Existence of many functions in D ( Hess )

for every𝑇 ∈L²(𝑇^⊗2M). Now Lemma 3.2.72, Lemma 3.2.54 as well as the continuity of Φallow us to replace the termsℎ_{𝑖 𝑗}ℎ⁰

𝑖 𝑗by arbitrary elements𝑘_{𝑖 𝑗} ∈Test(M), still retaining the identity (3.3.4) with𝑘_{𝑖 𝑗}in place ofℎ_{𝑖 𝑗}ℎ⁰

𝑖 𝑗,𝑖∈ {1, . . . , 𝑛}and 𝑗 ∈ {1, . . . , 𝑚}. In particular, by Definition 3.3.2, we deduce that 𝑓 ∈ D(Hess) and 𝐴⁰ =Hess 𝑓. By Proposition 3.2.19 again and (3.3.5), we obtain

kHess𝑓k_L²_{( (𝑇}∗)^⊗2M)=kΦk_L²_(𝑇⊗2M)⁰ ≤

√ 𝐶 , which is precisely what was left prove.

Remark 3.3.4. IfE2 is extended toL²(M) byE2(𝑓) := ∞ for 𝑓 ∈ L²(M) \F, it is unclear if the resulting functional isL²-lower semicontinuous. To bypass this issue in applications, one might instead use that by Theorem 3.3.3, the functional E₂^𝜀:L²(M) → [0,∞]given by

E2^𝜀(𝑓):=









 𝜀

ˆ

M

|∇𝑓|²d𝔪+ ˆ

M

Hess𝑓

2

HSd𝔪 if 𝑓 ∈D(Hess),

∞ otherwise

isL²-lower semicontinuous for every𝜀 > 0.

IfMis, say, a compact Riemannian manifold without boundary, one can easily prove using the Bochner identity that the (nonpositive) generator associated withE₂^𝜀 [FOT11, Thm. 1.3.1] is the Paneitz-type operator

−Δ²+𝜀Δ𝑓 +div(Ric^♭∇·).

Remark 3.3.5. In general, the Hessian is not the trace of the Laplacian in the sense of (3.2.12). This already happens on weighted Riemannian manifolds without boundary:

of course, the associated Laplacian Δ is defined by partial integration w.r.t. the reference measure [Gri09, Sec. 3.6], while the definition of Hessian only depends on the metric tensor. See also the second part of Example 3.2.48. Examples of abstract spaces for which thisisthe case — and which currently enjoy high research interest [BNS20, DPG18, HZ20] — arenoncollapsedRCD(𝐾 , 𝑁)spaces,𝐾∈Rand 𝑁∈ [1,∞)[DPG18, Thm. 1.12]. See also Remark 3.3.16 below.

Remark 3.3.6. In line with Remark 3.3.5, although a priori𝔪plays a role in Definition 3.3.2, we expect the Hessian to only depend on conformal transformations ofh·,·i, but not on drift transformations of𝔪. For instance, this is known on RCD^∗(𝐾 , 𝑁)spaces, 𝐾 ∈Rand𝑁∈ [1,∞), see e.g. [Han19, Prop. 3.11] or [HS21, Lem. 2.16], and it does not seem hard to adapt the arguments from [Han19] to more general settings.

Remark 3.3.7. As an alternative to Definition 3.3.2, one can defineD(Hess)as the finiteness domain of the r.h.s. of the duality formula in (iv) in Theorem 3.3.11. The Hessian of 𝑓 ∈D(Hess)is then well-defined by the same duality arguments as in the

proof of Theorem 3.3.11.

[ER⁺20, Sav14, Stu18a]. The key technical part (not only for Theorem 3.3.11, but in fact for Theorem 3.6.9 below as well) is contained in Lemma 3.3.9, where — loosely speaking and up to introducing the relevant objects later — we show that

|∇𝑋 :𝑇|² ≤ h

Δ^{2 𝜅}|𝑋|² 2

+

𝑋 ,( ®Δ𝑋^♭)^♯

−

(∇𝑋)_asym

2 HS

i 𝑇

2

HS 𝔪-a.e.

for 𝑋 , 𝑇 ∈ Test(𝑇M). Of course, neither we introduced the covariant derivative ∇, Definition 3.4.2 or the Hodge Laplacian Δ®, Definition 3.5.21, yet, nor in general we have|𝑋|² ∈ D(Δ^{2 𝜅})for𝑋 ∈Test(𝑇M). Reminiscent of Proposition 3.2.75 and [ER⁺20, Cor. 6.3], we instead rephrase the above inequality in terms of measures, and the involved objects∇𝑋 andΔ®𝑋^♭therein as the “r.h.s.’s of the identities one would expect for∇𝑋 andΔ®𝑋^♭for 𝑋 ∈Test(M)”, rigorously proven in Theorem 3.4.3 and Lemma 3.6.1 below. In particular, by optimization over𝑇 ∈Test(𝑇M),

(∇𝑋)_sym

2

HS ≤Δ^{2 𝜅}|𝑋|² 2

+

𝑋 ,( ®Δ𝑋^♭)^♯

−

(∇𝑋)_asym

2

HS 𝔪-a.e.,

which is the Bochner inequality for vector fields according to (3.2.9). For𝑋 :=∇𝑓, 𝑓 ∈Test(M), this essentially provides Theorem 3.3.11. Details about this inequality for general𝑋 ∈Reg(𝑇M), leading to Theorem 3.6.9, are due to Lemma 3.6.2.

