Representation Theorems for the Cauchy Transform in Weighted Bergman–Dirichlet Spaces

In this paper we analyse the integrability of Cauchy transforms of functions and measures. As an application of general Paley–Wiener theorems, we show that the integrability of the Cauchy transform of a function or measure is deeply related to the integrability of the Fourier transform of the corresponding function (measure). Our main results are representation theorems for the Cauchy transform in weighted spaces of Bergman–Dirichlet type.


Introduction
The classical Paley Wiener Theorem (see [12] or [15]) states that a function g which is analytic in the upper half-plane + = {z ∈ C : Im(z) > 0} is the Fourier-Laplace transform of some f ∈ L 2 (0, ∞), i.e. g(z) = 1 2π presented in [2] a Paley-Wiener theorem for Bergman spaces. In particular, they stated that the Fourier-Laplace transform establishes an isometric isomorphism between certain weighted L 2 spaces on (0, ∞) and corresponding weighted Bergman spaces on + . As a consequence, they deduced related results for weighted Dirichlet spaces. A few years later, Harper [5] generalized their result. While Duren et al. considered weights of the form ω(t) = t −β where β > 0, Harper only required that the weights should satisfy a special integrability condition of local type. This concept led to important sets of functions, so called Zen spaces. These spaces have been studied before in [4] and some of the most recent theory can be found in [7] and [5]. In 2016, Peloso and Salvatori [13] seized on the Theorem of Harper and proved a Paley-Wiener theorem for Zen spaces using the same technique.
Until now, the Paley-Wiener theorems have various applications, for example in control theory, signal analysis, complex dynamics and even for differential equations. Duren et al. gave in [2] an application concerning the shift operator on the unweighted Bergman space on the disc. Harper presented in [5] a relation to the Cauchy problem for the heat equation. Partington et al. [7] considered so called Laplace-Carleson embeddings and gave answers to the question when the Fourier-Laplace transform maps weighted L 2 spaces on (0, ∞) boundedly into weighted L 2 spaces on + . Kucik applied his results in [9] to control and observation operators for linear evolution equations.
In this paper, we want to present another application of these generalized Paley-Wiener theorems, the integrability of Cauchy transforms of measures and functions. This problem has already been considered before in [10], but from a different point of view. More precisely, the authors characterized the boundedness of the Cauchy transform as an operator from L 2 (R, σ ) to L 2 ( + , τ ) for measures σ on R and τ on + . Our approach is different. We will put our focus on the Cauchy transform of measures and functions belonging to L p (R) for p ∈ [1,2] and only partly investigate the problem of boundedness. Instead we answer the question what conditions on the measure or function imply the integrability of the Cauchy transform and conversely. In this context, we shall use an important relationship between the Cauchy transform and the real Fourier transform to our advantage. As a consequence, we deduce representation theorems for the Cauchy transform in weighted Bergman-Dirichlet spaces.
The paper is scheduled as follows: We start with some preliminaries and then formulate our main results in the third part. For the proofs we will need some preparation. In the fourth part we will therefore introduce Zen spaces which are related to weighted Bergman spaces. Furthermore, we will prove some facts in context with Cauchy transforms. In the fifth part we use general Paley-Wiener theorems to show integrability theorems for the Cauchy transform of measures and functions. We will focus on two important sets of measures on the upper half plane which have already been considered in [5] and [13]. In the sixth part we discuss the boundary behaviour of Cauchy transforms and deduce several lemmas which are crucial for the proofs of our main results presented in the last section. We conclude with an application of our results to certain inclusion mappings. For future research related to the topic of this paper the results in [6] may be a suitable connecting factor.

