Hardy spaces of differential forms on Riemannian manifolds · The case of Riemannian manifolds...
Transcript of 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 [email protected]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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).
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H2(ΛT∗M)Hp (ΛT∗M)
The decomposition into moleculesThe maximal characterization
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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).
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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
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Can give similar descriptions of H2(ΛT ∗M) in terms of ∆ := D2.
Emmanuel Russ [email protected]
Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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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.
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Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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H2(ΛT∗M)Hp (ΛT∗M)
The decomposition into moleculesThe maximal characterization
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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.
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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]
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) < +∞.
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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.
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Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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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.
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Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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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]
Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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H2(ΛT∗M)Hp (ΛT∗M)
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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]
Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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H2(ΛT∗M)Hp (ΛT∗M)
The decomposition into moleculesThe maximal characterization
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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.
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Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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H2(ΛT∗M)Hp (ΛT∗M)
The decomposition into moleculesThe maximal characterization
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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]
Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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H2(ΛT∗M)Hp (ΛT∗M)
The decomposition into moleculesThe maximal characterization
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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]
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
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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]
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]
Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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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]
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
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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.
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Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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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.
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H2(ΛT∗M)Hp (ΛT∗M)
The decomposition into moleculesThe maximal characterization
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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.
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Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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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]
Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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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 | < +∞.
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Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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H2(ΛT∗M)Hp (ΛT∗M)
The decomposition into moleculesThe maximal characterization
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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]
Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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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).
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Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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H2(ΛT∗M)Hp (ΛT∗M)
The decomposition into moleculesThe maximal characterization
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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.
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Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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H2(ΛT∗M)Hp (ΛT∗M)
The decomposition into moleculesThe maximal characterization
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One has:
Theorem
Assume (D). Then H1(ΛT ∗M) = H1max (ΛT ∗M).
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Riesz transforms and Hardy spaces in RnHardy spaces and tent spaces
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H2(ΛT∗M)Hp (ΛT∗M)
The decomposition into moleculesThe maximal characterization
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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.
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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]