We prove that the pointwise product of two holomorphic functions of the upper
half-plane, one in the Hardy space $\mathcal H^1$, the other one in its dual,
belongs to a Hardy type space. Conversely, every holomorphic function in this
space can be written as such a product. This generalizes previous
characterization in the context of the unit disc.Comment: C. R. Math. Acad. Sci. Paris (to appear

Bonami, Aline

Ky, Luong Dang

Comptes Rendus Mathematique

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 352 (2014) 817–821Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comHarmonic analysisFactorization of some Hardy-type spaces of holomorphic functionsFactorisation de fonctions holomorphes dans des espaces de type HardyAline Bonami a, Luong Dang Ky ba MAPMO–UMR 6628, Département de mathématiques, Université d’Orléans, 45067 Orléans cedex 2, Franceb Department of Mathematics, University of Quy Nhon, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Vietnama r t i c l e i n f o a b s t r a c tArticle history:Received 20 June 2014Accepted 8 September 2014Presented by Yves MeyerWe prove that the pointwise product of two holomorphic functions of the upper half-plane, one in the Hardy space H1, the other one in its dual, belongs to a Hardy-type space. Conversely, every holomorphic function in this space can be written as such a product. This generalizes a previous characterization in the context of the unit disc.© 2014 Published by Elsevier Masson SAS on behalf of Académie des sciences.r é s u m éNous démontrons que le produit ponctuel de deux fonctions holomorphes du demi-plan supérieur, l’une dans l’espace de Hardy H1, l’autre dans son dual, appartiennent à un espace de type Hardy. À l’inverse, chaque fonction holomorphe de cet espace peut s’écrire sous la forme d’un tel produit. Ceci généralise un résultat connu dans le cas du disque unité.© 2014 Published by Elsevier Masson SAS on behalf of Académie des sciences.1. IntroductionLet C+ be the upper half-plane in the complex plane. We recall that, for p > 0, the holomorphic Hardy space Hpa (C+)is defined as the space of holomorphic functions f such that‖ f ‖pHpa := supy>0+∞∫−∞∣∣ f (x + iy)∣∣p dx < ∞. (1.1)By Fefferman’s Theorem, the dual space of H1a(C+) is the space BMOA(C+). Here, we are not interested in the definition of the Hermitian scalar product 〈 f , g〉 when f is in H1a(C+) and g is in BMOA(C+), but by the pointwise product f g of the two holomorphic functions. We identify the space of such products. This has already been done in the case of the unit disc in [4], where one finds a Hardy–Orlicz space. The novelty here is the fact that one has also to take into account the behavior E-mail addresses: aline.bonami@univ-orleans.fr (A. Bonami), dangky@math.cnrs.fr (L.D. Ky).http://dx.doi.org/10.1016/j.crma.2014.09.0041631-073X/© 2014 Published by Elsevier Masson SAS on behalf of Académie des sciences.818 A. Bonami, L.D. Ky / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 817–821at infinity. The space under consideration belongs to the family of Hardy spaces of Musielak type. It has been introduced in the setting of real Hardy spaces in [2].Before stating the theorem, let us recall that g is in BMOA(C+) if and only if one of the two equivalent conditions is satisfied:(i) |g′(x + iy)|2 y dx dy is a Carleson measure,(ii) g can be written asg(x + iy) = P y ∗ g0(x), (1.2)where g0 belongs to BMO(R) and its Fourier transform is supported in [0, ∞).Here P y is the Poisson kernel. Next we define Llog(R), as in [2], as the space of measurable functions f such that∫R| f (x)|log(e + |x|) + log(e + | f (x)|) dx < ∞.This is a particular case of a Musielak–Orlicz space, defined as the space of measurable functions f such that∫Rθ(x,∣∣ f (x)∣∣)dx < ∞under adequate assumptions on θ (see [9] for details). For f ∈ Llog(R), we define the “norm” by‖ f ‖Llog := inf{λ > 0 :∫Rθ(x,∣∣ f (x)∣∣/λ)dx ≤ 1},with θ(x, t) = t(1 + log+(|x|) + 12 log+(t))−1. The choice of this particular function, whose ratio with log(e + |x|) + log(e +| f (x)|) is bounded above and below, guarantees that t → θ(x, t2) is a convex function, which will be useful later.We then define Hloga (C+) as the space of holomorphic functions f such that‖ f ‖Hloga = supy>0∥∥ f (· + iy)∥∥Llog < ∞.Our main result is the following one.Theorem 1.1. The product of f ∈ H1a(C+) and g ∈ BMOA(C+) belongs to Hloga (C+). Moreover, every function in Hloga (C+) can be written as such a product. In other words,H1a(C+).BMOA(C+) = Hloga (C+).As a consequence, by using standard methods, we can identify the class of holomorphic symbols b of Hankel operators Hb that extend into continuous antilinear operators on H1a (C+). For simplicity, we only consider symbols b that are bounded and defineHb( f ) = P (b f )with P the Cauchy operator, which extends the orthogonal projection from L2(R) onto the subspace of functions whose Fourier transforms are supported in [0, ∞). In what follows, I(x0, r) stands for the interval (x0 −r, x0 +r) in R and T (I(x0, r))for the tent over I(x0, r), that is, the set of points z = x + iy such that x ∈ I(x0, r) and y < r.Theorem 1.2. Be b a bounded holomorphic function in C+ , the operator Hb extends into a bounded antilinear operator on H1a(C+) if and only if b belongs to the space BMOAlog(C+), that is,supI(x0,r)| log r| + log(e + |x0|)r∫T (I(x0,r))∣∣∇b(x + iy)∣∣2 y dx dy < ∞, (1.3)where the supremum is taken over all intervals I(x0, r) ⊂R.A. Bonami, L.D. Ky / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 817–821 8192. The space Hloga (C+)In this section, we extend to the space Hloga (C+) those properties of Hardy spaces that we will need. Let us first define, more generally, spaces Lϕ(R) and Hϕa (C+), with the specific function θ replaced with ϕ . We will only use two other specific functions, namelyθ0(x, t) = θ(x, t2)θ1(x, t) = θ(x, t)2. (2.1)Both are convex functions, of upper and lower type 2. So Hθ0a (C+) and Hθ1a (C+) are Banach spaces, while Hloga (C+) is not a normed space.As usual, the operator M will be the classical Hardy–Littlewood maximal operator. The nontangential maximal function of a function f defined in C+ is given byf ∗(x) = supz∈Γ (x)∣∣ f (z)∣∣, (2.2)where Γ (x) = {z = u + iy ∈C+ : |u − x| < y}.The next theorem gives the characterization of the space Hloga (C+).Theorem 2.1. f ∈Hloga (C+) if and only if f ∗ ∈ Llog(R). Moreover,‖ f ‖Hloga ∼ ‖ f∗‖Llog .Proof. One implication is obvious. Let us prove the other one. We consider f ∈ Hloga (C+). We use the fact that g = | f |1/2is a sub-harmonic function and satisfies the inequalitysupy>0∥∥g(· + iy)∥∥Lθ0 = ‖ f ‖Hloga < ∞.It is easy to adapt to the present situation the classical theorems. Namely, norms ‖g(· + iy)‖Lθ0 are decreasing and there exists a boundary value g0 ∈ Lθ0 (R) such that g(x + iy) ≤ P y ∗ g0(x). So,(f ∗)1/2(x) ≤ supu+iy∈Γ (x)P y ∗ g0(u) ≤ C M(g0)(x). (2.3)By the Lθ0 -boundedness of M (see [10, Corollary 2.8]), we obtain that‖ f ∗‖Llog ≤ C‖g0‖Lθ0 ≤ C‖ f ‖Hlogafor some uniform constant related with the norm of M in Lθ0 (R). We will also need the next statement.Theorem 2.2. Let f be a function in Hloga (C+). Then f has nontangential limits almost everywhere. Moreover, if the boundary value functionf0(x) := limΓ (x)z→x f (z)belongs to the space L1(R), then f is in H1a(C+).Proof. The proof of the existence of a. e. nontangential limits is quite similar to the classical one (see Garnett’s book [7], see also [3,5] for more details). It will be omitted. Assume that f0 is in L1(R). Since H1a(C+) is a subspace of Hloga (C+), we can assume that f0 = 0. We proceed as above and consider the subharmonic function g = | f |1/2 whose boundary values are 0. This forces f to be 0. 