Propagation of low regularity for solutions of nonlinear PDEs on a Riemannian manifold with a sub-Laplacian structure
aa r X i v : . [ m a t h . A P ] O c t PROPAGATION OF LOW REGULARITY FOR SOLUTIONS OFNONLINEAR PDES ON A RIEMANNIAN MANIFOLD WITH ASUB-LAPLACIAN STRUCTURE
FR´ED´ERIC BERNICOT AND YANNICK SIRE
Abstract.
Following [8], we introduce a notion of para-products associated to a semi-group.We do not use Fourier transform arguments and the background manifold is doubling, endowedwith a sub-laplacian structure. Our main result is a paralinearization theorem in a non-euclideanframework, with an application to the propagation of regularity for some nonlinear PDEs.
Contents
1. Preliminaries : Riemannian structure with a sub-Laplacian operator 21.1. Structure of doubling Riemannian manifold 21.2. Framework of Semigroup 41.3. Framework of sub-Laplacien 52. Examples of such situations 62.1. Laplacian operators on Carnot-Caratheodory spaces 62.2. Lie groups 62.3. Carnot groups 72.4. Riemannian manifolds with a bounded geometry 73. The scale of Sobolev spaces 74. Paraproducts associated to a semigroup 114.1. Definitions and spectral decomposition 114.2. Boundedness of paraproducts in Sobolev and Lebesgue spaces 135. Linearization Theorem 146. Propagation of low regularity for solutions of nonlinear PDEs 17References 20The theory of paradifferential calculus was introduced by Bony in [9] and developed by manyothers, particularly Meyer in [28]. This tool that arises is quite powerful in nonlinear analysis.The key idea relies on Meyer’s formula for a nonlinarity F ( f ) as M ( x, D ) f + R where F is smoothin its argument(s), f belongs to a H¨older or Sobolev space, M ( x, D ) is a pseudodifferentialoperator (depending on f ) of type (1 ,
1) and R is more regular than f and F ( f ). This operationis called the “paralinearization”.Such an approach has given many important results (or improvements of existing results):Moser estimates, elliptic regularity estimates, Kato-Ponce inequalities, ... and is the basis ofmicrolocal analysis.The notion of paradifferential operators is built on appropriate functional calculus and sym-bolic representation, available on the Euclidean space. The Fourier transform is crucial by thispoint of view to study and define the symbolic classes. That is why this approach cannot beextended to Riemannian manifolds. Date : May 28, 2018.2000
Mathematics Subject Classification. ...
However, for the last years, numerous works deal with nonlinear PDEs on manifolds. So itseems important to try to extend this tool of “paralinearization” in a non-Euclidean situation.First, on specific situations, namely on a Carnot group it is possible to define a suitable Fouriertransform, involving irreductible representations. In this context, we can also define the notionof symbols and so of pseudo-differential calculus (see the survey [5] of Bahouri, Fermanian-Kammerer and Gallagher for Heisenberg groups and [21] of Gallagher and Sire for more generalCarnot groups). Excepted this particular setting, no Fourier transform are known.Following this observation, the aim of this current work is to define another suitable notionof paralinearization on a manifold, without requiring use of Fourier transform. Since (non-linear) PDEs on a manifold usually requires vector fields, we work on a manifold having asub-Riemannian structure. To define a suitable paralinearization, we use paraproducts definedvia the heat semigroup (introduced by Bernicot in [8], independently by Frey in [19, 20] andalready used by Badr, Bernicot and Russ in [4] to get Leibniz type estimates and algebra prop-erties for Sobolev spaces) and look for a paralinearization result. However, a new phenomenomappears (due to the lack of flexibility of the method), the classical paralinearization result holdsonly for low regularity. More precisely, we prove the following (see Section 3 for Sobolev spacesand Section 4 for the definition of the paraproduct Π): let consider Sobolev spaces associated toa Sub-Laplacian operator on a Riemannian manifold then (under usual assumptions), we have
Theorem 0.1.
Consider p ∈ (1 , ∞ ) , s ∈ ( d/p, and f ∈ W s + ǫ,p for some ǫ > (as small aswe want). Then for every smooth function F ∈ C ∞ ( R ) with F (0) = 0 , (1) F ( f ) = Π F ′ ( f ) ( f ) + w with w ∈ W s − d/p,p . As in the Euclidean situation, we are able to obtain some applications concerning propagationof the regularity for solutions of nonlinear PDEs.With respect to the well-known paralinearization results, the first point is that we have onlya gain of regularity at order s − d/p − ǫ and the main difference is that this result is only provedfor s <
1. This condition can be viewed as very strong but we will explain in Remark 5.4 how toallow larger regularity s > F ). This limitation s < k ... which seems to be very difficult.1. Preliminaries : Riemannian structure with a sub-Laplacian operator
In this section, we aim to describe the framework and the required assumptions, we will useafter. Let us precise the main hypothesis about the manifold M and the operator L .1.1. Structure of doubling Riemannian manifold.
In all this paper, M denotes a completeRiemannian manifold. We write µ for the Riemannian measure on M , ∇ for the Riemanniangradient, | · | for the length on the tangent space (forgetting the subscript x for simplicity) and k · k L p for the norm on L p := L p ( M, µ ), 1 ≤ p ≤ + ∞ . We denote by B ( x, r ) the open ball ofcenter x ∈ M and radius r > The doubling property.
Definition 1.1 (Doubling property) . Let M be a Riemannian manifold. One says that M satisfies the doubling property ( D ) if there exists a constant C > , such that for all x ∈ M, r > we have ( D ) µ ( B ( x, r )) ≤ Cµ ( B ( x, r )) . ARA-DIFFERENTIAL CALCULUS ON A MANIFOLD 3
Lemma 1.2.
Let M be a Riemannian manifold satisfying ( D ) and let d := log C . Then for all x, y ∈ M and θ ≥ µ ( B ( x, θR )) ≤ Cθ d µ ( B ( x, R )) There also exists c and N ≥ , so that for all x, y ∈ M and r > µ ( B ( y, r )) ≤ c (cid:18) d ( x, y ) r (cid:19) N µ ( B ( x, r )) . For example, if M is the Euclidean space M = R d then N = 0 and c = 1.Observe that if M satisfies ( D ) thendiam( M ) < ∞ ⇔ µ ( M ) < ∞ (see [1]) . Therefore if M is a complete Riemannian manifold satisfying ( D ) then µ ( M ) = ∞ . Theorem 1.3 (Maximal theorem) . ( [11] ) Let M be a Riemannian manifold satisfying ( D ) .Denote by M the uncentered Hardy-Littlewood maximal function over open balls of M definedby M f ( x ) := sup Q ball x ∈ Q | f | Q where f E := − Z E f dµ := 1 µ ( E ) Z E f dµ. Then for every p ∈ (1 , ∞ ] , M is L p bounded and moreoverof weak type (1 , .Consequently for s ∈ (0 , ∞ ) , the operator M s defined by M s f ( x ) := [ M ( | f | s )( x )] /s is of weak type ( s, s ) and L p bounded for all p ∈ ( s, ∞ ] . Doubling property allows us to control the growth of ball-volumes. However, it can be inter-esting to have a lower-bound too. So we will make this following assumption:
Assumption 1.4.
We assume that there exists a constant c > such that for all x ∈ M (4) µ ( B ( x, ≥ c. Due to the homogeneous type of the manifold M , this is equivalent to a below control of thevolume ( M V d )( M V d ) µ ( B ( x, r )) & r d for all < r ≤ . Poincar´e inequality.
