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Quaternionic Hardy spaces

The theory of slice regular functions of a quaternionic variable, introduced in 2006 by Gentili and Struppa, extends the notion of holomorphic function to the quaternionic setting. This fast growing theory is already rich of many results and has interesting applications. In this setting, the present paper is devoted to introduce and study the quaternionic counterparts of Hardy spaces of holomorphic functions of one complex variable. The basic properties of the theory of quaternionic Hardy spaces are investigated, and in particular a Poisson-type representation formula, the notions of outer function, singular function and inner function are given. A quaternionic (partial) counterpart of the classical $H^p$-factorization theorem is proved. This last result assumes a particularly interesting formulation for a large subclass of slice regular functions, where it is obtained in terms of an outer function, a singular function and a quaternionic Blaschke product.


Introduction
The theory of slice regular functions of a quaternionic variable (often simply called regular functions) was introduced in [22], [23], and represents a natural quaternionic counterpart of the theory of complex holomorphic functions. This recent theory has been growing very fast: a detailed presentation appears in the monograph [21], while an extension to the case of real alternative algebras is discussed in [26]. The theory of regular functions is presently expanding in many directions, and it has in particular been applied to the study of a non-commutative functional calculus, (see for example the monograph [13] and the references therein) and to contribute to the problem of the construction and classification of orthogonal complex structures in open subsets of the space of quaternions (see [20]).
Let H denote the skew field of quaternions. Set S = {q ∈ H : q 2 = −1} to be the 2−sphere of purely imaginary units in H, and for I ∈ S let L I be the complex plane R + RI, so that The natural domains of definition for slice regular functions are called slice symmetric domains (see [21]). In this paper we will consider slice regular functions defined on the open unit ball B = {q ∈ H : |q| < 1}, which is a particular example of slice symmetric domain. for all x + yI ∈ B I .
As shown in [23], if the domain of definition is the open unit ball B of H, the class of regular functions coincides with the class of convergent power series of type n≥0 q n a n , with all a n ∈ H. One of the most fertile chapters of the theory of complex holomorphic functions consists of the theory of Hardy spaces. This theory contains results of great significance that led to important general achievements and subtle applications. In this paper we define the quaternionic counterpart of complex Hardy spaces, and investigate their basic and fundamental properties. Our definition is the following. In Section 3 we study the main properties and features of the quaternionic H p -norms, we motivate the chosen definition and establish the initial properties of the quaternionic H p spaces. In Section 4 we investigate the boundary behavior of functions f in H p (B), obtaining that for almost every θ ∈ R, the limit exists for all I ∈ S and in this case it belongs to L p (∂B I ). We then investigate the properties of the boundary values of the * -product of two functions, each belonging to some H p (B) space. In section 5 possible analogues of outer and inner functions and singular factors on B are given, whose definitions (when compared with those used in the complex case) clearly resent of the peculiarities of the non commutative quaternionic setting. Our results here include factorization properties of H p functions. We are able to identify the Blaschke factor of a function f in H p (B), built from the zero set of f ; then we can exhibit a complete factorization result, in terms of an outer, a singular and a Blaschke factor, for a subclass of regular functions, namely for the one-slice-preserving functions. The cases of H 2 (B) (defined already in [1]) and H ∞ (B) are special cases, as in the complex setting. The Hardy spaces H 2 (B) have been also treated in [3,4,6,7], and the space H ∞ (B) briefly appears in [6]. The case of Hardy spaces H 2 defined on the half-space is considered in [4,7], where the aspect of the reproducing kernels is also investigated. Integral representations of slice hyperholomorphic functions have been recently studied in [17].
The "slicewise" approach has been also used in [14,15,16,18] to treat quaternionic Bergman spaces, and in [5,10] to introduce and discuss the analogs of Besov, Bloch, Dirichlet and Fock spaces. Hardy and Dirichlet spaces in the more general setting of Clifford valued monogenic functions are studied for example in [9].

Preliminaries
We recall in this section some preliminary definitions and results that are essential for the comprehension of the rest of the paper. All these preliminaries are stated, for the sake of simplicity, in the particular case of the open unit ball even if they hold in the more general setting of slice symmetric domains. The related proofs can be found in the literature, and in particular in [21]. Let us begin with a basic result that establishes the connection between the class of regular functions and the class of complex holomorphic functions of one complex variable.

