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Riesz transforms and Hardy spaces in R n Hardy spaces and tent spaces Hardy spaces of differential forms in R n The case of Riemannian manifolds H 2 T * M) H p T * M) The decomposition into molecules The maximal characterization Further results Hardy spaces of differential forms on Riemannian manifolds Emmanuel Russ Université Paul Cézanne Aix-Marseille III LATP With Pascal Auscher (Orsay) and Alan McIntosh (Canberra) AHPI, 06/04/2008 Emmanuel Russ [email protected]

Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

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Page 1: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Hardy spaces of differential forms on Riemannianmanifolds

Emmanuel Russ

Université Paul Cézanne Aix-Marseille IIILATP

With Pascal Auscher (Orsay) and Alan McIntosh (Canberra)

AHPI, 06/04/2008

Emmanuel Russ [email protected]

Page 2: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

I. Riesz transforms and Hardy spaces in Rn

Let n ≥ 1. Well-known in Rn:

‖|∇f |‖Lp(Rn) ∼∥∥∥(−∆)1/2f

∥∥∥Lp(Rn)

, 1 < p < +∞,

where, say, f ∈ D(Rn) and

|∇f | =n∑

i=1|∂i f | .

The Riesz transforms are the operators R1, ...,Rn given by

Rj = ∂j(−∆)−1/2.

For all 1 < p < +∞, Rj is Lp(Rn)-bounded. This follows from standardCalderón-Zygmund theory.

Emmanuel Russ [email protected]

Page 3: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

For p = 1, one has∥∥∥(−∆)1/2f∥∥∥

H1(Rn)∼ ‖∇f ‖H1(Rn)

:=n∑

i=1‖∂j f ‖H1(Rn) ,

where H1(Rn) is the (real) Hardy space.

A definition: f ∈ H1(Rn) iff f ∈ L1(Rn) and ∂j(−∆)−12 f ∈ L1(Rn) for all

1 ≤ j ≤ n. Set

‖f ‖H1(Rn) = ‖f ‖L1(Rn) +n∑

j=1

∥∥∥∂j(−∆)−12 f∥∥∥

L1(Rn).

Emmanuel Russ [email protected]

Page 4: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

There are many characterizations of this space.

One possible characterization of H1(Rn): first, an atom is a functiona ∈ L2(Rn) supported in a cube Q ⊂ Rn such that∫

Rna(x)dx = 0 and ‖a‖L2(Rn) ≤ |Q|−1/2

.

Emmanuel Russ [email protected]

Page 5: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

A function f is in H1(Rn) iff

f =∑

jλjaj

where the aj ’s are atoms and∑

j|λj | < +∞. One has

‖f ‖H1(Rn) ∼ inf∑j≥1

|λj | .

A consequence: if f ∈ H1(Rn), one has∫Rn

f (x)dx = 0.

Emmanuel Russ [email protected]

Page 6: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

II. Hardy spaces and tent spaces

We are still in Rn.

If F : Rn × (0,+∞) → R, define, for all x ∈ Rn,

SF (x) =

(∫∫|y−x |<t

1tn |F (y , t)|2 dy dt

t

) 12

.

If 1 ≤ p < +∞, say that F ∈ T p,2(Rn) (tent space) iff

‖F‖T p,2(Rn) := ‖SF‖Lp(Rn) < +∞.

Emmanuel Russ [email protected]

Page 7: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

If B = B(x , r) ⊂ Rn is a ball, define the tent over B as

T (B) := (y , t) ∈ Rn × (0,+∞); |y − x | < r − t .

An atomic decomposition for T 1,2(Rn): an atom is a functionA ∈ L2 (Rn × (0,+∞), dxdt/t) supported in T (B) for some ball B ⊂ Rn

and satisfying ∫∫T (B)

|A(x , t)|2 dx dtt ≤ 1

|B| .

An atom belongs to T 1,2(Rn) with a norm controlled by a constant.Every F ∈ T 1,2(Rn) has an atomic decomposition (Coifman , Meyer,Stein).

Emmanuel Russ [email protected]

Page 8: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

The link with Hardy spaces:

For f : Rn → C and x ∈ Rn, define

Sf (x) =

(∫∫Γ(x)

∣∣∣t√−∆e−t√−∆f (y)

∣∣∣2 dy dtt

)1/2

where Γ(x) = (t, y) ∈ (0,+∞)× Rn; |y − x | < t.