We start with a technical preparation. Given𝜇, 𝜈∈𝔐_f⁺(M), we define the Borel measure√

𝜇 𝜈∈𝔐_f⁺(M)as follows. Let𝜄∈𝔐⁺_f(M)with𝜇𝜄and𝜈𝜄be arbitrary, denote the respective densities w.r.t.𝜄by 𝑓 , 𝑔∈L¹(M, 𝜄), and set

√

𝜇 𝜈:=p 𝑓 𝑔 𝜄.

For instance, one can choose𝜄:=|𝜇| + |𝜈|[Hal50, Thm. 30.A] — in fact, the previous definition is independent of the choice of𝜄, whence√

𝜇 𝜈is well-defined.

The following important measure theoretic lemma is due to [Gig18, Lem. 3.3.6].

Lemma 3.3.8. Let𝜇₁, 𝜇₂, 𝜇₃∈𝔐^±_f(M)satisfy the inequality 𝜆²𝜇₁+2𝜆 𝜇₂+𝜇₃≥0 for every𝜆∈R. Then the following properties hold.

(i) The elements𝜇₁and𝜇₃are nonnegative, and

|𝜇₂| ≤√ 𝜇₁𝜇₃. (ii) We have𝜇₂ 𝜇₁,𝜇₂𝜇₃and

k𝜇₂k_TV≤p

k𝜇₁k_TVk𝜇₃k_TV.

(iii) The𝔪-singular parts(𝜇₁)⊥and(𝜇₃)⊥of𝜇₁and𝜇₃are nonnegative. Moreover, expressing the densities of the𝔪-absolutely continuous parts of 𝜇_𝑖 by 𝜌_𝑖 := d(𝜇_𝑖)/d𝔪∈L¹(M),𝑖∈ {1, 2, 3}, we have

|𝜌2|²≤𝜌₁𝜌₃ 𝔪-a.e.

In the subsequent lemma, all terms where𝑁⁰is infinite are interpreted as being zero. Similar proofs can be found in [Bra20, Gig18, Han18a].

Lemma 3.3.9. Let 𝑁⁰ ∈ [𝑁 ,∞], 𝑛, 𝑚 ∈ N, 𝑓 , 𝑔 ∈ Test(M)^𝑛 and ℎ ∈ Test(M)^𝑚. Define𝜇₁[𝑓 , 𝑔] ∈𝔐_f^±(M)as

𝜇₁[𝑓 , 𝑔]:=

𝑛

X

𝑖 ,𝑖⁰=1

e𝑔_𝑖

e𝑔_𝑖0𝚪^{2 𝜅}₂ (𝑓_𝑖, 𝑓_𝑖0) +2

𝑛

X

𝑖 ,𝑖⁰=1

𝑔_𝑖H[𝑓_𝑖] (𝑓_𝑖0, 𝑔_𝑖0)𝔪 + 1

2

𝑛

X

𝑖 ,𝑖⁰=1

h∇𝑓_𝑖,∇𝑓_𝑖0i h∇𝑔_𝑖,∇𝑔_𝑖0i + h∇𝑓_𝑖,∇𝑔_𝑖0i h∇𝑔_𝑖,∇𝑓_𝑖0i 𝔪

− 1 𝑁⁰

hX^𝑛

𝑖=1

𝑔_𝑖Δ𝑓_𝑖+ h∇𝑓_𝑖,∇𝑔_𝑖ii² 𝔪

As in Lemma 3.3.8, we denote the density of the 𝔪-absolutely continuous part of 𝜇₁[𝑓 , 𝑔]by𝜌₁[𝑓 , 𝑔]:=d𝜇₁[𝑓 , 𝑔]/d𝔪∈L¹(M). Then the𝔪-singular part𝜇₁[𝑓 , 𝑔]⊥

of𝜇₁[𝑓 , 𝑔]as well as𝜌₁[𝑓 , 𝑔]are nonnegative, and hX^𝑛

𝑖=1 𝑚

X

𝑗=1

h∇𝑓_𝑖,∇ℎ_𝑗i h∇𝑔_𝑖,∇ℎ_𝑗i +𝑔_𝑖H[𝑓_𝑖] (ℎ_𝑗, ℎ_𝑗)

− 1 𝑁⁰

𝑛

X

𝑖=1 𝑚

X

𝑗=1

𝑔_𝑖Δ𝑓_𝑖+ h∇𝑓_𝑖,∇𝑔_𝑖i

|∇ℎ_𝑗|²i²

≤𝜌₁[𝑓 , 𝑔]hX^𝑚

𝑗 , 𝑗⁰=1

h∇ℎ_𝑗,∇ℎ𝑗⁰i²− 1 𝑁⁰

hX^𝑚

𝑗=1

|∇ℎ𝑗|²i²i

𝔪-a.e.

