On the Characterization of Triebel–Lizorkin Type Spaces of Analytic Functions

We consider different characterizations of Triebel–Lizorkin type spaces of analytic functions on the unit disc. Even though our results appear in the folklore, detailed descriptions are hard to find, and in fact we are unable to discuss the full range of parameters. Without additional effort we work with vector-valued analytic functions, and also consider a generalized scale of function spaces, including for example so-called Q-spaces. The primary aim of this note is to generalize, and clarify, a remarkable result by Cohn and Verbitsky, on factorization of Triebel–Lizorkin spaces. Their result remains valid for functions taking values in an arbitrary Banach space, provided that the vector-valuedness “sits in the right factor”. On the other hand, if we impose vector-valuedness on the “wrong” factor, then the factorization theorem fails even for functions taking values in a separable Hilbert space.

1 Introduction Definition 1.1 Let X , X 1 and X 2 be normed linear spaces of functions on D. If for any f ∈ X there exists f 1 ∈ X 1 and f 2 ∈ X 2 such that f = f 1 f 2 and then we say that X ⊂ X 1 · X 2 . If for any f 1 ∈ X 1 and f 2 ∈ X 2 it holds that f 1 f 2 ∈ X and sup then we say that X 1 · X 2 ⊂ X . If X ⊂ X 1 · X 2 and X 1 · X 2 ⊂ X , then we say that X = X 1 · X 2 .
Throughout this paper, we let X and H respectively denote a general Banach space and a separable Hilbert space, both complex. By A (X ) we denote the space of analytic X -valued functions on the open unit disc D. For short, we write A = A (C), i.e. we suppress X = C. The same principle will apply to all function spaces discussed below.
We let T denote be the unit circle in C, and give it the parametrization x → ζ x , where ζ x = e 2πi x , x ∈ R. For p ∈ (0, ∞), we denote by L p (T, X ) the class of strongly measurable functions f : where we somewhat abusively write dx to indicate Lebesgue integration with respect to ζ x . We will often identify f ∈ L 1 (T, X ) with its Poisson extension P The duality of H 1 and B M O A is a celebrated result by Fefferman [7]. For a discussion on the vector-valued case, see for instance [3]. Given α ∈ R and f ∈ A (X ), we define the fractional derivative D α f by (1 + n) αf (n) w n , w ∈ D.
Consider the class D α H p := f ∈ A; D −α f ∈ H p equipped with the norm f D α H p := D −α f H p . The following result is due to Cohn and Verbitsky [4, Theorem 2]: Theorem 1.2 Let α > 0, 0 < p, p 1 , p 2 < ∞, and p −1 The present author's interest in the above result arose while studying the following type of bilinear forms appearing naturally in control theory, e.g. [12,18]: Given φ ∈ A and α > 0, we define the bilinear Hankel type form on analytic polynomials. The next result on H 2 -boundedness of H φ,α has several proofs in the literature, e.g. [12,18]. As an illustration, we prove it by applying Theorem 1.2: Proof Suppose that g, h ∈ H 2 , and let f ∈ H 1 be a suitable function such that The statement now follows from the Fefferman The primary aim of this paper is to consider vector-valued generalizations of Theorem 1.2. Given φ ∈ A (H) and α > 0, there are two natural analogues of (1): and The proof of Proposition 1.3 now leads us to the following questions: The first question will receive a positive answer. This yields that H φ,α is bounded if and only if D α φ ∈ H 1 (H) * = B M O A (H), a result also obtained in [18]. The second question receives a negative answer. Indeed, if the answer was positive, then H φ,α and H * φ,α would be simultaneously bounded. This would contradict the following result, essentially due to Davidson and Paulsen [5]. See also [18,Sect. 4]: Before stating our main results, we define the concept of an outer analytic function: Definition 1.5 Let k ∈ L 1 (T) be a real-valued function. A function of the form e u+iv , where u is the Poisson extension of k, and v is the harmonic conjugate of u, is called an outer analytic function.
For our purposes, the main feature of outer analytic functions is that they are never vanishing in D.
As in [4], we state our main result in the language of Triebel-Lizorkin spaces F s p,q . Their definition is quite elaborate, and we refer to Sect. 4, where we discuss different characterizations. Remarkably, our results hold for analytic functions taking values in an arbitrary Banach space: Moreover, the H p -factor can be constructed as an outer analytic function. Remark 1.7 The attentive reader will of course note that we are lacking a statement for 0 < q < 1. This is perhaps the main shortcoming of this paper, and it relates to the fact that for this range of parameters we are not able to define the corresponding spaces. Since Theorem 1.2 corresponds to the special case D α H p = F −α p,2 . For emphasis, we state a corollary: We also obtain a non-factorization result: Theorem 1.9 Let X be a complex Banach space, s < 0, 0 < p < ∞ and 1 ≤ q < ∞. Then H p (X ) · F s ∞,q ⊂ F s p,q (X ) . In general, this inclusion is strict. In particular, Proposition 1.4 shows that if H is an infinite-dimensional Hilbert space, then there exists f ∈ F s 1,2 (H), such that for any g ∈ H 1 (H) and h ∈ F s ∞,2 , f = gh. The first ingredient needed in order to generalize Theorem 1.2 is a factorization result for tent spaces, Theorem 3.1. We point out that the proofs from [4] go through also in the vector-valued case (replacing moduli with vector space norms). However, the scalar-valued result even implies the vector-valued one. We demonstrate how in Sect. 3.
We will also need some properties of Triebel-Lizorkin spaces of analytic functions on D. These appear in the literature (e.g. [4]) but I have not been able to find any stringent justification. The vector-valued setting that we consider does not require any additional effort. Nevertheless, this setting does not appear to have been considered before. For these reasons we dedicate Sect. 4 to establishing some rudimentary theory. We develop our theory in the language of the more general class of distribution spaces F s,τ p,q R d introduced in by Yang and Yuan [23,24]. Rather than increasing our efforts, F s,τ p,q (X ) unifies the spaces F s p,q (X ) where p < ∞, and F s ∞,q (X ), perhaps even decreasing the amount of work needed. Another motivation to study the generalized scale is that it encompasses more spaces, for example the so-called Q-spaces introduced by Aulaskari et al. [1].

