Quasianalytic functionals and ultradistributions as boundary values of harmonic functions

We study boundary values of harmonic functions in spaces of quasianalytic functionals and spaces of ultradistributions of non-quasianalytic type. As an application, we provide a new approach to H\"ormander's support theorem for quasianalytic functionals. Our main technical tool is a description of ultradifferentiable functions by almost harmonic functions, a concept that we introduce in this article. We work in the setting of ultradifferentiable classes defined via weight matrices. In particular, our results simultaneously apply to the two standard classes defined via weight sequences and via weight functions.


Introduction
The representation of functions and linear functionals as boundary values of harmonic functions is an important and useful idea in functional analysis.For analytic functionals such a representation follows e.g. from Bengel's work [1] (see also [7]) on formal boundary values of zero solutions of elliptic operators and leads to an elementary proof of the support theorem for analytic functionals.Moreover, it may be used to develop a harmonic function approach to the theory of hyperfunctions in several variables, which is reminiscent of the simple one variable theory.We refer to [10,12,13,22] for more information on this subject.In [12] Komatsu studied boundary values of harmonic functions in ultradistribution spaces of non-quasianalytic type (see [14] for the distribution case).
The main goal of this article is to enhance these results by developing a theory of boundary values of harmonic functions in spaces of quasianalytic functionals (= compactly supported quasianalytic ultradistributions) [9].Our ideas give rise to a new approach to the support theorem for quasianalytic functionals, originally shown by Hörmander in [9] (see also [8]).Hörmander's proof of this result is quite involved.We believe that the approach given here is conceptually simpler, supplying a description of the support (= minimal carrier) of a quasianalytic functional in terms of the harmonic continuation properties of its Poisson transform.Furthermore, in the non-quasianalytic case, we obtain alternative proofs of Komatsu's results [12].Our method allows us to work under much weaker assumptions on the defining weight sequence (see Remark 4.14 for details).
A simple but powerful method to ensure the existence of (ultra)distributional boundary values of holomorphic functions consists in combining Stokes' theorem (more precisely, the formula [10, Equation (3.1.9),p. 62]) with the notion of almost analytic extensions.This technique was used for the first time by Hörmander [10, p. 64] for distributions and was later extended to the ultradistributional setting by Petzsche and Vogt [18] (see also [17]).We mention that the characterization of Denjoy-Carleman classes by almost analytic extensions goes back to Dyn'kin [3,4].We refer to the recent article [6] for the newest generalizations of such results and an overview of the topic of almost analytic extensions.Here we develop a similar method to establish the existence of ultradistributional boundary values of harmonic functions.Our method combines Green's theorem with a novel description of ultradifferentiable functions by so-called almost harmonic functions.Therefore, the first part of this article is devoted to an almost harmonic function characterization of ultradifferentiable classes.
We work with the notion of ultradifferentiability defined via weight matrices, as introduced in [19].This leads to a unified treatment of ultradifferentiable classes defined via weight sequences [11] (Denjoy-Carleman approach) and via weight functions [2] (Braun-Meise-Taylor approach), but also comprises other spaces, e.g., the union and intersection of all Gevrey spaces.We point out that we shall infer the weight function case from the weight matrix case by employing the method from [6], which is based upon results from [19,20,21].
Finally, we fix some notation.Let Ω ⊆ R d be open.We write K ⊂ comp Ω to indicate that K is a compact subset of Ω.The notation Θ ⋐ Ω means that Θ is a relatively compact open subset of Ω.We write H(Ω) for the space of harmonic functions on Ω and endow it with the compact-open topology.Points of R d+1 = R d ×R will be denoted by (x, y) = (x 1 , . . ., x d , y).We will often identify R d with the subspace R d × {0} of R d+1 .If V ⊆ R d+1 is open and symmetric with respect to y, we write H − (V ) for the space of harmonic functions in V that are odd with respect to y.