Proof. We define𝜇₂[𝑓 , 𝑔, ℎ], 𝜇₃[ℎ] ∈𝔐_f^±(M)by 𝜇₂[𝑓 , 𝑔, ℎ]:=

𝑛

X

𝑖=1 𝑚

X

𝑗=1

h∇𝑓_𝑖,∇ℎ_𝑗i h∇𝑔_𝑖,∇ℎ_𝑗i +𝑔_𝑖H[𝑓_𝑖] (ℎ_𝑗, ℎ_𝑗) 𝔪

− 1 𝑁⁰

𝑛

X

𝑖=1 𝑚

X

𝑗=1

𝑔_𝑖Δ𝑓_𝑖+ h∇𝑓_𝑖,∇𝑔_𝑖i

|∇ℎ_𝑗|²𝔪,

𝜇₃[ℎ]:= hX^𝑚

𝑗 , 𝑗⁰=1

h∇ℎ_𝑗,∇ℎ_𝑗0i²− 1 𝑁⁰

hX^𝑚

𝑗=1

|∇ℎ_𝑗|²i²i 𝔪.

Both claims readily follow from Lemma 3.3.8 as soon as𝜆²𝜇₁[𝑓 , 𝑔] +2𝜆 𝜇₂[𝑓 , 𝑔, ℎ] + 𝜇₃[ℎ] ≥0for every𝜆∈R, which is what we concentrate on in the sequel.

Let𝜆 ∈ R and pick𝑎, 𝑏 ∈ R^𝑛 as well as 𝑐 ∈ R^𝑚. Define the function𝜑 ∈ C^∞(R^2𝑛+𝑚)through

𝜑(𝑥 , 𝑦, 𝑧):=

𝑛

X

𝑖=1

𝜆 𝑥_𝑖𝑦_𝑖+𝑎_𝑖𝑥_𝑖−𝑏_𝑖𝑦_𝑖 +

𝑚

X

𝑗=1

(𝑧_𝑗−𝑐_𝑗)²−𝑐²

𝑗

.

For every𝑖 ∈ {1, . . . , 𝑛} and every 𝑗 ∈ {1, . . . , 𝑚}, those first and second partial derivatives of𝜑which, do not always vanish identically read

𝜑_𝑖(𝑥 , 𝑦, 𝑧)=𝜆 𝑦_𝑖+𝑎_𝑖, 𝜑_𝑛+𝑖(𝑥 , 𝑦, 𝑧)=𝜆 𝑥_𝑖−𝑏_𝑖, 𝜑_2𝑛+𝑗(𝑥 , 𝑦, 𝑧)=2(𝑧_𝑗−𝑐_𝑗), 𝜑_{𝑖 , 𝑛+𝑖}(𝑥 , 𝑦, 𝑧)=𝜆,

𝜑_{𝑛+𝑖 ,𝑖}(𝑥 , 𝑦, 𝑧)=𝜆,

𝜑_{2𝑛+𝑗 ,2𝑛+𝑗}(𝑥 , 𝑦, 𝑧)=2.

For convenience, we write

A^{2 𝜅}(𝜆, 𝑎, 𝑏, 𝑐):=A^{2 𝜅}[𝜑◦𝑞], B(𝜆, 𝑎, 𝑏, 𝑐):=B[𝜑◦𝑞], C(𝜆, 𝑎, 𝑏, 𝑐):=C[𝜑◦𝑞], D(𝜆, 𝑎, 𝑏, 𝑐):=D[𝜑◦𝑞],

where the respective r.h.s.’s are defined as in Lemma 3.2.76 for𝛼 := 2𝑛+𝑚 and 𝑞 :=(𝑓 , 𝑔, ℎ). Using the same Lemma 3.2.76, we compute