Lemma 2.1 (Splitting Lemma
). If f is a regular function on B, then for every I ∈ S and for every J ∈ S, J orthogonal to I, there exist two holomorphic functions F, G : B I → L I , such that for every z = x+yI ∈ B I , we have f I (z) = F (z) + G(z)J.
We can recover the values of a regular function defined on B from its values on a single slice L I (see [12,11]).

Theorem 2.2 (Representation Formula
). Let f be a regular function on B and let I ∈ S. Then, for all x + yJ ∈ B with J ∈ S, the following equality holds In particular the function f is affine on each sphere of the form Thanks to the Representation Formula it is possible to prove that we can estimate the maximum modulus of a function with its maximum modulus on each slice, (see [25]). In the special case of a regular function f that maps a slice L I to itself, we obtain that the maximum (and minimum) modulus of f is actually attained on the preserved slice.

Proposition 2.3.
Let f be a regular function on B such that f (B I ) ⊂ L I for some I ∈ S. Then, for any Holomorphic functions defined on a disc B I , in the complex plane L I , extend uniquely to B.
extends f I to a regular function f : B → H. Moreover f is the unique extension and it is denoted by ext(f I ).
The pointwise product of two regular functions is not, in general, regular. To guarantee the regularity we have to use a different multiplication operation, the * -product, which extends the one classically defined for (polynomials and) power series with coefficients in a non-commutative field.
Its regular extension ext(f I * g I ) is called the regular product (or * -product) of f and g and it is denoted by f * g.
Notice that the * -product is associative but generally is not commutative. Its connection with the usual pointwise product is stated by the following result.
Proposition 2.6. Let f and g be regular functions on B. Then In particular f * g(q) = 0 if and only if f (q) = 0 or f (q) = 0 and g(f (q) −1 qf (q)) = 0.
The regular product coincides with the pointwise product for the special class of regular functions defined as follows.

Definition 2.7.
A regular function f : B → H such that f (B I ) ⊆ L I for all I ∈ S is called a slice preserving regular function.
Lemma 2.8. Let f, g be regular functions on B. If f is slice preserving, then f g is a regular function on B and f * g = f g = g * f .
The following operations are naturally defined in order to study the zero set of regular functions. Definition 2.9. Let f be a regular function on B and suppose that for z ∈ B I , the splitting of f with respect to J (J ∈ S orthogonal to I) is f I (z) = F (z) + G(z)J. Consider the holomorphic function Both f c and f s are regular functions on B.
We now recall a result (proven in [29]) that relates the norm of a regular function f : B → H and that of its regular conjugate.
where a ∈ B and u ∈ ∂B.
In particular, these transformations are regular bijections of the open unit ball onto itself and, by direct computation, it is possible to prove that they map the boundary of the unit ball to itself.

The quaternionic spaces H p (B)
In this section we give a definition of quaternionic Hardy spaces. The first natural step is to define an appropriate Hardy-type norm. Let p ∈ (0, +∞), r ∈ [0, 1), and I ∈ S. For any regular function f in the unit ball B, let (f I ) r be the function defined on the unit circle ∂B I by and let M p (f I , r) be the integral mean If p = +∞, set M ∞ (f, r) to be defined as for 0 < r < 1 (and |f (0)| for r = 0). Proof. The last part of the statement follows directly from the Maximum Modulus Principle for regular functions, [21]. To prove the first part, fix p ∈ (0, +∞) and I ∈ S. We will prove that the function is a subharmonic function, and then the statement will follow from classical results, see for instance Theorem 17.5 in [28]. In order to obtain the subharmonicity of g(z), we will first show that is a subharmonic function. Then, since g(z) = e ph(z) is the composition of an increasing convex function with a subharmonic function, we will conclude that also g is subharmonic (see e.g., Theorem 17.2 in [28]).
Suppose f ≡ 0 (otherwise the statement is trivially true). Then h(z) is an upper semicontinuous function. Moreover in the set {z ∈ B I | f I (z) = 0} the function h is subharmonic. In fact we have that where F, G are the splitting (holomorphic) functions of f I with respect to J ∈ S, J orthogonal to I. Hence (omitting the variable z) Schwarz inequality yields that h(z) is subharmonic where f I is non vanishing. It remains to show that h is still subharmonic in a neighborhood of each zero of f I . Recall that the zero set of a regular function intersected with a slice L I , is a discrete subset of L I (see Theorem 2.12). Then for any zero z 0 of f I , there exists a neighborhood U I ⊂ B I where z 0 is the only point where f I vanishes. Hence we have that for all r such that z 0 + B I (0, r) ⊂ U I the submean property is trivially satisfied Since this condition implies the subharmonicity of h near z 0 , we can conclude the proof.