Ultradifferentiable classes
2.1.Denjoy-Carleman classes.Let M = (M p ) p∈N be a sequence of positive numbers.We set m p = M p /M p−1 , p ∈ Z + .Furthermore, we define M * = (M p /p!) p∈N and m * p = M * p /M * p−1 = m p /p, p ∈ Z + .We will make use of the following conditions on a positive sequence M: We refer to [11] for the meaning of the standard conditions (M.1), (M.2) ′ , and (M.1) * .Condition (M.1) * w is inspired by [21,Lemma 8].A sequence M of positive numbers is called a weight sequence if M 0 = 1, lim p→∞ m p = ∞, and M satisfies (M.1).A weight sequence and quasianalytic otherwise.The relation N ⊂ M between two weight sequences M and N means that there are C, H > 0 such that N p ≤ CH p M p , p ∈ N. The stronger relation N ≺ M means that the latter inequality remains valid for every H > 0 and a suitable C = C H > 0. We write N ≈ M if both N ⊂ M and M ⊂ N hold.
We shall also use the following condition on a weight sequence M: Each non-quasianalytic weight sequence satisfies (NA).
The associated function of a positive sequence M is defined as Let M be a weight sequence and let Ω ⊆ R d be open.For h > 0 we write B M,h (Ω) for the Banach space consisting of all ϕ ∈ C ∞ (Ω) such that We define and The space E {p!} (Ω) coincides with the space A(Ω) of real analytic functions in Ω.
From now on we shall write [M] instead of (M) or {M} if we want to treat both cases simultaneously.In addition, we shall often first state assertions for the Beurling case (= (M)-case) followed in parentheses by the corresponding ones for the Roumieu case (= {M}-case).
Given two weight sequences M and N, we have that Let M be a weight sequence.For K ⊂ comp R d and h > 0 we write Classes defined by weight matrices.Following [6] (see also [19]), we define a weight matrix as a non-empty family M of weight sequences that is totally ordered with respect to the pointwise order relation ≤ on sequences.We will make use of the following conditions on a weight matrix M: The conditions (M.2) ′ and {M.2} ′ are denoted by (M (dc) ) and (M {dc} ) in [19], respectively.The conditions (M.1) * w and {M.1} * w were introduced in [21] but no name was given to them there.A weight matrix M is said to be non-quasianalytic if each M ∈ M is non-quasianalytic.
The relation N(⊂)M (N{⊂}M) between two weight matrices M and N means that We write In the Beurling case, it is clear that N = N also satisfies (NA) (is non-quasianalytic, respectively).In the Roumieu case, there exists N 0 ∈ N that satisfies (NA) (is nonquasianalytic, respectively).The result then holds for Let M be a weight matrix and let Ω ⊆ R d be open.We define and Given two weight matrices M and N, we have that Let M be a weight matrix and let K ⊂ comp R d .We set For Ω ⊆ R d open we define The space ), is convex.We refer to [2] for the meaning of these conditions.A weight function ω is called non-quasianalytic if ∞ 0 ω(t) 1 + t 2 dt < ∞ and quasianalytic otherwise.Each non-quasianalytic weight function ω satisfies ω(t) = o(t).We also consider the following condition on a weight function ω : (α 0 ) ∃C > 0 ∃t 0 > 0 ∀λ ≥ 1 ∀t ≥ t 0 : ω(λt) ≤ Cλω(t).By [19,Theorem 6.3] (see also the proof of [18, Proposition 1.1]), a weight function ω satisfies (α 0 ) if and only if there is a concave weight function σ such that ω ≍ σ (meaning that ω(t) = O(σ(t)) and σ(t) = O(ω(t))).
Let ω be a weight function and let Ω ⊆ R d be open.For h > 0 we write B ω,h (Ω) for the Banach space consisting of all ϕ ∈ C ∞ (Ω) such that We define and For ω(t) = max{t − 1, 0} the space E {ω} (Ω) coincides with A(Ω).
Let ω be a weight function.For K ⊂ comp R d and h > 0 we write D ω,h K for the Banach space consisting of all ϕ For Ω ⊆ R d open we define The space D [ω] (Ω) is non-trivial if and only if ω is non-quasianalytic.Given a weight function ω, we associate to it the weight matrix M ω = (M h ω ) h>0 , where the weight sequence M h ω = (M h ω,p ) p∈N is defined by We , and (NA).For a weight function ω satisfying ω(t) = o(t) we define The function ω ⋆ is decreasing and convex.Given h > 0, we set h ω(t) = hω(t) and ω h (t) = ω(ht).Then, For a weight sequence M satisfying (NA) it holds that [18, Proof of Lemma 5.6] (cf. [20, Lemma 3.10]) Lemma 2.4.Let ω be a weight function satisfying ω(t) = o(t).
(i) For all M ∈ M ω and h > 0 there are C, k > 0 such that (ii) For all k > 0 there are M ∈ M ω and C, h > 0 such that (2.3) holds.
(iii) For all M ∈ M ω and h > 0 there are C, k > 0 such that (iv) For all k > 0 there are M ∈ M ω and C, h > 0 such that (2.4) holds.

