Entire monogenic functions of given proximate order and continuous homomorphisms

Infinite order differential operators appear in different fields of mathematics and physics. In the past decade they turned out to play a crucial role in the theory of superoscillations and provided new insight in the study of the evolution as initial data for the Schr\"odinger equation. Inspired by the infinite order differential operators arising in quantum mechanics, in this paper we investigate the continuity of a class of infinite order differential operators acting on spaces of entire hyperholomorphic functions. Precisely, we consider homomorphisms acting on functions in the kernel of the Dirac operator. For this class of functions, often called monogenic functions, we introduce the proximate order and prove some fundamental properties. As important application we are able to characterize infinite order differential operators that act continuously on spaces of monogenic entire functions.


Introduction
Infinite order differential operators have been studied already since a long time.In the recent years they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial data for the Schrödinger equation.Superoscillatory functions arise in several areas of science and technology, for example in quantum mechanics they are the outcome of Aharonov's weak values, see [1,8].To study their time evolution as initial data of quantum field equations represents an important problem in quantum mechanics.
The study of the evolution of superoscillatory functions under the Schrödinger equation is highly nontrivial.A natural functional analytic setting is the space of entire functions with certain growth conditions.In fact, the Cauchy problem for the Schrödinger equation with superoscillatory initial data leads to infinite order differential operators of the type where the coefficients u m (t, z) depend on the Green's function of the time dependent Schrödinger equation for a given potential V , t is the time variable and z is the complexification of the space variable.For more details, see for example the monograph [6] and [2,3,4,7,14,16,17,30].
Another application of infinite order differential operators within the scope of the theory of superoscillatory functions is the extension of this theory to several superoscillating variables, see [5].
For p ≥ 1 the natural spaces on which operators such as U(t, z; ∂ z ) act are the spaces of entire functions with growth order either order lower than p or equal to p and finite type.In other words, they consist of entire functions f for which there exist constants B, C > 0 such that |f (z)| ≤ Ce B|z| p .
The problem to extend infinite order differential operators to the hypercomplex setting is treated in the recent paper [9] where the authors investigate the continuity of a class of infinite order differential operators acting on spaces of entire hyperholomorphic functions that include monogenic functions.In [9] we find the hypercomplex version of some results obtained in [10,13]; the class of monogenic functions is the most delicate case to investigate.Even though the classical exponential function is not in the kernel of the Dirac operator, monogenic functions with exponential bounds play a crucial role in the study of continuity of a class of infinite order differential operators in the hypercomplex settings.The hypercomplex setting is non-trivial and requires some efforts because of the structure of monogenic functions and of the fact that they admit series expansions in terms of the so called Fueter polynomials.Precisely, the Fueter's polynomials V k (x), see for instance formula (4) of [21] page 794 or elsewhere, are defined by where k represents a multi-index.They play the role of the sequence of the complex polynomials (z n ) n when z is a complex variable.In order to preserve the monogenicity we need a special product, called Cauchy-Kowalewski product, for short CK-product, that does not coincide with the ordinary pointwise product.The CK-product of two left entire monogenic f and g is denoted by f ⊙ L g is given in Definition 2.4 in terms of the Fueter polynomials.A similar definition is given for right monogenic functions.
In the paper [9] we obtained the following result regarding monogenic functions.Let p ≥ 1 and set N 0 = N ∪ {0}.Let (u m ) m∈(N0) n : R n+1 → R n be left entire monogenic functions such that for every ε > 0 there exist B ε > 0, C ε > 0 for which where 1/p + 1/q = 1 and where we set 1/q := 0 when p = 1, and m is a multi-index.We considered the formal infinite order differential operator for left entire monogenic functions f where ∂ m x := ∂ m1 x1 . . .∂ mn xn and ⊙ L denotes the CK-product.Then for p ≥ 1, we proved that the operator U L (x, ∂ x ) acts continuously on the space of left monogenic functions with the condition |f (x)| ≤ Ce B|x| p .
In this paper we characterize the continuous homomorphisms of type (1.2) acting on monogenic functions.In order to do this we introduce proximate orders for monogenic functions and we study some fundamental properties.After that we investigate the monogenic counterpart of the differential operator representation of continuous homomorphisms between the spaces of entire functions of given proximate order proved by T. Aoki and co-authors in [12,11].