A^{2 𝜅}(𝜆, 𝑎, 𝑏, 𝑐)=

𝑛

X

𝑖 ,𝑖⁰=1

(𝜆

e𝑔_𝑖+𝑎_𝑖) (𝜆

e𝑔_𝑖0+𝑎_𝑖0)𝚪^{2 𝜅}₂ (𝑓_𝑖, 𝑓_𝑖0) +other terms, B(𝜆, 𝑎, 𝑏, 𝑐)=4

𝑛

X

𝑖 ,𝑖⁰=1

(𝜆 𝑔_𝑖+𝑎_𝑖)𝜆H[𝑓_𝑖] (𝑓_𝑖0, 𝑔_𝑖0) +4

𝑛

X

𝑖=1 𝑚

X

𝑗=1

(𝜆 𝑔_𝑖+𝑎_𝑖)H[𝑓_𝑖] (ℎ_𝑗, ℎ_𝑗) +other terms, C(𝜆, 𝑎, 𝑏, 𝑐)=2

𝑛

X

𝑖 ,𝑖⁰=1

𝜆²

h∇𝑓_𝑖,∇𝑓_𝑖0i h∇𝑔_𝑖,∇𝑔_𝑖0i + h∇𝑓_𝑖,∇𝑔_𝑖0i h∇𝑔_𝑖,∇𝑓_𝑖0i +8

𝑛

X

𝑖=1 𝑚

X

𝑗=1

𝜆h∇𝑓_𝑖,∇ℎ_𝑗i h∇𝑔_𝑖,∇ℎ_𝑗i +4

𝑚

X

𝑗 , 𝑗⁰=1

h∇ℎ_𝑗,∇ℎ_𝑗0i²+other terms, D(𝜆, 𝑎, 𝑏, 𝑐)=

𝑛

X

𝑖 ,𝑖⁰=1

(𝜆 𝑔_𝑖+𝑎_𝑖) (𝜆 𝑔_𝑖0+𝑎_𝑖)Δ𝑓_𝑖Δ𝑓_𝑖0

+4

𝑛

X

𝑖 ,𝑖⁰=1

𝜆(𝜆 𝑔_𝑖+𝑎_𝑖)Δ𝑓_𝑖h∇𝑓_𝑖0,∇𝑔_𝑖0i +4

𝑛

X

𝑖 ,𝑖⁰=1

𝜆²h∇𝑓_𝑖,∇𝑔_𝑖i h∇𝑓_𝑖0,∇𝑔_𝑖0i +4

𝑛

X

𝑖=1 𝑚

X

𝑗=1

(𝜆 𝑔_𝑖+𝑎_𝑖)Δ𝑓_𝑖|∇ℎ_𝑗|² +8

𝑛

X

𝑖=1 𝑚

X

𝑗=1

𝜆h∇𝑓_𝑖,∇𝑔_𝑖i |∇ℎ_𝑗|² +4

hX^𝑚

𝑗=1

|∇ℎ_𝑗|²i²

+other terms.

Here, every “other term” contains at least one factor of the form𝜆e𝑓_𝑖−𝑏_𝑖oreℎ_𝑗−𝑐_𝑗 for some𝑖∈ {1, . . . , 𝑛}and 𝑗 ∈ {1, . . . , 𝑚}.

By Lemma 3.2.76 and Proposition 3.2.75 with the nonnegativity of D(𝜆, 𝑎, 𝑏, 𝑐)as well as the trivial inequality1/𝑁 ≥1/𝑁⁰,

A^{2 𝜅}(𝜆, 𝑎, 𝑏, 𝑐) +h

B(𝜆, 𝑎, 𝑏, 𝑐) +C(𝜆, 𝑎, 𝑏, 𝑐) − 1

𝑁⁰D(𝜆, 𝑎, 𝑏, 𝑐)i 𝔪 ≥0.

By the arbitrariness of𝑎, 𝑏 ∈R^𝑛 and𝑐∈ R^𝑚, for every Borel partition (𝐸_𝑝)_𝑝∈Nof M, every Borel set𝐹 ⊂Mand all sequences(𝑎_𝑘)𝑘∈Nand(𝑏_𝑘)𝑘∈NinR^𝑛as well as (𝑐_𝑘)𝑘∈NinR^𝑚,

1_𝐹X

𝑘∈N

1_𝐸

𝑘

h

A^{2 𝜅}(𝜆, 𝑎_𝑘, 𝑏_𝑘, 𝑐_𝑘) +h

B(𝜆, 𝑎_𝑘, 𝑏_𝑘, 𝑐_𝑘) +C(𝜆, 𝑎_𝑘, 𝑏_𝑘, 𝑐_𝑘)

− 1

𝑁⁰D(𝜆, 𝑎_𝑘, 𝑏_𝑘, 𝑐_𝑘)i 𝔪i

≥0.

(3.3.6)

We now choose the involved quantities appropriately. Let(𝐹_𝑘)𝑘∈Nbe anE-nest with the property that the restrictions of e𝑓,e𝑔 andeℎ to𝐹_𝑘 are continuous for every 𝑘 ∈ N, and set𝐹:=S

𝑘∈N𝐹_𝑘. Since𝐹^c is anE-polar set and thus not seen by𝔪 and𝚪^{2 𝜅}₂ (𝑓_𝑖, 𝑓_𝑖0),𝑖, 𝑖⁰∈ {1, . . . , 𝑛}, its contribution to the subsequent manipulations is ignored. For𝑙 ∈Nwe now take a Borel partition(𝐸^𝑙