Preliminaries
In order to formulate our main results we shall need some notation. Let H ( ) denote the space of all analytic functions on an open set ⊂ C and for a function F ∈ H ( ) we denote by F (n) the n-th derivative of F. As above, we write + for the upper half plane and − for the lower half-plane. For p ∈ [1, ∞), H p ( + ) denotes the Hardy space on + which consists of all functions F ∈ H ( + ) such that Here, λ 1 denotes the Lebesgue measure on R. The space H p ( − ) is defined similarly. In this paper, we will often consider the Banach space For p ∈ [1, ∞), a measurable subset A ⊂ R and a positive Borel measure μ on R we denote by L p (A, μ) the subspace of all functions in L p (R, μ) which have support in A. Moreover, 1 A denotes the indicator function on A, i.e.
M(R) is the set of all complex Borel measures on R and μ denotes the total variation norm of a complex Borel measure μ which is by definition finite. If μ ∈ M(R), we write μ for the Fourier transform of μ, i.e.
Clearly, μ is a continuous and bounded function. If μ = f λ 1 where f ∈ L 1 (R), then we simply write f instead of f λ 1 and speak of the Fourier transform of f .
If f ∈ L 2 (R), then its Fourier transform f is defined by Plancherel's theorem and we have f ∈ L 2 (R). Moreover, if f ∈ L p (R) for some p ∈ (1, 2), then there are functions f 1 ∈ L 1 (R) and f 2 ∈ L 2 (R) such that f = f 1 + f 2 and in this case, the Fourier transform of f is simply defined by f := f 1 + f 2 , see e.g. [11], p. 273. The Hausdorff-Young theorem (see [11], p. 273) implies that f ∈ L q (R) where q is the conjugate exponent of p.
Let μ ∈ M(R). Then, the function Cμ : is called the Cauchy transform of μ. Moreover, we say that a measurable function f : In this case, the function C f : is called the Fourier-Laplace transform of f . There is a strong connection betwen Fourier-Laplace transforms and Cauchy transforms. The corresponding result can be found in [11], p. 226 and p. 275.

Theorem 2.1
The following statements hold:

Main Results
In the following, we say that a positive measure ν belongs to class M exp if it has support in [0, ∞) and for n ∈ N 0 . We shall consider two important subsets of M exp . The first one is the class M eps consisting of all measures ν = ωλ 1 where ω : [0, ∞) → [0, ∞] satisfies the ε-condition, that is ω > 0 almost everywhere and for each 0 This condition appears in [5] and is there a crucial condition on the occuring weights, see Theorem 5.4 below. The second subset of M exp that we consider is the set M 2 which consists of all positive measures ν which satisfy the 2 -condition, i.e.
Of particular interest to us is the subclass M 2 ,0 which is the set of all ν ∈ M 2 such that ν({0}) = 0. The 2 -condition has already been studied earlier in harmonic analysis and partial differential equations, see [7], p. 4. It will also fit for our purposes since it has found a frequent use in context of Fourier-Laplace transforms, see [13].
It is important to notice that we neither have , then ωλ 1 satisfies the 2 -condition, but not the ε-condition. Conversely, the function ω(x) = 1 x (x ∈ (0, ∞)) satisfies the ε-condition, but the measure ωλ 1 does not fulfill the 2 -condition since it is not locally finite.
The following two theorems are our main results. It should be remarked that it is also possible to consider integrability conditions on the lower half plane − by modifying the weights w ν,n in an appropriate way.
In this case, f is unique and we have , w ν,n λ 1 ) and in this case In this case, f is unique and we have , w ν,n λ 1 ) and in this case The proofs of Theorem 3.1 and Theorem 3.2 are given simultaneously in Sect. 7 and are consequences of several results in the subsequent sections.

Zen Spaces
The following useful characterization of Zen spaces can be found in [13]:

and in this case
and in this case Proof Let μ ∈ M(R). Since μ1 (0,∞) ∈ E 1,+ ∩ E 2,+ (notice that μ is bounded), Theorem 2.1, Lemma 4.4 and the monotone convergence theorem imply that The second statement is therefore clear for p = 1. If f ∈ L 2 (R), then f 1 (0,∞) ∈ E 1,+ ∩ E 2,+ by Hölder's inequality and the boundedness of the mapping t → e −at 1 (0,∞) for each a > 0. If f ∈ L p (R) where p ∈ (1, 2), then we write . Since E 1,+ and E 2,+ are vector spaces over C we see that f 1 (0,∞) ∈ E 1,+ ∩ E 2,+ as well. The rest of the proof is similar to the case of measures.