Ultradifferentiable classes via almost harmonic functions
Let Ω ⊆ R d be open and let ϕ 0 , ϕ 1 : Ω → C. The Cauchy-Kovalevski theorem implies that ϕ 0 , ϕ 1 ∈ A(Ω) if and only if for all Θ ⋐ Ω there exist The goal of this section is to characterize the classes E [M] (Ω) and D [M] (Ω) in a similar way by almost harmonic functions.Namely, we shall show the following two results.
The proofs of Theorem 3.1 and Theorem 3.2 are divided into several intermediate results.
Proposition 3.3.Let M, N, and Q be three weight sequences satisfying (NA).Suppose that Q satisfies (M.1) * and Moreover, there is C > 0 such that for all ϕ 0 , ϕ For ϕ 0 , ϕ 1 ∈ B {p!} (Θ) the series (cf.[10, p. 330]) The idea of the proof of Proposition 3.3 is to suitably modify the series in (3.3).This approach is inspired by Petzsche's construction of almost analytic extensions by means of modified Taylor series [17, Proposition 2.2].Furthermore, in our estimates we follow the same technique as in [6, Proposition 3.12], which is essentially due to Dyn'kin [3,4].
We claim that: where δ j,k denotes the Kronecker delta.These properties imply that belongs to C 2 (Θ × R) and satisfies (i) and (ii).We now prove the above claims.In the rest of the proof C will denote a positive constant that is independent of ϕ but may vary from place to place.We introduce the following auxiliary function Fix 0 < t ≤ 1/q * 1 .Then, p ≤ Γ(t) if and only if tq * p < 1 for all p ∈ Z + .Hence, the function p → t p Q * p is decreasing for p ≤ Γ(t) and increasing for p ≥ Γ(t).Consequently, ) .We start by showing (3.5).Note that ∆Φ j = S 1 + S 2 + S 3 , where For all (x, y) ∈ Θ × (R\{0})) with |y| small enough, we have that Likewise, for all (x, y) ∈ Θ × (R\{0})) with |y| small enough, one gets Next, we show (3.6) and (3.7).We only treat the case j = 0 as the case j = 1 is similar.Let α ∈ N d , |α| ≤ 2, be arbitrary.For all (x, y) ∈ Θ × R\{0} with |y| small enough it holds that Similarly, for all α ∈ N d , |α| ≤ 1, and (x, y) ∈ Θ × R with |y| small enough, we have Finally, (3.2) follows from an inspection of the estimates in the proofs of (3.5)-(3.7).
Proposition 3.4.Let M, N, and Q be three non-quasianalytic weight sequences satisfying (3.1).Suppose that Q satisfies (M.1) * .There is A > 0 such that for all K ⊂ comp R d and ε, h > 0 the following holds: For all ϕ 0 , ϕ Moreover, there is C > 0 such that for all ϕ 0 , ϕ 1 ∈ D M,h K max{|||Φ(ϕ 0 , ϕ 1 )|||, max Proposition 3.5.Let M be a weight sequence satisfying (NA).There is A > 0 such that for all V ⊆ R d+1 open and h > 0 the following holds: Let Φ ∈ C 2 (V ) be such that We need some preparation for the proof of Proposition 3.5.Consider the following fundamental solution of the Laplacian and for d > 1 where c d+1 denotes the area of the unit sphere in R d+1 .The Poisson kernel is given by We need the following bounds for the derivatives of E and P .
(i) There are C, H > 0 such that and for d > 1 (ii) There are C, H > 0 such that Proof.We only show (ii) as (i) can be treated similarly.We will use the following property of harmonic functions (cf.[5, p. 29, Thm.7]): There are C, H > 0 such that for all w ∈ R d+1 and r > 0 for all functions U that are harmonic in a neighborhood of B(w, r).Fix (x, y) ∈ R d+1 \{0} and α ∈ N d .By applying the above inequality to w = (x, y), r = |(x, y)|/2, β = (α, 0) and U = P , we obtain The result now follows from the inequality Proof of Proposition 3.5.We only show the statement for ∂ y Φ as the one for Φ can be shown similarly.Set V ∩ R d = Ω and ϕ = ∂ y Φ |Ω .Fix an arbitrary x 0 ∈ Ω and choose r > 0 such that B d+1 (x 0 , r) ⋐ V .Note that for some U ∈ H(B d+1 (x 0 , r)).Hence, Since U ∈ H(B d+1 (x 0 , r)), we have that Lemma 3.6(ii) and (3.9) yield that ψ ∈ C ∞ (B d (x 0 , r)) with and that there are C, H > 0 such that for all α ∈ N d and x ∈ B d (x 0 , r).Since x 0 was arbitrary, this proves the result.