Preliminaries and Notation
We use the standard notation Z, R, and C for the respective rings of integers, real numbers, and complex numbers. In addition, D = {w ∈ C; |w| < 1} and T = {ζ ∈ C; |ζ | = 1}. We will often identify T with R/Z, using the map x → e 2πi x . Subsets of R/Z and T are identified in a similar way. In particular, we let the set of dyadic arcs D (T) be the image of the set D ([0, 1)) = 2 − j k, 2 − j (k + 1) ; j ∈ N 0 , 0 ≤ k ≤ 2 j − 1 . We define the rank of I ∈ D(T) as rk(I ) = − log 2 |I |. Note that in general, an arc I ⊂ T may correspond to the union of two intervals in [0, 1). We use the letters x, y, z to denote generic points on R.
By ζ x we denote the point e 2πi x ∈ T. The arc-wise distance between ζ x and ζ y is denoted by |ζ x − ζ y |. A Euclidean ball with radius r and center w is denoted B (w, r ). Given two parametrized sets of nonnegative numbers {A i } i∈I and {B i } i∈I , we use the notation A i B i , i ∈ I to indicate the existence of a positive constant C such that A i ≤ C B i whenever i ∈ I . Sometimes we allow ourselves to not mention the index set I and instead let it be implicit from the context.
For a background on the Bochner-Lebesgue classes L p (T, X ) we refer to [6]. Given a strongly measurable function f : T → X , we define the corresponding Hardy-Littlewood maximal function by The following periodic analogue of the vector-valued maximal theorem follows easily from [8, Theorem 1]: is the Poisson kernel for D. We also write P[μ](w) = T P r ζ x−y dμ(y), whenever w = r ζ x , and μ is a measure of finite total variation on T. By geometric summation,P r (n) = r |n| . It is well-known that If v is subharmonic and extends continuously to T, then v is majorized by the Poisson extension of its boundary values, i.e.

