, we believe that RCD spaces (or more general spaces with “lower Ricci bounds” [BHS21, ER+20, Stu20]) with the tensor language of [Gig15, Gig18] and Chapter 3 are the correct framework to develop nonsmooth notions of stochastic differential geometry, e.g. (damped) stochastic parallel transports, a project which, as mentioned above, currently lacks in the nonsmooth setting and which we attack in the future. Therein, in establishing Bismut–Elworthy–Li-type derivative formulas for the functional heat flow(P𝑡)𝑡≥0([Bis84a, EL94b], see also Chapter 2) — which in turn are expected to provide further regularity information about it — a good understanding of its1-form counterpart(H𝑡)𝑡≥0is essential [DT01]. Lastly, in smooth contexts, heat kernel methods for1-forms are useful in many important applications, all of which could lead to RCD analogues that, however, are not addressed in this thesis. Exemplary, let us quote
• a deeper understanding of the Hodge theorem (see [Gig18] for the RCD result and Theorem 3.5.23 below) by the study of the heat kernel [MR51],
• a proof variant of index theorems in Riemannian geometry [Bis84b, Cha84, Hsu02a], along with introducing a working notion of nonsmooth Dirac operators on RCD spaces,
• the study of boundedness of the Riesz transform, see e.g. [CS08, Cou13, CDS20, Dev14, MO20] and the references therein, and
• the study of its short-time asymptotics playing dominant roles in theoretical physics and quantum gravity [Avr00, MP49, Ros97].
Logarithmic Sobolev inequalities The first consequence of Theorem 4.1.1 we discuss arelogarithmic Sobolev inequalitiesfor(H𝑡)𝑡≥0. Such inequalities for functions and their connections to the functional heat flow(P𝑡)𝑡≥0, initiated in [Gro75], have been an active field of research in past decades. For an overview over the vast literature on this subject, see [BGL14, Dav89]. In a similar manner, in this chapter we relate logarithmic Sobolev inequalities for1-forms to certain further integral properties of (H𝑡)𝑡≥0described below. There is some ambiguity in formulating the former depending on whether one regards1-forms as vector fields or really as contravariant objects. For brevity, we only outline Definition 4.3.4, where we say that a sufficiently regular vector field𝑋overMobeys the2-logarithmic Sobolev inequality LSI2(𝛽, 𝜒)with parameters 𝛽 > 0and𝜒∈Rif
ˆ
M
|𝑋|2log|𝑋|d𝔪≤𝛽
∇𝑋
2
L2(𝑇⊗2M)+𝜒 𝑋
2
L2(𝑇M)+ 𝑋
2
L2(𝑇M)logk𝑋kL2(𝑇M). The advantage of this form is that it follows from logarithmic Sobolev inequalities for functions, known to hold in various cases [CM17, Vil09], via Kato’s inequality Lemma 3.4.13, see Lemma 4.3.8. It also implies its contravariant pendant from Definition 4.3.5 for arbitrary exponents, see Proposition 4.3.10.
The integral properties of(H𝑡)𝑡≥0to be derived are the following. We call(H𝑡)𝑡≥0
• hypercontractiveif there exist𝑇 ∈ (0,∞]and a strictly increasing C1-function 𝑝: [0, 𝑇) → (1,∞)such thatH𝑡is bounded fromL𝑝(0)(𝑇∗M)toL𝑝(𝑡)(𝑇∗M) for every𝑡∈ (0, 𝑇), and
• ultracontractiveif there exist𝑝0∈ (1,∞)and𝑇 > 0such thatH𝑇 is bounded fromL𝑝0(𝑇∗M)toL∞(𝑇∗M).
In great generality, in Theorem 4.3.12 we study when certain logarithmic Sobolev inequalities imply hyper- or ultracontractivity of (H𝑡)𝑡≥0. We also treat a partial converse in Theorem 4.3.16.
Read in concrete applications, according to all these discussions and the known functional examples from [CM17, Vil09], we deduce the following hypercontractiv-ity. (According to [CM17, Vil09], if 𝐾 > 0in either case, then the constant 𝛽in Theorem 4.1.2 can be chosen to be(𝑁−1)/𝐾 𝑁or1/𝐾, respectively.)
Theorem 4.1.2. On any compactRCD∗(𝐾 , 𝑁)space with𝑁 ∈ (1,∞)or, for𝐾 > 0, anyRCD(𝐾 ,∞)space, there exists a constant𝛽 > 0such that for every𝑝0∈ (1,∞), H𝑡is bounded from L𝑝0(𝑇∗M)to L𝑝(𝑡)(𝑇∗M)with operator norm no larger thane−𝐾 𝑡 for every𝑡 > 0, where the function𝑝: [0,∞) → (1,∞)is given by
𝑝(𝑡):=1+ (𝑝0−1)e2𝑡/𝛽.
Many of our arguments for Theorem 4.1.2 are inspired by the functional treatise [Dav89]. In the case of non-weighted Riemannian manifolds, logarithmic Sobolev inequalities for1-forms have been studied with similar results in [Cha07].