Corollary 3.8. Let ω be a non-quasianalytic weight function satisfying
Then, ϕ 0 , ϕ 1 ∈ D [ω] (Ω) if and only if for all h > 0 (for some h > 0) there exists ) such that K is a carrier of f .The space A ′ (K) may be characterized in terms of harmonic functions, as we now proceed to explain.We follow Hörmander's exposition [10, Section 9.1] (see also [12,13,22]).
Let K ⊂ comp R d .The Poisson transform of f ∈ A ′ (K) is defined as Recall that H − (R d+1 \K) stands for the space of harmonic functions in R d+1 \K that are odd with respect to y.We denote by H 0,− (R d+1 \K) the space consisting of all F ∈ H − (R d+1 \K) such that F (x, y) → 0 as (x, y) → ∞.
Remark 4.3.Let K ⊂ comp R d and let V be an open R d+1 -neighborhood of K that is symmetric with respect to y.Let F ∈ H − (V \K) and consider the associated f ∈ A ′ (K) from Corollary 4.2.By Green's theorem (cf. the proof of Proposition 4.4 below), we have ) and ρ ∈ D(V ) even with respect to y such that ρ ≡ 1 on an R d+1 -neighborhood of K. Hence, f may be interpreted as the boundary value of F in A ′ (K) and we write f = bv(F ).(Ω) the strong dual of E [M] (Ω).We have once again that the space of entire functions is dense in E [M] (Ω) (cf.[9, Proposition 3.2]), we therefore obtain that E ′ [M] (Ω) may be viewed as a subspace of A -carrier of f .We have the following canonical isomorphism of vector spaces We endow E ′ [M] (K) with the projective limit topology induced by this isomorphism.Suppose that M is non-quasianalytic.Given Ω ⊆ R d open, we denote by D ′ [M] (Ω) the strong dual of D [M] (Ω).

Boundary values of harmonic functions in E
′ [M] (K).Let M be a weight sequence satisfying (NA).Let V ⊆ R d+1 be open and symmetric with respect to y and let where d S (x, y) denotes the distance from (x, y) to S. We set Next, let M be a weight matrix satisfying (NA).Let K ⊂ comp R d and let V be an open R d+1 -neighborhood of K that is symmetric with respect to y. Choose a sequence (V n ) n∈N of relatively compact open sets in R d+1 that are symmetric with respect to This definition is independent of the chosen sequences (V n ) n∈N and (K n ) n∈N .For two weight matrices M and N with M[≈]N we have that − (V \K) as locally convex spaces.
Let K ⊂ comp R d and let V be an open R d+1 -neighborhood of K that is symmetric with respect to y. Recall from Remark 4.3 that we employ the notation f = bv(F ) for the analytic functional corresponding to a harmonic function F ∈ H − (V \K) via the relation (4.1).We now show that the elements of