The plan of the paper.Section 1 provides an introduction.In Section 2 we state some preliminary results on monogenic functions.In Section 3 we study entire monogenic functions where the growth is determined by a proximate order.We establish some important properties on the proximate order of monogenic functions.In Section 4 we then apply these results to characterize the continuous homomorphisms.We observe that Section 3 and Section 4 closely follow Section 3 and Section 4 of [11].The points where the adaptation to the case of monogenic functions are not straightforward are Theorem 3.4, Theorem 4.4 and Theorem 4.5.In particular, in the last two theorems we have used the new Lemmas 3.12 and 3.13.

Preliminary results on monogenic functions
In this section we recall some results on monogenic functions, whose proofs can be found in [15].We recall that R n is the real Clifford algebra over n imaginary units e 1 , . . ., e n .The element (x 0 , x 1 , . . ., x n ) ∈ R n+1 will be identified with the paravector x = x 0 + x = x 0 + n ℓ=1 x ℓ e ℓ and the real part x 0 of x will also be denoted by Re(x).An element y in R n , is called a Clifford number.If A is an element in the power set P (1, . . ., n), then the element e i1 . . .e ir can be written as e i1...ir or, in short, e A .Thus, we can write a Clifford number as y = A y A e A .Possibly using the defining relations e 2 i = −1, e i e j + e j e i = 0, i, j ∈ {1, . . ., n}, i = j, we will order the indices in A as i 1 < . . .< i r .When A = ∅ we set e ∅ = 1.The Euclidean norm of an element y ∈ R n is given by |y| We will restrict ourselves to consider only the set of the left monogenic functions since for right monogenic functions analogous computations hold.
We will denote the set of left entire monogenic function by M L (R n+1 ).The Cauchy formula for monogenic functions and their derivatives, is computed on the boundary of U ⊂ R n+1 where U is contained in the set of monogenicity of f .
where A n+1 is the n-dimensional surface area of the n + 1-dimensional unit ball, (−1) j e j dξ j and q 0 (x If f is a left monogenic function in a ball x < R, then for all x < r with 0 < r < R, its Taylor series expansion is given by where V m are the Fueter polynomials and where q m (x) = ∂ m1 x1 . . .∂ mn xn q 0 (x).We have the following Cauchy inequality m! and M (r, f ) = sup x =r f (x) .In particular, setting m := (m 0 , . . ., m n ) ∈ N n+1 0 , we have Using their Taylor series representation and we define We mention that a different product, that is defined just for the subclass of axially monogenic function (see [23] and [24]), can also be used to define infinite order differential operators in the monogenic setting.
Next, following [21], we recall the definition of the standard growth order of an entire monogenic function: Definition 2.5.Let f : R n+1 → R n be an entire left monogenic function.Then its growth order is said to be where log + (r) = max{0, log(r)}.
It may occur that 0 ≤ ρ ≤ +∞.For the case where 0 < ρ < +∞ Constales et al. defined in [19] the growth type of an entire monogenic function of growth order ρ by As shown in [18] the growth order of a monogenic function can be directly computed by its Taylor coefficients, namely by Similarly, as shown in [19] the growth type can be expressed by Remark 2.6.To also get a finer classification of functions with slow growth ρ = 0 and fast growth ρ = ∞ Seremeta [31] and Shah [32] introduced the notion of generalized growth in the complex analysis setting which has been generalized to the monogenic setting in [22,26,27,33].More precisely the authors considered functions α(•), β(•) and γ(•) satisfying particular conditions mentioned concisely for example in Definition 1 of [22] and introduced the notions of the generalized growth and type in the way and (see for instance Definition 2 of [22] or [27]).In the particular case where α(r) = log(r) and where β is the identity function one re-obtains the classical definition of the growth order and growth type.Another generalization of the classical growth order that includes the classical growth order as a special case, and even the generalized growth orders under particular conditions, is the definition of the proximate growth order.
We observe that for any proximate order function ρ(r) there exists a positive constant r 0 > 0 such that for any r > r 0 the function r ρ(r) is strictly increasing and tending to +∞.Definition 2.8.For any proximate order function ρ(r) we can always take another proximate order function, called the normalization of the proximate order function ρ(r), ρ(r) such that there exists a constant r 1 > 0 for which ρ(r) = ρ(r) for any r ≥ r 1 and r ρ(r) is strictly increasing on r > 0 and maps the interval (0, +∞) to (0, +∞).
