Research

RESEARCH PROJECT:ESTIMATE FOR THE SMALL HANKEL OPERATOR ON THE BIDISC AND RELATED TOPICS

BENOˆIT SEHBA

The guiding thread of this program concerns new difficulties when passing from one parameter analysis to multiparameter analysis. When considering analysis of spaces o f holomorphic functions in the unit ball, the boundary values of a function are obtained by letting one parameter go to 0, namely the distance to the boundary. The same kind of problems in the bidisk D×D, considered now with its distinguishd boundary 2, involves two parameters when approaching it, corresponding to each coordinate. One passes from the by now classical Calder ́on-Zygmund analysis to the much more difficult and less known two parameter analysis. The aim of this project is to push further a certain number of this methods and understand better some questions related to Hankel operators. .

The aim of this project is to understand some still open but related ques- tions in multi-parameter harmonic analysis. If in the unit ball B^n of C^n and related domains all the ingredients of real analysis are available, the case of the polydisc required the development of more elaborated techniques.

A fundamental operator in this area is the Hilbert transform. Boundedness of the Hilbert transform on L^p, p > 1 was proved by M. Riesz in 1927. This operator is the standard example of a class of singular integral operators known as the Calder ́on-Zygmund operators which are all bounded on L^p for 1 < p < ∞. Our interest here is for the endpoints behavior of related operators, i.e when p = 1. S. Pertermichl in ([14]) was the first to observe that the Hilbert transform can be approximated by the so called “dyadic shift”. This allows to transfer questions from the continuous side to the dyadic one. This is now a quite standard technique in harmonic analysis and more recently, this type of approach known as sparse domination has been applied in analytic function spaces and their operators (see for example [9]). Note that I used this technique in some of my works starting from my Ph.D thesis [4, 5, 6, 16, 17, 18, 19, 20, 21, 22, 23]. In fact the ideas of Petermichl were extended by T. Hyt ̈onen to general Calder ́on-Zygmund operators [10] with an application to the proof of the A2-conjecture.

Let me recall that the real Hardy H1(R2) is the following set of functions: {f : f ∈ L^1,H1f ∈ L^1,H2f ∈ L^1,H1H2f ∈ L^1}.

Here Hj is the Hilbert transform in the j-th variable. Its dual space is the so called Chang-Fefferman or product BMO space [3].

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Let H^1(R^2_+ × R^2_+) be the biholomorphic analogue of H1(R) = {f ∈ L^1(R), Hf ∈ L^1(R)}. The product Hilbert-Hardy is denoted H^2(R^2_+ × R^2_+). The little Hankel operator hb with symbol b can be understood as the oper- ator defined on H^2(R^2_+ × R^2_+) by

⟨hb(f),g⟩ = ⟨b,fg⟩
for f and g in H^2(R^2_+ ×R^2_+). Here ⟨, ⟩ denotes the inner product of H^2(R^2_+ ×

R^2_+ ).

The study of the above operator is somehow equivalent to the one of the iterated commutator [[Mb, H1], H2], where Mb is the multiplication by b. Ferguson and Sadosky proved in [8] that the operator norm on L^2(R^2) of [[Mb, H1], H2] is controlled by the BMO-norm of the symbol b. The converse was claimed in [7] but pretty recently, a gap in the proof was found by Volberg (see [11]). We note a proof of the converse, would imply a weak factorization of elements in H1(R2+ ×R2+) (see [7]) which is equivalent to the boundedness of the little Hankel operator on H^2(R^2_+ × R^2_+). In fact weak factorizations are very useful in the study of Hankel operators.

The research project: For the analogue spaces in the bidisc D × D, we would like to find estimates for the small Hankel operator from H^1(D × D) to itself. This question implies study of product of elements in this space and elements in its dual space and weak factorization for associated spaces to be specified.

In relation with the above problem and on the dual side, there is the question of the multipliers (the multiplication Mb appears in the iterated commutator) of the product BMO or the dyadic BMO which involves the one of paraproducts that can be found in the work of M. Lacey, S. Treil, S. Volberg and their collaborators. These paraproducts arise naturally as non-diagonal terms in the product of the expansion of functions in the Haar basis. The estimates of these paraproducts were obtained in [15, 22]. Note that iterated commutators as defined above were also studied in [15], more precisely a sufficient condition for their boundedness on BMO was obtained.