Emmanuel Russ [email protected]

Page 9: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

f ∈ H1(Rn) iff Sf ∈ L1(Rn) (Fefferman, Stein, 1972). This means that

(y , t) 7→ t√−∆e−t

√−∆f (y)

belongs to T 1,2(Rn).

Emmanuel Russ [email protected]

Page 10: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Let f ∈ H1(Rn) ∩ L2(Rn), and set

F (y , t) = t√−∆e−t

√−∆f (y) ∈ T 1,2(Rn).

Thenf = c

∫ +∞

0t√−∆e−t

√−∆F (·, t)dt

t(Calderón reproducing formula).

Emmanuel Russ [email protected]

Page 11: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Moreover, if F ∈ T 1,2(Rn) ∩ T 2,2(Rn), then, if

f := c∫ +∞

0t√−∆e−t

√−∆F (t, .)dt

t ,

one hasf ∈ H1(Rn).

Emmanuel Russ [email protected]

Page 12: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Sketch of the proof:• enough to assume that F is an atom,• then, f is a “molecule”, i.e. has good L2 decay,• this is due to the precise knowledge of t

√−∆e−t

√−∆, but actually

L2 off-diagonal estimates are enough: if d(E ,F ) = d and f issupported in E , then, for all N ≥ 1,∥∥∥t

√−∆e−t

√−∆f

∥∥∥L2(F )

≤ CN

(td

)N‖f ‖L2(E) .

Emmanuel Russ [email protected]

Page 13: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Can replace the Poisson semigroup by ϕ(t√−∆

)whenever ϕ is analytic

in some sector in C containing the positive real axis and has appropriatedecay: some flexibility is alllowed. For instance,

ϕ(z) = zN(1 + z2)−α

with 2α > N, orϕ(z) = zN exp

(−z2) .

Emmanuel Russ [email protected]

Page 14: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

III. Hardy spaces of differential forms in Rn

Introduced by Lou and McIntosh. Let 0 ≤ l ≤ n and f : Rn → Λl adifferential l-form. It can be decomposed as

f =∑

I=(i1,...,il )

fIeI

whereeI = e i1 ∧ ... ∧ e il .

Say that f ∈ H1(Rn,Λl) if each fI ∈ H1(Rn). Then, define

H1d (Rn,Λl) =

f ∈ H1(Rn,Λl); f = dgfor some g ∈ D′(Rn,Λl−1)

.

When l = n, this is H1(Rn).

Emmanuel Russ [email protected]

Page 15: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

There is a characterization by tent spaces in the same spirit as forfunctions.

An atomic decomposition: an atom in H1d (Rn,Λl) is an a ∈ L2(Rn,Λl)

such that there exists a cube Q ⊂ Rn and b ∈ L2(Rn,Λl−1) supported inQ such that

a = db and ‖a‖L2(Rn,Λl ) ≤ |Q|−1/2,

‖b‖L2(Rn,Λl−1) ≤ l(Q) |Q|−1/2.

Here the cancellation for a (meaningless since a is a form) is replaced bythe fact that a = db.

Emmanuel Russ [email protected]

Page 16: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Then:

Theorem

(Lou, McIntosh, 2005) f ∈ H1d (Rn,Λl) iff f =

∑jλjaj where the aj ’s are

atoms and∑

j|λj | < +∞.

Emmanuel Russ [email protected]

Page 17: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

IV. The case of Riemannian manifolds

Let M be a connected complete Riemannian manifold . Denote by• ρ the Riemannian metric,• µ the Riemannian measure,• d the exterior differentiation,• ∆ the Laplace-Beltrami operator.

Emmanuel Russ [email protected]

Page 18: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Here, the are no (global) coordinates, the Riesz transforms ∂j∆− 1

2 do notexist. The form-valued Riesz transform is d∆− 1

2 .

Question (first raised by Strichartz, 1982): does one have

‖|df |‖Lp(M) ∼∥∥∥∆1/2f

∥∥∥Lp(M)

, 1 < p < +∞ ?