Remark 3.2.
The previous result is the analogue of the first statement of the Hardy convexity Theorem in the complex setting, see Theorem 1.5 in [19].
for some k ∈ L I . In this case F and G are exponential (or constant) functions, Thanks to Proposition 3.1, we can give the following Definition.

Remark 3.5.
Notice that ||f || ∞ is the uniform norm of f on B. Moreover, if we set for any I ∈ S The set where the uniform norm is taken will be B I when considering the restriction f I (or its splitting components), and B when considering the function f . p ≥ 1 the function || · || p satisfies the triangle inequality, and hence it is a norm on H p (B). Moreover, the same relations of inclusions that hold for complex H p spaces, hold in the quaternionic setting. In fact, for any p, q such that 0 < p < q ≤ +∞, thanks to the classical Jensen inequality we have that The inclusion is continuous for 1 ≤ p < q ≤ +∞.
In analogy with the complex case, the space H 2 (B) is special. Indeed the 2-norm turns out to be induced by an inner product (see [2]).
n≥0 q n a n be its power series expansion. Then the 2-norm of f , By polarization we obtain that, for any b n a n = f, g that recalls the classic Hermitian product of the space H 2 (D).
Looking at the definition of complex Hardy spaces, one could wonder why, instead of the integral mean M p (f I , r) defined in (2), taken on a circle, we did not choose the classical integral mean taken on a 3dimensional sphere. In fact consider where σ 3 (rS 3 ) is the usual hypersurface measure of the 3-dimensional sphere rS 3 and 2π 2 r 3 is the 3dimensional volume of the 3-dimensional sphere rS 3 . If we then set it turns out that the class of regular functions f such that N p (f ) is finite does not coincide with the space H p (B) defined earlier. Indeed, using an appropriate change of variable in the integral appearing in (5), it is possible to prove that N p (f ) ≤ ||f || p for any regular function f and any To show that the two classes of functions do not coincide, it is possible to exhibit explicit examples. For instance, the function This example suggests us that a definition of a quaternionic H p space that relies upon the N p norm is not convenient. In fact, the function (1 − q) −1 has real coefficients, and therefore, its restriction to hence it would be weird if its regular extension were in the quaternionic H p space. Recently, a normalized integral mean on 3-spheres has been introduced to define a norm on H 2 (B), [8].
Our choice (2) of the p-integral mean allows the slicewise approach to the theory of H p spaces. Let us begin by studying how ||f || p of a regular function f is related with the value of ||f I || p on the slice L I .
Proof. Let p ∈ (0, +∞) and suppose first that ||f || p < +∞. Then trivially, for any I ∈ S, To see the other implication, let J ∈ S be such that ||f J || p is finite, and use the Representation Formula 2.2 to write where the last integral does not depend on I ∈ S. If p ≥ 1, taking into account the convexity of the map x → x p on the positive real axis, we get that On the other hand, if 0 < p < 1, taking into account the subadditivity on the positive real axis of the map x → x p (it is concave and maps 0 to 0), we have Let p = +∞. On one side, if f is bounded, then for any I ∈ S On the other side, using the Representation Remark 3.10. In particular, we get that if ||f J || p < +∞ for some J ∈ S, we have the following inequalities The key fact that allows us to apply classical results to the splitting components of a function in H p (B) is the following. Proof. Let I ∈ S and consider, for any z ∈ B I , the splitting f I (z) = F (z) + G(z)J. Then, for any z ∈ B I , Hence, for p ∈ (0, +∞), Notice that Remark 3.10 and Proposition 3.11 imply that, for all p ∈ [1, +∞], the space H p (B) is a Banach space.
The natural guess that if a function is in H p (B) then its regular conjugate is in H p (B) as well, is in fact true. Proof. If p = +∞ the proof follows directly by Proposition 2.10. Consider then p ∈ (0, +∞). For any I ∈ S, if f splits on B I as f (z) = F (z) + G(z)J, then, recalling Definition 2.9, we get that on the same slice f c can be written as Therefore, for 0 ≤ r < 1, If 0 < p < 2, thanks to the subadditivity on the positive real axis of the map x → x p/2 , we get Proposition 3.11 yields that both F and G belong to H p (B I ), hence Hence, as before, For the symmetrization of a function in H p (B) the following result holds true. Proof. Let f ∈ H p (B) for some 0 < p < +∞. For any r ∈ [0, 1), I ∈ S (such that f (re Iθ ) = 0) we have for some J ∈ S. Hence, recalling Proposition 2.10, for a suitable K(r, θ) ∈ S which depends on r, θ but does not depend on I (since S is compact). For such a K(r, θ) inequality (8) implies Let now L be any (fixed) imaginary unit independent of K(r, θ). Thanks to the Representation Formula 2.2 we can write and hence we have Then, if p ∈ (0, 1), we get that In both cases, taking the limit for r → 1 − , and the supremum in I ∈ S, we get that f s ∈ H p 2 (B). If instead f ∈ H ∞ (B), we observe that inequality (8) directly implies that, for any q ∈ B, The Hölder inequality leads to the following result concerning the * -product of regular functions.
For the remaining cases (p = 1, q = +∞ and p = +∞, q = 1) the proofs follow the same lines. Proof. If 1 p + 1 q = 1 and p ≥ 2, then 1 ≤ q ≤ 2. Therefore H p (B) ⊂ H q (B) and hence g belongs to H q (B). Theorem 3.14 leads to the conclusion.