v (w)
For proofs of these claims, we refer to [11]. If f ∈ A (X ), then for any 0 < p < ∞, The functional A ∞ is a classical tool in the characterization of H p , and proves useful also in the X -valued setting: Proof 1 The statement is obvious for p = ∞, as is the inclusion For the reverse inclusion, suppose that f ∈ H p (X ). We may also assume that f is not identically 0. By analyticity, log f X is subharmonic, as is f p X , c.f. [16,Theorem 4.2.A]. A theorem by Littlewood [14] implies that the limit u (ζ Let log + t = max{0, log t}, and note that log By Fatou's theorem, h(ζ x ) = lim w→ζ x h(w) exists as a nontangential limit for Lebesgue a.e. ζ x ∈ T. Moreover, h ∈ L 1 (T), and dμ = h(ζ x ) dx + dλ + − dλ − , where λ + , λ − ≥ 0 are singular measures with respect to dx.
Chose (some) H ∈ A with real part h, and define g = exp(H ). Then f X ≤ |g|, and so f H p (X ) ≤ g H p . An application of Jensen's inequality (with the probability measure where we have used, in turn, Minkowski's inequality and Fatou's lemma. The non-tangential maximal characterization of H p (X ) now follows from the scalar case, since The square function of f ∈ A (X ) is given by It is a famous result by Fefferman and Stein [9] if and only if S f ∈ L p (T). In general, H p (X ) may fail to have a square function characterization, e.g. [5,Remark 4.11].

The duality between H 1 (H) and B M O A (H)
is a celebrated theorem by Fefferman [7], adapted to analytic functions on D (e.g. [11, Exercise VI.5]), with values in H (e.g. [3]). H). [16,Chap. 4]. For this reason we will typically not distinguish between a function f ∈ H p (H), and its boundary values b f ∈ L p (H).
The convolution of If f and g are Poisson extensions of integrable functions, then so is f * g, and Given a smooth function ϕ : R → C, let ϕ (k) denote its (classical) derivative of order k ∈ N 0 . The Schwartz space S is the class of functions ϕ : R → C for which all derivatives decay faster than any rational function, i.e. ϕ ∈ S if and only if for any We will be interested in the 1-periodization of ϕ given by It is easy to see that if N ≥ 2, then

Tent Spaces
Given a subset E ⊂ T, we define the "tent" over E as Let f : D → C be a measurable function. For q ∈ (0, ∞), we define the functional A q by Note that the functional A ∞ was defined in the previous section. For p, q ∈ (0, ∞), is a Carleson measure. For p, q in the range discussed above, we define T p,q (X ) to be the set of functions f : D → X such that f X ∈ T p,q . We equip this space with the obvious metric structure.
The main result of [4] can be explained in three steps. First, the authors obtain the inclusion T p,∞ · T ∞,q ⊂ T p,q . This holds in particular when the first factor is analytic, but this is not a requirement. Second, the inclusion T p,q ⊂ T p,∞ · T ∞,q is proved. Here the first factor may be chosen to be outer analytic. Third, the authors use the fact that H p = A ∩ T p,∞ , i.e. the scalar version of Lemma 2.2. Put together, this reads that T p,q = H p · T ∞,q , where the factor in H p may be constructed to be an outer analytic function. Using the first two steps together with Lemma 2.2, this result easily carries over to the X -valued setting, provided that we take care which one of the factors is X -valued: Theorem 3.1 Let X be a complex Banach space, and 0 < p, q < ∞. Then T p,q (X ) = H p · T ∞,q (X ) , and H p (X ) · T ∞,q ⊂ T p,q (X ) .

If X = H is an infinite-dimensional Hilbert space, then the inclusion is strict.
Proof If f ∈ T p,q (X ), then by the scalar-valued result f X = g H, where g ∈ H p is outer analytic, H ∈ T ∞,q , and g H p H T ∞,q f T p,q (X ) . Define h = f g . Then h X = |H | ∈ T ∞,q . By definition h ∈ T ∞,q (X ), and f = gh. This proves that T p,q (X ) ⊂ H p · T ∞,q (X ). The reverse inclusion H p · T ∞,q (X ) ⊂ T p,q (X ) also follows from the scalar-valued result: Let g ∈ H p and h ∈ T ∞,q (X ). Then The statement that H p (X ) · T ∞,q ⊂ T p,q (X ) follows similarly. For infinitedimensional Hilbert spaces, this inclusion must be strict in order to not contradict Theorem 1.9.