𝑘)𝑘∈NofMand sequences(𝑎^𝑙

𝑘)𝑘∈N

and(𝑏^𝑙

𝑘)𝑘∈NinR^𝑛as well as(𝑐^𝑙

𝑘)𝑘∈NinR^𝑚with sup

𝑘 ,𝑙∈N

|𝑎^𝑙

𝑘| + |𝑏^𝑙

𝑘| + |𝑐^𝑙

𝑘|

<∞ in such a way that

𝑙→∞lim X

𝑘∈N

1𝐸^𝑙 𝑘

𝑎^𝑙

𝑘 =𝜆 e𝑔,

𝑙→∞lim X

𝑘∈N

1𝐸^𝑙 𝑘

𝑏^𝑙

𝑘 =𝜆e𝑓 ,

𝑙→∞lim X

𝑘∈N

1𝐸^𝑙 𝑘

𝑐^𝑙

𝑘 =eℎ

pointwise on𝐹. Thus, the l.h.s. of (3.3.6) with(𝐸_𝑘)𝑘∈N,(𝑎_𝑘)𝑘∈N,(𝑏_𝑘)𝑘∈Nand(𝑐_𝑘)𝑘∈N

replaced by(𝐸^𝑙

𝑘)𝑘∈N,(𝑎^𝑙

𝑘)𝑘∈N,(𝑏^𝑙

𝑘)𝑘∈Nand(𝑐^𝑙

𝑘)𝑘∈N,𝑙 ∈N, respectively, converges w.r.t.k · k_TVas𝑙 → ∞. In fact, in the limit as𝑙→ ∞every “other term” above becomes zero, and the prefactors𝜆

e𝑔_𝑖+ (𝑎^𝑙

𝑘)𝑖 become2𝜆

e𝑔_𝑖,𝑖∈ {1, . . . , 𝑛}. We finally obtain 4𝜆²

𝑛

X

𝑖 ,𝑖⁰=1

e𝑔_𝑖

e𝑔_𝑖0𝚪^{2 𝜅}₂ (𝑓_𝑖, 𝑓_𝑖0) +8𝜆²

𝑛

X

𝑖 ,𝑖⁰=1

𝑔_𝑖H[𝑓_𝑖] (𝑓_𝑖0, 𝑔_𝑖0)𝔪+8𝜆

𝑛

X

𝑖=1 𝑚

X

𝑗=1

𝑔_𝑖H[𝑓_𝑖] (ℎ_𝑗, ℎ_𝑗)𝔪 +2𝜆²

𝑛

X

𝑖 ,𝑖⁰=1

h∇𝑓_𝑖,∇𝑓_𝑖0i h∇𝑔_𝑖,∇𝑔_𝑖0i + h∇𝑓_𝑖,∇𝑔_𝑖0i h∇𝑔_𝑖,∇𝑓_𝑖0i 𝔪 +8𝜆

𝑛

X

𝑖=1 𝑚

X

𝑗=1

h∇𝑓_𝑖,∇ℎ_𝑗i h∇𝑔_𝑖,∇ℎ_𝑗i𝔪 +4

𝑚

X

𝑗 , 𝑗⁰=1

h∇ℎ_𝑗,∇ℎ_𝑗0i²𝔪

−4𝜆² 𝑁⁰

hX^𝑛

𝑖=1

𝑔_𝑖Δ𝑓_𝑖+ h∇𝑓_𝑖,∇𝑔_𝑖ii² 𝔪

− 8𝜆 𝑁⁰

𝑛

X

𝑖=1 𝑚

X

𝑗=1

𝑔_𝑖Δ𝑓_𝑖+ h∇𝑓_𝑖,∇𝑔_𝑖i

|∇ℎ_𝑗|²𝔪

− 4 𝑁⁰

hX^𝑚

𝑗=1

|∇ℎ_𝑗|²i²

≥0.

Dividing by4and sorting terms by the order of𝜆yields the claim.

We note the following consequence of Lemma 3.3.9 that is used in Theorem 3.3.11 below as well, but becomes especially important in Subsection 3.3.3.

Remark 3.3.10. The nonnegativity of 𝜇₃[ℎ] from Lemma 3.3.9 can be translated into the following trace inequality, compare with [Bra20, Rem. 2.19] and the proof of [Han18a, Prop. 3.2]. With the pointwise trace defined as in (3.2.12), we have

𝑚

X

𝑗=1

∇ℎ_𝑗⊗ ∇ℎ_𝑗

2 HS ≥ 1

𝑁trhX^𝑚

𝑗=1

∇ℎ_𝑗⊗ ∇ℎ_𝑗 i²

𝔪-a.e.

for every𝑚∈Nand everyℎ∈Test(M)^𝑚.

Theorem 3.3.11. Every 𝑓 ∈Test(M)belongs toD(Hess)and satisfies

Hess𝑓(∇𝑔₁,∇𝑔₂)=H[𝑓] (𝑔₁, 𝑔₂) 𝔪-a.e. (3.3.7) for every𝑔₁, 𝑔₂ ∈Test(M). Moreover, denoting by𝛾^{2 𝜅}

2 (𝑓) ∈L¹(M)the density of the 𝔪-absolutely continuous part of𝚪^{2 𝜅}₂ (𝑓), we have

