Lp-estimates for Riesz Transforms on Forms in the Poincaré Space

Using hyperbolic form convolution with doubly isometry-invariant kernels, the explicit expression of the inverse of the de Rham laplacian ∆ acting on m-forms in the Poincaré space Hn is found. Also, by means of some estimates for hyperbolic singular integrals, Lp-estimates for the Riesz transforms ∇i∆−1, i ≤ 2, in a range of p depending on m,n are obtained. Finally, using these, it is shown that ∆ defines topological isomorphisms in a scale of Sobolev spaces Hs m,p(H) in case m ≠ (n± 1)/2, n/2. 1. STATEMENT OF RESULTS AND PRELIMINARIES 1.1. The main object of study in this paper is the Hodge-de Rham Laplacian ∆ acting on m-forms in the Poincaré hyperbolic space (Hn, g). The aim is to prove that ∆ defines topological isomorphisms in a range Hs m,p(H) of Sobolev spaces of forms defined as follows. For 0 ≤ m ≤ n, 1 ≤ p < ∞ and s ∈ N, the Sobolev space Hs m,p(H) is the completion of the space Dm(Hn) of smooth m-forms with compact support with respect to the norm ‖η‖p,s = s ∑ i=0 ‖∇η‖p. Here ∇(i) means the i-th covariant differential of η, and for a covariant tensor α


1.1.
The main object of study in this paper is the Hodge-de Rham Laplacian ∆ acting on m-forms in the Poincaré hyperbolic space (H n , g).The aim is to prove that ∆ defines topological isomorphisms in a range H s m,p (H n ) of Sobolev spaces of forms defined as follows.For 0 ≤ m ≤ n, 1 ≤ p < ∞ and s ∈ N, the Sobolev space H s m,p (H n ) is the completion of the space D m (H n ) of smooth m-forms with compact support with respect to the norm Here ∇ (i) means the i-th covariant differential of η, and for a covariant tensor α , |α| being the pointwise norm of α with respect to the metric g and dµ the volume-invariant measure on H n given by g.The space H s m,p (H n ) can be alternatively defined in terms of weak derivatives.The main result of this paper is the following theorem.
Theorem A. ∆ is a topological isomorphism from H s+2 m,p (H n ) to H s m,p for p ∈ (p 1 , p 2 ) with in case m ≠ (n ± 1)/2, n/2.
In the exceptional case m = (n±1)/2, ∆ is one to one but is not a topological isomorphism for any p.For this case we obtain as well some weighted estimates.If m = n/2, ∆ is known to have a non-trivial kernel.Of course, Sobolev spaces H s m,p can be considered for non integer s as well, and the same results hold by interpolation.
For the Sobolev spaces for p = 2, H s m,2 (H n ), another proof of the theorem, based on energy methods and valid for an arbitrary complete hyperbolic manifold, is given in [1].The motivation for the theorem, as with [1], comes from mathematical physics, where most operators exhibit ∆ as their principal part, and results like the above become essential to establish existence and uniqueness theorems.
Our method of proof is simply to construct an explicit inverse L for ∆ on D m (H n ) and show that there is a gain of two covariant derivatives Lη p,s+2 ≤ const η p,s .
Thus Lη plays the role of the classical Riesz transform in the Euclidean setting.The most delicate part is of course Riesz-type operators such as ∇∆ −1/2 , ∇ (2) ∆ −1 have extensively been studied in different contexts, for the case of functions.On symmetric spaces, they are bounded in L p , 1 < p < ∞ and of weak type (1,1).This was shown in [2] for the first order ones in some spaces, and later extended to all symmetric spaces in [3].The L p -boundedness holds as well for higher order Riesz transforms in symmetric spaces, but not generally the weak type (1, 1) estimate.In more general contexts, this has been shown in [6], [7], [8], among others.In case of m-forms, 0 < m < n, as far as we know, there are much less known results, and is for those that our result is new.In [12], [13] some aspects of harmonic analysis of forms are developed; in particular, the exact expression for the heat kernel is given, and it is very likely that from it one can get as well an explicit expression for ∆ −1 .Strictly speaking, to prove the result, an exact expression of ∆ −1 is not needed, it is enough having estimates for the resolvent both local and at infinity.In [8], estimates of this kind are obtained and applied to Sobolev-type inequalities for forms, and they might work for this purpose too. 1 However, we feel that our approach, that we next describe, is more elementary and might be interesting in itself.
The de Rham Laplacian ∆ is invariant by all isometries ϕ of H n .These form a group that we denote here by Iso(H n ).Denoting by ϕ * (η)(x) = η(ϕ(x)) the pull-back of a form η by ϕ, this means that ∆ and ϕ * commute, for all ϕ ∈ Iso(H n ).Therefore the inverse L of ∆ should commute too with Iso(H n ).Among all isometries of H n , the hyperbolic translations Tr(H n ) constitute a (noncommutative) subgroup, in one to one correspondence with H n itself.In Section 2 we do some harmonic analysis for forms in H n and introduce hyperbolic convolution of forms to describe all operators acting on m-forms and commuting with Tr(H n ).In a second step (Subsection 2.2) we characterize the hyperbolic convolution kernels k(x, y) corresponding to operators commuting with the full group Iso(H n ).
Once the general expression of an operator commuting with Iso(H n ) has been found, we look for our L among these.This corresponds to L having a kernel k(x, y) which is a fundamental solution of ∆ in a certain sense, and having the best decay at infinity.This kernel turns out to be unique for m ≠ (n ± 1)/2, n/2, we call it the Riesz kernel for m-forms in H n , it is found in Subsection 3.1 and estimated in Subsection 3.2.Section 4 is devoted to the proof of the L pestimates.Here we use standard techniques in real analysis (Haussdorf-Young inequalities, Schur's lemma, etc.).For the second-order Riesz transform, to show its boundedness in the specified range (p 1 , p 2 ) needs considering some notion of "hyperbolic singular integral."There exist some references dealing with this, e.g.[9], [11], and giving some criteria for L p -boundedness that might apply; however, as the singular integral arises locally, we have found it easier and more elementary to treat it with the classical Euclidean Calderón-Zygmund theory as a local model, and patch it in a suitable way to infinity.

