MIXED NORM SPACES OF ANALYTIC FUNCTIONS AS SPACES OF GENERALIZED FRACTIONAL DERIVATIVES OF FUNCTIONS IN HARDY TYPE SPACES

The aim of the paper is twofold. First, we present a new general approach to the deﬁnition of a class of mixed norm spaces of analytic functions A q ; X ( D ), 1 (cid:2) q < ∞ on the unit disc D . We study a problem of boundedness of Bergman projection in this general setting. Second, we apply this general approach for the new concrete cases when X is either Orlicz space or generalized Morrey space, or generalized complementary Morrey space. In general, such introduced spaces are the spaces of functions which are in a sense the generalized Hadamard type derivatives of analytic functions having l q summable Taylor coeﬃcients.


Introduction
Let X(I) ⊆ L 1 (I), I = (0, 1), denote any Banach space of functions f on interval I containing step functions, and let · X(I) stand for the norm. Given a function f (z) = f (r, e iα ) on D or in general a distribution c 2017 Diogenes Co., Sofia pp. 1106-1130 , DOI: 10.1515/fca-2017-0059 on D we denote by f n its distributional Fourier coefficients (see Section 7 for definition). We introduce the mixed norm space L q;X (D), 1 q < ∞, as the space of distributions on D such that the (distributional) Fourier coefficients f n are regular functions f n (r) ∈ X(I) and n∈Z f n q X(I) is finite. The q − th root from this sum gives the norm in L q;X (D). The mixed norm Bergman space A q;X (D), 1 q < ∞, is defined as the subspace in L q;X (D) of functions analytic in D.
In [20,21] we studied special cases for the space X(I). The variable exponent Lebesgue space X(I) = L p(·) (I), 1 p(r) ∞, was treated in [20], and the cases of classical Morrey space X(I) = L p,λ (I), 0 λ < 1, 1 p < ∞, and complementary Morrey space X(I) = L p,λ (I), 0 λ < p − 1, 1 < p < ∞, were studied in [21]. This research was inspired by the evident fact that introduction of the mixed norm in the unit disc allows to distinguish between radial and angular behavior of functions, and, hence, to specify the boundary behaviour with more accuracy. In such a way one can reveal the behaviour of a function using variety of norms, including norms of the so called spaces of functions of non standard growth.
A special motivation to introduce new spaces in this paper is in the fact that, for instance, in the case q = 2 these Bergman type spaces may be precisely characterize as the range of certain generalized fractional differentiation operator over the Hardy space H 2 (D) (see Theorem 4.1). Moreover, the introduced spaces, in general, are the spaces of functions which are the generalized fractional derivatives of analytic functions with l q summable Taylor coefficients. This important fact sheds a light on the nature of the introduction of the spaces via conditions on Fourier coefficients, which is different from the usual mixed norm space setting. The notion of generalized derivatives of Hadamard type is a wide generalization of the used in the theory of analytic functions so called radial derivatives, Flett's derivatives, etc. We discuss this notion in Section 4. In the particular cases of Orlicz and generalized Morrey type spaces the corresponding form of Hadamard derivative can be explicitly seen from the characterization of functions in such spaces, see Theorems 5.2, 6.2. For a general theory of fractional derivatives and integrals we refer to [23], [36].
In the last two decades the theory of new spaces arising in harmonic analysis of functions with non standard growth and the theory of operators of harmonic analysis in various general spaces with non-standard growth have been intensively developed. These spaces include in particular variable exponent Lebesgue, Hölder, Sobolev spaces, Lorentz spaces, and Orlicz, Morrey-Campanato type, Herz spaces, and others. Within these spaces there were widely considered singular integrals, Riesz and Bessel potentials, maximal and fractional operators, some other classical operators of Brought to you by | De Gruyter Trial Portugal 2018 Authenticated Download Date | 7/24/19 5:52 PM harmonic analysis. As a matter of fact, major attention was paid to real variable settings. We refer to the books [7,8,25,26] (see also review paper [34]).
Therefore it seems natural and fruitful to make use of widely developed methods in the area mentioned above in the case of new spaces of analytic functions. Indeed, the investigation in such a direction already presents very new effects and interesting results. For instance, such a variety of spaces include classical Hardy space and Bergman type space, these very different spaces of functions even within one scale of spaces, i.e. when X(I) = L p(·) (I). Depending on the growth of p(r) as r → 1 we may obtain both Hardy and Bergman type spaces as particular cases of A q;X (D) with X(I) = L p(·) (I).