We denote by ϕ : (0, +∞) → (0 + ∞) the inverse function of the function t = r ρ(r) .Moreover, we set Let ρ(r) be a proximate order for a positive order ρ > 0. For any σ > 0, we consider the Banach space with the norm • ρ,σ .Moreover, we use the following notation: for any m = (m 1 , . . ., m n ) ∈ N n we write We recall some basic properties of proximate orders that will be used in the following.The proofs of these results can be found in Section 2 of [11] and in the references therein.Lemma 2.9.There exist constants k > 0 and B > 0 depending only on ρ such that Precisely speaking, we can choose k depending only on the order ρ = lim r→+∞ ρ(r) Lemma 2.10.The sequence {G p } p is supermultiplicative, that is, G p G q ≤ G p+q , for any p, q ∈ N.
Lemma 2.11.For every δ > 0 with δ < 1 ρ , there exists T 0 > 0 such that if t ≥ T 0 , we have Lemma 2.12.For u, t, σ > 0 we define Then for any σ ′ with 0 < σ ′ < σ, there exists T 1 such that Keeping in mind the above results we can now introduce the notion of monogenic functions of proximate order and we can study some properties.

Some properties of monogenic functions of proximate order
In the following we will use some results on monogenic entire functions contained in [18,19,20].We prove some new properties of entire slice monogenic functions that appear here for the first time to the best of the knowledged of the authors.Some of the difficulties in proving our results relay on in the series expansion of these functions in terms of the Fueter polynomials.
Proof.We will show that B := {f ∈ A ρ,σ1 : f ρ,σ1 ≤ 1} is relatively compact in A ρ,σ2 , i.e., any sequence {f j } j∈N ⊂ B has an accumulation point with respect to the norm of A ρ,σ2 .
First we will prove that any sequence {f j } j∈N ⊂ B admits a convergent subsequence in the uniform convergence topology to an entire monogenic function.By the Arzelá-Ascoli Theorem it is sufficient to prove that {f j } j∈N is equicontinuous and uniformly bounded in any compact convex set K ⊆ R n+1 .We fix a compact convex subset K of R n+1 .Since {f j } j∈N ⊂ B, the sequence is uniformly bounded int K.Moreover, we have f j (x) − f j (y) ≤ C j x − y , where ∇ is the usual gradient, C j = sup x∈K ∇f j (x) and x, y ∈ K.We choose r large enough in a such way that K ⊂ B(0, r).Thus, there exists a positive constant C K which only depends on K such that for any x ∈ K and for any j ∈ N, we have where C 1 and C 2 are suitable positive constants while in the first inequality we have used (2.1).In particular, for any x, y ∈ K and for any j ∈ N we have i.e., {f j } j∈N is equicontinuous.After taking a subsequence if necessary, we can suppose that the sequence {f j } j∈N converges to f ∈ M L (R n+1 ) in the topology of the uniform convergence.Now we will prove that the sequence {f j } j∈N is a Cauchy sequence in A ρ,σ2 .We fix δ > 0 and we observe that for any R > 0 we have With respect to the second supremum, we can choose R > 0 large enough such that: Thus, since f j , f ℓ ∈ B, we have that Moreover, with respect to the first supremum, by the uniform convergence of the sequence {f j } j∈N on the compact subset of R n+1 , there exists a positive integer N such that for any j, ℓ ≥ N we have Thus we have proved that the sequence {f j } j∈N is a Cauchy sequence.
Remark 3.3.The spaces A ρ and A ρ coincide with each other and they share the same locally convex topologies as well, and the same holds for A ρ,σ+0 and A ρ,σ+0 .
For any fixed f ) be a left monogenic entire function of finite order ρ > 0 and of proximate order ρ(r).Then its type σ with respect to ρ(r) is given by Proof.First we prove that and suppose that w q ∈ R n+1 be such that P q (w q ) = K q and w q = 1.There exists b ∈ R n (see [25,Theorem 3.20]) such that where z i ∈ R n+1 and z i = 1 for any i = 1, . . .r, • w q := b(xe 1 ) b for some x ∈ R with |x| ≤ 1 where b := zr • • • z1 .The function g(x) := bf (bx b) is a left monogenic entire function (see [29]) whose Taylor series is We define Pq (x) : Thus we have where m q = (q, 0, . . ., 0).If σ > σ then for r large we have: and ln(K q ) ≤ ln(c(n, m q )) + σr ρ(r) − q ln(r).
If q is large enough, then we define r q to be the real number such that q = σ • ρ • (r q ) ρ(rq) and we have ϕ q σρ = r q .Thus, for q large enough, dividing by q and summing log(ϕ(q)) to both side of inequality (3.2), we have By [28, (1) Theorem 1.23] we have Moreover, since lim q→+∞ (c(n, m q )) 1 q = 1, taking the lim sup to both side of (3.3), we have lim sup q→+∞ ln ϕ(q)K 1 q q ≤ ln (eσρ) for any σ > σ.Thus we have proved that .