Lemma 4.6
Let p ∈ (1, ∞), r ∈ [1, 2] and n ∈ N 0 . Then, the following statements hold: by Jensen's inequality. But by the substitution x = t − ys we see that The second statement is therefore clear for p = 1. If f ∈ L 2 (R), then we use Theorem 2.1 and Lemma 4.4 and see that for a > 0 we have Thus, (C f ) (n) ∈ H 2 (ia + + ) for each n ∈ N 0 and a > 0. If f ∈ L r (R) where r ∈ (1, 2), then there are functions f 1 ∈ L 1 (R) and f 2 ∈ L 2 (R) such that f = f 1 + f 2 and we conclude by the linearity of the Cauchy transform that (C f ) (n) ∈ H 2 (ia + + ) for each a > 0.
The following corollary is crucial for the subsequent section.

Integrability Theorems for Weighted Bergman and Dirichlet Spaces
We start with a lemma which is an analogue of a result in [9], p. 4.
The latter observations are crucial to illustrate the relation between Cauchy transforms and real Fourier transforms. One could ask for a converse of the previous result. If we put extra conditions on the measure, then a converse statement holds. One of the main ingredients of the proof is a theorem of Paley-Wiener type which is due to Harper and can be found in [5], p. 236-237.

Theorem 5.4 (Harper) Let
In this case, g is unique and
The following theorem of Paley-Wiener type is due to Peloso and Salvatori and can be found in [13], p. 62.
In this case, g is unique and Remark 5.8 Theorem 5.7 states in particular that for α > −1 But due to the injectivity of L and Remark 5.6 we conclude that A 2 μ α ( + ) = Z 2 μ α ( + ) for all α > −1. In particular, the unweighted Bergman space coincides with the unweighted Zen space. Now, we are ready to prove converse statements to Corollary 5.3.

Boundary Behaviour of Cauchy Transforms and Applications
For the proofs of our main theorems we shall need several lemmas which consider important properties of Cauchy transforms. The first one attaches the boundary behaviour of Cauchy transforms and can partly be found in [11], p. 318. Lemma 6.1 (Plemelj formulae) Let p ∈ [1, ∞) and f ∈ L p (R). Then for almost every x ∈ R. In particular, for almost every x ∈ R. Thus, the Cauchy transformation is an injective operator on p≥1 L p (R).
defines a complete norm on H + p (R) ⊕ H − p (R). Thus, Lemma 6.5 implies that for p ∈ (1, ∞) the norms · ⊕, p and · p are equivalent and the same holds for · ⊕,1 and · H 1 (R) .

Proof of Theorem 3.1 and Theorem 3.2
The uniqueness is clear from the injectivity of the Cauchy transform, see Lemma 6.1.
. Lemma 6.5 gives us the necessity. Conversely, fix F ∈ H p (C \ R). By Lemma 6.2, there exist functions . Therefore, and the surjectivity is clear by Lemma 6.5.
Due to Theorem 3.1, for p ∈ (1, 2], the Cauchy transform is also a bijective map between the space A similar statement can be said for the case p = 1. It is natural to ask if it is an isomorphism as well, i.e., if both spaces are complete (and in which sense). This is indeed the case.
Proof We pick a Cauchy sequence (F k ) k∈N which is Cauchy with respect to trhe given norm. Then, it is Cauchy with respect to · H p (C\R) and hence there is some In particular, F (n) k → F (n) local uniformly on C\R, see [11], p. 250. On the other hand, since (F k ) n∈N is Cauchy with respect to · A 2 ν ( + ) , we know that there isF ∈ A 2 ν ( + ) such that F (n) k −F A 2 ν ( + ) → 0 (here we used Lemma 4.3 for the case that ν n ∈ M 2 ). In particular, F (n) k →F almost everywhere on + , at least for a subsequence, and we conclude that F (n) | + ∈ A 2 ν ( + ). Remark 7.2 Let n ∈ N 0 and ν ∈ M eps ∪ M 2 ,0 . Then, Theorem 3.1 and Theorem 3.2 imply in particular that the following assertions hold: Banach space with respect to the norm f p + f L 2 ((0,∞),w ν,n λ 1 ) . 2. The space f ∈ H 1 (R) : f 1 (0,∞) ∈ L 2 ((0, ∞), w ν,n λ 1 ) is a Banach space with respect to the norm f H 1 (R) + f L 2 ((0,∞),w ν,n λ 1 ) .
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