Starting with the papers [4], [17], [18], the Bergman spaces, called sometimes the Bergman-Jerbashian spaces, and other spaces of analytic functions attracted attention of many researchers, see the books [5,10,16,39,40] and references therein. In particular, an important role is played by the boundedness of the Bergman projection. Besides the above cited books we also refer to [1,3,9] with respect to such boundedness and related questions. There are known result on the mixed norm Bergman spaces with integral mixed norm with integration in angular and radial variables (boundedness of the Bergman projection, and some functional space properties of the mixed norm Bergman spaces, such as duality, interpolation etc.). We refer to the papers [12,15,19,29,30] (see also references therein).
The spaces that we study are different from such mixed integration norm spaces. In some cases embedding of our spaces into such spaces may be traced under some concrete choice of the space X(I) (see Theorem 4.2).
Introducing the spaces A q;X (D), 1 q < ∞, we in fact suggest a new general approach to the definition of a class of mixed norm Bergman spaces on the unit disc D. We provide some general assumption on the space X(I) under which the Bergman projection is bounded from L q;X (D) onto A q;X (D) and reveal the importance of the asymptotical behaviour of the norms r n X(I) as n → ∞ for the characterization of the spaces A q;X (D). This general scheme may be considered as a useful guide. However, the main difficulties start when we treat concrete cases: we have to overcome various problems to verify the above mentioned assumption on the space X(I) and the asymptotic of r n X(I) as n → ∞ in the case of this or other concrete case of X(I). Exactly at this step we often have to use very specific properties of the space X(I) or even to obtain new ones, as for instance in [20]. We also present a short review of results previously obtained in [20,21], i.e. we summarize these results under a general point of view. We apply this general approach for the new concrete cases where X(I) is either the Orlicz space L Φ (I) or the generalized Morrey space L p,ϕ (I), or the generalized complementary Morrey space L p,ϕ (I). The fundamentals of Young functions and Orlicz spaces appeared to be very useful in this study. This is a promising area of research, and we plan to explore new cases, for instance, via introducing Besov type norms (see [22]), as well as we keep in mind to study Toeplitz type operators on such new spaces.
The paper is organized as follows. Section 2 contains necessary preliminaries on classical function spaces: Bergman, Morrey and Orlicz. In Section 3 we develop our general approach. In Section 3.1 we give our basic definition and prove the completeness of L q;X (D), and in Section 3.2 we give general condition on the boundedness of Bergman projection from L q;X (D) onto A q;X (D). In Section 3.3 we briefly discuss Toeplitz operators on our new spaces. In Section 4 we discuss the characterization of our spaces in terms of generalized fractional differentiation. A special attention is paid to the space A 2;X (D) which coincides with the range of certain differential operator over the classical Hardy space H 2 (D). We also outline certain results previously obtained in [20,21]. In Sections 5, 6 we realize our approach for X(I) being Orlicz space or generalized Morrey and generalized complementary Morrey space where we manage to find the asymptotic for the norm r n X(I) which also leads to the characterization of the spaces under consideration and to the corresponding results on boundeddness of the Bergman projection. In Section 7 we pay a special attention how we interpret distributional Fourier coefficients of traces of functions. For the references, see [10,16,39,40]. Let dA(z) stand for the area measure on D normalized so that the area of D is 1. As usual A p (D) stands for the Bergman space of analytic in D functions f that belong to L p (D) = L p (D; dA(z)). The corresponding Bergman projection B D which is defined on f ∈ L 1 (D) as For more details on the Morrey type spaces, we refer to [13,28,32,33,37]. In the definition below we naturally assume that the function ϕ : I → R + is increasing on I and ϕ(t) > 0 for t > 0, ϕ(0) = 0. We treat the Lebesgue space L p (I), 1 ≤ p < ∞, as equipped with the measure 2rdr. Let 1 p < ∞. The generalized Morrey space L p,ϕ (I) over the interval I is defined as the set of functions f measurable on I such that Let 1 < p < ∞. The generalized complementary Morrey space L p,ϕ (I) over the interval I is defined as the set of functions f measurable on I such that The p-th root from the expressions above provides the corresponding norm in L p,ϕ (I) and in L p,ϕ (I). The space The spaces L p,ϕ (I), L p,ϕ (I) are non separable. The embedding L p,ϕ (I) → L p (I), 1 p < ∞ is obvious. For the embedding L p,ϕ (I) → L 1 (I), 1 < p < ∞, we need to assume additional condition on ϕ provided by the following result.