3. Factorization of the space Hloga (C+)Since we consider functions in BMOA(C+) and not only equivalence classes, we define a norm on this space. For a function f ∈ BMO(R), following [4], we define the norm by‖ f ‖BMO+ := ‖ f ‖BMO +∫ ∣∣ f (x)∣∣ dx.I(0,1)820 A. Bonami, L.D. Ky / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 817–821Let f be a function in BMOA(C+) given by (1.2), with f0 ∈ BMO(R). We define:‖ f ‖BMOA+ := ‖ f0‖BMO+ .Here and in future, we denote by mB f the average value of f over the ball B . Constants C may vary from line to line.The next lemma gives a bound of norms in BMO(R) on lines that are parallel to the x-axis.Lemma 3.1. Let f be a function in BMOA(C+). Then there exists some constant C such that, for all y > 0,∥∥ f (· + iy)∥∥BMO+ ≤ C log(e + y)‖ f ‖BMOA+ . (3.1)Proof. Let us first prove (3.1). Since BMO(R) is invariant by translation, we already know that∥∥ f (· + iy)∥∥BMO = ‖P y ∗ f0‖BMO ≤ C‖ f0‖BMO.So it is sufficient to prove that∫I(0,1)∣∣ f (x + iy)∣∣dx =∫I(0,1)∣∣P y ∗ f0(x)∣∣dx ≤ C log(e + y)‖ f0‖BMO+ . (3.2)Remark that y2 + |x − u|2 ∼ a2 + |u|2 for x ∈ I(0, 1), u /∈ I(0, a), with a := 2 max{1, y}. We cut f0 into f0χI(0,a) + f0χI(0,a)c . Since the convolution by the Poisson kernel has norm 1 in L1(R), the first L1-norm is bounded by mI(0,a)(| f0|). So∫I(0,1)∣∣P y ∗ f0(x)∣∣ dx ≤ mI(0,a)(| f0|) + Ca∫R| f0(u)|a2 + |u|2 du.We then use the standard inequalities, valid for all functions g ∈ BMO(R),a∫R|g(u) − mI(0,a) g|a2 + |u|2 dt ≤ C‖g‖BMOand|mI(0,a) g| ≤ |mI(0,1)g| + C log a‖g‖BMOto conclude that∫I(0,1)∣∣P y ∗ f0(x)∣∣ dx ≤ C log(e + y)‖ f0‖BMO+since a = 2 max{1, y} ≤ 2(e + y). This ends the proof. With this lemma we conclude for one side of the theorem.Proposition 3.1. There exists a constants C such that for every f ∈ H1a(C+) and g ∈ BMOA(C+), the pointwise product f g is in Hloga (C+) and satisfies‖ f g‖Hloga ≤ C‖ f ‖H1a ‖g‖BMOA+ . (3.3)Proof. We start by an a priory estimate. From Corollary 3.3 in the book of Garnett [7], we know that continuous functions f on C+ such thatA( f ) := supz∈C+(1 + |z|)3∣∣ f (z)∣∣ < ∞are dense. Using this last assumption and (3.1), we find that∫R∣∣ f (x + iy)∣∣∣∣g(x + iy)∣∣ dx ≤ A( f )1 + y∫R|g(x + iy)|1 + x2 dx ≤ C A( f )‖g‖BMOA+ .So the product h := f g is in H1a(C+) and h(· + iy) = P y ∗ h0, where h0 is the boundary value of h. By the same argument as in the proof of Theorem 2.1, and using the fact that θ0(·, t) is convex, we prove that h(· + iy) = P y ∗ h0 has decreasing norms in Hloga (C+). So‖h‖Hloga (C+) = sup∥∥h(· + iy)∥∥Llog .y≤1A. Bonami, L.D. Ky / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 817–821 821Now we use the fact that the product of a function in L1(R) with a function in BMO(R) is bounded (see [2]) to conclude about the a priori estimate (3.3).Multiplication by h extends to the whole space H1a (C+) by continuity. It remains to prove that the extension coincides with the multiplication by h. This can be done by a routine argument: convergence in Hloga (C+) implies uniform conver-gence on compact sets. Proof of Theorem 1.1. We have to prove that for every h ∈Hloga (C+), there exist