Boundary values of regular functions
A very important result, that in the classic case is quite laborious to reach, states that all functions in H p (B) have radial limit along almost any ray. Proposition 4.1. Let f ∈ H p (B) for some p ∈ (0, +∞]. Then for any I ∈ S, the limit exists for almost every θ ∈ [0, 2π).
Proof. Let I ∈ S. Write the splitting of f on B I as f I (z) = F (z) + G(z)J, then Proposition 3.11 yields that the holomorphic functions F and G are both in H p (B I ). Classical results in the theory of H p spaces (see e.g. [27]) yield that the radial limits exist respectively for almost every θ and for almost every φ. Therefore the radial limit exists for almost every θ ∈ [0, 2π).

Remark 4.2.
The previous result is slightly stronger than its complex counterpart: in fact, if f ∈ H p (B) for some p ∈ (0, +∞], then on each slice, the radial limit of f exists along almost any ray. From now on, we will denote byf the radial limit of a function f ∈ H p (B). As it happens in the complex case, the functionf is measurable on ∂B.
exists for every I ∈ S. Namely, the radial limit of f exists at all points of the sphere cos θ + (sin θ)S (contained in the boundary of B) for almost every θ ∈ [−π, π).
Proof. Choose I, J ∈ S and use the Representation Formula for f Since for almost every θ ∈ [−π, π) the radial limitf exists both at e Iθ and at e −Iθ , then formulã leads to the conclusion.
Moreover, we will show that radial limits of a (non identically zero) function in H p (B) can not vanish on a subset of positive measure of the boundary of the ball. Proof. Choose any J ∈ S orthogonal to I. If f splits on B I as f I (z) = F (z) + G(z)J, then the splitting components F and G are in H p (B I ) and (thanks to the Identity Principle for regular functions, see [21]) at least one of them is not identically vanishing. Suppose that F ≡ 0 on B I . The classical result stated, e.g., in Theorem 17.18, [28], yields that lim r→1 − F (re Iθ ) = F (e Iθ ) = 0 for almost every θ ∈ [−π, π). Thanks to the orthogonality of I and J, we easily conclude that for almost every θ ∈ [−π, π), f (e Iθ ) = F (e Iθ ) + G(e Iθ )J = 0.
This easy consequence of the previous result will be used in the sequel.
Let us define the * -product and the * -inverse for radial limits. To this aim we prove the following statement.
Proof. Proposition 4.4 yields that if f is not vanishing identically, then for any I ∈ S,f (e Iθ ) = 0 for almost every θ ∈ [−π, π), and the same holds for f c . Set Thanks to Remark 4.5, T (re Iθ ) = f (re Iθ ) −1 re Iθ f (re Iθ ) is well defined for any I ∈ S, for almost every θ ∈ [−π, π), and for 1 − r sufficiently small. Moreover the radial limit exists for any I ∈ S and almost every θ ∈ [−π, π). Given any I ∈ S, if f (re Iθ ) = 0 set J(r, θ) = f (re Iθ ) −1 If (re Iθ ) ∈ S; then, for almost every θ the radial limit exists and belongs to S. Hence we can writẽ Using the Representation Formula 2.2 twice, for almost every θ we can write Hence, recalling Proposition 2.6, we have The same arguments apply also to the proof for the regular quotient.
We are now ready to give the announced definitions. for almost every θ. In particular, if g ≡ 1, we obtain the definition of the * -inverse off .
Thanks to the existence of the radial limit it is possible to obtain integral representations for functions in H p (B) for p ∈ [1, +∞].
Proof. The proof is an application of the corresponding results for holomorphic functions to the splitting components of f I .
Our next goal is to show that, for any p ∈ (0, +∞], the radial limitsf I of the restrictions of a function f in H p (B), are L p functions on the circle ∂B I .