Definition 1.5 (Poincar´e inequality on M ) . We say that a complete Riemannian manifold M admits a Poincar´e inequality ( P q ) for some q ∈ [1 , ∞ ) if there exists a constant C > suchthat, for every function f ∈ W ,qloc ( M ) (the set of compactly supported Lipschitz functions on M )and every ball Q of M of radius r > , we have ( P q ) (cid:18) − Z Q | f − f Q | q dµ (cid:19) /q ≤ Cr (cid:18) − Z Q |∇ f | q dµ (cid:19) /q . Remark 1.6.
By density of C ∞ ( M ) in W ,qloc ( M ) , we can replace W ,qloc ( M ) by C ∞ ( M ) . Let us recall some known facts about Poincar´e inequalities with varying q .It is known that ( P q ) implies ( P p ) when p ≥ q (see [24]). Thus, if the set of q such that ( P q )holds is not empty, then it is an interval unbounded on the right. A recent result of S. Keithand X. Zhong (see [26]) asserts that this interval is open in [1 , + ∞ [ : FR´ED´ERIC BERNICOT AND YANNICK SIRE
Theorem 1.7.
Let ( X, d, µ ) be a complete metric-measure space with µ doubling and admittinga Poincar´e inequality ( P q ) , for some < q < ∞ . Then there exists ǫ > such that ( X, d, µ ) admits ( P p ) for every p > q − ǫ . Assumption 1.8.
We assume that the considered manifold satisfies a Poincar´e inequality ( P ) .Indeed we could just assume a Poincar´e inequality ( P σ ) for some σ < and all of our resultswill remain true for Lebesgue exponents bigger than σ Framework of Semigroup.
Let us recall the framework of [17, 18].Let ω ∈ [0 , π/ C by S ω := { z ∈ C , | arg( z ) | ≤ ω } ∪ { } and denote the interior of S ω by S ω . We set H ∞ ( S ω ) for the set of bounded holomorphicfunctions b on S ω , equipped with the norm k b k H ∞ ( S ω ) := k b k L ∞ ( S ω ) . Then consider a linear operator L . It is said of type ω if its spectrum σ ( L ) ⊂ S ω and for each ν > ω , there exists a constant c ν such that (cid:13)(cid:13) ( L − λ ) − (cid:13)(cid:13) L → L ≤ c ν | λ | − for all λ / ∈ S ν .We refer the reader to [17] and [27] for more details concerning holomorphic calculus of suchoperators. In particular, it is well-known that L generates a holomorphic semigroup ( A z := e − zL ) z ∈ S π/ − ω . Let us detail now some assumptions, we made on the semigroup.Assume the following conditions: there exists a positive real m > δ > Assumption 1.9. • For every z ∈ S π/ − ω , the linear operator A z := e − zL is given by akernel a z satisfying (5) | a z ( x, y ) | . µ ( B ( x, | z | / )) (cid:18) d ( x, y ) | z | / (cid:19) − d − N − δ where d is the homogeneous dimension of the space (see (2)) and N is the other dimensionparameter (see (3)); N ≥ could be equal to . • The operator L has a bounded H ∞ -calculus on L . That is, there exists c ν such that for b ∈ H ∞ ( S ν ) , we can define b ( L ) as a L -bounded linear operator and (6) k b ( L ) k L → L ≤ c ν k b k ∞ . • The Riesz transform R := ∇ L − / is bounded on L p for every p ∈ (1 , ∞ ) . Remark 1.10.
The assumed bounded H ∞ -calculus on L allows us to deduce some extra prop-erties (see [18] and [27] ) : • Due to the Cauchy formula for complex differentiation, pointwise estimate (5) still holdsfor the kernel of ( tL ) k e − tL with t > . • For any holomorphic function ψ ∈ H ( S ν ) such that for some s > , | ψ ( z ) | . | z | s | z | s , thequadratic functional f → (cid:18)Z ∞ | ψ ( tL ) f | dtt (cid:19) / is L -bounded. Remark 1.11.
Concerning a square estimate on the gradient of the semigroup, it follows thatfor every integer k ≥ the square functional (7) f → (cid:18)Z ∞ (cid:12)(cid:12)(cid:12) t / ∇ ( tL ) k e − tL ( f ) (cid:12)(cid:12)(cid:12) dtt (cid:19) / is bounded on L . Indeed, this is just a direct consequence of the boundedness of the previoussquare functions, by making appear the Riesz transform and uses its L -boundedness. Remark 1.12.
We claim that Assumption (7) is satisfied under the L -boundedness of the Riesztransform R := ∇ L − / .Indeed if R is L -bounded, then it admits L -valued estimates, which yield (cid:18)Z ∞ (cid:12)(cid:12)(cid:12) R ( tL ) k +1 / e − tL ( f ) (cid:12)(cid:12)(cid:12) dtdµt (cid:19) / ≤ kRk L → L (cid:18)Z ∞ (cid:12)(cid:12)(cid:12) ( tL ) k +1 / e − tL ( f ) (cid:12)(cid:12)(cid:12) dtdµt (cid:19) / . This gives the desired result (cid:18)Z ∞ (cid:12)(cid:12)(cid:12) t / ∇ ( tL ) k e − tL ( f ) (cid:12)(cid:12)(cid:12) dtdµt (cid:19) / . k f k L , thanks to Remark 1.10. About different square functions, we have the following proposition:
Proposition 1.13.
Under these assumptions, we know that the square functionals in Remark1.10 or in (7) are L p -bounded for every p ∈ (1 , ∞ ) .Proof. Let T be one of the square functions in Remark 1.10. We also already know that it is L bounded, by holomorphic functional calculus. Then consider the “oscillation operator” at thescale t : B t := 1 − A t = 1 − e − tL = − Z t Le − sL ds. Then, by using differentiation of the semigroup, it is classical that
T B t satisfies L − L off-diagonal decay at the scale t /m , since the semigroup e − tL is bounded by Hardy Littlewoodmaximal function M s − . So we can apply interpolation theory (see [7] for a very general exposi-tion of such arguments) and prove that T is bounded on L p for every p ∈ ( s − ,
2] (and then for p ∈ [2 , ∞ ) by applying a similar reasoning with the dual operators).Then consider a square function U of type (7). Then by using the Riesz transform, it yields U ( f ) = (cid:18)Z ∞ |R ψ ( tL ) f | dtt (cid:19) / with ψ ( z ) = z /m φ ( z ). Since R is supposed to be L p -bounded, it verifies ℓ -valued inequalitiesand so the L p -boundedness of U is reduced to the one of a square functional of previous type,which was before proved. (cid:3) Framework of sub-Laplacien.
We will only consider operators L which are sub-Laplaciensand generating a semigroup ( e − tL ) t> satisfying the above assumptions. Let us first explain whata sub-Laplacian means :We assume that there exists X = { X k } k =1 ,...,κ a finite family of real-valued vector fields (so X k is defined on M and X k ( x ) ∈ T M x ) such that(8) L = − κ X k =1 X k . We identify the X k ’s with the first order differential operators acting on Lipschitz functionsdefined on M by the formula X k f ( x ) = X k ( x ) · ∇ f ( x ) , and we set Xf = ( X f, X f, · · · , X κ f ) and | Xf ( x ) | = κ X k =1 | X k f ( x ) | ! / , x ∈ M. FR´ED´ERIC BERNICOT AND YANNICK SIRE
We define also the higher-order differential operators as follows : for I ⊂ { , ..., κ } k , we set X I := Y i ∈ I X i . We assume the following:
Assumption 1.14.
For every subset I , the I th-local Riesz transform R I := X I (1 + L ) −| I | / andits adjoint R ∗ I := (1 + L ) −| I | / X I are bounded on L p for every p ∈ (1 , ∞ ) . Remark 1.15.