P r o o f. The proof is straightforward: use dyadic decomposition, which is the standard tool when working with Morrey type norm (see, for instance, [6,31]) for the interval I and apply Hölder inequality. We have Brought to you by | De Gruyter Trial Portugal 2018 Authenticated Download Date | 7/24/19 5:52 PM We refer, for instance, to Lemma 3.2 from [6] for the estimation of the infinite sum via integral. 2 Note that in the classical case of complementary Morrey space when t β decreases on I, then the condition (2.2) is satisfied for such ϕ.
In what follows when considering the Morrey space L p,ϕ (I) and complementary Morrey space L p,ϕ (I), we always assume the mentioned above natural assumptions on ϕ : the function ϕ : I → R + is increasing on I and ϕ(t) > 0 for t > 0, ϕ(0) = 0. Additionally, for the Morrey space we suppose that ϕ(t) Ct, when t → 0, and for the complementary Morrey space L p,ϕ (I) we assume validity of (2.2).
Remark 2.2. In the definition of the Morrey space one may want to use a more classical way, writing ϕ(4rh) instead of ϕ(h), since the measure of the interval (r − h, r + h) with respect to the 2rdr equals to 4rh. Such introduced mixed norm space, let us denote it as L q;ϕ * (D), will be different from L q;ϕ (D). However, the corresponding subspaces of analytic functions coincide up to norm equivalence. For instance, the proof of the Theorem 6.3 explicitly shows that we will have similar results in that another setting. We leave the details for the reader. Also note that the usual definition of the Morrey space deals with the supremum over all r, h ∈ I and intervals I ∩ (r − h, r + h). For our goals, for the definition of L p,ϕ (I) we admit only intervals (r − h, r + h) ⊂ I.
From the convexity and Φ(0) = 0 it follows that any Young function is increasing. To each Young function Φ one identifies the complementary function Ψ, which possesses the same properties, by the rule , t > 0. This Δ 2 condition is usually referred as doubling condition. A Young function Φ is said to satisfy the ∇ 2 condition, denoted also by Φ ∈ ∇ 2 , if for some k > 1 one has Φ(kt) 2kΦ(t), t 0. Let, as usual, L Φ (I) be the Orlicz space of functions f measurable on I such that I Φ(k|f (r)|)2rdr < ∞ for The following analog of Hölder inequality is valid: (2.5) The following result may be found in [24], however we present it here, with slight modification, for the sake of completeness. A function ϕ is said to be almost decreasing for t > 0 if there exists C > 0 such that (2) . Then given any β log 2 C (2) we have (1) and (2) are equivalent up to the constants in the inequalities. Let us prove the second one. Naturally we assume that C (2) Hence it follows that given any β . . , be a Cauchy sequence. Then for each n ∈ Z the sequence of Fourier coefficients f k n , k = 0, 1, . . . , is Cauchy sequence in X(I) and converges in X(I) to some element f n ∈ X(I).
From the inequality q < ∞, and, hence, converges to some numerical sequence {a n } n∈Z ∈ l q . Obviously, a n = f n X(I) , and therefore the distribution f = n∈Z f n (r)e inα is the limit of the sequence f k ∈ L q;X (D), k = 0, 1, . . . , and its distributional Fourier coefficients are nothing but f n , by Lemma 7.1. Hence, f ∈ L q;X (D) and the space L q;X (D) is complete. 2 We introduce the mixed norm Bergman space A q;X (D), 1 q < ∞, as the space of functions from L q;X (D) which are analytic in D. Hence, the norm of a function f ∈ A q;X (D) is given by Remark 3.1. From the definition of the space A q;X (D), it can be derived that the Fourier coefficients f n = f n (r), n ∈ Z, of a function f in Bergman space A q;X (D) may be represented as It is evident that the multipliers r n −1 X(I) , n ∈ Z + , in (3.2) characterize the functions in A q;X (D). Their behaviour when n → ∞ is a crucial point in the whole study. This behavior depends only on the choice of the space X(I). So to characterize the introduced space in each particular case of X(I) we should examine the asymptotic behavior of the numbers r n −1 This is a quite difficult issue for spaces of functions with special norm, such as, for instance, variable exponent Lebesgue norm, Morrey space norm, etc.