1.2.
We collect here several notations and known facts about H n .We will use both the unit ball model B n with metric g = 4(1 − |x| 2 ) −2 i dx i dx i and the half-space model R n + = {x n > 0} with metric g = x −2 n i dx i dx i .Both models are connected via the Cayley transform ψ : R n + → B n given in coordinates by We denote by e ∈ H n the point (0, 0, . . ., 1) ∈ R n + or 0 ∈ B n .The metric g defines a pointwise inner product (α, β)(x) between forms at x, for every x ∈ H n , and a volume measure dµ.In the ball model dµ is written the half-space model.We denote by , the pairing between forms that makes We write |α| and α for the pointwise and global norms, respectively, of the form α. In terms of the Hodge star operator * the inner product can be written too The group Tr(H n ) of hyperbolic translations is in one to one correspondence x T x with H n through the equation T x (e) = x.The equations of z = T x y are better described in the half-space model by It is easily checked that indeed Tr(H n ) is a (non-commutative) group.The inverse transformation of T x will be denoted S x .Another explicit isometry ϕ x mapping e to x, satisfying ϕ −1 x = ϕ x , is given in the ball model by (1.1) Since the isotropy group of 0 is the orthogonal group O(n), the general expression The hyperbolic (or geodesic) distance between x, y ∈ H n is written d(x, y).We will rather use the pseudohyperbolic distance r = r (x, y), related to d by the formula d(x, y) = 2 arctanh r (x, y).The explicit expression of r (x, y) 2 in the R n + model and the B n model is respectively (1.2a) (1.2b) Associated to the group of translations we have the basis of orthonormal translation-invariant vector fields X i (x) = (T x ) * (X i (e)), such that X i (e) = ∂/∂x i .They satisfy X i (u • T x ) = (X i u) • T x for every smooth function u.We will denote by w i (x) the dual basis of X i , which accordingly is orthonormal and translation invariant too: T * x w i = w i .Their expression in the R n + model is simply Because of their translation-invariance property, the (X i , w i ) are more suitable than the (X i , η i ) defined in the ball model B n by For an increasing multiindex I of length |I| = m we write w I = w i 1 ∧w i 2 ∧• • •∧ w i m , and similarly dx I or η I .The {w I } I is an orthonormal translation-invariant basis of m-forms.
Recall that the de Rham Laplacian is defined as ∆ = dδ + δd, where δ is the adjoint of d with respect to , .Although strictly speaking not needed, the following expression of ∆ in w I -coordinates will simplify the analysis at some points.If α = I α I w I , a computation shows that in case n ∈ J Here Jk means the multiindex obtained replacing k by n.In case n ∈ J, where J means the multiindex obtained replacing n by .For a function In the ball model, with usual coordinates, (1.5)