It is easy to check that this last assumption is implied by the boundedness ofeach local-Riesz transform R i and R ∗ i in Sobolev spaces W k,p for every p ∈ (1 , ∞ ) and k ∈ N .Indeed for I = { i , ..., i n } , we have kR ∗ I f k L p ≤ kR ∗ i ( X i ...X i n f ) k W | I |− ,p . kR ∗ i ,...,i n f k L p . Repeating this reasoning, we obtain that the Sobolev boundedness of the Riesz transforms and ofits adjoint implies the previous Assumption.
From now on, we will consider a doubling Riemannian manifold M satisfying Poincar´e in-equality ( P ), lower bound of the volume Assumption (1.4) and a structure of sub-Riemannianlaplacian associated to a semigroup satisfying Assumption (1.9) and with bounded Riesz trans-forms (Assumption (1.14)). 2. Examples of such situations
In this section, we would like to give two examples of situations where all these assumptionsare satisfied.2.1.
Laplacian operators on Carnot-Caratheodory spaces.
Let Ω be an open connectedsubset of R d and Y = { Y k } κk =1 a family of real-valued, infinitely differentiable vector fields. Definition 2.1.
Let Ω and Y be as above. Y is said to satisfy H¨ormander’s condition in Ω ifthe family of commutators of vector fields in Y ( Y i , [ Y i , Y j ] , ....) span R d at every point of Ω . Suppose that Y = { Y k } Mk =1 satisfies H¨ormander’s condition in Ω . Let C Y be the family ofabsolutely continuous curves ζ : [ a, b ] → Ω , a ≤ b, such that there exist measurable functions c j ( t ) , a ≤ t ≤ b, j = 1 , · · · , M, satisfying P Mj =1 c j ( t ) ≤ ζ ′ ( t ) = P Mj =1 c j ( t ) Y j ( ζ ( t )) foralmost every t ∈ [ a, b ] . If x, y ∈ Ω define ρ ( x, y ) = inf { T > ζ ∈ C Y with ζ (0) = x and ζ ( T ) = 1 } . The function ρ is in fact a metric in Ω called the Carnot-Carath´eodory metric associated to Y. This allows us to equipped the space Ω of a sub-Riemannian structure.2.2.
Lie groups.
Let M = G be a unimodular connected Lie group endowed with its Haarmeasure dµ = dx and assume that it has a polynomial volume growth. Recall that “unimodular”means that dx is both left-invariant and right-invariant. Denote by L the Lie algebra of G .Consider a family X = { X ..., X κ } of left-invariant vector fields on G satisfying the H¨ormandercondition, which means that the Lie algebra generated by the X i ’s is L . By “left-invariant,”one means that, for any g ∈ G and any f ∈ C ∞ ( G ), X ( τ g f ) = τ g ( Xf ), where τ g is the left-translation operator. As previously, we can build the Carnot-Carath´eodory metric on G . Theleft invariance of the X i ’s implies the left-invariance of the distance d . So that for every r ,the volume of the ball B ( x, r ) does not depend on x ∈ G and also will be denoted V ( r ). It iswell-known (see [23, 29]) that ( G, d ) is then a space of homogeneous type. Particular case areCarnot groups, where the vector fields are given by a Jacobian basis of its Lie algebra and satisfyH¨ormander condition. In this situation, two cases may occur : either the manifold is doublingor the volume of the balls admit an exponential growth [23]. For example, nilpotents Lie groupssatisfies the doubling property ([16]).
ARA-DIFFERENTIAL CALCULUS ON A MANIFOLD 7
We refer the reader to [30, Thm5.14] and [14, Section 3, Appendix 1] where properties ofthe heat semigroup are studied: in particular the heat semigroup e − tL satisfies gaussian upper-bounds and Assumption 1.9 on the higher-order Riesz transforms (Assumption (1.14) is satisfiedtoo.2.3. Carnot groups.
Particular cases of nilpotents Lie groups are the Carnot groups (if itadmits a stratification). A stratification on a Lie group G (whose g is its Lie algebra) is acollection of linear subspaces V , ..., V r of G such that g = V ⊕ ... ⊕ V r which satisfy [ V , V i ] = V i +1 for i = 1 , ..., r − V , V r ] = 0. By [ V , V i ] we mean the subspaceof G generated by the elements [ X, Y ] where X ∈ V and Y ∈ V i . Consider n i the dimension of V i , n + · · · + n r = d and dilations { δ λ } λ> of the form δ λ ( x ) = ( λ x (1) , λ x (2) , · · · , λ s x ( r ) ) , x ( i ) ∈ V i . The couple G = ( G, δ λ ) is said to be a homogeneous Carnot group (of step r and n generators)if δ λ is an automorphism of G for every λ > n elements of the Jacobian basisof g , say Z , · · · , Z n , satisfy(9) rank(Lie[ Z , · · · , Z n ]( x )) = d, for all x ∈ G, where Lie[ Z , · · · , Z n ] is the Lie algebra generated by the vector fields Z , · · · Z n . The number Q = P ri =1 i n i is called the homogeneous dimension of G .As for example the different Heisenberg groups, H d is a Carnot group of dimension Q = 2 d +2.We refer the reader to [21] for an introduction of pseudodifferential operators in this contextusing a kind of Fourier transforms involving irreductible representations (and to [5] for a completework about pseudo-differential calculus on Heisenberg groups).2.4. Riemannian manifolds with a bounded geometry.
We shall say that a Riemannianmanifold M has a bounded geometry if • the curvature tensor and all its derivatives are bounded • Ricci curvature is bounded from below • and M has a positive injectivity radius.In such situations, we know that there exists a collection of smooth vector fields X , ..., X κ such that ∆ = − κ X i =1 X i . Moreover Assumptions 1.9 and 1.14 are satisfied (see [14] and [34]).3.
The scale of Sobolev spaces
We use the Bessel-type Sobolev spaces, adapted to the operator L : Definition 3.1.
For p ∈ (1 , ∞ ) and s ≥ , we set W s,p = W s,pL := n f ∈ L p , (1 + L ) s/ ( f ) ∈ L p o . First, we have this characterization:
Proposition 3.2.
For all p ∈ (1 , ∞ ) and s > , we have the following equivalence k f k L p + k L s/ ( f ) k L p ≃ k (1 + L ) s/ f k L p . FR´ED´ERIC BERNICOT AND YANNICK SIRE
Proof.
Set α = s/ α = k + θ with k ∈ N and θ ∈ [0 , L ) α withthe semigroup as following(1 + L ) α f = Z ∞ e − t e − tL (1 + L ) t − θ dtt (1 + L ) k ( f )= Z ∞ e − t e − tL t − θ dtt (1 + L ) k ( f ) + Z ∞ e − t e − tL ( tL ) − θ dtt L θ (1 + L ) k ( f ) . The first integral-operator is easily bounded on L p since the semigroup e − tL is uniformlybounded. The second integral operator is bounded using duality: h Z ∞ e − t e − tL ( tL ) − θ ( u ) dtt , g i = Z ∞ e − t h e − tL/ ( tL ) − θ ( u ) , e − tL ∗ / ( tL ∗ ) − θ g i dtt ≤ Z (cid:18)Z ∞ (cid:12)(cid:12)(cid:12) e − tL/ ( tL ) − θ ( u ) (cid:12)(cid:12)(cid:12) dtt (cid:19) / (cid:18)Z ∞ (cid:12)(cid:12)(cid:12) e − tL ∗ / ( tL ∗ ) − θ ( g ) (cid:12)(cid:12)(cid:12) dtt (cid:19) / dµ. Since (1 − α ) / >
0, then the two square functionals are bounded in L p and L p ′ (by Proposition1.13) and that concludes the proof of k (1 + L ) α f k L p . k (1 + L ) k f k L p + k L θ (1 + L ) k ( f ) k L p . Then, developing (1 + L ) k , it follows a finite sum of k L z ( f ) k L p with z ∈ [0 , α ]. We decompose L z ( f ) = Z ∞ e − tL ( tL ) α ( f ) t − z dtt = Z e − tL ( tL ) α ( f ) t − z dtt + Z ∞ e − tL ( tL ) α t − z dtt . The first quantity in L p is controlled by k L α ( f ) k L p and the second one by k f k L p , which concludesthe proof of k (1 + L ) s/ f k L p . k f k L p + k L s/ ( f ) k L p . Let us now check the reverse inequality. As previously, for u = 0 or u = α we write L u f = Z ∞ e − t (1+ L ) (1 + L ) L u t α dtt (1 + L ) α f. By producing similar arguments as above, the operator R ∞ e − t (1+ L ) (1 + L ) L u t α dtt is easilybounded on L p (splitting the integral for t ≤ t ≥
1) and we can also conclude to k L u ( f ) k L p . k (1 + L ) α f k L p , which ends the proof. (cid:3) Corollary 3.3.