Boundedness of Bergman projection from
The proof of the following result is straightforward (see for instance [20]).
where the constant C 0 comes from condition (3.4) and does not depend on f. Making use of the Banach-Steinghaus theorem we finish the proof. 2 For an analytic function f (z) = n∈Z + c n z n in A q;X (D) the following formula is true: (T a f ) n (r) = γ a (n)c n r n , n ∈ Z + , and (T a f ) n (r) = 0, n ∈ Z \ Z + , where {c n r n X(I) } n∈Z + ∈ l q + , and γ a (n) = (n+1) I τ 2n a(τ ) 2τ dτ, n ∈ Z + . From Remark 3.1 and the definition of the norm in A q;X (D) it is follows that the operator T a is bounded on A q;X (D) if and only if the sequence {γ a (n)} n∈Z + is bounded. The operator T a is compact on A q;X (D) if and only if γ a (n) → 0, n → ∞.
There are known many sufficient and, in some cases, necessary conditions for boundedness and vanishing of the sequence {γ a (n)} n∈Z + . These conditions were obtained (see, for instance, [14]) in terms of behavior of some means (averages) of the symbol a when r → 1. There also many examples of badly behaved oscillating and unbounded symbols that generate even compact operators. We refer to the book [38], and also references therein, for recent development of the theory of Toeplitz operators with special non standard symbols on classical weighted Bergman spaces over unit disc and half plain. We call attention to the fact that in the setting of our spaces A q;X (D) the boundedness and compactness conditions do not depend on the choice of the space X(I). We suppose to give more consideration to the study of Toeplitz operators in another paper.

The spaces A q;X (D) and the range of generalized fractional differentiation operator over Hardy type space
We will use the notion of the Hadamard product composition. Let the functions b(z) = k∈Z + b k z k , g(z) = n∈Z + g n z n be analytic in D. Consider the expression D(b, g)(z) = b • g(z) = n∈Z + b n g n z n , known as Hadamard product composition of functions b and g. This general notion includes in particular operation of fractional integro-differentiation. It generalizes fractional differentiation of analytic function g if b n → ∞, when n → ∞. For g analytic in D, We refer to [23], [36] for instance. We will use the following operator defined in terms of Hadamard product composition. The case q = 2 is of a special interest. Basing on Remark 3.1 for the case q = 2 we arrive at the following theorem. In view of Theorem 4.1, the distinction between A 2;X (D) and H 2 (D) is determined by the behavior of r n −1 X(I) when n → ∞. In our general case these asymptotics may be quite different varying, for instance, from very slow logarithmical to very high exponential type. As we showed in [20], we may even meet the non trivial situation when r n X(I) = 1. In that case A 2;X (D) = H 2 (D). This situation is realized for the case of variable exponent Lebesgue space X(I) = L p(·) (I) when p = p(r) grows to infinity fast enough, so that If again X(I) = L p(·) (I) and p(r) is still growing but slower, then in (4.1), like p(r) C 1 ln α 1 1−r , in a neighborhood of the point r = 1, for some 0 < α < 1, C 1 > 0, then r n X(I) At least for 1 q 2 we can provide the information of embedding of A q;X (D) into mixed integral norm space. Following [11] we introduce the mixed norm space H(s, t, γ), s > 0, t > 0, γ > 0 of measurable on D functions with the norm: Most resent information about such spaces, including embedding theorems, may be found in [2]. By D α , α > 0, we denote the Flett's fractional derivative whose action on analytic function is defined by the multiplication by (n + 1) α of its nth Taylor coefficient.