and this quasianalytic functional may be represented as follows: For all
where χ ∈ D(Ω) is such that χ ≡ 1 on a neighborhood of K.Moreover, the boundary value mapping bv : As stated in the introduction, we shall show Proposition 4.4 by combining Green's theorem with our description of ultradifferentiable functions by almost harmonic functions (Proposition 3.3).This method is suggested by (4.1) (see Remark 4.3).
Proof of Proposition 4.4.We only consider the Beurling case as the Roumieu case can be treated similarly.By Lemma 2.1, we may assume that each M ∈ M satisfies (M.1) * .
Fix an arbitrary open subset Ω ⊆ V ∩ R d with K ⊂ comp Ω and let χ ∈ D(Ω) be such that χ ≡ 1 on a neighborhood of K. Choose Θ ⋐ Ω with piecewise smooth boundary such that supp χ ⊂ comp Θ.Let r > 0 be such that Θ × (−r, r) ⋐ V .Pick L ⊂ comp Θ such that K ⊂ comp int L and χ ≡ 1 on a neighborhood of L. It suffices to show that for all N ∈ M and k > 0 there is M ∈ M such that (4.2) bv where bv Ω (F ), ϕ = lim By applying Green's theorem to the pair (F y , ρΦ) on the region Θ × (0, r), we obtain Note that there is C > 0 (independent of ϕ) such that and there is C > 0 such that . Therefore, (3.2) implies that the mapping in (4.2) is well-defined and continuous.

Our next goal is to study the Poisson transform of elements of E
′ [M] (K).To this end, we need to introduce some additional spaces of harmonic functions.Let M be a weight sequence satisfying (NA).Let K ⊂ comp R d .Recall that H 0,− (R d+1 \K) stands for the space of harmonic functions on R d+1 \K that are odd with respect to y and vanish at infinity.For h > 0 we write H M,h ∞,0,− (R d+1 \K) for the Banach space consisting of all F ∈ H 0,− (R d+1 \K) such that Next, let M be a weight matrix satisfying (NA).Let This definition is independent of the chosen sequence (K n ) n∈N .
Proof.We only consider the Beurling case as the Roumieu case is similar.It suffices to show that for all Ω ⋐ R d with K ⊂ comp Ω and all M ∈ M there is N ∈ M such that ∞,0,− (R d+1 \Ω) is well-defined and continuous.Choose N ∈ M such that for some C 0 , H 0 > 0. By Theorem 4.1(i), we have that ∞,0,− (R d+1 \Ω).By the Banach-Steinhaus theorem, there are for all f ∈ B. Lemma 3.6(ii) therefore implies that for all (x, y for all f ∈ B. We are ready to prove the main result of this article.(i) Let V be an open R d+1 -neighborhood of K that is symmetric with respect to y.
Then, the sequence is exact.Moreover, the boundary value mapping is continuous and it has the Poisson transform − (V \K) as a continuous linear right inverse.(ii) The boundary value mapping bv : is a topological isomorphism whose inverse is given by the Poisson transform (ii) This follows from part (i), Proposition 4.5, and Liouville's theorem for harmonic functions.
We set Let K ⊂ comp R d and let V be an open R d+1 -neighborhood of K that is symmetric with respect to y.The spaces H [ω] − (V \K) and H [ω] 0,− (R d+1 \K) are defined in the natural way.Corollary 4.7.Let ω be a weight function satisfying (α 0 ) and ω(t) = o(t).Let K ⊂ comp R d .
(i) Let V be an open R d+1 -neighborhood of K that is symmetric with respect to y.
Then, the sequence is exact.Moreover, the boundary value mapping is continuous and it has the Poisson transform − (V \K) as a continuous linear right inverse.
(ii) The boundary value mapping bv : is a topological isomorphism whose inverse is given by the Poisson transform 0,− (R d+1 \K).Proof.This follows from Lemma 2.3, Lemma 2.4, and Theorem 4.6.