TRANSLATION INVARIANT AND ISOMETRY INVARIANT OPERATORS ON FORMS
2.1.We are interested in finding the general expression of an operator acting on m-forms, and isometry-invariant.In a first step we consider translationinvariant operators acting on m-forms; these are described by what we might call hyperbolic convolution as follows.Let k(x, y) be a double m-form in x, y and define Using the translation-invariant basis of m-forms w I we see that the general ex- where k I (x, y) are doubly-invariant functions, that is, of the form k I,J (x, y) = a I,J (S y x) for some function (or distribution) a I,J .If δ 0 denotes the Delta-mass at e and δ(x, y) = I,J δ 0 (S y x)w I (x) ⊗ w J (y), then formally If P is an operator on m-forms commuting with the T y , S y , we will thus have and α(x) = α I (x)w I (x), then C k α has in the basis w I (x) coefficients given by Thus in the basis w I everything reduces of course to convolution of functions.For a function convolution kernel a(S y x) and a test function u ∈ D(H n ) we may think of as an infinite linear combination of inverse translates a(S y x) of a(x).Since the vector fields X i commute with translations, it follows that, whenever everything makes sense, (2.1) We point out that this convolution is not commutative; C a u is in general different from C u a. Correspondingly, X i C a u − C a X i u is in general not zero; in fact one can easily show ([1, Lemma 3.1]) that these commutators are linear combinations of other convolution operators built from a(S y x).

2.2.
Let P be a generic translation-invariant operator acting on m-forms.We have seen in the previous subsection that we can associate to P a doublytranslation invariant kernel k(x, y) so that P = C k .By the same argument as before, P will be isometry invariant if and only if k(ϕx, ϕy) = k(x, y) ∀ ϕ ∈ Iso(H n ), in which case we say that k is doubly isometry-invariant.Working in the ball model and since every ϕ ∈ Iso(H n ) is the composition of a translation with some U ∈ O(n), the additional requirement on the kernel k(x, y) = a I,J (S y x)w I (x) ⊗ w J (y) amounts to k(Ux, U 0) = k(x, 0), that is, Thus we are interested in describing those k(x, 0)-which are m-forms at 0 whose coefficients are m-forms in x-that are doubly invariant by all U ∈ O(n) in the sense above.Once the k(x, 0) having this property are known, k(x, y) = k(S y x, 0) defines the general doubly isometry invariant m-form.For m = 0 the k(x, 0) are simply the radial functions a(|x|), and a(|S y x|) = a(|ϕ y x|) is the general doubly isometry invariant function.For m ≠ 0 their general expression is not so simple.We find it more convenient to use the usual basis dx I so we look at k(x, 0) in the form and we must impose is easily seen to be doubly O(n)-invariant, and so is (here we use the symbol ∧ to denote as well the exterior product of double forms defined by x i dx i (0) .