Theorem 4.2. Let 1 q 2 and there exist C > 0 and α > 0 such that r n −1

X(I)
Cn α , n → ∞. Then the continuous embedding P r o o f. We provide a sketch of proof. Due to A q;X (D) → A 2;X (D), 1 q 2 it suffices to prove the theorem for q = 2. According to Flett's result (see [11], Theorem B and Theorem 6), we have the esti- , which shows that the fractional derivative g = D α f of a function f ∈ H 2 (D) belongs to the weighted mixed norm space H(s, t, 1 2 − 1 s + α) and proves the embedding A 2;X (D) → H(s, t, 1 2 − 1 s +α). To prove H(s, t, 1 2 − 1 s +α) → L 1 (D) one has to repeatedly use the Hölder inequality under the condition on parameter s : Note that in Theorem 4.2 there is no restriction on t except 2 t ∞. Though it is useful to note that the minimal space in the scale H(s, t, 1 2 − 1 s + α) with respect to the parameter t, 2 t ∞, is achieved when t = 2 (see [2]).
In the following theorem we use the notation of the following classical fractional derivative of analytic functions: The condition r n −1 X(I) ∼ Cn α , n → ∞, is satisfied with some α > 0, for instance, for the following cases of the space X(I) considered in [20], [21]: (1) α = 1 p(1) for the case of the variable exponent Lebesgue space X(I) = L p(·) (I), p(1) = lim r→1 − p(r) < ∞, p satisfies logarithmic decay condition in a neighborhood of r = 1, i.e. there exist δ > 0, p for the case of the classical complementary Morrey space

Mixed norm Bergman -Orlicz space A q;Φ (D)
Let Φ be a Young function and X(I) = L Φ (I). Here instead of writing L q;L Φ (I)) (D), as we did previously for an abstract space X(I), we use L q;Φ (D) for simplicity. The same applies to A q;Φ (D).

Characterization of functions in
. We first provide the description of functions in A q;Φ (D) in terms of Taylor coefficients and in terms of fractional derivatives.
Theorem 5.1. Let Φ be a Young function, Φ ∈ Δ 2 ∩ ∇ 2 . Then for an analytic in D function f (z) = n∈Z + c n z n ∈ A q;Φ (D), z ∈ D the norm Theorem 5.2. Under the conditions of Theorem 5.1, each function f ∈ A q;Φ (D) has the form f = D X g with some analytic function g(z) = n∈Z + g n z n , such that {g n } n∈Z + ∈ l q + , where D X g(z) = n∈Z + Φ −1 (n)g n z n , z ∈ D.
Theorem 5.2 follows form Theorem 5.1. The later, in view of Remark 3.1, is the corollary of the following result which will be also crucial for the proof of the boundedness of the Bergman projection in the next section.
If, in addition, Φ satisfies the Δ 2 doubling condition with the constant C (2) , P r o o f. Due to the convexity of Φ : so the estimate (5.1) follows by the definition of the norm in L Φ (I).
Remark 5.1. Note that Theorems 5.1, 5.2 are obtained under the condition: Φ ∈ Δ 2 ∩ ∇ 2 , which eliminates such Young functions which have exponential type growth. This case undoubtedly presents a great interest and we formulate as an open problem: to prove similar results for the mixed norm Bergman-Orlicz type spaces A q;Φ (D) when the condition Φ ∈ Δ 2 ∩∇ 2 is either omitted or at least weakened.  2 g L Φ (I) t n L Ψ (I) . Therefore,

The ultimate inequality follows by (2.3). 2
As an immediate consequences of Theorem 3.2 we obtain the following results. Remark 5.2. Note that Theorem 5.5 holds for an arbitrary Young function, i.e. it admits, in particular, exponential growth of Φ, like Φ(t) = e t −1. In this case the norms r n L Φ (I) will be bounded by log-type estimate: 2 ln(n+2) .

Mixed norm Bergman-Morrey type spaces
Here and below the space X(I) is either the generalized Morrey space L p,ϕ (I), or the generalized complementary Morrey space L p,ϕ (I). Similarly to the previous section, if X(I) = L p,ϕ (I), instead of writing L q;L p,ϕ (D), A q;L p,ϕ (D) we use the notations L q;p,ϕ (D), A q;p,ϕ (D) for simplicity. Also we will denote by L q;p,ϕ (D), A q;p,ϕ (D) the spaces that correspond to X(I) = L p,ϕ (I).
6.1. Characterization of functions in A q;p,ϕ (D) and A q;p,ϕ (D). We first provide the description of functions in A q;p,ϕ (D) and A q;p,ϕ (D) in terms of Taylor coefficients and in terms of fractional derivatives.