The other side of the inequality follows as in [28, (3) Theorem 1.23] by the properties of ϕ.
Proof.We have that f ∈ A ρ,σ+0 if and only if for any ǫ > 0 there exists a D ǫ > 0 such that
Proof.We have Combining the previous estimate with Lemma 3.5 we get that for any ǫ > 0 there exists q 0 > 0 such that for any q > q 0 we have Conversely, if we have (3.5), then for any ǫ > 0 there exists C > 0 and q 0 > 0 such that for any q > q 0 we have Thus, we have lim sup The following proposition is a direct consequence of the previous corollary.
We now need some estimates of norms of Fueter polynomials.
Note that for any given σ ′ > 0, we can take T 2 ≥ T 1 : Thus we have for |m| = q ≥ T 2 and x ≤ ϕ(T 1 ).By (3.6) and (3.7) for any x ∈ R n+1 and for any |m| = q ≥ T 2 , we have We know that there exists C > 0 such that V m (x)e −σr ρ(r) ≤ C for any |m| < T 2 and for any x ∈ R n+1 .Thus V m (x)e −σr ρ(r) G −1 ρ,q ≤ C ′ (σ ′ ) −q/ρ where Lemma 3.9.There exists a constant k depending only on ρ for which the following statement holds: For any σ > 0, we can take C(σ) such that for any f ∈ A ρ,σ , and any m ∈ N n 0 , the inequality holds.Here we write q = |m|.
Proof.There exists a constant k > 0 which depends on ρ and there exists a constant B > 0 which depends on ρ(r) such that (r + s) ρ(r+s) ≤ k(r ρ(r) + s ρ(s) ) + B for all r, s > 0.
The Cauchy estimates give us, for x ≤ r and |m| = q the chain of inequalities ≤ e Bσ f ρ,σ exp kσr ρ(r) c(n, m) ρ/q e q/ρ ϕ(q) q ϕ(q) ϕ(q/(kσρ)) where the last two inequalities are obtained as in [11,Lemma 3.8].In particular we have which implies the inequality in the statement.= n there exists a constant C(n) which depends only on n such that for any m ∈ N n we have Thus, in Lemma 3.9 the estimate can be rewritten in the following way In view of the above stated properties, we can now prove the following crucial results.
Proposition 3.11.For an entire left monogenic function f (x) belonging to A ρ,σ+0 , its Taylor expansion m∈N n V m (x)a m converges to f (x) in the space A ρ,n ρ σ+0 .In particular, the set of Fueter polynomials is dense in A ρ,+0 and also dense in A ρ .
Proof.For the former statement, it suffices to show that is finite for any ǫ > 0. By Lemma 3.8 we have On the other hand, by Corollary 3.6, we have max |m|=q a m G ρ,q ≤ C 1 (n ρ (σ + ǫ/4)) q/ρ .Therefore, we have , where the last series is convergent.For the latter statement in the case f ∈ A ρ,+0 , it follows from the former one with σ = 0 that lim in the space A ρ,+0 .In the case f ∈ A ρ , there exists σ > 0 such that f ∈ A ρ,σ+0 .Then the same convergence holds in the space A ρ,n ρ σ+0 and therefore also in the space A ρ .
Lemma 3.12.Let f ∈ A ρ,σ and f = ∞ |m|=0 V m (x)f m .Let s be a real positive number.Then, for any η > 0 there exists C η > 0 such that for any m ∈ N n 0 , we have Proof.We have that where in the first inequality we have used the maximum modulus principle, in the second inequality we have used the Cauchy inequalities in the ball centered at x with radius s x , in the third inequality we have used the fact that all the balls centered at x with x = r of radius s x are contained in the ball centered at 0 with radius (s + 1)r and in the last inequality we have used the fact that for any η > 0 there exists a positive constant C η such that . Let δ be a positive constant then for any η > 0 there exists C(n, η, τ 1 , τ 2 ) > 0 such that Proof.Choosing s = (n + δ) in Lemma 3.12, we have where in the last inequality we used the fact that The results of this section will be used to characterize continuous homomorphisms in terms of differential operators in the sense that will be specified in the next session.

Differential operators, representations of continuous homomorphisms
Before studying continuous homomorphisms between A ρi (i = 1, 2) and those between A ρi,+0 (i = 1, 2), we define a differential operator representation of homomorphisms from We define the space of formal right linear differential operators of infinite order with coefficients in R n [[x]] by , by the correspondence Proposition 4.1.There are two left linear isomorphisms: where the first and second mappings are given by respectively.
