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 fails even for separable Hilbert spaces.


Introduction
Definition 1.1. Let X, X 1 and X 2 be normed linear spaces of analytic 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 f1∈X1\{0},f2∈X2\{0} 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). 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 2πix , x ∈ R. For p ∈ (0, ∞), we denote by L p (T, X ) the class of strongly measurable functions f : T → X such that f p L p (T,X ) = ζx∈T f (ζ x ) p X dx < ∞, 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 [f ] : D → X . Under this identification, the Fourier coefficientsf (n) = T f (ζ x ) ζ n x dx are the Taylor coefficients of P [f ]. For short, we typically write f in place of P [f ]. We denote the nth Taylor coefficient of a general function f ∈ A (X ) byf (n), even though f is not necessarily the Poisson extension of an integrable function.
We define the Hardy space H p (X ) as the class of functions f ∈ A (X ) such that f H p (X ) = sup 0<r<1 f r L p (T,X ) < ∞, where f r : w → f (rw). In the case where X = H, we have the so-called square function characterization of H p (H); in the language of Section 4, H p (H) = F 0 p,2 (H). If p ≥ 1, then H p (H) also coincides with the space of f ∈ L p (T, H) such thatf (n) = 0 for n < 0. We A characterization of BM OA (H) relevant to this paper is, in the language of Section 4, that BM OA (H) = F 0 ∞,2 (H). This is the so-called Carleson measure characterization of BM OA (H).
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. [12,17]: 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. [12,17]. 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 H 1 − BM OA duality theorem.
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) * = BM OA (H), a result also obtained in [17]. 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 [17,Section 4]: , and define the forms H φ,α and H * φ,α by (2) and (3) respectively. If H φ,α is bounded, then H * φ,α is also bounded. The converse does not hold.
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 Section 4, where we discuss different characterizations. Remarkably, our results hold for analytic functions taking values in an arbitrary Banach space: Theorem 1.5. Let X be a complex Banach space, s < 0, 0 < p < ∞ and 1 ≤ q < ∞. Then F s p,q (X ) = H p · F s ∞,q (X ) . Moreover, the H p -factor can be constructed as an outer analytic function. Remark 1.6. 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 adheres to the fact that for this range of parameters we are not able to define the corresponding spaces.
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 1.8. 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 scalarvalued result even implies the vector-valued one. We demonstrate how in Section 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 Section 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 [22,23]. 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, Xiao and Zhao [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 2πix . 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 . 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 2πix ∈ T. The arcwise 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 ≤ CB 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. 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.
It is well-known that f ∈ H p if and only if f ∈ A and A ∞ f ∈ L p . This so-called nontangential maximal characterization of H p carries over to the general X -valued setting. This is known, e.g. [2, Lemma 1.1], but for the reader's convenience we provide a short indication of a proof based on the Szegö-Solomentsev theorem [15,Appendix 2]. It is Leth be the harmonic conjugate of h, withh (0) = 0, and define g = exp(h + ih). Then f X ≤ |g|, and so f H p (X ) ≤ g H p . A standard application of Jensen's inequality shows that . 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] 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 BM OA (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]). [15,Chapter 4]. For this reason we will typically not distinguish between a function f ∈ H p (H), and its boundary values bf ∈ L p (H).
The convolution of f ∈ A and g ∈ A (X ) is defined as f * g ∈ A (X ) with (f * g)ˆ(n) = f (n)ĝ (n). 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 k, N ∈ N 0 , sup x∈R (1 + |x|) N |ϕ (k) (x) | < ∞. The Fourier transform of ϕ ∈ S is given bŷ We will be interested in the 1-periodization of ϕ given by It is easy to see that if N ≥ 2, then MoreoverΦ (n) =φ (n), n ∈ Z.