Definition 4.9. Let g be a (quaternion valued) function defined (almost everywhere) on ∂B I and such that
|g| is measurable. If p ∈ (0, +∞) we set ||g|| L p to be the integral mean |g(e Iθ )|.
For any p ∈ (0, +∞], we denote by L p (∂B I ) the standard L p space, Let us now point out that, if (f I ) r (e Iθ ) = f (re Iθ ), then for all r ∈ [0, 1) and all I ∈ S.  ||f I || L p = ||f I || p .
Proof. The proof is a direct consequence of Proposition 4.10 and of the Poisson integral representation stated in Theorem 4.8.

Factorization theorems
In the classical setting it is possible to decompose a holomorphic function in H p (D) into its inner and outer factors, see Chapter 5 of [27]. The quaternionic counterparts are defined as follows. In the complex setting, the definition of outer function can be given equivalently in terms of a never-vanishing holomorphic function, expressed by means of an integral representation (see [27], Chapter 5). This correspondence fails to be true for regular functions, since in general we can not reproduce the same construction.

Definition 5.2. A regular function I ∈ H ∞ (B)
is an inner function if |I(q)| ≤ 1 for any q ∈ B and | I(q)| = 1 for almost any q ∈ ∂B.
We recall that, in the complex setting, each inner function can be factored into a product of two distinct types of inner functions, namely a Blaschke product and a singular function. Let us define the quaternionic analogues of singular functions, and then study the analogues of Blaschke products.