For all p ∈ (1 , ∞ ) and ≤ t ≤ s , we have the following inequality k L t f k L p . k (1 + L ) t f k L p ≃ k f k W t,p . Let us then describe classical Sobolev embeddings in this setting (see [4] for a more generalframework):
Proposition 3.4.
Under Assumption 1.4 (lower bound on the ball-volumes), let s ≥ t ≥ befixed and take p ≤ q such that q − td > p − sd . Then, we have the continuous embedding W s,p ֒ → W t,q . We refer the reader to [4, Proposition 3.3] for a precise proof. The proof is based on aspectral decomposition, to write the resolvant with the semigroup and then to use the off-diagonal estimates (here the pointwise estimates on the heat kernel).
Corollary 3.5.
Under the previous assumption, W s,p ֒ → L ∞ as soon as s > dp . ARA-DIFFERENTIAL CALCULUS ON A MANIFOLD 9
We now recall a result of [4], where a characterization of Sobolev spaces is obtained, involvingsome fractional functionals.
Proposition 3.6. [4, Thm 5.2]
Under Poincar´e inequality ( P ) , for s ∈ (0 , we have thefollowing characterization : a function f ∈ L p belongs to W s,p if and only if S ρ,locs f ( x ) = Z r s µ ( B ( x, r )) Z B ( x,r ) | f ( y ) − f ( x ) | ρ dµ ( y ) ! /ρ drr belong to L p , for some ρ < min(2 , p ) . This characterization can be extended for s >
1, using the sub-Laplacian structure. Indeed,we have this first Lemma:
Lemma 3.7.
For every integer k and p ∈ (1 , ∞ ) , k f k W k,p ≃ X I ⊂{ ,...,κ } k k X I ( f ) k L p . Proof.
As point out in [14], this is consequence of Assumption 1.14 about the local Riesz trans-forms. Indeed, for k ≥ I a subset, we have assumed that the I -th Riesz transform R I arebounded on L p , which is equivalent to k X I ( f ) k L p . k f k W | I | / ,p . Moreover, making appear the adjoint of the Riesz transforms and the resolvant (which are allbounded on L p ) as follows(1 + L ) / = (1 + L ) − / (1 + L ) = (1 + L ) − / + κ X i =1 (1 + L ) − / X i = (1 + L ) − / + κ X i =1 R ∗ i X i , we conclude to the reverse inequality and so we have proved the desired result for k = 1. Welet the details for k ≥ (cid:3) We also deduce the following characterization (see Proposition 19 in [14]):
Proposition 3.8.
Let s := k + t > (with k an integer and t ∈ (0 , ), then f ∈ W s,p ⇐⇒ f ∈ L p and ∀ I ⊂ { , ..., κ } k , X I ( f ) ∈ W t,p ⇐⇒ f ∈ L p and ∀ I ⊂ { , ..., κ } k , S ρt ( X I ( f )) ∈ L p . We also deduce the following chain rule (see Theorem 22 in [14] for a proof by induction on k ): Proposition 3.9. If F ∈ C ∞ with F (0) = 0 and let s := k + t > dp (with p ∈ (1 , ∞ ) , k aninteger and t ∈ (0 , ). Then k F ( f ) k W s,p . k f k W s,p + k f k kW s,p . If F (0) = 0 , we still have such inequalities with localized Sobolev spaces. We refer the reader to [14] for a proof by induction on k . Here for completeness, we produceanother direct proof. Proof.
We use the previous characterization of the Sobolev space with S ρt for s = k + t . First usingthe differentiation rule, it comes X i ( F ( f )) = X i ( f ) F ′ ( f ), then X j X i ( F ( f )) = X j X i ( f ) F ′ ( f ) + X i ( f ) X j ( f ) F ′′ ( f ) ... By iterating the reasoning, for I ⊂ { , ..., κ } k , estimating X I ( F ( f )) in W t,p is reduced to estimate quantities as h := " l Y α =1 X i α ( f ) F ( n ) ( f ) where i α ⊂ I , n ≤ k and P | i α | = | I | ≤ k . Then for x, y , we have | h ( x ) − h ( y ) | ≤ X β | X i β ( f )( x ) − X i β ( f )( y ) | Y α = β sup z = x,y | X i α ( f )( z ) |k F ( n ) ( f ) k L ∞ + Y α sup z = x,y | X i α ( f )( z ) || F ( n ) ( f )( x ) − F ( n ) ( f )( y ) | . By this way, since ρ ≤ p let us choose exponents ρ α , p α such that1 ρ = X α ρ α , ρ α ≤ p α and 1 p = X α p α . Moreover we require that(10) 1 p α − | i α | + td > p − sd . This is possible since P α | i α | = | I | ≤ s − t and s > d/p (indeed we let the reader to check that p α = | I | + t | i α | + t p is a good choice). Moreover, we chose exponents ρ α , ρ , p α and p such that1 ρ = X α ρ α + 1 ρ , ρ α ≤ p α and ρ ≤ ρ ≤ p with 1 p = X α p α + 1 p . As previously, we require (10) with p α instead of p α and(11) 1 p > p − sd . Such exponents can be chosen by perturbing the previous construction with a small parametersince s > d/p . By this way (with H¨older inequality), we deduce that S ρt ( h ) . X β S ρ β t ( X i β ( f )) Y α = β M ρ α [ X i α ( f )] k F ( n ) ( f ) k L ∞ + Y α M ρ α [ X i α ( f )] S ρt ( F ( n ) ( f ) . Since F is supposed to be bounded in C ∞ , then F ( n ) is Lipschitz and so, we finally obtain S ρt ( h ) . X β S ρ β t ( X i β ( f )) Y α = β M ρ α [ X i α ( f )] k F ( n ) ( f ) k L ∞ + Y α M ρ α [ X i α ( f )] S ρt ( f ) . Then applying H¨older inequality, we get k S ρt ( h ) k L p . X β (cid:13)(cid:13) S ρ β t ( X i β ( f )) (cid:13)(cid:13) L pβ Y α = β kM ρ α [ X i α ( f )] k L pα k F ( n ) ( f ) k L ∞ + Y α kM ρ α [ X i α ( f )] k L pα (cid:13)(cid:13)(cid:13) S ρt ( f ) (cid:13)(cid:13)(cid:13) L p . Since (10) with Sobolev embeddings (Proposition 3.4), we have kM ρ α [ X i α ( f )] k L pα . k f k W | iα | ,pα . k f k W s,p and (cid:13)(cid:13) S ρ β t ( X i β ( f )) (cid:13)(cid:13) L pβ . k f k W | iβ | + t,pβ . k f k W s,p . ARA-DIFFERENTIAL CALCULUS ON A MANIFOLD 11
So with (10) and (11), we finally obtain k S ρt ( h ) k L p . k f k W s,p + k f k kW s,p , where we used s > d/p and the Sobolev embedding W s,p ⊂ L ∞ with the smoothness of F tocontrol k F ( n ) ( f ) k L ∞ .Since F (0) = 0 and F is Lipschitz, we also deduce that F ( f ) belongs to L p , which allows usto get the expected result k F ( f ) k W s,p . k F ( f ) k L p + X I k S ρt ( F ( f )) k L p . k f k W s,p + k f k kW s,p . (cid:3) Remark 3.10. If F ∈ C ∞ with F (0) = 0 and s > d/p then to obtain k F ( f ) k W s,p . k f k W s,p + k f k kW s,p , it is sufficient to assume that F is locally bounded in C ∞ and then the implicit constant willdepend on k f k L ∞ . Indeed, using Sobolev embedding, we know that as soon as s > d/p , W s,p iscontinuously embedded in L ∞ . Paraproducts associated to a semigroup
Our aim is to describe a kind of “paralinearization” results. In the Euclidean case, this isperformed by using paraproducts (defined with the help of Fourier transform). Here, we cannotuse such powerful tools, so we require other kind of paraproducts, defined in terms of semigroup.These ones were introduced by the first author in [8], already used in [4] and more recently wasextended in [19, 20]. Let us recall these definitions.4.1.