Triebel-Lizorkin Type Spaces
The so called Triebel-Lizorkin spaces F s p,q R d , s ∈ R, 0 < p, q ≤ ∞, are wellstudied objects. An extensive treatment is given in Triebel's monographs [19][20][21]. We also mention papers by Liang et al. [13], Peetre [15], Rychkov [17], 2 and Ullrich [22], all of which give a more direct introduction to many of the ideas to be used in this paper.
In this section we investigate some Triebel-Lizorkin type spaces of X -valued analytic functions on D. This setting is analogous to the corresponding theory of distributions on R d , but requires some new ideas. For instance, our spaces cannot be defined by mere finiteness of the appropriate norm, c.f. Definition 4.1 and Proposition 4.4. Moreover, the kernels for the local means, as defined for example in [22], poorly reflect the fact that the functions we deal with are analytic. A kernel that efficiently reflects the property of analyticity is the Poisson kernel. However, this kernel is not sufficiently regular for us to apply the typical techniques for analyzing F s p,q (R d ). Instead we need to use an intermediate substitute, c.f. the discussion following Theorem 4.3. We remark that the X -valuedness does not present any major obstacles, since the proofs boil down to estimates of the scalar-valued integral kernels. Finally, for the convenience of the reader not yet familiar with the theory of F s p,q (R d ), or its generalization discussed below, we attempt to give reasonably complete proofs, the more involved parts of which are postponed to Sect. 4.1.
The claim of this section is that the above definition is unambiguous, i.e. that it does not depend on the parameter α. Moreover, the respective topologies defined for different choices of α are equivalent. This claim was made in [4], also for q < 1, and obtaining a proof of this is one of the main motivations for this paper.
If we for the moment accept the unambiguity of Definition 4.1, then the proof of Theorem 1.6 is indeed short:  R d introduced by Aulaskari et al. [1]. This is in fact a motivation in [23]. We chose to work with the more general scale of F s,τ p,q -spaces, since this requires no additional effort.
A fundamental tool in the study of Triebel-Lizorkin spaces is the so-called Peetre maximal function: For a > 0 and f ∈ A (X ), we define Definition 4.2 Let (r l ) l≥0 be a sequence such that 0 ≤ r l < 1 and 2 l (1 − r l ) ≈ 1.
Furthermore, let 0 < p, q < ∞, s ∈ R, τ ≥ 0 and a > max 1 p , 1 q . Define then following (quasi-)norms for f ∈ A (X ): If we wish to indicate the values of p, q, s and τ , then we use the notation f | s,τ p,q k , The following theorem is analogous to [13,Theorem 3.2]. Weaker theorems of the same form go back at least to Triebel [19].

Theorem 4.3 The (quasi-)norms in Definition 4.2 are comparable for f ∈ A (X ).
Let ϕ ∈ S be a function such that for ξ ≥ 0,φ (ξ ) = e −ξ . With (r l ) l≥0 as in Definition 4.2, let t l = log 1 r l , set ϕ l (x) = 1 t l ϕ x r l , and let l denote the corresponding periodization. If f ∈ A (X ), then l * f = P r l * f , and so This expression is a verbatim analogue of the defining (quasi-)norm for F s,τ p,q R d . If s < 0, then imposing finiteness of the above expression indeed gives us a space with the natural properties. However, for general s ∈ R, such a definition would be severely flawed: Proof We consider first the case where p = q. By interchanging orders of integration, q X dx is increasing, and so the above right-hand side is infinite, unless f ≡ 0. The general case follows in the same way, with some simple modifications: If p > q, then we first apply Hölder's inequality to the integral: If p < q, then we instead use Hölder's inequality on a partial sum: Integrating the above inequality over T, since s ≥ 0, we may again argue by subharmonicity to conclude that The result now follows by letting N → ∞.
A related observation is that if s < 0 and f ∈ A (X ), then Definition 4.5 Let 0 < p < ∞, 1 ≤ q < ∞, s ∈ R, 0 ≤ τ ≤ 1 p , and α > s. We define the Triebel-Lizorkin type space F s,τ p,q (D, X ) as the space of functions f ∈ A (X ) such that X ). Definition 4.5 is justified by the following lemma: Then By Proposition 4.4, the above lemma is trivial if s ≥ 0. Assume therefore that s < 0, and let α > s. In particular (7) holds. Moreover, −α > s − α. By another application of Lemma 4.6, we obtain Combined with Theorem 4.3, this yields that whenever s < 0 and s − α < 0. This implies that if α 1 , α 2 > s, then and thus F s,τ p,q (D, X ) is well-defined, with topology independent of α.

Proofs
In 4.1.1 we quantify the rigidity of analytic functions in a certain way (Lemma 4.7). We refer to this as "the first stability property". This will imply that f 3 is essentially independent of (r l ) l≥0 , and also that f 1 f 2 f 3 . The proof that f 3 f 1 is simpler. In 4.1.2, we deduce a similar stability property (the second) for the Peetre maximal function (Lemma 4.8). It follows that f 4 ≈ f 5 . The estimate f 3 f 4 is trivial, since f l (ζ x ) ≤ f * l,a (ζ x ). We dedicate 4.1.3 to obtaining the reverse maximal control, the most involved part of this paper. In 4.1.4 we prove Lemma 4.6.