Emmanuel Russ [email protected]

Page 19: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

OK for p = 2, since ∥∥∥∆ 12 f∥∥∥2

2= 〈∆f , f 〉L2(M)

= ‖|df |‖22 .

Emmanuel Russ [email protected]

Page 20: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

What about p 6= 2 ?

A general fact: if∣∣∣d∆− 1

2

∣∣∣ is Lp(M)-bounded, i.e.

‖|df |‖Lp(M) ≤ Cp

∥∥∥∆1/2f∥∥∥

Lp(M)

for some 1 < p < +∞ and all f , then∥∥∥∆1/2f∥∥∥

Lp′(M)≤ Cp′ ‖|df |‖Lp′(M)

with 1/p + 1/p′ = 1. The converse is false.

Emmanuel Russ [email protected]

Page 21: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Some assumptions on M. Say that M has the doubling property if

∃C > 0 ∀x ∈ M ∀r > 0 V (x , 2r) ≤ CV (x , r). (D)

This means that (M, d , µ) is a space of homogeneous type (in the senseof Coifman and Weiss). Suitable framework for harmonic analysis:• covering lemmata,• Lp boundedness of Hardy-Littlewood maximal function

(1 < p ≤ +∞),• Calderón-Zygmund decomposition,• T1,Tb theorems...

OK when Ric M ≥ 0 (Bishop comparison theorem).

Emmanuel Russ [email protected]

Page 22: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Another assumption: Let pt be the kernel of e−t∆, i.e.

e−t∆f (x) =

∫M

pt(x , y)f (y)dµ(y).

Say that pt has Gaussian upper bound if

pt(x , y) ≤ CV (x ,

√t)

exp(−c ρ

2(x , y)

t

). (GUE )

Emmanuel Russ [email protected]

Page 23: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

When M = Rn, then

pt(x , y) =1

(4πt) n2

exp(−|x − y |2

4t

).

More generally, (GUE ) holds when Ric M ≥ 0 (Li-Yau).

Emmanuel Russ [email protected]

Page 24: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

(D) +(GUE ) is equivalent to a Faber-Krahn inequality: there existC , ν > 0 such that, for all ball B = B(x , r) and all smooth openΩ ⊂ B(x , r),

λ1(Ω) ≥ Cr2

(V (x , r)µ(Ω)

)ν,

where λ1(Ω) is the principal eigenvalue of ∆ on Ω under Dirichletboundary condition:

λ1(Ω) = infϕ∈D(Ω)

∫Ω|∇ϕ(x)|2 dµ(x)∫Ωϕ2(x)dµ(x)

.

Emmanuel Russ [email protected]

Page 25: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Then:

Theorem(Coulhon, Duong, 1999) Assume that (D) and (GUE) hold. Then,

‖|df |‖Lp(M) ≤ Cp

∥∥∥∆1/2f∥∥∥

Lp(M)

for all 1 < p ≤ 2 and all f .

False for p > 2, but OK for p > 2 under stronger assumptions (Auscher,Coulhon, Duong, Hofmann (2004)).

Emmanuel Russ [email protected]

Page 26: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Want an H1(M) estimate.

Define H1(M) via atoms, assuming (D). An atom is a function asupported in a ball B with

‖a‖2 ≤ V (B)−12 and

∫a(x)dµ(x) = 0.

Then f ∈ H1(M) iff f =∑

j λjaj ...

Emmanuel Russ [email protected]

Page 27: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

First, there is a H1(M)− L1(M) estimate under further assumptions. Saythat M satisfies a scaled L2-Poincaré inequality on balls if there existsC > 0 such that∫

B|f (x)− fB |2 dµ(x) ≤ Cr2

∫B|df (x)|2 dµ(x) (P)

for any ball B ⊂ M and any f ∈ C∞(2B).

Holds when Ric M ≥ 0 (Buser).

Emmanuel Russ [email protected]

Page 28: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Then, one has:

Theorem

(R, 2001) Assume that (D) and (P) hold. Then,

‖|df |‖L1(M) ≤ Cp

∥∥∥∆1/2f∥∥∥

H1(M)

for all f ∈ D(M). In other words, d∆−1/2 is H1(M)− L1(M) bounded.