Proof.We can easily see that both mappings are injective left linear mappings between vector spaces and that their composition is given by Therefore, it suffices to show that the relations induce a bijection This follows from the fact that the relation (4.1) can be inverted as In fact, we can calculate the p-element of the image of (u m (x)) m∈N n 0 by composition of (4.2) and (4.1) as which implies that the composition is the identity.We can similarly show that the composition of (4.1) and (4.2) is the identity.Now we study continuous homomorphism from A ρ1 and A ρ2 and those from A ρ1,+0 and A ρ2,+0 where ρ i (r) (i = 1, 2) are two proximate orders for positive orders ρ i = lim r→∞ ρ i (r) > 0, satisfying Definition 4.2.Let ρ i (i = 1, 2) be two proximate orders for orders ρ i > 0 satisfying (4.3).We take normalization ρ1 of ρ 1 as in Definition 2.8 and G ρ1,q by (2.2).We denote by D ρ1→ρ2 and by D ρ1→ρ2,0 the sets of all formal right linear differential operator P of the form where the multisequence (u m (x)) and respectively.Note that in the latter case, each u m belongs to A ρ2,+0 .
For the following remark see also [11,Remark 4.4].
Since a proximate order and its normalization define equivalent norms, we have Proof.We can replace ρ i by ρi (i = 1, 2) as in Remark 4.3, and we may assume from the beginning that ρ i (i = 1, 2) are normalized orders satisfying ) for any f and τ > 0.
(1) We fix η > 0 and C η > 0 in a such a way that ≤ C(n, η, σ(ǫ), kτ ) In view of Remark 3.10, the last sum converges if ǫ < (C(n)(2kτ ) 1/ρ1 ) −1 .For such a choice of ǫ > 0 depending on ρ 1 and τ , we set and Then, |m| u m (x) ⊙ L ∂ m x f converges in A ρ2,τ ′ (τ ) and defines an element Since f ∈ A ρ1,τ was chosen arbitrarily, this implies the well-definedness and the continuity of P : A ρ1,τ → A ρ2,τ ′ (τ ) .Also since τ > 0 was chosen arbitrarily, we get the well-definedness and the continuity of P : A ρ1 → A ρ2 by the definition of the inductive limit of locally convex spaces.
(2) Let F : A ρ1 → A ρ2 be a continuous right linear homomorphism.Then, thanks to the theory of locally convex spaces, we can conclude, using Lemma 3.1, that for any τ > 0, there exists τ ′ = τ ′ (τ ) > 0 such that F (A ρ1,τ ) ⊂ A ρ2,τ ′ (τ ) and that is continuous.Therefore, we can in particular take C(τ ) depending on τ > 0 for which Let us define a multi-sequence of entire functions (u m (x)) m∈N n 0 by whose convergence in A ρ2 will be proved together with their estimates.We define a formal differential operator P ∈ D of infinite order by First we show that P ∈ D ρ1→ρ2 .For any fixed τ 0 and τ 1 with 0 < τ 0 < τ 1 , we have

.10)
Here we used Lemma 3.13 at the first inequality, (4.6) and (4.8) at the second inequality, Lemma 3.8 with 0 < τ 0 < τ 1 at the third inequality, and Lemma 2.10 at the fourth inequality.For a given ǫ > 0, we can take τ 0 > 0 large enough such that Then, by choosing τ 1 as τ 1 > τ 0 and by putting we have for any m, which implies P ∈ D ρ1→ρ2 .Now we show that P f = F f for any f ∈ A ρ1 .First we show that the equality holds for f ∈ R n [x].This will imply the equality holds for any f ∈ A ρ1 since R n [x] is dense in A ρ1 by Proposition 3.11 and P and F are continuous Theorem 4.5.Let ρ i (r) (i = 1, 2) be two proximate orders for orders ρ i > 0 satisfying (4.3). ( converges and P f ∈ A ρ2,+0 .Moreover, f → P f defines a continuous homomorphism P : A ρ1,+0 → A ρ2,+0 .(2) Let F : A ρ1,+0 → A ρ2,+0 be a continuous right linear homomorphism.Then there is a unique P ∈ D ρ1→ρ2,0 such that F f = P f holds for any f ∈ A ρ1,+0 .
Proof.Again we can make a substitution of proximate orders as in Remark 4.3, and we may assume from the beginning that ρ i (i = 1, 2) are normalized proximate orders satisfying (4.6).