Definition 5.3. An inner function f ∈ H ∞ (B) is a singular function if f is non-vanishing on B.
In the complex setting there are two possible approaches to the factorization of an H p function (compare [19,27]). The first one is to begin by the extraction of the outer factor, thus obtaining the inner one. At this point, extracting the zeros one separates the Blaschke product and the singular part. The other possibility is to start with the extraction of the zeros, thus identifying the Blaschke product, and then separate the outer factor from the singular one. In the quaternionic setting, since we can not reproduce the construction of the outer factor, let us begin with the extraction of the zeros of a function f in H p (B) for p ∈ (0, +∞]. Thanks to the characterization of the zero set of regular functions (see Theorem 2.12), recalling Definitions 2.13 and 2.14, we can build a sequence representing the zeros of f . In analogy with the complex case, we can give the following two definitions (previously introduced in [2]). converges uniformly on compact subsets of B, then B is called Blaschke product, and it defines a regular function on B (see [21]).
Quaternionic Blaschke products are also treated in [3,4,7]. Blaschke products are examples of inner functions.
be a Blaschke product. Then |B(q)| ≤ 1 for any q ∈ B and | B(q)| = 1 for almost any q ∈ ∂B.
Proof. For k ∈ N consider the finite (regular) product B k , where T 0 (q) = q and T ℓ is defined iteratively (outside the zero set of B ℓ−1 ), in view of Proposition 2.6, by M aj (T j (q)).
for any ℓ = 1, . . . , k. Since |T n (q)| = |q| < 1 for any q ∈ B and since each factor M an is bounded in modulus by 1 on B (see Proposition 2.17), we get that we can bound each finite product, |M an (T n (q))| < 1.
Thanks to the uniform convergence on compact subsets of the finite products to B we get that |B(q)| ≤ 1 for any q ∈ B.
Hence B ∈ H ∞ (B) and therefore for any I ∈ S, for almost any θ ∈ [−π, π) there exists the radial limit The same clearly holds true for any finite product B k ∈ H ∞ (B). For any k ∈ N, the following regular function is a Blaschke product (and hence a bounded regular function) and uniformly on compact subsets of B. Observe that, for any k ∈ N, the finite regular product B k (as well as its regular conjugate B c k ) is regular up to the closure of B. Moreover, both B k and B c k only have finitely many zeros in the interior of B. Hence (see Proposition 2.15) the function is a diffeomorphism of a neighborhood (for instance a spherical shell) of ∂B onto itself, it maps the boundary of B onto itself and it has inverse given by Let I ∈ S. Proposition 2.15 yields that we can write Since B k is a finite Blaschke product, it maps ∂B to itself and for any θ. Hence, for any ε > 0 there exists r(ε) such that for any r(ε) ≤ r < 1 Therefore, for any ε > 0 there exists r sufficiently close to 1 such that where we use Proposition 3.1 and the fact that so that τ −1 k (re Iθ ) = r cos θ + (r sin θ)J k (r, θ). Using the Representation Formula 2.2 we can then write Since B k is a diffeomorphism of (a neighborhood of) ∂B onto itself, we get that for every θ the limit lim r→1 − J k (r, θ) =J k (θ) exists. Hence, recalling Proposition 4.6 and 4.11, applying twice the Representation Formula 2.2, we get whereτ −1 k is the radial limit of τ −1 k (see Proposition 4.6). Recalling inequality (12), we get then that for any Now we want to take the limit for k → +∞ of the previous inequality. For almost every θ, the limit which implies that J k (θ) converges for almost every θ. Using again the Representation Formula 2.2, we have then that Recalling that B − * k * B converges uniformly on compact sets to the function constantly equal to 1, and that J k (θ) converges, we obtain lim Therefore we get that, for any ε > 0, that finally implies B (e Iθ ) = 1 for almost every θ.
In order to show that the sequence of zeros of a function f in H p (B) is such that the Blaschke product associated with it is convergent, we will use classical results in the theory of complex H p spaces, that apply to the symmetrization of f .