Emmanuel Russ [email protected]

Page 29: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

• the assumptions of Theorem (4.2) are stronger than these ofCoulhon and Duong’s result, since (D) + (P) imply (GUE ) and also

pt(x , y) ≥ cV (x ,

√t)

exp(−C d2(x , y)

t

). (GLE )

(Saloff-Coste)• cannot replace here L1(M) by H1(M).

Emmanuel Russ [email protected]

Page 30: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

One obtains an H1(M) boundedness by some kind of “linearization”. Fixa harmonic function u on M such that

|u(x)| ≤ C(1 + d(x0, x))

for some x0 ∈ M and all x ∈ M. Then:

Theorem(Marias, R, 2003) Under assumptions (D) and (P), the operatorf 7→ du · d∆−1/2f is H1(M) bounded.

Want to say that d∆− 12 f is H1 bounded: this requires Hardy spaces of

differential forms .

Emmanuel Russ [email protected]

Page 31: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

• Strategy: use tent spaces on M, very easy to handle,• Difficulty: this involves the Hodge-de-Rham Laplacian :

∆ = (d + d∗)2 = dd∗ + d∗d ,

and the semigroup generated by ∆. Very little is known on thissemigroup.

• Everything can be done just with L2 off-diagonal estimates for thissemigroup, which hold in any complete Riemannian manifold.

• Fairly general idea, also used by Hofmann-Mayboroda for Hardyspaces in Rn associated with second order elliptic operators indivergence form.

Emmanuel Russ [email protected]

Page 32: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

We will define Hp for all 1 ≤ p ≤ +∞.

Emmanuel Russ [email protected]

Page 33: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

V. H2(ΛT ∗M)

M Riemannian manifold. For all x ∈ M, denote by ΛT ∗x M the complex

exterior algebra over the cotangent space T ∗x M. Let

ΛT ∗M = ⊕0≤k≤dim MΛkT ∗M

be the bundle over M whose fibre at each x ∈ M is given by ΛT ∗x M, and

let L2(ΛT ∗M) be the space of square integrable sections of ΛT ∗M.Furthermore:• d is the exterior differentiation,• d∗ is the adjoint of d on L2(ΛT ∗M),• D = d + d∗ is the Hodge-Dirac operator,• ∆ = D2 = dd∗ + d∗d is the Hodge-de-Rham Laplacian.

Emmanuel Russ [email protected]

Page 34: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

L2 Hodge decomposition:

L2(ΛT ∗M) = R(d)⊕R(d∗)⊕N (∆),

and the decomposition is orthogonal.

Emmanuel Russ [email protected]

Page 35: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Define

H2(ΛT ∗M) = R(D)

= Du ∈ L2(ΛT ∗M); u ∈ L2(ΛT ∗M).

Note that

L2(ΛT ∗M) = R(D)⊕N (D) = H2(ΛT ∗M)⊕N (D).

Emmanuel Russ [email protected]

Page 36: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

A description in terms of quadratic functionals: if θ ∈(0, π2

), set

Σ0θ+ = z ∈ C \ 0 ; |arg z | < θ ,

Σ0θ = Σ0

θ+ ∪(−Σ0

θ+

).

Emmanuel Russ [email protected]

Page 37: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Let H∞(Σ0θ) be the algebra of bounded holomorphic functions on Σ0

θ.

For σ, τ > 0, let

Ψσ,τ (Σ0θ) =

ψ ∈ H∞(Σ0

θ);

|ψ(z)| ≤ C inf|z |σ , |z |−τ

for some C > 0 and all z ∈ Σ0θ.

Emmanuel Russ [email protected]

Page 38: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Examples: if N, α are integers such that 1 ≤ N < α, then

ψ(z) = zN(1± iz)−α ∈ ΨN,α−N(Σ0θ).

If N, β are intergers with 1 ≤ N < 2β,

ψ(z) = zN(1 + z2)−β ∈ ΨN,2β−N(Σ0θ)

If N ∈ N, thenψ(z) = zN exp(−z2) ∈ ΨN,τ (Σ

0θ)

for all τ > 0.

Emmanuel Russ [email protected]

Page 39: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

DefineH = L2

((0,+∞), L2(ΛT ∗M),

dtt

)equipped with the norm

‖F‖H =

(∫ +∞

0

∫M|F (x , t)|2 dx dt

t

)1/2

.