If we use now U ∈ O(n) permuting the first two axes, we see that
where x = (x 2 , . . ., x n ).If we impose the invariance under the permutation of the first two axes as before, it is clear that k must be zero.
Having proved that S(n) holds for all n, let now k(x, 0) be as in (2.2), doubly O(n)-invariant.Clearly k(x, 0) is then determined by its values k( r , 0), where r = (r , 0, 0, . . ., 0).Fixed r, k( r , 0) may be regarded as a double (m, m)-form with constant coefficients, which is invariant by all U ∈ O(n) fixing r , that is, of type (2.4).We write now the decomposition of k( r , 0) in terms of k 1 (r , 0), k 2 (r , 0), k 3 (r , 0), and k 4 (r , 0) as before, and applying S(n) we get (2.5) the last term is zero and the first is γ m ), which we write Finally, with fixed x, we choose U such that Ux = r , r = |x|, and use the invariance of k, τ, γ to find (2.3)

Ë
To find the general expression of a doubly isometry invariant kernel k(x, y) we must translate k(x, 0) to an arbitrary point: k(x, y) = k(S y x, S y y).We may use any isometry mapping y to 0, for instance we may use ϕ y given by (1.1) instead of S y .We introduce the basic forms α, β, τ, and γ α = α(x, y) The lemma gives part (a) of the following theorem.Part (b) gives other equivalent general expressions, which are intrinsic, that is, independent of the model of H n at use.

Theorem 2.2. (a)
The general expression of an (m, m)-form k(x, y) doubly isometry-invariant in H n , in the ball model, is where D denotes an arbitrary function of the geodesic distance d(x, y).(c) All such k(x, y) are symmetric in x, y ∈ H n .
Proof.Part (a) has been already proved.For (b) note first that it is enough to consider one function of d: we choose D = r (x, y) 2 , which in the ball model equals |ϕ y (x)| 2 .Then d x D = 2α, and using (1.1), (1.2a) one finds This gives and and so (b) follows.Part (c) is a consequence of (b).

Ë
We will need the expression of the generators τ, γ in terms of the invariant basis w i .We obtain these using formula (1.2a) for r 2 (x, y) in the half-space model.First In the following we write w ij = w i (x) ⊗ w j (y).We have where the P ij (x, y) are certain homogeneous polynomials.As we know, everything can be written in terms of z = S y x: for instance , and say for i, j < n Therefore we may write For γ = d x β we obtain a similar expression Again this can be written Notice that and hence

RIESZ FORMS AND RIESZ FORM-POTENTIALS IN H n
3.1.Our next objective is now to find an explicit left-inverse L for ∆ on D m (H n ).Since ∆ is invariant by all isometries, L should be too.By what has been discussed in Section 2, L should have a kernel k m (x, y), doubly invariant by all isometries.Alternatively, notice that if k is some kernel such that (3.1) (which formally exists because ∆η = 0, η ∈ D m (H n ) imply η = 0), then its average over the unitary group O(n) with respect to the normalized left-invariant measure dµ(U), still satisfies (3.1), and it is doubly invariant by O(n).If ϕ x is an isometry mapping x to 0, k 2 (x, y) = k 1 (ϕ x x, ϕ x y) is independent of ϕ x , satisfies (3.1), and is doubly invariant by all isometries.
Anyway, we look for a doubly isometry-invariant kernel k m for which (3.1) holds, and then consider the operator L defined by k m as above.Taking for granted by now that this operator L is well defined on D m (H n ) and maps D m (H n ) into locally integrable m-forms, notice that (3.1) and the symmetry of k m together imply that L is a right-inverse too, that is, ∆Lα = α for α ∈ D m (H n ) in the weak sense: We work in the ball model.By Theorem 2.2, k m (x, y) is of type x (y) (notice that we are exchanging x, y, using (c) in Theorem 2.2).Condition (3.1) implies ∆ y k m (x, y) = 0 in y ≠ x (while ∆Lw = w implies ∆ x k m (x, y) = 0 in x ≠ y).In fact, (3.1) amounts to requiring ∆ y k m (x, y) = δ x in a sense to be described below.