Spectral behavior of Hodge’s Laplacian As indicated in Corollary 3.6.28, Kato’s inequality Lemma 3.4.13 also connects the spectra of the Hodge and the (negative) functional Laplacian. The study of the former is our goal in Section 4.4.
The following is first shown in full generality in Theorem 4.4.3 and Corollary 4.4.4.
Theorem 4.1.3. If a positive real number belongs to the spectrum of−Δ, then it is also contained in the spectrum ofΔ. Similar inclusions hold between the respective point® and essential spectra. In particular,
inf𝜎(−Δ+𝐾) ≤inf𝜎( ®Δ) ≤inf𝜎( ®Δ) \ {0} ≤inf𝜎(−Δ) \ {0}.
The stated spectral inclusions are known in the non-weighted Riemannian setting by [CL19]. Our proof of the former adopts a similar strategy, relying on a suitable variant of Weyl’s criterion. The first stated spectral gap inequality follows by basic spectral theory and is well-known in the smooth setting. See e.g. [Gün17a] for a more general smooth treatise and further references.
On compact RCD∗(𝐾 , 𝑁)spaces, as in the case of functions, the spectrum ofΔ® can be characterized much better. A key tool towards an explicit understanding of it in this case is the following Rellich-type compact embedding theorem, Theorem 4.4.8.
Theorem 4.1.4. If(M,d,𝔪) is a compactRCD∗(𝐾 , 𝑁) space, the formal operator Δ®−1is compact.
For Ricci limit spaces, i.e. noncollapsed mGH-limits of sequences of non-weighted Riemannian manifolds with uniformly lower bounded Ricci curvatures, Theorem 4.1.4 is due to [Hon17, Hon18a]. In the very recent work [HZ20], Theorem 4.1.4 has been proven independently in a different way using so-called𝛿-splitting maps.
The proof of Theorem 4.1.4 uses several powerful properties of(H𝑡)𝑡≥0on compact RCD(𝐾 , 𝑁)spaces. Using that(P𝑡)𝑡≥0admits a heat kernel which obeys Gaussian bounds [JLZ16, Stu95, Tam19], together with Theorem 4.1.1 and Bishop–Gromov’s inequality, we see in Theorem 4.3.3 that the heat operatorH𝑡mapsL𝑝(𝑇∗M)boundedly intoL∞(𝑇∗M)for every𝑡 > 0and every 𝑝 ∈ [1,∞]. In particular,H𝑡 is a Hilbert–
Schmidt operator onL2(𝑇∗M), and Theorem 4.1.4 as well as expected properties of the spectrum ofΔ®stated in Theorem 4.4.12 are then deduced by abstract functional analysis. We also establishL∞-estimates on eigenforms ofΔ®, with an explicit growth rate for positive eigenvalues. See Corollary 4.4.13 and Proposition 4.4.14.
The last part of Section 4.4, especially Theorem 4.4.18, is devoted to the proof of the independence of theL𝑝-spectrum ofΔ® on𝑝∈ [1,∞], provided(M,d,𝔪)is an RCD∗(𝐾 , 𝑁)space satisfying, for every𝜀 > 0, the volume growth condition
sup
𝑥∈M
ˆ
M
e−𝜀d(𝑥 , 𝑦)𝔪[𝐵1(𝑥)]−1/2𝔪[𝐵1(𝑦)]−1/2d𝔪(𝑦)<∞.
On non-weighted Riemannian manifolds, this is shown in [Cha05]. Our proof, based on a perturbation argument, Theorem 4.1.1 and functional heat kernel bounds, is inspired by similar results for the functional Laplacian [HV86, HV87, SC92, Stu93]. See also [CF12, DL+10, KS14, Tak07, TT09] for further works in this direction for Markov processes and Feynman–Kac semigroups.
Heat kernel Up to now no general result ensuring theexistence of a heat kernel for(H𝑡)𝑡≥0was known in the setting of [Gig18]. Outside the scope of noncompact, even weighted Riemannian manifolds [Gün17a, Pat71, Ros97], there are only few metric measure constructions under restrictive structural (existence of acontinuous covector bundle with constant fiber dimensions) and volume doubling assumptions [CS08, Sik04]. Our axiomatization and existence proof of a heat kernel for(H𝑡)𝑡≥0
on RCD(𝐾 ,∞)spaces is hoped to push forward research in the above areas on such spaces. Our general study applies to non-locally compact or non-doubling, possibly infinite-dimensional RCD spaces.
Let us motivate our axiomatization via the heat kernelpof(P𝑡)𝑡≥0from [AGS14b], cf. Section 4.2 for details. Slightly abusing notation, it induces a mapp: (0,∞) × L0(M)2 → L0(M2) sending 𝑡 > 0and (𝑔, 𝑓) ∈ L0(M)2 to the𝔪⊗2-measurable function given byp𝑡[𝑔, 𝑓] (𝑥 , 𝑦):=p𝑡(𝑥 , 𝑦)𝑔(𝑥) 𝑓(𝑦)such that for a sufficiently large class of functions 𝑓 , 𝑔∈L0(M), we havep𝑡[𝑔, 𝑓] ∈L1(M2)as well as
𝑔P𝑡𝑓 = ˆ
M
p𝑡[𝑔, 𝑓] (·, 𝑦)d𝔪(𝑦) 𝔪-a.e.