Definitions and spectral decomposition.
We consider a sub-Laplacian operator L sat-isfying the assumptions of the previous sections. We write for convenience c for a suitablychosen constant, ψ ( x ) = c x N e − x (1 − e − x ) and so ψ t ( L ) := c ( tL ) N e − tL (1 − e − tL ) , with a large enough integer N > d/
2. Let φ be the function φ ( x ) := − c Z ∞ x y N e − y (1 − e − y ) dy, e φ ( x ) := − c Z ∞ x y N − e − y (1 − e − y ) dy, and set φ t ( L ) := φ ( tL ). Then we get a “spectral” decomposition of the identity as follows(choosing the appropriate constant c ), we have f = − Z ∞ φ ′ ( tL ) f dtt . So for two smooth functions, we have f g := − Z s,u,v> φ ′ ( sL ) (cid:2) φ ′ ( uL ) f φ ′ ( vL ) g (cid:3) dsdudvsuv . Since φ ′ ( x ) = ψ ( x ) := c x N e − x (1 − e − x ) and x e φ ′ ( x ) = φ ′ ( x ), it comes that (by integratingaccording to t := min { s, u, v } ) f g := − Z ∞ ψ ( tL ) h e φ ( tL ) f e φ ( tL ) g i dtt − Z ∞ e φ ( tL ) h ψ ( tL ) f e φ ( tL ) g i dtt − Z ∞ e φ ( tL ) h e φ ( tL ) f ψ ( tL ) g i dtt . (12) Let us now focus on the first term in (12) : I ( f, g ) = Z ∞ ψ ( tL ) h e φ ( tL ) f e φ ( tL ) g i dtt . Since
N >>
1, let us write ψ ( z ) = z ˜ ψ with ˜ ψ (still vanishing at 0 and at infinity). Then usingthe structure of the sub-Laplacian L , the following algebra rule holds L ( f g ) = L ( f ) g + f L ( g ) + h Xf · Xg i , where X is the collection of vector fields Xf := ( X f, ..., X κ f ). Hence, we get I ( f, g ) = Z ∞ ˜ ψ ( tL )( tL ) h e φ ( tL ) f e φ ( tL ) g i dtt = Z ∞ ˜ ψ ( tL ) h tL e φ ( tL ) f e φ ( tL ) g i dtt + Z ∞ ˜ ψ ( tL ) h e φ ( tL ) f tL e φ ( tL ) g i dtt + Z ∞ ˜ ψ ( tL ) t h X e φ ( tL ) f · X e φ ( tL ) g i dtt . Combining with (12), we define the paraproduct as follows :
Definition 4.1.
With the previous notations, we define the paraproduct of f by g , by Π g ( f ) := − Z ∞ ˜ ψ ( tL ) h tL e φ ( tL ) f e φ ( tL ) g i dtt − Z ∞ e φ ( tL ) h ψ ( tL ) f e φ ( tL ) g i dtt . Remark 4.2.
We first want to point out the difference with the initial definition in [8] . There,general semigroup was considered and the previous operation on the term I can be performed bymaking appear the “carr´e du champ” introduced by Bakry and ´Emery (see [6] for details) Γ( f, g ) := L ( f g ) − L ( f ) g − f L ( g ) instead of the vector field X . However in [8] , the paraproducts was only defined by the secondterm. This new definition comes from the following observation: considering the quantity I ( f, g ) and distributing the Laplacian as we have done (or make appearing the “carr´e du champ”),it comes three terms. The term ˜ ψ ( tL ) h tL e φ ( tL ) f e φ ( tL ) g i has the same regularity propertiesas e φ ( tL ) h ψ ( tL ) f e φ ( tL ) g i (in the sense that tL e φ ( tL ) can be considered as ψ ( tL ) ). This alsolegitimate to add this extra term in the definition of the paraproducts.By this way, as we will see in the next properties and in Remark 5.4, this new paraproduct isthe “maximal” (in a certain sense) part of the product f g , where the regularity is given by theregularity of f . It naturally comes the following decomposition :
Corollary 4.3.
Let f, g be two smooth functions, then we have f g = Π g ( f ) + Π f ( g ) + Rest ( f, g ) where the “rest” is given by Rest ( f, g ) := − Z ∞ ˜ ψ ( tL ) h t / X e φ ( tL ) f, t / X e φ ( tL ) g i dtt . ARA-DIFFERENTIAL CALCULUS ON A MANIFOLD 13
Boundedness of paraproducts in Sobolev and Lebesgue spaces.
Concerning esti-mates on these paraproducts in Lebesgue spaces, we refer to [8]:
Theorem 4.4 (Boundedness in Lebesgue spaces) . For p, q ∈ (1 , ∞ ] with < r := p + q then ( f, g ) → Π g ( f ) is bounded from L p × L q into L r . Let us now describe boundedness in the scale of Sobolev spaces.