The First Stability Property
Given I ∈ D (T), we use the notation I n = I + n|I |, for 1 − 1 2|I | ≤ n ≤ 1 2|I | . For other n ∈ Z, we let I n be the empty set. Furthermore, we set I L = ∪ |n|≤L I n . X be a Banach space, α ≥ 0 and c 1 , c 2 , c 3 , c 4 > 0. Then there exists constants K > 0 and L ∈ N with the following property:

Lemma 4.7 Let
Let I ∈ D (T). If r, r ∈ [0, 1) satisfy Proof By subharmonicity it holds that for any L ∈ N By the non-tangential maximal control of Poisson extensions, Provided that L is sufficiently big, it follows from (4) that f 3 is independent of (r l ) l≥0 : Let (r l ) l≥0 and r l l≥0 be sequences in [0, 1) such that Chose M ∈ N such that 2 M−1 c ≥ C . For any l ∈ N 0 it then holds that r l < r l+M , and 2 −1−l c ≤ r l+M − r l ≤ 2 −l C. If we want to prove that f | (r l ) l≥0 1 f | r l l≥0 1 , then by a shift of index we may assume that M = 0. In this case our sequence has the asymptotic behavior Let δ < min { p, q} and I ∈ D (T). It follows from Lemma 4.7 (with α = 0) that for l ≥ rk (I ) and x ∈ I . We now exploit the fact that p/q = δ/q· p/δ. By Minkowski's inequality, By Theorem 2.1,

By Minkowski's and Jensen's inequalities
Obviously, where θ r ≈ 1 − r . Let r = 1+r 2 . By Lemma 4.7, Moreover, these balls may be chosen so that they are disjoint and d l 2 −l . By subharmonicity, By (9) and (11), we see that if N > a + 1, then The statement follows.
f 4 is independent of (r l ) l≥0 : Once again, we may assume that (9) holds. The statement follows immediately from Lemma 4.8.
f 4 ≈ f 5 : We may chose (r l ) l≥0 so that r l = 1 − 2 −l . Note that For an arbitrary sequence r l l≥0 such that r l+2 ∈ r l , r l+1 , (9) holds. By an application of Lemma 4.8, we obtain r l+1 It follows that f 5 f 4 . The reverse estimate is similar.

Reverse Maximal Control
It will be convenient to work with the sequence given by r l = e −2π 2 −l . Let ϕ t and r be as in the proof of Lemma 4.8, and set ϕ m = ϕ 2 −m and m = r m . Then m is the 1-periodization of ϕ m . Now choose {W m } ∞ m=1 ⊂ S such that suppŴ 1 ⊂ − 1 2 , 2 , Furthermore, for any l ∈ N 0 we have that Define λ m,l ∈ S byλ m,l (ξ ) =ψ m 2 −l ξ , ξ ∈ R, and let m,l denote the corresponding 1-periodization. We thus obtain ∞ m=1ˆ m,l (n)ˆ m+l (n) = 1, l, n ∈ N 0 .
This is a so-called Calderon reproducing type formula for analytic functions. We will need the following technical lemma: Lemma 4.9 If N ∈ N, then there exists a K > 0 such that for all m ∈ N and l ∈ N 0 .
Proof By (5), it suffices to show that We consider the case m ≥ 2. By elementary properties of the Fourier transform, Sinceφ l (ξ )λ m,l (ξ ) = e −2π 2 −l −2 −l−m ξŴ 2 2 2−l−m ξ , we may use Leibniz's rule together with the support ofŴ 2 to obtain This proves the statement for m ≥ 2, since for any k ∈ N 0 , The case m = 1 is similar.
The proof that · 2 · 1 is based on the following lemma: Proof By (13), By the triangle inequality and Lemma 4.9, we have that If δ > 1, then we proceed as follows: Since N is arbitrary, clearly Applying Hölder's inequality twice we obtain Now use that a ≤ N , divide by 1 + 2 l |ζ z − ζ x | aδ and use (12) with b = 2 l get that This completes the proof for δ > 1.
If δ ≤ 1, then we instead do the following: By a shift of the index l in (14) we see that for each k ∈ N 0 , Using (12) again, we have that Introduce the maximal function