3.2.
In a first step we look for conditions on the A 1 , A 2 , so that ∆ y k m (x, y) = 0 in y ≠ x.A lengthy computation will show that the general harmonic k m depends on four parameters.By the invariance of k m , we may assume x = 0, in which case, writing r = |y|, with γ = dx i (0) ⊗ dy i , τ = α ⊗ β, α = y i dx i (0), β = r dr .Since * x * y k m (x, y) is again doubly invariant, it must have an analogous expression with m replaced by n − m.Indeed, it is easily checked that for 0 < m < n.
Moreover, since * commutes with ∆, it is natural to require as well that * x * y k m = k n−m , that is, we may assume from now on that 0 ≤ m ≤ n/2.For m = 0, using (1.5) we find from which it follows that A (r ) = c 0 (1 − r 2 ) n−2 r 1−n and We start now computing ∆ y k m (0, y) for 0 < m ≤ n/2, using that on m-forms The double form ∆ y k m (x, y) is also doubly invariant, and therefore it must have the same expression as k m with A 1 , A 2 replaced by other functions B 1 , B 2 to be found.In the computations we will use besides (3.2) the equations which are easily checked as well.First, d y k m (0, y) = (A 1 − r A 2 ) dr ∧ γ m , so by the equations above By analogous computation, applying d y to (3.2)

It follows finally that
with Therefore, ∆ y k(0, y) = 0 is equivalent to the system B 1 = 0, B 2 = 0.It easily follows from this that A 3 satisfies the equation Replacing in the equation B 1 = 0, A 4 by its expression in terms of A 1 and A 2 , and then A 2 by its expression in terms of A 1 and A 3 , we find that A 1 satisfies the inhomogeneous equation The change of variables A 1 (r ) = G(x), A 3 (r ) = H(x), x = r 2 , transforms these into the hypergeometric equations This system is equivalent to ∆ y k m (x, y) = 0 in y ≠ x, whence the general doubly-invariant k m harmonic in y ≠ x depends on four parameters.Note that for m = n/2, the homogeneous equations are the same and can be solved explicitly: the general solution is H = as −n/2 + b and For m < n/2, a fundamental family for the equation (3.5) is given by The hypergeometric function in u 1 is a polynomial in x of degree m with positive coefficients, 1 + x if m = 1.A fundamental family for the equation (3.6) is given by The hypergeometric function in u 3 is a polynomial of degree m − 1 with positive coefficients (see [5] for all these facts).The wronskian w(x) for this second equation is, by Liouville's formula, It follows from this that the parametrization for G is given by where c(x), d(x) satisfy, with Once A 1 (r ) = G(r 2 ) and A 3 (r ) = H(r 2 ) are known, the kernel k m (x, y) is completely known, because by the definition of A 3 in (3.3), The choice a = 0, c(0) = 0 (a = c = 0 in the parametrization (3.7) for m = n/2) gives all doubly invariant k m (x, y) which are globally harmonic, with no singularity, and they are therefore spanned by the forms corresponding to the choice G = u 4 and to the choice a = 0, b = 1, c(0) = 0, d(0) = 0, As a particular case, note that for m = n/2, γ m is harmonic in H 2m , and it is the simplest example of a non-zero harmonic m-form in L 2 (H 2m ).