Theorem 4.5 (Boundedness in Sobolev spaces) . For p, q, r ∈ (1 , ∞ ) with r := p + q and s ∈ (0; 2 N − then ( f, g ) → Π g ( f ) is bounded from W s,p × L q into W s,r .Proof. It is sufficient to prove the following homogeneous estimates : for every β ∈ [0 , N − k L β Π g ( f ) k L r . k L β ( f ) k L p k g k L q . For β = 0, this is the previous theorem so it remains us to check it for β ∈ (0 , N − g ( f ) = − Z ∞ ˜ ψ ( tL ) h tL e φ ( tL ) f e φ ( tL ) g i dtt − Z ∞ e φ ( tL ) h ψ ( tL ) f e φ ( tL ) g i dtt , giving rise to two quantities, Π g ( f ) and Π g ( f ). Indeed, applying L β to the paraproduct Π g ( f ),it yields L β Π g ( f ) = Z ∞ L β e φ ( tL ) h ψ ( tL ) f e φ ( tL ) g i dtt = Z ∞ ψ ( tL ) h t − β ψ ( tL ) f e φ ( tL ) g i dtt = Z ∞ ψ ( tL ) h e ψ ( tL ) L β f e φ ( tL ) g i dtt , where we set ψ ( z ) = z β φ ( z ) and e ψ ( z ) = z − β ψ ( z ). So if the integer N in φ and ψ is takensufficiently large, then ψ and e ψ are still holomorphic functions with vanishing properties at 0and at infinity. As a consequence, we get L β Π g ( f ) = Π g ( L β f )with the new paraproduct Π built with ψ and e ψ . We also apply the classical reasoning aimingto estimate this paraproduct. By duality, for any smooth function h ∈ L r ′ we have h L β Π g ( f ) , h i = Z Z ∞ ψ ( tL ∗ ) h e ψ ( tL )( L β f ) e φ ( tL ) g dtt dµ ≤ Z (cid:18)Z ∞ | ψ ( tL ∗ ) h | dtt (cid:19) / (cid:18)Z ∞ | e ψ ( tL )( L β f ) | dtt (cid:19) / sup t | e φ ( tL ) g | dµ. From the pointwise decay on the semigroup (5), we know thatsup t | e φ ( tL ) g ( x ) | ≤ M ( g )( x )and so by H¨older inequality (cid:12)(cid:12)(cid:12) h L β Π g ( f ) , h i (cid:12)(cid:12)(cid:12) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ | ψ ( tL ∗ ) h | dtt (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L r ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ | e ψ ( tL )( L β f ) | dtt (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p kM g k L q . Since ψ and e ψ are holomorphic functions vanish at 0 and having fast decays at infinity, we knowfrom (1.13) that the two square functions are bounded on Lebesgue spaces. We also concludethe proof by duality, since it comes (cid:12)(cid:12)(cid:12) h L β Π g ( f ) , h i (cid:12)(cid:12)(cid:12) . k h k L r ′ (cid:13)(cid:13)(cid:13) L β f (cid:13)(cid:13)(cid:13) L p k g k L q . We let the reader to check that the same arguments still holds for the first part Π g ( f ) and sothe proof is also finished. (cid:3) Linearization Theorem
Theorem 5.1.
Consider s ∈ ( d/p, and f ∈ W s + ǫ,p for some ǫ > . Then for every smoothfunction F ∈ C ∞ ( R ) with F (0) = 0 , (13) F ( f ) = Π F ′ ( f ) ( f ) + w with w ∈ W s − d/p,p . We follow the proof in [10, 28, 9].
Proof.
Let us refer the reader to the operators φ ( tL ) and ψ ( tL ), defined in Subsection 4.1: ψ ( x ) = c x N e − x (1 − e − x ), φ is its primitive vanishing at infinity. Let us write e ψ ( z ) = z − ψ ( z )and e φ its primitive vanishing at infinity. Moreover, these functions are normalized by the suitableconstant c such that e φ (0) = 1.It comes f = lim t → e φ ( tL )( f )and so we decompose F ( f ) = e φ ( L ) F ( e φ ( L ) f ) − Z ddt e φ ( tL ) F ( e φ ( tL ) f ) dt. Since t ddt e φ ( tL ) F ( e φ ( tL ) f ) = tL e φ ′ ( tL ) F ( e φ ( tL ) f ) + e φ ( tL ) h ( tL e φ ′ ( tL ) f ) F ′ ( e φ ( tL ) f ) i = φ ′ ( tL ) F ( e φ ( tL ) f ) + e φ ( tL ) h ( φ ′ ( tL ) f ) F ′ ( e φ ( tL ) f ) i , we get F ( f ) = e φ ( L ) F ( e φ ( L ) f ) − Z e φ ′ ( tL ) tL [ F ( e φ ( tL ) f )] + e φ ( tL ) h ( φ ′ ( tL ) f ) F ′ ( e φ ( tL ) f ) i dtt = e φ ( L ) F ( e φ ( L ) f ) − Z e φ ′ ( tL ) h F ′′ ( e φ ( tL ) f ) | t / X e φ ( tL ) f | + F ′ ( e φ ( tL ) f ) tL e φ ( tL ) f i + e φ ( tL ) h ( φ ′ ( tL ) f ) F ′ ( e φ ( tL ) f i dtt , where we used the differentiation rule for the composition with the vector fields X = ( X , ..., X κ ).We also set w := I + II + III + IV + V with I := e φ ( L ) F ( e φ ( L ) f ) ,II := − Z e φ ′ ( tL ) h F ′′ ( e φ ( tL ) f ) | t / X e φ ( tL ) f | i dtt ,III := Z e φ ′ ( tL ) h(cid:16) e φ ( tL ) F ′ ( f ) − F ′ ( e φ ( tL ) f ) (cid:17) tL e φ ( tL ) f i dtt ,IV := Z e φ ( tL ) h ( φ ′ ( tL ) f ) (cid:16) e φ ( tL ) F ′ ( f ) − F ′ ( e φ ( tL ) f ) (cid:17)i dtt , ARA-DIFFERENTIAL CALCULUS ON A MANIFOLD 15 and V := Z ∞ ˜ ψ ( tL ) h tL e φ ( tL ) f e φ ( tL ) F ′ ( f ) i dtt + Z ∞ e φ ( tL ) h ψ ( tL ) f e φ ( tL ) F ′ ( f ) i dtt in order that (13) is satisfied. It remains us to check that each term belongs to W s − d/p,p . Step 1:
Term I .Since f ∈ W s + ǫ,p then e φ ( L ) f belongs to W ρ,p for every ρ ≥ s + ǫ and so Proposition 3.9 yieldsthat (cid:13)(cid:13)(cid:13) e φ ( L ) F ( e φ ( L ) f ) (cid:13)(cid:13)(cid:13) W s − d/p,p . k f k W s + ǫ,p . Step 2:
Term V .We only treat the first term in V (the second one can be similarly estimated). Using duality,we have with some g ∈ L p ′ and for α ∈ { , s − d/p } since α ≥ t ≥ k L α/ V k L p ≤ Z Z ∞ (cid:12)(cid:12)(cid:12) ( tL ) α/ ˜ ψ ( tL ∗ ) g (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) tL e φ ( tL ) f e φ ( tL ) F ′ ( f ) (cid:12)(cid:12)(cid:12) dtdµt . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ (cid:12)(cid:12)(cid:12) tL e φ ( tL ) f e φ ( tL ) F ′ ( f ) (cid:12)(cid:12)(cid:12) dtt (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . k f k L p sup t ≥ k e φ ( tL ) F ′ ( f ) k L ∞ , where we used the boundedness of the square functional. Then we conclude since e φ ( tL ) F ′ ( f )is uniformly bounded by k F ′ ( f ) k L ∞ which is controlled by k f k W s,p (due to Sobolev embeddingwith s > d/p and Proposition 3.9).Indeed our problem is to gain some extra regularity (from s to 2 s − d/p ) so the main difficultyrelies on the study of the “high frequencies” and not on the lower ones. Step 3:
Term II .By duality and previous arguments, we get k II k W s − d/p,p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z t − s + d/p | t / X e φ ( tL ) f | dtt (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z | t / X e φ ( tL ) f | dtt (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p where we decomposed the norm with its homogeneous and its inhomogeneous parts and thenused uniform boundedness of F ′′ ( e φ ( tL ) f ). Since (using L p -boundedness of the Riesz transforms,see Assumption 1.14 and Sobolev embedding) k X e φ ( tL ) f |k L ∞ . k X e φ ( tL ) f k W d/p + ǫ,p . k L / e φ ( tL ) f k L p + k L / d/ p + ǫ/ e φ ( tL ) f k L p . t s/ − / k f k W s,p + t s/ − d/ p − ǫ/ − / k f k W s,p . t s/ − d/ p − ǫ/ − / k f k W s,p where we used t <
1. Finally it comes, k II k W s − d/p,p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z t − s − ǫ | t / X e φ ( tL ) f | dtt (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z | ( tL ) / − ( s + ǫ ) / e φ ( tL ) L ( s + ǫ ) / f | dtt (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . k f k W s + ǫ,p , where we used s < Step 4:
Terms
III and IV .For these terms, we follow the reasoning of the Appendix of [10]. Using the finite increments Theorem, we have (cid:12)(cid:12)(cid:12) e φ ( tL ) F ′ ( f ) − F ′ ( e φ ( tL ) f ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ( e φ ( tL ) − I ) F ′ ( f ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( e φ ( tL ) − I ) f (cid:12)(cid:12)(cid:12) . So using similar arguments as previously, we get (with h = F ′ ( f ) and h = f ) k III k W s − d/p,p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z (cid:12)(cid:12)(cid:12) e φ ( tL ) F ′ ( f ) − F ′ ( e φ ( tL ) f )) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) tL e φ ( tL ) f (cid:12)(cid:12)(cid:12) t − s + d/ (2 p ) dtt (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z (cid:12)(cid:12)(cid:12) ( e φ ( tL ) − I ) h (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) tL e φ ( tL ) f (cid:12)(cid:12)(cid:12) t − s + d/p dtt (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z (cid:12)(cid:12)(cid:12) ( e φ ( tL ) − I ) h (cid:12)(cid:12)(cid:12) t − s dtt (cid:19) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e φ ( tL ) − I )( tL ) s L s h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dtt / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . k L s h k L p . k h k W s,p where we used s <
2, which yields(14) (cid:13)(cid:13)(cid:13) tL e φ ( tL ) f (cid:13)(cid:13)(cid:13) L ∞ ≤ (cid:13)(cid:13)(cid:13) ( tL ) − s/ e φ ( tL )( tL ) s/ f (cid:13)(cid:13)(cid:13) L ∞ . t − d/ (2 p ) (cid:13)(cid:13)(cid:13) ( tL ) s/ f (cid:13)(cid:13)(cid:13) L p . t − d/ (2 p )+ s/ k f k W s,p and the boundedness of the square functional associated to the function e φ ( z ) − z s which is holo-morphic and vanishing at 0 and at ∞ (see Proposition 1.13). We conclude the estimate of III since h = f or h = F ′ ( f ) belongs to W s,p . The term IV is similarly estimated. (cid:3) Corollary 5.2.