Remark 5.8.
Recall that the symmetrization f s of a regular function f behaves exactly as a holomorphic function on any slice L I . Hence, if f s is in H p (B) (and therefore in H p (B I )) for some p ∈ (0, +∞], and f s ≡ 0, classical results (see e.g. Theorem 15.23 in [28]) yield that, if {a I n } n≥0 is the sequence of zeros of f s in B I , listed according to their multiplicity, then the Blaschke condition (1 − |b n |) < +∞.
Proof. Let us consider the symmetrization of f , f s . Thanks to Proposition 3.13 and Remark 5.8, for any I ∈ S, if {a I n } n∈N is the sequence of zeros of f s on B I , then it satisfies the Blaschke contidion. If b n = b 1,n + b 2,n I n , let us set b I n = b 1,n + b 2,n I for all n ∈ N. Then |b n | = |b I n | for all n ∈ N, and {b I n } n∈N ⊆ {a I n } n∈N . Therefore (recalling that |a I n | < 1) The previous result implies that the Blaschke product built from the zeros of a regular function f ∈ H p (B) for some p ∈ (0, +∞] does converge uniformly on compact sets (compare with [2]).
Proposition 5.10. Let f be in H p (B) for some p ∈ (0, +∞] and let {a n } n≥0 be its sequence of zeros. If M an (q) denotes the Blaschke factor associated with a n , M an (q) = (1 − qa n ) − * * (a n − q) a n |a n | (M an (q) = q if a n = 0), then the Blaschke product converges uniformly on compact sets of B. Moreover, the function B is regular on B.
Proof. In [21] the convergence of infinite quaternionic *products of regular functions is presented in detail.
In particular the convergence of B(q) is equivalent to the convergence of n≥0 |1 − M an (q)|. With this in mind, the proof can be found in [2].
We point out that the convergence of B(q) depends only on the moduli |a n |, n ∈ N. This means that we can build a Blaschke product B(q) having exactly the same sequence of zeros {a n } n∈N of a given regular function f ∈ H p (B) (see also [2]). In fact, in order to build such a Blaschke product B(q), we have to consider the product of Blaschke factors associated with suitable conjugates of the points a n , lying on the same 2-spheres x n + y n S generated by a n , taking into account Proposition 2.6. Proof. We will give the proof in the case in which all the zeros (both isolated and spherical) have multiplicity 1. In this case, we can assume, without loss of generality, that a j = a k for all j, k ∈ N. Our aim is now to build a sequence {b n } n∈N , where each b n is a conjugate of a n , such that the Blaschke product associated with it has {a n } n∈N as its sequence of zeros. The convergence of the Blaschke product is guaranteed by Proposition 5.10. Since the regular multiplication does not conjugate the zeros of the first function in the * -product, the first term of the sequence will be equal to a 0 , For the second term, we need to find b 1 such that vanishes at q = a 1 . Notice that, for any k ∈ N, and, thanks to Proposition 2.6, we can write Therefore, if we want that this product vanishes at a 1 , we can set well defined since a 1 = a 0 . We can iterate this process, setting, for any n ≥ 1, vanishes at q = a n , we have to set b n = T n (a n ), well defined since a n = a n−1 . In the case in which some of the zeros of f have multiplicities greater than 1, the proof follows the same lines; one has only to take into account that: has the same modulus of β n , and each β n is contained in the sequence of zeros of a function in H p (B).
is a regular function that has only spherical zeros, and its sequence of zeros is {β n , β n } n∈N . Therefore, if we set then we can write for some function h, never vanishing and regular on B. To prove the regularity of h, it suffices to observe that since B β has real coefficients, it is regular and it has exactly the same zeros of f β * B β (q), then the regular quotient is well defined (and regular) on the entire ball B. Consider the regular conjugate of B β , and * -multiply on the right by B c β all terms of equality (13). We obtain that can also be written as Now notice that B s β (q) = B β (q) because they both are Blaschke products associated to the same spherical zeros. Therefore we infer . Setting g(q) = B α * B c β (q) leads to the conclusion of the proof.
Once "extracted" the zeros of a function in H p (B), we would like to identify its outer factor and its singular part. Proposition 5.14. Let f ∈ H p (B) for some p ∈ [1, +∞] be such that f − * ∈ H q (B) where 1 p + 1 q = 1. Then f is an outer function.