3.3.
Besides being harmonic in y ≠ x, the singularity at y = x must be such that (3.1) holds.Again, we may assume x = 0; we check this property using second's Green identity, whose version for general forms we recall now.
The operator δ being the adjoint of d, one has, for a smooth domain Ω ⊂ B n and α, β smooth forms on Ω with deg α Given two m-forms η, ω, applying this with α = δη, β = ω, next with α = ω, β = dη and subtracting, one gets the first Green's identity for m-forms Permuting ω, η and subtracting again gives the second Green's identity We apply this to In case m = 0, the terms in δk m , δη are of course zero; to get a term in η(0) on the right when ε → 0, we need dk m of the order of ε 1−n and k m of the order of ε 2−n in |y| = ε.That makes k m locally integrable too, and (3.1) is obtained letting ε → 0. This means that, for m = 0, k is unique and is given by the well-known Green's function (3.10) for an appropriate choice of c n .In case m > 0, again we need |k m (0, y)| = o(r 1−n ) as r → 0, so that the first and third terms on the right have limit 0 as ε → 0; then k m is integrable in y, and the integral on the left converges to ∆η ∧ * k m .Using the expression for * dk m in (3.3), we find By Stoke's theorem, and since α = O(r ), the last integral equals Using (3.4) for δk m = (−1) n(m+1)+1 * d * , and proceeding in the same way, But by the equation and hence the limit of the above expression is −c n m!a 0 mη(0).Altogether, we conclude that if . Now look at the general expression of H, G in (3.8).The condition H(x) ∼ c 0 x −n/2 fixes a = a 0 ; then near x = 0, c (x) is bounded and d (x) behaves like x −n/2 .Since u 4 (x) is bounded, the term d(x)u 4 (x) behaves like x 1−n/2 .So, we must normalize c(x) by c(0) = 0, so that c(x) = O(x), and the other term c(x)u 3 (x) will behave like In conclusion, all this discussion shows that the doubly invariant kernels k m (x, y) satisfying (3.1) constitute a two parameter family described by H = a 0 u 1 (x) + bu 2 (x), c(0) = 0.The two parameters are b and the constant of integration for d(x) in (3.8).Equivalently, they are obtained by adding to the form corresponding to H = a 0 u 1 (x), c(0) = 0, and say d( 12 ) = 0 the general globally smooth one described before.

3.4.
In order to produce the best estimates, in a sense we need to choose the best of the kernels k m .Naturally enough, we choose the k m having the best behaviour at infinity, x = 1, that is, so that G, H have the best decrease in size as x → 1.In case m = n/2, where we already have the normalization c = 0, a = a 0 , the choice b = −a gives the best growth

The hypergeometric function u 3 behaves like
In case 2m < n − 1, however, we can choose the constant d 0 so that d(1) = 0, and then It remains to estimate the growth of A 2 (r ) near r = 1.Recall that the definition but a cancellation occurs.The functions u 1 , u 3 are C ∞ at 1 and have developments whence u 4 has a finite derivative at 1 and a development which gives ; the second term d(x)u 4 (x) satisfies the required bound, while the first c(x)u 3 (x) has a development As However, for m = n/2, this no longer holds.Indeed, from (3.7), where We point out that all this can be obtained, in loose terms, working directly with the hypergeometric equations relating G, H, and using asymptotic developments.
, identifying the lower order terms in both sides gives, When H ≡ 0, one must have either j = 0 (corresponding to u 4 ) or j = n−2m+1 (corresponding to u 3 ).For the inhomogeneous equation, if j ≠ 0, j ≠ n+1+2m (that is, G contains no contribution from u 3 , u 4 ), one We summarize the results in this and the previous subsections in the following theorem.
Theorem 3.1.For |n − 2m| > 1, there is a unique doubly invariant kernel for which (3.1) holds, and satisfying moreover For m = (n ± 1)/2, there is a one-parameter family of such kernels satisfying For m = n/2, there is a one-parameter family of such kernels satisfying

In all cases
For |n − 2m| > 1, we call k m (x, y) the Riesz kernel for m-forms in H n , and the Riesz potential of η, whenever this is defined.From (2.8) we see that (3.11) With the notations used before, the function A 3 (r ) = H(r 2 ) is bounded with bounded derivatives near r = 1.Then (3.3) and symmetry imply (3.12) By construction, one has L∆η = η for η ∈ D m (H n ).We will need the following generalization of this fact.
Proof.In (3.9) we would get an extra term Estimates (3.11), (3.12) and (3.13) imply that, with x fixed and |y| = R 1, 2 ) m we see that this extra term vanishes as R 1.