The diagonal term
Rest (defined in Corollary 4.3) is bounded from L p × L q into L r as soon as p, q ∈ (1 , ∞ ] with < r := p + q . Moreover for p ∈ (1 , ∞ ) , ǫ > as smallas we want and s > d/p (with s < ( N − / ), then k Rest ( f, g ) k W s − d/p,p . k f k W s + ǫ,p k g k W s + ǫ,p . Proof.
Apply Theorem 5.1 to the quantities f + g and f − g with F ( u ) := u . Then thepolarization formulas give that Rest ( f, g ) has the same regularity has w in Theorem 5.1. (cid:3) Remark 5.3.
Usually, we have a gain of regularity of order s − d/p for this quantity. Here wehave a gain of s − d/p − ǫ for every ǫ > , as smal as we want. Remark 5.4.
Of course, the assumption s ∈ ( d/p, can be seen as very constraining. Wewant to explain here how it seems to us possible to weaken that point. First we point that this“technical” difficulty is new since it does not appear in the Euclidean situation.Legitimating the definition of the paraproducts (just before Definition 4.1 and Remark 4.2), wehave developed ( tL ) ( e φ ( tL ) f e φ ( tL ) F ′ ( f )) using the Leibniz rule of the Laplacian. Now for M < By this way, we may define a new kind of paralinearization and prove that for f ∈ W s + ǫ,p with large s > d/p then F ( f ) = X j Z ∞ T j,t ( e φ ( tL ) f, e φ ( tL ) F ′ ( f ) , ..., e φ ( tL ) F (2 M − ( f )) dtt + w with w ∈ W s − d/p,p . By this way, the rest w should be decomposed as previously, in the corre-sponding term II , it will appear | t / X e φ ( tL ) f | M − such that the corresponding square functionalwill be bounded as soon as ( s − d/p )( M − > s − . The other terms may be bounded (as we didin Step 4) since they will have quantities as | e φ ( tL ) F ( k ) ( f ) − F ( k )( e φ ( tL ) f ) | and other differentialoperators on e φ ( tL ) f . The key idea is that now the multilinearity of the operator T j,t will besufficiently high to involve sufficiently such differential terms, each of them bringing a positivepower of t as shown in (14).By this way, it is also possible to get a paralinearization result for high regularity s > d/p and s << N (by taking a large exponent N ), by defining new multilinear operators involving thederivatives F ( k ) ( f ) . As explained in [10] (see its Appendix I.3, theorem 38), a vector-valued version of the precedingresult allows us to prove the following one: Theorem 5.5. Consider s ∈ ( d/p, , f ∈ W s + k,p and a smooth function F ( x, u , · · · u N ) ∈ C ∞ ( M × R N ) with F ( x, , · · · , 0) = 0 . Then by identifying { , · · · , N } with a set of multi-indices { α , · · · α N } (and | α i | ≤ k ), we can build (15) x ∈ M → F ( x, X α f ( x ) , · · · X α N f ( x )) which belongs to W s,p . Moreover, (16) F ( x, X α f ( x ) , · · · , X α N f ( x )) = N X i =1 Π [ ∂ ui F ]( x,X α f ( x ) , ··· ,X αN f ( x )) ( X α i f )( x ) + w ( x ) with w ∈ W s − d/p,p . Propagation of low regularity for solutions of nonlinear PDEs As in the Euclidean case, paralinearization is a powerful tool to study nonlinear PDEs andto prove the propagation of regularity for solutions of such PDEs. Let us try to present someresults in this direction with this new setting of Riemannian manifold.Let us consider a specific case of nonlinear PDEs for simplifying the exposition : let F ( x, u , · · · u κ +1 ) ∈ C ∞ ( M × R κ +1 ) be a smooth function with F ( x, , · · · , 0) = 0. Then by identifying { , · · · , κ +1 } with a set of multi-indices { , , · · · κ } , we deal with the function(17) F ( f, Xf ) := x ∈ M → F ( x, f ( x ) , X f ( x ) , · · · X κ f ( x ))for some function f . That corresponds to the case N = κ + 1, k = 1 with α = 0 and α i = X i − for i = 2 , · · · N + 1 in (17). Theorem 6.1. Consider s ∈ ( d/p, , f ∈ W s +1 ,p and a smooth function (as above) F ( x, u , · · · u N ) ∈ C ∞ ( M × R N ) with F ( x, , · · · , 0) = 0 and assume that f is a solution of F ( f, Xf )( x ) = 0 . Consider the vector field Γ( x ) := κ +1 X i =2 [ ∂ u i F ]( x, f ( x ) , X f ( x ) , · · · , X κ f ( x )) X i . Then, locally around each point x ∈ M in “the direction Γ ”, the solution f has a regularity W s +1+ ρ for every ρ > such that ρ < min { , s − d/p } . In the sense that U ( f ) := κ +1 X i =2 [ ∂ u i F ]( x, f ( x ) , X f ( x ) , · · · , X κ f ( x )) L ( s + ρ ) / X i ( f ) ∈ L p . Such results can be seen as a kind of directional “Implicit function theorem”, where theregularity of F ( f, Xf ) implies some directional regularity for f (in the suitable direction, wherewe can regularly “invert” the nonlinear equation). Proof. The previous paralinearization result yields that κ X i =1 Π [ ∂ ui +1 F ]( f,Xf ) ( X i ( f )) ∈ W s + ρ,p , which gives T F ( f ) := κ X i =1 e Π [ ∂ ui +1 F ]( f,Xf ) ( L α X i f ) ∈ L p , where e Π is another paraproduct. Indeed e Π b ( a ) = − Z ∞ ( tL ) α ˜ ψ ( tL ) h ( tL ) − α e φ ( tL ) a e φ ( tL ) b i dtt − Z ∞ ( tL ) α e φ ( tL ) h t − α ψ ( tL ) a e φ ( tL ) b i dtt , where we have taken the notations of the definition for the initial paraproduct Π (see Definition4.1). Then, we want to compare this quantity to the main one : U ( f ). So let us examine thedifference. Since for every constant c , we have cf = Π c ( f ) = L α Π c ( L − α f ) = e Π c ( f ) , it