Theorem 6.2. Under the conditions of Theorem 6.1, each func- is represented as f = D X g with some analytic function g(z) = n∈Z + g n z n , Theorem 6.2 follows form Theorem 6.1. The latter, in view of Remark 3.1, is the corollary of the estimates obtained below for the norms r n L p,ϕ (I) and r n L p,ϕ (I) , which will be also crucial for the proof of the boundedness of the Bergman projection in the next section.
By assumption, the function tϕ 1 t is concave and increasing for t > 1, so for t > 1 we set now Φ −1 (t) = tϕ 1 t , t > 1. We have to take care about a possibility of concave continuation of Φ −1 to the interval I = (0, 1) with the condition Φ −1 (0) = 0. Note that the behavior of ϕ when t approaches t = 1 is not of importance for the Morrey space up to the equivalence of norms. Therefore, if needed, we may modify the function ϕ in a left-sided neighbourhood of t = 1 so that the function Φ −1 will always have a concave continuation for t 1. To this end, it suffices to change the function ϕ so that ϕ ∈ C 1 (1 − δ, 1], and ϕ (1) > 0.
We claim that The function ϕ is concave and increasing on I, so we set now Φ −1 (t) = ϕ(t), 0 < t < 1. Again, we need to guarantee the existence of a concave continuation of Φ −1 for t 1 with the condition Φ −1 (∞) = ∞. As in the previous case we free to modify ϕ in a left sided neighbourhood of t = 1 so that ϕ ∈ C 1 (1 This finishes the proof. P r o o f. In is obvious that Since ϕ is increasing on I, then sup 0<h< 1 n ϕ 1 n , and since there exists β ∈ (0, p − 1) such that ϕ(t) t β is decreasing on I, then We find it convenient for further references to outline the following bilateral estimates which are proved in Theorems 6.3, 6.4: (1) Let 1 p < ∞. Let the function ϕ be concave on I, the function ϕ(t) t be decreasing on I, lim t→0 ϕ(t) t = ∞, and the function tϕ( 1 t ) be concave for t > 1. Then where C 1 , C 2 are some positive constants which do not depend on n.

Appendix: On distributional Fourier coefficients
Given a function f (z) = f (r, e iα ) ∈ L 1 (D) its Fourier coefficients f n (r) = 1 2π 2π 0 f (r, e iα )e −inα dα, n ∈ Z, exist for almost all r ∈ I. The distribution of Fourier series of 2π periodic functions, i.e., Fourier analysis on the unit circle, is well known. For our goals we need the notion of distributional Fourier coefficient of a distribution f on D. This coefficient are treated as a distribution on the interval I. We do not touch the study of the distributional Fourier transform in full extend and rather restrict ourselves to necessary facts for our needs.
We define the test function space S = S(D) as the set of functions ω = ω(r, e iα ) ∈ C ∞ (D) such that Λ m 1 ,m 2 (ω) = sup r,α ∂ m 1 r ∂ m 2 α ω(r, e iα ) < ∞, m 1 , m 2 ∈ Z + . The set S is a linear topological space with the topology defined by the countable set of seminorms Λ m 1 ,m 2 (·). For any γ > 0 Fourier coefficients of ω ∈ S satisfy the estimates sup Let now σ = σ(I) be the set of test functions v ∈ C ∞ (I) such that λ m (v) = sup r∈I v (m) (r) < ∞, m ∈ Z + . Thus the space of test functions σ is a linear topological space with the topology defined by the countable set of seminorms λ m (·). By σ = σ (I) we denote the space of linear continuous functionals (distributions) on σ. Similarly, g, v will represent the corresponding bilinear form in the case of "nice" functionals g : g, v = 1 0 g(r) v(r) 2rdr. Given a distribution f ∈ S we define its distributional Fourier coefficient f n ∈ σ by the rule f n , v = (f, ve inα ), v ∈ σ, n ∈ Z. (7. 2) The function ve inα belongs to S so the right side of this equality is well defined for any f ∈ S . If f ∈ L 1 (D) then the equality (7.2) is valid in the regular sense when both sides replaced with corresponding integrals. This fact justifies our definition (7.2). The distributional coincidence of nth Fourier coefficients of f with f n is then obvious. 2