Emmanuel Russ [email protected]

Page 40: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Let ψ ∈ Ψ(Σ0θ) for some θ > 0. Define the operator

Qψ : L2(ΛT ∗M) → H by

(Qψh)t = ψt(D)h , t > 0

where ψt(z) = ψ(tz).

Emmanuel Russ [email protected]

Page 41: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Since D is self-adjoint on L2(ΛT ∗M), Qψ is bounded and

‖Qψf ‖H ∼ ‖f ‖2

for all f ∈ H2(ΛT ∗M).

Emmanuel Russ [email protected]

Page 42: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Define Sψ : H → L2(ΛT ∗M) by

SψH =

∫ +∞

0ψt(D)Ht

dtt

where the limit is in the L2(ΛT ∗M) strong topology. This operator isalso bounded, since Sψ = Qψ

∗ where ψ is defined by ψ(z) = ψ(z).

Emmanuel Russ [email protected]

Page 43: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

If ψ ∈ Ψ(Σ0θ) is chosen to satisfy∫ ∞

0ψ(±t)ψ(±t)dt

t = 1,

then one has:S eψQψf = SψQ eψf = f

for all f ∈ R(D) and hence for all f ∈ H2(ΛT ∗M) (a version of Calderónreproducing formula).

Emmanuel Russ [email protected]

Page 44: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

SψQ eψ is the orthogonal projection of L2(ΛT ∗M) onto H2(ΛT ∗M). Itfollows that R(Sψ) = H2(ΛT ∗M) and that

‖f ‖2 ∼ inf ‖H‖H ; f = SψH

for all f ∈ H2(ΛT ∗M).

Emmanuel Russ [email protected]

Page 45: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Can give similar descriptions of H2(ΛT ∗M) in terms of ∆ := D2.

Emmanuel Russ [email protected]

Page 46: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

The Riesz transform on M is D∆−1/2 : H2(ΛT ∗M) → H2(ΛT ∗M). It is abounded operator.

Emmanuel Russ [email protected]

Page 47: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

SetH2

d (ΛT ∗M) = R(d), H2d∗(ΛT ∗M) = R(d∗)

so that by the Hodge decomposition

H2(ΛT ∗M) = H2d (ΛT ∗M)⊕ H2

d∗(ΛT ∗M)

and the sum is orthogonal. The orthogonal projections are given by dD−1

and d∗D−1.

Emmanuel Russ [email protected]

Page 48: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

The Riesz transform D∆−1/2 splits naturally as

D∆−1/2 = d∆−1/2 + d∗∆−1/2.

Sinced∆−1/2 = (dD−1)(D∆−1/2)

andd∗∆−1/2 = (d∗D−1)(D∆−1/2),

d∆−1/2 and d∗∆−1/2 extend to bounded operators on H2(ΛT ∗M).Furthermore,

d∆−1/2 : H2d∗(ΛT ∗M) → H2

d (ΛT ∗M),

d∗∆−1/2 : H2d (ΛT ∗M) → H2

d∗(ΛT ∗M)

are bounded and invertible and are inverse to one another.

Emmanuel Russ [email protected]

Page 49: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

VI. Hp(ΛT ∗M)

From now on, we always assume that M satisfies (D).

The definition of Hardy spaces relies on tent spaces. For all x ∈ M, define

Γ(x) = (y , t) ∈ M × (0,+∞) ; y ∈ B(x , t) .

Let F = (Ft)t>0 be a family of measurable sections of ΛT ∗M. WriteF (y , t) := Ft(y) for all y ∈ M and all t > 0 and assume that F ismeasurable on M × (0,+∞). Define then, for all x ∈ M,

SF (x) =

(∫∫Γ(x)

|F (y , t)|2 dyV (x , t)

dtt

)1/2

,

and, if 1 ≤ p < +∞, say that F ∈ T p,2(ΛT ∗M) if

‖F‖T p,2(ΛT∗M) := ‖SF‖Lp(M) < +∞.