Hess𝑓

2 HS ≤𝛾^{2 𝜅}

2 (𝑓) 𝔪-a.e. (3.3.8) Proof. Recall that indeed𝛾^{2 𝜅}

2 (𝑓) ∈L¹(M)by Proposition 3.2.79. Let𝑔, ℎ₁, . . . , ℎ_𝑚∈ Test(M),𝑚∈N. Applying Lemma 3.3.9 for𝑁⁰:=∞and𝑛:=1then entails

hX^𝑚

𝑗=1

h∇𝑓 ,∇ℎ_𝑗i h∇𝑔,∇ℎ_𝑗i +𝑔H[𝑓] (ℎ_𝑗, ℎ_𝑗)i²

≤ h 𝑔²𝛾^{2 𝜅}

2 (𝑓) +2𝑔H[𝑓] (𝑓 , 𝑔) + 1 2

|∇𝑓|²|∇𝑔|²+ 1 2

h∇𝑓 ,∇𝑔i²i

×

𝑚

X

𝑗 , 𝑗⁰=1

h∇ℎ_𝑗,∇ℎ_𝑗0i²

= h

𝑔²𝛾^{2 𝜅}

2 (𝑓) +𝑔

∇|∇𝑓|²,∇𝑔 + 1

2

|∇𝑓|²|∇𝑔|²+ 1 2

h∇𝑓 ,∇𝑔i²i

×

𝑚

X

𝑗 , 𝑗⁰=1

h∇ℎ_𝑗,∇ℎ_𝑗0i² 𝔪-a.e.

In the last identity, we used the definition (3.2.22) of H[𝑓] (𝑓 , 𝑔). Using the first part of Lemma 3.2.73 and possibly passing to subsequences, this𝔪-a.e. inequality extends to all𝑔 ∈ F∩L^∞(M). Thus, successively setting𝑔 :=𝑔_𝑛,𝑛 ∈ N, where (𝑔_𝑛)_𝑛∈Nis the sequence provided by Lemma 3.2.6, together with the locality of∇from Proposition 3.2.37, and by the definition (3.2.7) of the pointwise Hilbert–Schmidt norm ofL²(𝑇^⊗2M), we obtain

𝑚

X

𝑗=1

H[𝑓] (ℎ_𝑗, ℎ_𝑗) ≤𝛾^{2 𝜅}

2 (𝑓)^1/2

𝑚

X

𝑗=1

∇ℎ_𝑗 ⊗ ∇ℎ_𝑗

_HS 𝔪-a.e. (3.3.9) This implies pointwise𝔪-a.e. off-diagonal estimates as follows. Given any𝑚⁰∈Nand ℎ_𝑗, ℎ⁰

𝑗 ∈Test(M), 𝑗 ∈ {1, . . . , 𝑚⁰}, since

𝑚⁰

X

𝑗=1

H[𝑓] (ℎ_𝑗, ℎ⁰

𝑗)= 1 2

𝑚⁰

X

𝑗=1

H[𝑓] (ℎ_𝑗+ℎ⁰

𝑗, ℎ_𝑗+ℎ⁰

𝑗) −H[𝑓] (ℎ_𝑗, ℎ_𝑗) −H[𝑓] (ℎ⁰

𝑗, ℎ⁰

𝑗)

holds𝔪-a.e., applying (3.3.9), using that 1

2

𝑚⁰

X

𝑗=1

∇(ℎ_𝑗+ℎ⁰_𝑗) ⊗ ∇(ℎ_𝑗+ℎ⁰_𝑗) − ∇ℎ_𝑗⊗ ∇ℎ_𝑗− ∇ℎ⁰_𝑗 ⊗ ∇ℎ⁰_𝑗

= 1 2

𝑚⁰

X

𝑗=1

∇ℎ_𝑗 ⊗ ∇ℎ⁰_𝑗 + ∇ℎ⁰_𝑗⊗ ∇ℎ_𝑗

= h^𝑚

0

X

𝑗=1

∇ℎ_𝑗⊗ ∇ℎ⁰_𝑗 i

sym

and finally employing that|𝑇_sym|_HS≤ |𝑇|_HSfor every𝑇 ∈L²(𝑇^⊗2M), we get

𝑚⁰

X

𝑗=1

H[𝑓] (ℎ_𝑗, ℎ⁰

𝑗) ≤𝛾^{2 𝜅}

2 (𝑓)^1/2 1 2

𝑚⁰

X

𝑗=1

∇ℎ_𝑗⊗ ∇ℎ⁰

𝑗+ ∇ℎ⁰

𝑗 ⊗ ∇ℎ_{𝑗 HS}

≤𝛾^{2 𝜅}

2 (𝑓)^1/2

𝑚⁰

X

𝑗=1

∇ℎ_𝑗 ⊗ ∇ℎ⁰

𝑗

_HS 𝔪-a.e.