f ∈ H1a(C+) and g ∈ BMOA(C+) for which h = f g . The proof is very similar to the one of [4]. Let h0 be the boundary value function of h. By the Coifman–Rochberg theorem [6], we have:b := log(e + | · |) + log(e + M(|h0|1/2)) ∈ BMO(R).Let H be the Hilbert transform in R. One knows that it can be defined as a continuous operator on BMO(R). We define g as the Poisson integral of b + iHb, so that g belongs to BMOA(C+) and has b + iHb as a boundary value function. We claim thatf = h/g ∈ H1a(C+).Indeed, since b ≥ 1 and h ∈ Hloga (C+), we obtain that f ∈ Hloga (C+). Moreover, f has the boundary value function f0 =h0(b + iHb)−1. We write:| f0(x)| ≤ h∗(x)log(e + |x|) + log(e + M(|h0|1/2)(x))≤ C h∗(x)log(e + |x|) + log(e + (h∗)1/2(x)) ,where we have used the inequality (2.3). We use Theorem 2.1 to conclude that f0 is in L1(R), then Theorem 2.2 to conclude that f ∈H1a(C+). This ends the proof. 4. Hankel operators and conclusionLet us now give a sketch of the proof of Theorem 1.2. It is equivalent to prove that the Hankel form defined byHb( f , g) := 〈b, f g〉is bounded on H1a (C+) × BMOA(C+) if and only if b belongs to BMOAlog(C+). This is a straightforward consequence of the main theorem once one knows that BMOAlog(C+) is the dual of the space Hloga (C+). The duality has been proved in [9] for the real Hardy space Hlog(R), as well as the continuity of the Hilbert transform, which implies the required duality result. We refer to [2,8,9] for the definitions and their applications in studying commutators of singular integral operators.The main theorem implies also that, on the real line, the embedding of products of functions in H1(R) and BMO(R) in L1(R) + Hlog(R) is sharp: any real function that can be written as the sum of an integrable function and of a function in Hlog(R) can also be written as a sum f1 g1 + f2 g2, with f1 and f2 in H1(R), g1 and g2 in BMO(R). The proof is the same as for the unit disc in [4] (see also [1]).One may ask whether these results can be generalized to the Siegel domain that is holomorphically equivalent to the unit ball. This is the case for Proposition 3.1. But the converse, with the construction of a function in BMOA(C+) from its real part, cannot be generalized in higher dimension.References[1] A. Bonami, S. Grellier, Hankel operators and weak factorization for Hardy–Orlicz spaces, Colloq. Math. 118 (1) (2010) 107–132.[2] A. Bonami, S. Grellier, L.D. Ky, Paraproducts and products of functions in BMO(Rn) and H1(Rn) through wavelets, J. Math. Pures Appl. (9) 97 (3) (2012) 230–241.[3] A. Bonami, S. Grellier, L.D. Ky, Hardy spaces of Musielak–Orlicz type on the half-plane, preprint.[4] A. Bonami, T. Iwaniec, P. Jones, M. Zinsmeister, On the product of functions in BMO and H1, Ann. Inst. Fourier (Grenoble) 57 (5) (2007) 1405–1439.[5] J. Cao, D.-C. Chang, D. Yang, S. Yang, Riesz transform characterizations of Musielak–Orlicz–Hardy spaces, Trans. Amer. Math. Soc. (to appear) or arXiv:1401.7373.[6] R.R. Coifman, R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (2) (1980) 249–254.[7] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.[8] L.D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc. 365 (6) (2013) 2931–2958.[9] L.D. Ky, New Hardy spaces of Musielak–Orlicz type and boundedness of sublinear operators, Integral Equ. Oper. Theory 78 (1) (2014) 115–150.[10] Y. Liang, J. Huang, D. Yang, New real-variable characterizations of Musielak–Orlicz Hardy spaces, J. Math. Anal. Appl. 395 (1) (2012) 413–428.

Factorization of some Hardy type spaces of holomorphic functions

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