Emmanuel Russ [email protected]

Page 50: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Let ψ ∈ Ψ(Σ0θ) for some θ > 0, set ψt(z) = ψ(tz). Recall the operators

Sψ : T 2,2(ΛT ∗M) −→ L2(ΛT ∗M) defined by

SψH =

∫ +∞

0ψt(D)Ht

dtt

and Qψ : L2(ΛT ∗M) −→ T 2,2(ΛT ∗M) by

(Qψh)t = ψt(D)h

for all h ∈ L2(ΛT ∗M) and all t > 0.

Emmanuel Russ [email protected]

Page 51: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

DefineE p

D,ψ(ΛT ∗M) = Sψ(T p,2(ΛT ∗M) ∩ T 2,2(ΛT ∗M))

with semi-norm

‖h‖HpD,ψ(ΛT∗M) = inf

‖H‖T p,2(ΛT∗M) ;

H ∈ T p,2(ΛT ∗M) ∩ T 2,2(ΛT ∗M),SψH = h.

Emmanuel Russ [email protected]

Page 52: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

This space is actually independent of ψ and can be described in terms ofthe functional Q:Lemme

Let 1 ≤ p < 2. If ψ, ψ ∈ Ψβ,2(Σ0θ) and ˜ψ ∈ Ψ1,β+1(Σ

0θ), then

E pD,ψ(ΛT ∗M) = E p

D, eψ(ΛT ∗M)

=

h ∈ H2(ΛT ∗M) ;

Qeeψh ∈ T p,2(ΛT ∗M)

with norm‖h‖Hp

D,ψ(ΛT∗M) ∼ ‖h‖HpD, eψ(ΛT∗M)

∼ ‖Qeeψh‖T p,2(ΛT∗M).

β > 0 only depends on M.

Proof: relies on off-diagonal L2 estimates.Emmanuel Russ [email protected]

Page 53: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

PropositionWith the notation of Lemma 6.1, h ∈ R(D) ; ‖Qeeψh‖T p,2(ΛT∗M) <∞

is dense in E pD,ψ(ΛT ∗M) for all ˜ψ ∈ Ψ1,β+1(Σ

0θ).

Emmanuel Russ [email protected]

Page 54: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Finally, HpD,ψ(ΛT ∗M) is the completion of E p

D,ψ(ΛT ∗M) under any of theprevious equivalent norms. Denote this space by Hp

D(ΛT ∗M).

There is a similar procedure when p > 2.

Emmanuel Russ [email protected]

Page 55: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Using even functions, one defines similarly Hp∆(ΛT ∗M), and one has

HpD(ΛT ∗M) = Hp

∆(ΛT ∗M) := Hp(ΛT ∗M) for all 1 ≤ p < +∞.

Emmanuel Russ [email protected]

Page 56: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

For all 1 ≤ p < +∞,

(Hp(ΛT ∗M))′ ∼ Hp′(ΛT ∗M).

Define H∞(ΛT ∗M) as the dual of H1(ΛT ∗M).

Emmanuel Russ [email protected]

Page 57: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Interpolation:

Theorem

Let 1 ≤ p0 < p < p1 ≤ +∞ and θ ∈ (0, 1) such that1/p = (1− θ)/p0 + θ/p1. Then[Hp0(ΛT ∗M),Hp1(ΛT ∗M)]θ = Hp(ΛT ∗M).

Emmanuel Russ [email protected]

Page 58: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Riesz transforms:

Theorem

For all 1 ≤ p ≤ +∞, the Riesz transform D∆−1/2, initially defined onR(∆), extends to a Hp(ΛT ∗M)-bounded operator. More precisely, onehas ∥∥∥D∆−1/2h

∥∥∥Hp(ΛT∗M)

∼ ‖h‖Hp(ΛT∗M) .

Emmanuel Russ [email protected]

Page 59: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

More generally, Hp(ΛT ∗M) has a functional calculus:

Theorem

For all 1 ≤ p ≤ +∞, f (D) is Hp(ΛT ∗M)-bounded for all f ∈ H∞(Σ0θ)

with ‖f (D)h‖Hp(ΛT∗M) ≤ C ‖f ‖∞ ‖h‖Hp(ΛT∗M) .

Emmanuel Russ [email protected]

Page 60: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Can define Hpd (ΛT ∗M) and Hp

d∗(ΛT ∗M), and also Hp(ΛkT ∗M), withboundedness results for d∆−1/2 and d∗∆−1/2.