Application:
The support theorem for quasianalytic functionals.A fundamental result in the theory of analytic functionals states that each f ∈ A ′ (R d ) has a unique minimal carrier, called the support of f and denoted by supp A ′ f .Martineau [15] (see also [16]) showed this by using cohomological properties of the sheaf of germs of analytic functions.Theorem 4.1 may be used to give a simpler proof of the existence of supp A ′ f (cf.[10,Theorem 9.1.6]).In fact, by Theorem 4.1, a compact set K in R d is a carrier of f if and only if its Poisson transform P [f ] can be continued as a harmonic function to R d+1 \K.Hence, supp A ′ f is given by the compact set K ⊂ R d with the property that R d+1 \K is the largest open set in R d+1 on which P [f ] has a harmonic extension and, in particular, this notion is well-defined.
The existence of a unique minimal carrier can also be established for quasianalytic functionals, but the only known treatment in the literature, due to Hörmander [9], turns out to be much harder.Howeover, in view of Theorem 4.6, we can now repeat the simple reasoning involving the harmonic continuation of the Poisson transform to directly infer the ensuing support theorem for E It should be noted that Theorem 4.8 contains the corresponding support theorem for E where ω is a weight function satisfying (α 0 ) and ω(t) = o(t), which was earlier obtained by Heinrich and Meise in [8] via the method from [9] (without the assumption (α 0 )).We end this subsection with two remarks.Remark 4.9.Hörmander [9] showed the support theorem for E ′ {M } (R d ), where M is a weight sequence satisfying (M.2) ′ and (NA).His technique can be adapted to show that Theorem 4.8 is still valid if one removes the hypothesis [M.1] * w from its statement.We omit details since it is out of the scope of this article.Remark 4.10.Suppose that M is a non-quasianalytic weight matrix.The assignment This definition is independent of the chosen sequence (V n ) n∈N .For two weight matrices M and N with M[≈]N we have that H − (V \Ω) as locally convex spaces.We now show that the elements of H Then, bv(F ) belongs to D ′ [M] (Ω).Moreover, the boundary value mapping bv : Proof.This can be shown in a similar way to Proposition 4.4 but by using Proposition 3.4 instead of Proposition 3.3.
Next, we show an ultradistributional version of the Schwarz reflection principle.
Then, F extends to a harmonic function on V .
Proof.Let Θ ⋐ Ω be arbitrary and choose r > 0 such that Θ × (−r, r) ⋐ V .It suffices to show that F extends to a harmonic function on Θ × (−r, r).Since ∆ is elliptic, it is enough to show that there is F ∈ D We now show the claim.We only consider ( F + η ) 0<η<ε as ( F − η ) 0<η<ε can be treated similarly.We have lim Proof.The boundary value mapping is well-defined and continuous by Proposition 4.11, while Proposition 4.12 yields that ker bv = H − (V ).Next, we show that the boundary value mapping is surjective.To this end, we shall use some basic facts about the derived projective limit functor (see the book [24] for more information).Choose a sequence (V n ) n∈N of relatively compact open sets in R d+1 such that V n ⋐ V n+1 , V n+2 \V n has no connected component that is relatively compact in V n+2 and V = n∈N V n .Set Ω n = V n ∩ R d .We need to show that the mapping bv : H . . . . . . . . .
The boundary value mapping and the Poisson transform are well-defined and continuous by Proposition 4.4 and Proposition 4.5, respectively.Theorem 4.1(i) and Remark 4.3 yield that P is a right inverse of bv.Finally, the equality ker bv = H − (V ) follows from Corollary 4.2 and Remark 4.3.

Theorem 4 .
6 particularly applies to E ′ [M ] (K), where M is a weight sequence satisfying (M.2) ′ , (M.1) * w , and (NA).Finally, we give two representations of E ′ [ω] (K) by boundary values of harmonic functions.Let ω be a weight function satisfying ω(t) = o(t).Let V ⊆ R d+1 be open and symmetric with respect to y and let S ⊆ V ∩ R d be closed in V .For h > 0 we write H ω,h ∞,− (V \S) for the Banach space consisting of all

Theorem 4 . 8 .
Let M be a weight matrix satisfying [M.1] * w , [M.2] ′ , and (NA).For each K). Hence, there exists a unique minimal [M]carrier for each f ∈ E ′ [M] (R d ), which is well-known to coincide with supp A ′ f (cf.[11, Lemma 7.4]), a fact that also follows from Theorem 4.13 below.4.5.Boundary values of harmonic functions in D ′ [M] (Ω).Let M be a nonquasianalytic weight matrix.Let Ω ⊆ R d be open and let V ⊆ R d+1 be open and symmetric with respect to y such that V ∩ R d = Ω.Choose a sequence (V n ) n∈N of relatively compact open sets in R d+1 that are symmetric with respect to y such that

Proposition 4 . 12 .
Let M be a non-quasianalytic weight matrix satisfying [M.2] ′ .Let Ω ⊆ R d be open and let V ⊆ R d+1 be open and symmetric with respect to y such that
In such a case, we can find a non-quasianalytic weight matrix N ⊆ M such that M{≈}N and, thus, D {M} (Ω) = D {N} (Ω).