4.1.
Once the Riesz form k m (x, y) has been found, our aim is now to prove that the corresponding convolution for m ≠ (n ± 1)/2, n/2, and p in the range p 1 (m) = (n − 1)/(n − 1 − m) < p < (n − 1)/m = p 2 (m), and for a compactly supported m-form η (recall that we are assuming without loss of generality that m ≤ n/2).Since these are dense in the Sobolev spaces and we already know that ∆L m η = L m ∆η = η, this will prove the theorem for m ≠ (n ± 1)/2, n/2.The case m = (n ± 1)/2 will be commented later.
We work in the translation invariant basis w I .Taking into account formulas (2.6) and (2.7) for γ, τ, the Riesz form is written in the R n + model where each coefficient a I,J has an expression, with z = S y x, a I,J (z) = Ψ I,J (r ) p I,J (z) Here p I,J (z) is a certain polynomial in z 1 , . . ., z n , Ψ I,J is C ∞ in (0, 1) with Ψ I,J (r ) ∼ c 0 r 2−n as r 0, Ψ I,J (r ) = O(1 − r 2 ) n−m−1 as r 1.The term q I,J (z) = p I,J (z)/(|z| 2 + 2z n + 1) 2m is bounded.
If η = I η I (y)w I (y), the coefficient (Lη) I (x) of Lη in the basis w I is a finite linear combination of hyperbolic convolutions Ψ I,J (r )q I,J (z)η J (y) dµ(y).
By ellipticity of ∆, Lη is a smooth form.Moreover, since η has compact support, we see from (1.2a) and (3.11), (3.12), (3.13) that, in the ball model, which amounts to (4.2) We claim that for second-order derivatives we have too Notice that since we already know that ∆Lη = η, from the expression of ∆ in the basis w I given in (1.3)-(1.5) it follows that it is enough to show that for j < n.We will see below (equation (4.7) and invariance of the X i ) that each of the functions a(z) = Ψ I,J (r )q I,J (z) satisfies from which (4.3) follows as before.In fact, the discussion that follows will show that We continue the proof of (4.1).We claim first that it is enough to prove (4.1) for s = 0.For a smooth form η = η I w I , let X i η denote here the m-form that for each i there is an operator P i of order two in the X 1 , . . ., X n such that Applying this to Lη, which is smooth by the ellipticity of ∆, we get But X i Lη satisfies, by (4.2) and (4.3) and hence by Proposition 3.2, L∆ = Id on it.We conclude that for all η ∈ Assume that (4.1) has been proved up to s, so that by density it holds for α ∈ H s m,p (H n ) too, and let γ be a multiindex of length |γ| ≤ s.For i = 1, . . ., n and so using twice the induction hypothesis ≤ const ( η p,s+1 + η p,s ), proving (4.1) for s + 1. Proving (4.1) for s > 0 means proving (Lη) I p , X i (Lη) I p , X j X i (Lη) I p ≤ const η p .
As before, using that we already know that ∆Lη = η, we see that for the secondorder derivatives we may assume j < n.In the following we delete the indexes I, J and denote by a(z) = ψ(r )Q(z) a convolution kernel with ψ, Q as above, and proceed to prove that the convolution where in the last case we may assume that j < n.The fields X i are invariant, and therefore X i C a α, X j X i C a α are obtained, respectively, by convolution with Z i a, Z j Z i a (by (2.1)).Recall that In order to estimate Z i a, Z i Z j a, we collect first some auxiliary estimates.We claim that (4.5) The first two are routinely checked, for instance, when differentiating the denominator in Q, will still be bounded.All other terms can be treated similarly.Differentiating 1 − r 2 = 4z n /(1 + |z| 2 + 2z n ), we get These imply (4.5) because The estimates (4.5) imply We will call a convolution kernel b(z) m-admissible if |b(z)| = O(r 1−n ) as r 0 and, moreover, |b(z)| = O(1 − r 2 ) n−m−1 as r 1.We will prove later (Theorem 4.2) that a hyperbolic convolution with m-admissible kernels defines a bounded operator in L p (H n ) for the range (p 1 (m), p 2 (m)), as specified in the statement of the main result.From the estimates (4.6) we see that a and Z i a are m-admissible kernels, and so (4.4) will be proved for them.As |Z j Z i a(z)| = O(r −n ) has the