Emmanuel Russ [email protected]

Page 61: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

VII. The decomposition into molcules

Corresponds to the classical atomic decomposition for Hardy spaces.• Molecules do not have compact support but suitable L2 decay,• Cancellation is replaced by the fact that a molecule is an exact form.

Emmanuel Russ [email protected]

Page 62: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Fix C > 0. If B ⊂ M is a ball with radius r and if (χk)k≥0 is a sequenceof nonnegative C∞ functions on M with bounded support, say that(χk)k≥0 is adapted to B if χ0 is supported in 4B, χk is supported in2k+2B \ 2k−1B for all k ≥ 1,∑

k≥0χk = 1 on M and ‖|∇χk |‖∞ ≤ C

2k r , (1)

where C > 0 only depends on M.

Emmanuel Russ [email protected]

Page 63: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Let N be a positive integer. If a ∈ L2(ΛT ∗M), a is called an N-moleculeif and only if there exists a ball B ⊂ M with radius r , b ∈ L2(ΛT ∗M)such that a = DNb, and a sequence (χk)k≥0 adapted to B such that, forall k ≥ 0,

‖χka‖L2(ΛT∗M) ≤ 2−kV−1/2(2kB),

‖χkb‖L2(ΛT∗M) ≤ 2−k rNV−1/2(2kB).(2)

Emmanuel Russ [email protected]

Page 64: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

For N ≥ N0 only depending on M, f ∈ H1(ΛT ∗M) iff

f =∑

jλjaj

where the aj ’s are molecules and∑

j|λj | < +∞.

Emmanuel Russ [email protected]

Page 65: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

As a corollary, we obtain:

Corollaire

(a) For 1 ≤ p ≤ 2, Hp(ΛT ∗M) ⊂ R(D) ∩ Lp(ΛT ∗M)Lp(ΛT∗M)

.

(b) For 2 ≤ p < +∞, R(D) ∩ Lp(ΛT ∗M)Lp(ΛT∗M)

⊂ Hp(ΛT ∗M).

Emmanuel Russ [email protected]

Page 66: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

VIII. The maximal characterization

If x ∈ M and 0 < r < t, let

B((x , t), r) = B(x , r)× (t − r , t + r) .

For all x ∈ M and all α > 0, set

Γα(x) = (y , t) ∈ M × (0,+∞); y ∈ B(x , αt) .

Let 0 < α. Fix c > 0 small enough such that, for all x ∈ M, whenever(y , t) ∈ Γα(x), B((y , t), ct) ⊂ Γ2α(x).

Emmanuel Russ [email protected]

Page 67: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

For f ∈ L2(ΛT ∗M) and all x ∈ M, define

f ∗α,c(x)2 =

sup(y ,t)∈Γα(x)

1tV (y , t)

∫∫B((y ,t),ct)

∣∣∣e−s2∆f (z)∣∣∣2 dzds.

Define H1max (ΛT ∗M) as the completion of

f ∈ R(D); f ∗α,c ∈ L1(M)

for that norm. This space is actually independent from α, c.

Emmanuel Russ [email protected]

Page 68: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

One has:

Theorem

Assume (D). Then H1(ΛT ∗M) = H1max (ΛT ∗M).

Emmanuel Russ [email protected]

Page 69: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

IX. Further results

Specializing to 0-forms, i.e. functions, one has:

H1d∗(Λ

0T ∗M) ⊂ H1CW (M)

with a strict inclusion in general. If one assumes furthermore (P), theconverse inclusion also holds.

Emmanuel Russ [email protected]

Page 70: Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds H2(ΛT∗M) Hp(ΛT∗M) The decomposition into molecules The maximal characterization

Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces

Hardy spaces of differential forms in RnThe case of Riemannian manifolds

H2(ΛT∗M)Hp (ΛT∗M)

The decomposition into moleculesThe maximal characterization

Further results

Assuming Gaussian upper bounds, one has

Theorem

Let 1 < p < 2. Under pointwise Gaussian upper estimates, one has

Hp(ΛT ∗M) = R(D) ∩ Lp(ΛT ∗M)Lp(ΛT∗M)

.

Emmanuel Russ [email protected]