comes U ( f )( x ) = κ X i =1 e Π [ ∂ ui +1 F ]( f ( x ) ,Xf ( x )) ( L α X i f )( x ) , hence T F ( f )( x ) − U ( f )( x ) = κ X i =1 e Π λ i,x ( X i L α f )( x )with λ i,x ( · ) = [ ∂ u i +1 F ]( f, Xf ) − [ ∂ u i +1 F ]( f ( x ) , Xf ( x )). It remains us to check that for eachinteger i , the function x → e Π λ i,x ( X i (1 + L ) α f )( x ) belongs to L p . Let us recall that e Π λ i,x ( X i (1 + L ) α f )( x ) = − Z ∞ ( tL ) α ˜ ψ ( tL ) h ( tL ) − α e φ ( tL ) X i L α f e φ ( tL ) λ i,x i ( x ) dtt − Z ∞ ( tL ) α e φ ( tL ) h t − α ψ ( tL ) X i L α f e φ ( tL ) λ i,x i ( x ) dtt . Let us study only the first term I (the second one beeing similar): I := (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ ( tL ) α ˜ ψ ( tL ) h ( tL ) − α e φ ( tL ) X i L α f e φ ( tL ) λ i,x i ( x ) dtt (cid:12)(cid:12)(cid:12)(cid:12) . Z ∞ Z M µ ( B ( x, t − / )) (cid:18) d ( x, y ) t − / (cid:19) − d − δ (cid:12)(cid:12)(cid:12) ( tL ) − α e φ ( tL ) X i L α f ( y ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) e φ ( tL ) λ i,x ( y ) (cid:12)(cid:12)(cid:12) dµ ( y ) dtt . Z ∞ X j ≥ − jδ − Z C ( x, j t − / ) (cid:12)(cid:12)(cid:12) ( tL ) − α e φ ( tL ) X i L α f ( y ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) e φ ( tL ) λ i,x ( y ) (cid:12)(cid:12)(cid:12) dµ ( y ) dtt . ARA-DIFFERENTIAL CALCULUS ON A MANIFOLD 19 where we set C ( x, j t − / ) = B ( x, j +1 t − / ) \ B ( x, j t − / ) and by convention | C ( x, j t − / ) | = | C ( x, j t − / ) | . Now, for y ∈ B ( x, j t − / ), we have (cid:12)(cid:12)(cid:12) e φ ( tL ) λ i,x ( y ) (cid:12)(cid:12)(cid:12) . µ ( B ( y, t / )) Z (cid:18) d ( y, z ) t / (cid:19) − d − δ (cid:12)(cid:12) [ ∂ u i +1 F ]( f, Xf )( z ) − [ ∂ u i +1 F ]( f, Xf )( x ) (cid:12)(cid:12) dµ ( z ) . X k ≥ jd − δk − Z C ( y, k + j t / ) | H ( z ) − H ( x ) | dµ ( z ) . X k ≥ jd − δk − Z ˜ C ( x, k + j t / ) | H ( z ) − H ( x ) | dµ ( z )with H := [ ∂ u i +1 F ]( f, Xf ) and ˜ C another systems of coronas. So we get I . X k,j ≥ − kδ + j ( d − δ ) Z ∞ − Z B ( x, j t − / ) (cid:12)(cid:12)(cid:12) ( tL ) − α e φ ( tL ) X i L α f ( y ) (cid:12)(cid:12)(cid:12) dµ ( y ) ! − Z B ( x, k + j t / ) | H ( z ) − H ( x ) | dµ ( z ) ! dtt . X k,j ≥ − kδ + j ( d − δ ) Z ∞ M h t / (cid:12)(cid:12)(cid:12) ( tL ) − α e φ ( tL ) X i L α f (cid:12)(cid:12)(cid:12)i ( x ) t − / − Z B ( x, k + j t / ) | H ( z ) − H ( x ) | dµ ( z ) ! dtt . Using Cauchy-Schwartz inequality, we also have I . X k,j ≥ − kδ + j ( d − δ ) (cid:18)Z ∞ M h t / ( tL ) − α e φ ( tL ) X i L α f i ( x ) dtt (cid:19) / Z ∞ t − − Z B ( x, k + j t / ) | H ( z ) − H ( x ) | dµ ( z ) ! dtt / . X k,j ≥ − k ( δ − − j ( d − δ − (cid:18)Z ∞ M h t / ( tL ) − α e φ ( tL ) X i L α f i ( x ) dtt (cid:19) / Z ∞ t − − Z B ( x,t / ) | H ( z ) − H ( x ) | dµ ( z ) ! dtt / . (cid:18)Z ∞ M h t / ( tL ) − α e φ ( tL ) X i (1 + L ) α f i ( x ) dtt (cid:19) / Z ∞ t − − Z B ( x,t / ) | H ( z ) − H ( x ) | dµ ( z ) ! dtt / , where we have used a change of variables and δ > d + 1. So using exponents q, r > p (laterchosen) such that p = q + r , boundedness of the square functional on the one hand and on theother hand Fefferman-Stein inequality for the maximal operator, it comes k I k L p . k f k W α,q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ∞ t − − Z B ( x,t / ) | H ( z ) − H ( x ) | dµ ( z ) ! dtt / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L r . Then, using the characterization of Sobolev norms (using this functional, see Proposition 3.6),we conclude to k I k L p . k f k W α,q k H k W ,r . We also chose exponents q, r such that1 q − αd > p − s + 1 d and 1 r − d > p − sd , which is possible since p < s − ρd because of the condition on ρ . Then Sobolev embedding(Proposition 3.4) yields that W s +1 ,p ֒ → W α,q and W s,p ֒ → W ,r . Finally, the proof is alsoconcluded since we obtain k I k L p . k f k W s +1 ,p k H k W s,p , which is bounded by f ∈ W s +1 ,p (due to H := [ ∂ u i +1 F ]( f, Xf ) with Proposition 3.9). (cid:3) We let the reader to write the analog results for higher order nonlinear PDEs. Remark 6.2. Let us suppose that the geometry of the manifold allows us to use the followingproperty: For α > , the commutators [ X i , (1 + L ) α ] is an operator of order α , which meansthat for all p ∈ (1 , ∞ ) and s > , [ X i , (1 + L ) α ] is bounded from W s +2 α,p to W s,p .This property holds as soon as we can define a suitable pseudo-differential calculus with sym-bolic rules : in particular, this is the case of H-type Lie groups, using a notion of Fouriertransforms based on irreductible representations, see [5, 21] .Under this property, we can commute the vector field X with any power of the Laplacian andso with the same statement than in the previous theorem, we obtain that Γ(1 − L ) s + ρ f ∈ L p . 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Triebel, Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Ark. Mat. (1986),299–337. Fr´ed´eric Bernicot - CNRS - Universit´e Lille 1, Laboratoire de math´ematiques Paul Painlev´e,59655 Villeneuve d’Ascq Cedex, France E-mail address : [email protected] Yannick Sire - LATP-UMR6632-Universit´e Paul C´ezanne, 13397 Marseille, France E-mail address ::