Tent spaces
Given a subset E ⊂ T, we define the "tent" over E as Note that the functional A ∞ was defined in the previous section. For p, q ∈ (0, ∞), we 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] is that T p,q = H p · T ∞,q . This result easily carries over to the X -valued setting, provided that we take care which one of the factors is X -valued: 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 = gH, where g ∈ H p is outer analytic, H ∈ T ∞,q , and g H p H T∞,q f Tp,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 infinite-dimensional Hilbert spaces, this inclusion must be strict in order to not contradict Theorem 1.8.

Triebel-Lizorkin type spaces
The so called Triebel-Lizorkin spaces F s p,q R d , s ∈ R, 0 < p, q ≤ ∞, are well-studied objects. An extensive treatment is given in Triebel's monographs [18][19][20]. We also mention papers by Liang, Sawano, Ullrich, Yang and Yuan [13], Peetre [14], Rychkov [16] 1 , and Ullrich [21], which give a more direct introduction to many of the ideas to be used in this paper. In this section we investigate Triebel-Lizorkin type spaces of X -valued analytic functions on D. The more involved proofs are postponed to Subsection 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. If we for the moment accept this as a fact, then the proof of Theorem 1.5 is indeed short: The remainder of this section is dedicated to the justification of Definition 4.1. In [22,23], Yang and Yuan introduced the spaces F s,τ p,q R d , s ∈ R, τ ≥ 0, p ∈ (0, ∞), q ∈ (0, ∞]. These include the standard Triebel-Lizorkin spaces: In contrast to F s p,q R d , the spaces F s,τ p,q R d are not always distinct for different choices of parameters. On the other hand, they include for example the spaces Q α R d = F α,1/2−α/d 2,2 R d introduced by Aulaskari, Xiao and Zhao [1]. This is in fact a motivation in [22]. 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 ,

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 Φ l denote the corresponding periodization. If f ∈ A (X ), then Φ 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. Assume for simplicity that p = q. Then, by interchanging orders of integration, By subharmonicity, the right-hand side is infinite, unless f ≡ 0. The general case follows in the same way, with some simple modifications: If p > q, the we first apply Hölder's inequality to the integral: If p < q, then we instead use Hölder's inequality on a partial sum: . Now integrate over T, let N → ∞, and argue by subharmonicity.

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 . 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 .
Lemma 4.7. Let 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: 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 inequality, and the change of variables r → r ′ , To complete the proof, we now proceed as from (10), with the obvious modifications. f 2 f 3 : It will prove convenient to work with r l = 1 − 2 −l . For any l ∈ N 0 and r ∈ [r l , r l+1 ] it holds that By Lemma 4.7, it follows that for I ∈ D (T), x ∈ I, l ≥ rk (I) and r ∈ [r l , r l+1 ]. Since Once again, we proceed as from (10).
We work with the sequence given by r l = 1 − 2 −1−l . Given x ∈ I ∈ D (T), for each l ≥ rk (I) there exists a ball B l = B (r l ζ x , d l ) ⊂ Γ I (x). Moreover, these balls may be chosen so that they are disjoint and d l 2 −l . By subharmonicity, 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 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 −2π2 −l . Let ϕ t and Φ r be as in the proof of Lemma 4.8, and set ϕ m = ϕ 2 −m and Φ m = Φ rm . Then Φ m is Then ∞ m=1ψ m (ξ)φ m (ξ) = 1 for ξ ≥ 0.
This is a so-called Calderon reproducing type formula for analytic functions. We will need the following technical lemma: For notational simplicity, we assume that 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: for f ∈ A (X ), l ∈ N 0 and x ∈ T.
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 , (1 + 2 l |ζ x − ζ y |) N dy.
We treat the right-hand side as in the proof of Lemma 4.7.  It suffices to consider the cases where α > 0 is large, and when α = −1. The general case then follows from the diagram Applying Lemma 4.13, with h (ρ) = ½ [r0,1) (ρ) f ρ (ζ x ) X and µ = s + 1, we obtain These estimates together show that D −1 : F s,τ p,q (D, X ) → F s+1,τ p,q (D, X ) is bounded under the conditions in Theorem 4.3. This concludes the proof.