critical non-integrable singularity at r = 0, Z j Z i a(z) is not an m-admissible kernel.Notice however from (4.6), (4.7) that the last three terms ψ (r )Z i r Z j Q, ψ (r )Z j r Z i Q, ψ(r )Z j Z i Q are indeed m-admissible.Moreover, the estimate |Z i Q| ≤ const implies that Q is Lipschitz with respect to the hyperbolic metric, in particular for small r .This means that replacing Q by Q − Q(e) in the first two terms leads to an m-admissible kernel again.All this leaves us with the kernel in an m-admissible kernel.By the same reason, we may replace φ (r ), φ (r ) respectively by (r , that is to say we must deal with the convolution kernel (4.8) We introduce a class of singular hyperbolic convolution kernels to deal with the later.For this purpose it is more convenient to work in the ball model, so now b is defined in B n , and r = |z|.We replace the integrable singularity r 1−n by a typical Calderón-Zygmund singularity (see e.g.[14]).Thus, we will call b a m-Calderón-Zygmund singular kernel if it has the form where Ω is say a Lipschitz function on S n−1 satisfying the cancellation condition (4.9) In Theorem 4.2 below we prove that m-Calderón-Zygmund singular kernels define bounded operators in the same range of p.With the following proposition, applied to φ 2 (z) = |z| 2−n , this will end the proof of the main result.The proposition is the analogue of the well-known statement that for φ smooth and homogeneous of degree 1 − n in R n , ∂φ/∂x i defines a Calderón-Zygmund kernel; it is homogeneous of degree −n, and the cancellation condition (4.9) is automatically satisfied, because sum of (m − 1)-admissible and (m − 1)-Calderón-Zygmund singular kernels.
Proof.We replace the Z j by Y j = (1 − r 2 )∂/∂z j ; we have so in all cases we get an extra factor (1 − r 2 ).Besides, ∂φ 1 /∂z i and ∂ 2 φ 2 /∂z i ∂z j are, as noted before, homogeneous of degree −n, and satisfy the cancellation condition (4.9).We will make use of the following well-known Schur's lemma for boundedness in L p of an integral operator with positive kernel.By homogeneity (y n = x n t) this reduces to which holds whenever m < αq < n − 1 − m.By symmetry, for (4.11) we need as well m < αp < n − 1 − m.Therefore, a choice of α is possible whenever m max(1/p, 1/q) < (n − 1 − n) min(1/p, 1/q), and this gives the range   Since |x − y| 1−n A (n−1)/2 = r 1−n , the kernel K differs from in a m-admissible kernel, so we keep this one.We write it as the sum of and another K 2 (x, y), which we estimate by = O(r 2−n (1 − |x| 2 ) n/p (1 − |y| 2 ) n/q A −n/2 ).
Write K Ω for the (euclidean) Calderón-Zygmund convolution operator with kernel |x − y| −n Ω((x − y)/(|x − y|)), which as it is well-known, satisfies an L p (dV )-estimate.Notice that and therefore, using the L p -boundedness of K Ω For K 2 , we can ignore the integrable singularity r 2−n and arguing as we just did with K 1 , we need to show that the integral operator satisfies L p (dV )-estimates for all p, 1 < p < ∞.To see this, just check that the criteria in Lemma 4.3 holds, with h(x) = (1 − |x| 2 ) −1/(pq) .
Ë Notice that in case m = 0 a m-Calderón-Zygmund kernel defines a bounded operator in all L p (H n ), 1 < p < ∞: this is the right analogue of the euclidian kernels, because (1 − r 2 ) n−1 is the typical growth at infinity of a weak L 1 (dµ) function in H n .

4.3.
Finally we make some comments, with no proofs, on the critical case m = (n − 1)/2 in the main theorem.

Ë 4 . 2 .Theorem 4 . 2 .
It remains to prove the following result.Both m-admissible and m-Calderón-Zygmund kernels define, by hyperbolic convolution, bounded operators in L p (H n ) for

|y|≤1 1 ( 1
− |x| + |x − y|) n f (y) dV (y) 3.1) will hold for an appropriate choice of a 0 .Taking into account the definition of A 3 in (3.3) and that |k m | |A 1 | + r 2 |A 2 |, we see from (3.7) that if m = n/2, this is accomplished by the choice