Let us turn to1-forms. Recall that a heat kernel for(H𝑡)𝑡≥0in the smooth, possibly weighted setting is a jointly smooth maph: (0,∞) ×M2→ (𝑇∗M)∗𝑇∗M— i.e. for every𝑡 > 0and every(𝑥 , 𝑦) ∈M2,h𝑡(𝑥 , 𝑦)is a homomorphism mapping𝑇∗
𝑦Mto𝑇∗
𝑥M
— satisfying, for every𝜔∈L2(𝑇∗M), H𝑡𝜔=
ˆ
M
h𝑡(·, 𝑦)𝜔(𝑦)d𝔪(𝑦) 𝔪-a.e. (4.1.1) The heat kernel for1-forms has first been constructed on compact spaces by [Pat71] using the so-calledparametrix construction. See also [Gün17a, Ros97]. Since RCD(𝐾 ,∞) spaces a priori do neither come with any covector bundle nor with a smooth structure, the fiberwise notion (4.1.1) is be replaced by “testing the identity (4.1.1) pointwise
against sufficiently many1-forms”. Motivated by our functional considerations, we understand a mapping h: (0,∞) ×L0(𝑇∗M)2 → L0(M2) to be a heat kernel for (H𝑡)𝑡≥0if, for every𝑡 > 0,h𝑡isL0-bilinear, and for all sufficiently regular1-forms 𝜔, 𝜂∈L0(𝑇∗M), we haveh𝑡[𝜂, 𝜔] ∈L1(M2)with the identity
h𝜂,H𝑡𝜔i= ˆ
M
h𝑡[𝜂, 𝜔] (·, 𝑦)d𝔪(𝑦) 𝔪-a.e.
Theorem 4.1.5. The heat kernel for(H𝑡)𝑡≥0in the indicated sense exists and is unique.
The proof strategy for this result, see Theorem 4.5.5 for the precise formulation, is the following. Motivated by similar functional results [Stu95, SC10], a crucial tool to obtain integral kernels for certain operators is aDunford–Pettis-type theorem [DP40, DS58], a very generalL∞-module version of which we prove in Theorem 4.5.3.
Boiled down to the1-form setting, it states that any linear operator which is bounded fromL1(𝑇∗M)toL∞(𝑇∗M)in the Banach sense admits an integral kernel, the concept of which is similar to the axiomatization of the1-form heat kernel. Now for𝑡 > 0, the heat operatorH𝑡is not bounded fromL1(𝑇∗M)toL∞(𝑇∗M)in this generality. But by [Tam19], given any𝜀 > 0there exist constants𝐶1, 𝐶2 > 0with
p𝑡(𝑥 , 𝑦) ≤𝔪 𝐵√
𝑡(𝑥)−1/2 𝔪
𝐵√
𝑡(𝑥)−1/2 exph
𝐶1 1+𝐶2𝑡−d
2(𝑥 , 𝑦) (4+𝜀)𝑡 i
for every𝑡 > 0and𝔪⊗-a.e.(𝑥 , 𝑦) ∈M2. By Theorem 4.1.1, the perturbed operator A𝑡 :=𝜙𝑡H𝑡𝜙𝑡,
where𝜙𝑡(𝑥):= 𝔪[𝐵√
𝑡(𝑥)]1/2, is thus bounded fromL1(𝑇∗M)toL∞(𝑇∗M)and there-fore admits an integral kernel — formally multiplyingA𝑡by𝜙−1
𝑡 from both sides then yields the desired integral kernelh𝑡forH𝑡. Note that for this argument, it is essential that(P𝑡)𝑡≥0has a heat kernel. (This explains best our restriction to uniform lower Ricci bounds, a more general result is not available up to now.)
Having existence ofhat our disposal, further properties ofhsuch as symmetry, Hess–Schrader–Uhlenbrock’s inequality for the “pointwise operator norm”|h𝑡|⊗ofh𝑡,
|h𝑡|⊗(𝑥 , 𝑦) ≤e−𝐾 𝑡p𝑡(𝑥 , 𝑦)
for𝔪⊗2-a.e.(𝑥 , 𝑦) ∈M2holds for every𝑡 > 0, and Chapman–Kolmogorov’s formula are stated in Theorem 4.5.7 below.
Two further results are then finally given on the class of RCD(𝐾 , 𝑁)spaces. In Theorem 4.5.11, for every𝑡 > 0we first prove the trace inequality
trH𝑡≤ (dimd,𝔪M)e−𝐾 𝑡 trP𝑡.
Here, dimd,𝔪M, a positive integer not larger than 𝑁, is theessential dimensionof (M,d,𝔪)in the sense of [BS20, MN19]. This generalizes similar results on possibly weighted Riemannian manifolds [Gün17a, HSU80, Ros88]. Furthermore, our spectral analysis for Δ® from Theorem 4.1.4 entails a spectral resolution identity forh𝑡 in Theorem 4.5.13 as soon asMis also compact.