Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds...

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 emmanuel.russ@univ-cezanne.fr

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.

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(−∆)−12 f∥∥∥

L1(Rn).

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

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.

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

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

‖F‖T p,2(Rn) := ‖SF‖Lp(Rn) < +∞.

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

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).

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

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

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).

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).

Further results

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

f := c∫ +∞

0t√−∆e−t

√−∆F (t, .)dt

one hasf ∈ H1(Rn).

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

)N‖f ‖L2(E) .

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) .

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 )

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).

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.

Further results

Theorem

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

∑jλjaj where the aj ’s are

atoms and∑

j|λj | < +∞.

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.

Further results

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

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

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

‖|df |‖Lp(M) ∼∥∥∥∆1/2f

∥∥∥Lp(M)

, 1 < p < +∞ ?

Further results

OK for p = 2, since ∥∥∥∆ 12 f∥∥∥2

2= 〈∆f , f 〉L2(M)

= ‖|df |‖22 .

Further results

What about p 6= 2 ?

A general fact: if∣∣∣d∆− 1

∣∣∣ is Lp(M)-bounded, i.e.

‖|df |‖Lp(M) ≤ Cp

∥∥∥∆1/2f∥∥∥

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.

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).

Further results

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

e−t∆f (x) =

pt(x , y)f (y)dµ(y).

Say that pt has Gaussian upper bound if

pt(x , y) ≤ CV (x ,

exp(−c ρ

2(x , y)

). (GUE )

Further results

When M = Rn, then

pt(x , y) =1

(4πt) n2

exp(−|x − y |2

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

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)

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)

Further results

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

‖|df |‖Lp(M) ≤ Cp

∥∥∥∆1/2f∥∥∥

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)).

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 ...

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).

Further results

Then, one has:

Theorem

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

‖|df |‖L1(M) ≤ Cp

∥∥∥∆1/2f∥∥∥

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

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 ,

exp(−C d2(x , y)

). (GLE )

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

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 .

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.

Further results

We will define Hp for all 1 ≤ p ≤ +∞.

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.

Further results

L2 Hodge decomposition:

L2(ΛT ∗M) = R(d)⊕R(d∗)⊕N (∆),

and the decomposition is orthogonal.

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).

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

Further results

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

For σ, τ > 0, let

Ψσ,τ (Σ0θ) =

ψ ∈ H∞(Σ0

|ψ(z)| ≤ C inf|z |σ , |z |−τ

for some C > 0 and all z ∈ Σ0θ.

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

for all τ > 0.

Further results

DefineH = L2

((0,+∞), L2(ΛT ∗M),

)equipped with the norm

‖F‖H =

(∫ +∞

∫M|F (x , t)|2 dx dt

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).

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).

Further results

Define Sψ : H → L2(ΛT ∗M) by

SψH =

∫ +∞

0ψt(D)Ht

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).

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).

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).

Further results

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

Further results

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

Further results

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.

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.

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)

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

‖F‖T p,2(ΛT∗M) := ‖SF‖Lp(M) < +∞.

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

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.

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.

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 emmanuel.russ@univ-cezanne.fr

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

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.

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 < +∞.

Further results

For all 1 ≤ p < +∞,

(Hp(ΛT ∗M))′ ∼ Hp′(ΛT ∗M).

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

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).

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) .

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) .

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.

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.

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.

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)

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 | < +∞.

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).

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).

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.

Further results

One has:

Theorem

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

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.

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)

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