We replaceℎ_𝑗by𝑎_𝑗ℎ_𝑗,𝑗 ∈ {1, . . . , 𝑚⁰}, for arbitrary𝑎₁, . . . , 𝑎_𝑚0 ∈Q. This gives

𝑚⁰

X

𝑗=1

𝑎_𝑗H[𝑓] (ℎ_𝑗, ℎ⁰

𝑗) ≤𝛾^{2 𝜅}

2 (𝑓)^1/2

𝑚⁰

X

𝑗=1

𝑎_𝑗∇ℎ_𝑗 ⊗ ∇ℎ⁰

𝑗

_HS 𝔪-a.e. (3.3.10) In fact, sinceQis countable, we find an𝔪-negligible Borel set𝐵 ⊂M on whose complement (3.3.10) holds pointwise for every𝑎₁, . . . , 𝑎_𝑚0 ∈Q. Since both sides of (3.3.10) are continuous in𝑎₁, . . . , 𝑎_𝑚0, by density ofQinRwe deduce that (3.3.10) holds pointwise on𝐵^cfor every𝑎₁, . . . , 𝑎_𝑚0∈R. Therefore, given any𝑔₁, . . . , 𝑔_𝑚0∈Test(M), up to possibly removing a further𝔪-negligible Borel set𝐶⊂M, for every𝑥∈ (𝐵∪𝐶)^c we may replace𝑎_𝑗 by𝑔_𝑗(𝑥),𝑗 ∈ {1, . . . , 𝑚⁰}, in (3.3.10). This leads to

𝑚⁰

X

𝑗=1

𝑔_𝑗H[𝑓] (ℎ_𝑗, ℎ⁰_𝑗)

2

≤𝛾^{2 𝜅}

2 (𝑓)^1/2

𝑚⁰

X

𝑗=1

𝑔_𝑗∇ℎ_𝑗 ⊗ ∇ℎ⁰_𝑗

_HS 𝔪-a.e. (3.3.11) We now define the operatorΦ: Test(𝑇^⊗2M) →L⁰(M)by

Φ

𝑚⁰

X

𝑗=1

𝑔_𝑗𝑔⁰_𝑗∇ℎ_𝑗⊗ ∇ℎ⁰_𝑗 :=

𝑚⁰

X

𝑗=1

𝑔_𝑗𝑔⁰_𝑗H[𝑓] (ℎ_𝑗, ℎ⁰_𝑗). (3.3.12) From (3.3.11) and the algebra property of Test(M), it follows thatΦis well-defined, i.e. the value ofΦ(𝑇) does not depend on the specific way of representing a given element 𝑇 ∈ Test(𝑇^⊗2M). Moreover, the map Φ is clearly linear, and for every 𝑔 ∈Test(M)and every𝑇 ∈Test(𝑇^⊗2M),

Φ(𝑔 𝑇)=𝑔Φ(𝑇). (3.3.13)

Since the𝔪-singular part𝚪^{2 𝜅}₂ (𝑓)⊥of𝚪^{2 𝜅}₂ (𝑓)is nonnegative, by (3.2.23) we get ˆ

M

𝛾^{2 𝜅}

2 (𝑓)d𝔪 ≤𝚪^{2 𝜅}₂ (𝑓) [M]= ˆ

M

(Δ𝑓)²d𝔪− 𝜅

|∇𝑓|²

. (3.3.14)

After integrating (3.3.11) and employing Cauchy–Schwarz’s inequality, kΦ(𝑇) k_L1(M) ≤hˆ

M

(Δ𝑓)²d𝔪− 𝜅

|∇𝑓|²i^1/2

k𝑇k_L2(𝑇^⊗2M)

holds for every𝑇 ∈Test(𝑇^⊗2M). Thus, by density of Test(𝑇^⊗2M)inL²(𝑇^⊗2M)and (3.3.13),Φ uniquely extends to a (non-relabeled) continuous,L^∞-linear map from L²(𝑇^⊗2M)intoL¹(M), whenceΦ∈L²( (𝑇^∗)^⊗2M)by definition of the latter space.

To check that𝑓 ∈D(Hess)andΦ =Hess𝑓, first note that by the continuity ofΦand Lemma 3.2.72, we can replace𝑔_𝑗𝑔⁰

𝑗 by arbitrary𝑘_𝑗 ∈Test(M),𝑗 ∈ {1, . . . , 𝑚⁰}, still retaining the identity (3.3.12). Therefore, slightly changing the notation in (3.3.12), let 𝑔₁, 𝑔₂, ℎ ∈Test(M)and use (3.3.13), the definition (3.2.22) of H[𝑓]and Lemma 3.2.54 to derive that

2 ˆ

M

ℎΦ(∇𝑔₁,∇𝑔₂)d𝔪

= ˆ

M

ℎ

∇𝑔₁,∇h∇𝑓 ,∇𝑔₂id𝔪+ ˆ

M

ℎ

∇𝑔₁,∇h∇𝑓 ,∇𝑔₂id𝔪

− ˆ

M

ℎ

∇𝑔₁,∇h∇𝑓 ,∇𝑔₂i d𝔪

=− ˆ

M

h∇𝑓 ,∇𝑔₂idiv(ℎ∇𝑔₁)d𝔪− ˆ

M

h∇𝑓 ,∇𝑔₂idiv(ℎ∇𝑔₁)d𝔪

− ˆ

M

ℎ

∇𝑔₁,∇h∇𝑓 ,∇𝑔₂id𝔪, which is the desired assertion 𝑓 ∈D(Hess)andΦ =Hess𝑓.