Since f − * ∈ H q (B) then f has no zeros and T f is a diffeomorphism of B, yielding |g(q)| ≤ |f (q)| for any q ∈ B. We thus conclude that f is an outer function. For regular functions that preserve a slice L I , the factorization can be done in a stronger and more satisfactory fashion, that very much resembles its complex counterpart. namely the outer factor of F I . From factorization results in the complex setting, (see for instance [27]), we know that we can write F I (z) = E I (z)I I (z) where I I (z) is the inner factor of F I . In particular, since both E I and I I map B I to L I , we can also write where the last equality is due to the Identity Principle for regular functions, [21]. Let us set E(q) = ext(E I )(q) and I(q) = ext(I I )(q). Since E I (z) = 0 for all z ∈ B I , Proposition 2.11 yields that also E is never vanishing on B. To estimate the modulus of I, recall that a function that maps the slice L I to itself has the following properties (see Proposition 2.3) and min J∈S |I(x + yJ)| = min{|I(x + yI)|, |I(x − yI)|} = min{|I I (x + yI)|, |I I (x − yI)|} for all x, y such that x + yI ∈ B I . By classical results, the inner function I I is bounded in modulus by 1 and its uniform norm equals 1, therefore we get, thanks to (14) ||I|| ∞ = sup q∈B |I(q)| = sup z∈BI |I I (z)| = ||I I || ∞ = 1.
Therefore, for almost every x + yJ such that x 2 + y 2 = 1, max J∈S I(x + yJ) = min J∈S I(x + yJ) = 1 namely for almost every q ∈ ∂B, | I(q)| = 1. To obtain the wanted properties of the modulus of E, let us denote by T the transformation T (q) = (E c (q)) −1 qE c (q).
Since E c is non-vanishing on B, T is a diffeomorphism of B with inverse T −1 (q) = (E(q)) −1 qE(q).
The study of the behavior of the modulus | E| at the boundary, requires some more effort. First of all, classical results on outer functions (see e.g. [27]) imply that E I ∈ H p (B I ). Then, thanks to Proposition 3.9 we get that E ∈ H p (B). Therefore, for any J ∈ S, the function E has radial limit E for almost any x + yJ ∈ ∂B J . Hence, also the transformation T −1 (q) = (E(q)) −1 qE(q) does. This allows us to prove that T −1 maps almost every 2-sphere x + yS ⊂ ∂B one-to-one onto itself. In fact, for almost every sphere x+ yS ⊂ ∂B the function E is defined at all points of x+ yS (see Proposition 4.3). For such a sphere x+ yS, thanks to the Representation Formula 2.2 and to the fact that E preserves L I , there exist b, c ∈ L I such that E(x + yJ) = b + Jc for any J ∈ S.
Take x + yK ∈ x + yS, with K = I. We are going to show that we can find the only J ∈ S such that T −1 (x + yJ) = x + yK. This is possible if and only if E(x + yJ) solves our problem. Thanks to this property of the map T −1 , we get that 1 = | I( T −1 (q))| for almost every q ∈ ∂B. The radial limit version of equation (16), holding for almost every q ∈ ∂B, 1 = | I( T −1 (q))| = | E(q)| −1 |f (q)| leads to the equality | E(q)| = |f (q)| for almost every q ∈ ∂B, which leads also to ||E|| p = || E|| L p = ||f || L p = ||f || p . We want to show now that E is an outer function. To this aim, let g ∈ H p (B) be such that | E(q)| = |g(q)| for almost all q ∈ ∂B. If we restrict E and g to B I , and recall the definition of E I , we can write (see Proposition 4.3) log |E I (re Iθ )| = 1 2π π −π P r (θ − t) log |f I (e It )|dt = 1 2π π −π P r (θ − t) log |g I (e It )|dt ≥ log |g I (re Iθ )| where P r (t) is the Poisson kernel and where last inequality is due to the subharmonicity of log |g I | (see the proof of Proposition 3.1). As a consequence we get that 1 ≥ |E I (z)| −1 |g I (z)| = |E − * I * g I (z)| = |(E − * * g) I (z)| for all z ∈ B I . Since E − * * g is regular on B, using Proposition 3.9 we obtain that E − * * g belongs to H ∞ (B). Then the radial limit of E − * * g exists at almost every point of ∂B. Equation (18) and Proposition 4.6 guarantee that for every J ∈ S lim r→1 − |E − * * g(re Jθ )| = | E − * * g(e Jθ )| = 1 for almost every θ ∈ [−π, π). Hence, by Proposition 4.11, Recalling now that T −1 (q) = (E(q)) −1 qE(q) is a diffeomorphism of B, thanks to Proposition 2.15 we obtain 1 ≥ E − * * g(T −1 (q)) = E −1 (q)g(q) = E(q)| −1 |g(q) i.e., |E(q)| ≥ |g(q)| for all q ∈ B. This concludes the proof. for all q ∈ B.
Proof. Theorem 5.16 allows us to factor f = E * I as a * -product of an outer function E and an inner function I. Theorem 5.13 guarantees now the existence of a Blaschke product B and a non-vanishing function S such that I = S * B. Following the lines of the proof of Theorem 5.16 that led us to show that the outer factor of f belongs to H p (B), one can prove that S belongs to H ∞ (B) and hence is a singular function.
We point out that the Beurling-Lax type Theorem in [2], and the Krein-Langer type factorization theorem that appears in [3,4], are naturally connected to our factorization results.