The same argument gives (3.3.7), while the inequality (3.3.8) is due to (3.3.11), the density of Test(𝑇^⊗2M)inL²(𝑇^⊗2M)as well as the definition (3.2.7) of the pointwise Hilbert–Schmidt norm.

Theorem 3.3.11 implies the following qualitative result. A quantitative version of it, as directly deduced in [Gig18, Cor. 3.3.9] from [Gig18, Thm. 3.3.8], is however not yet available only with the information collected so far. See Remark 3.3.13 below.

Corollary 3.3.12. Every 𝑓 ∈ D(Δ)belongs to the closure ofTest(M) inD(Hess), and in particular toD(Hess). More precisely, let𝜌⁰ ∈ (0, 1) and𝛼⁰ ∈ Rbe as in Lemma 3.2.60 for𝜇:=𝜅⁻. Then for every 𝑓 ∈D(Δ), we have 𝑓 ∈D(Hess)with

ˆ

M

Hess𝑓

2

HSd𝔪≤ 1 1−𝜌⁰

ˆ

M

(Δ𝑓)²d𝔪+ 𝛼⁰ 1−𝜌⁰

ˆ

M

|∇𝑓|²d𝔪.

Proof. Since (M,E,𝔪) satisfies BE¹(𝜅,∞)by [ER⁺20, Thm. 6.9], it also trivially obeys BE¹(−𝜅⁻,∞), see also [ER⁺20, Prop. 6.7]. As in the proof of Proposition 3.2.79,

E^−𝜅−|∇𝑓|

≤ ˆ

M

(Δ𝑓)²d𝔪 holds for every 𝑓 ∈Test(M). Hence, using (3.2.19) we estimate

𝜅⁻ |∇𝑓|²

≤𝜌⁰E |∇𝑓| +𝛼⁰

ˆ

M

|∇𝑓|²d𝔪

=𝜌⁰E^−𝜅− |∇𝑓| +𝜌⁰

𝜅⁻ |∇𝑓|²

+𝛼⁰ ˆ

M

|∇𝑓|²d𝔪

≤𝜌⁰ ˆ

M

(Δ𝑓)²d𝔪+𝜌⁰ 𝜅⁻

|∇𝑓|² +𝛼⁰

ˆ

M

|∇𝑓|²d𝔪.

The claim for 𝑓 ∈Test(M)now follows easily. We already know from Theorem 3.3.11 that 𝑓 ∈D(Hess). Integrating (3.3.8) and using (3.3.14) thus yields

ˆ

M

Hess𝑓

2 HSd𝔪≤

ˆ

M

(Δ𝑓)²d𝔪− 𝜅

|∇𝑓|²

(3.3.15)

≤ ˆ

M

(Δ𝑓)²d𝔪+ 𝜅⁻

|∇𝑓|²

≤ 1

1−𝜌⁰ ˆ

M

(Δ𝑓)²d𝔪+ 𝛼⁰ 1−𝜌⁰

ˆ

M

|∇𝑓|²d𝔪.

Finally, given 𝑓 ∈D(Δ), let 𝑓_𝑛 :=max{min{𝑓 , 𝑛},−𝑛} ∈L²(M) ∩L^∞(M),𝑛∈N. Note that^P𝑡𝑓_𝑛 ∈Test(M)for every𝑡 > 0and every𝑛∈N, and that^P𝑡𝑓_𝑛 →P𝑡𝑓 inF as well as, thanks to Lemma 3.2.12,ΔP𝑡𝑓_𝑛→ΔP𝑡𝑓 inL²(M)as𝑛→ ∞. Moreover P𝑡𝑓 → 𝑓inFas well asΔP𝑡𝑓 =P𝑡Δ𝑓 →Δ𝑓 inL²(M)as𝑡→0. These observations imply that 𝑓 belongs to the closure of Test(M)inD(Hess), whence 𝑓 ∈D(Hess)by Theorem 3.3.3, and the claimed inequality, with unchanged constants, is clearly stable under this approximation procedure.

Remark 3.3.13. The subtle reason why we still cannot deduce (3.3.15) for general 𝑓 ∈ D(Δ) is that we neither know whether the r.h.s. of (3.3.15) makes sense — which essentially requires|∇𝑓| ∈ F— nor, in the notation of the previous proof, whether

𝜅

|∇P𝑡𝑓_𝑛|²

→

𝜅

|∇𝑓|² as𝑛 → ∞ and𝑡 → 0. (Neither we know if E |∇P𝑡𝑓_𝑛|

→E |∇𝑓| as𝑛 → ∞and𝑡 → 0.) Both points are trivial in the more restrictive RCD(𝐾 ,∞)case from [Gig18, Cor. 3.3.9],𝐾 ∈ R. In our setting, solely Lemma 3.2.60 does not seem sufficient to argue similarly. Instead, both points will follow from Lemma 3.4.13 and Lemma 3.6.2, see Corollary 3.4.14 and Corollary 3.6.3.

Im Dokument Heat flow aspects of synthetic Ricci bounds in the extended Kato class (Seite 123-131)