Few questions naturally follow from the above results on BMO ([15, 22]). I would like to answer the following questions

• How do the above results extend to the n-torus? What are the multipliers of BMO(T^n)?

• Can the n-torus be replaced by the unbounded domain Rn?

• How could we deduce from these results, the boundedness of Hankel
  operators on H^1(D × D)?
  It seems natural to start with the question of unbounded domains, where I developed recently new results for the one parameter jointly with A. Bonami and S. Grellier ([1, 2]), namely necessary and sufficient conditions for non negative functions outside a compact set to belong to the one parameter Hardy space H1(Rn). A related question is the one of estimations with loss for the Hankel operator, i.e from Hp to Hq with 1q = p1 + 1r . This question was answered in [13] for the Bergman spaces but the case p > 1 and q = 1 is still open. An answer to this question might require extension of the type of weak factorization in [2].

PROJECT IHES 3

References

• [1]  Bonami, A., Grellier, S., Sehba, B. F.: Global Stein Theorem on Hardy spaces. Anal. Math. (2023), to appear.

• [2]  Bonami, A., Grellier, S., Sehba, B. F.: Avatars Stein’s Theorem in the complex setting. Revista Uni. Mat. Argent. (2023), to appear.

• [3]  Chang, S-Y. and Fefferman, R.:A continuous version of duality of H1 with BMO on the bidisc. Ann. of Math. (2) 112 (1980), no. 1, 1791.

• [4]  Tanoh Dje, J. M.; Sehba, Benoˆıt F., Two-weight inequalities for multilinear max- imal functions in Orlicz spaces. Banach J. Math. Anal. 17 (2023), no. 2, Paper No. 30, 47 pp.

• [5]  Tanoh Dje, J. M.; Sehba, Benoˆıt F., Carleson embeddings for Hardy-Orlicz and Bergman-Orlicz spaces of the upper-half plane. Funct. Approx. Comment. Math. 64 (2021), no. 2, 163–201.

• [6]  Tanoh Dje, J. M.; Feuto, J.; Sehba, Benoˆıt F., A generalization of the Carleson lemma and weighted norm inequalities for the maximal functions in the Orlicz setting. J. Math. Anal. Appl. 491 (2020), no. 1, 124248, 24 pp.

• [7]  Ferguson, S., Lacey, M. T.: A characterization of product BMO by commutators. Acta Math. 189 (2002), no. 2, 143–160.

• [8]  Ferguson, S., Sadosky, C., Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures. J. Anal. Math. 81 (2000), 239–267.

• [9]  Hu, B., Li, S., Shi, Y., Wick, B., Sparse domination of weighted composition oper- ators on weighted Bergman spaces. J. Funct. Anal. 280, Issue 6 (2021), 108897.

[10] Hyto ̈nen, T.: The sharp weighted bound for general Caldero ́n-Zygmund operators, to appear in Ann. Math., also available at arXiv:1007.4330

[11] Lacey, M. T.; Petermichl, S.; Pipher, J. C.; Wick, Brett D.; Notification of error: multiparameter Riesz commutators. Amer. J. Math. 143 (2021), no. 2, 333–334. 42B20 (47B06)

[12] Lacey, M. T., Terwilleger, E.:Hankel operators in several complex variables and product BMO. Houston J. Math. 35 (2009), no. 1, 159–183.

[13] Pau, J.; Zhao, R., Weak factorization and Hankel forms for weighted Bergman spaces on the unit ball. Math. Ann. 363 (2015), no. 1-2, 363–383.

[14] Petermichl, S.: Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol. C. R. Acad. Sci. Paris S. I Math. 330 (2000), no. 6, 45500.

[15] Pott, S.; Sehba B. F., Logarithmic mean oscillation on the polydisc, endpoint results for multi-parameter paraproducts, and commutators on BMO. J. Anal. Math. 117 (2012), 1–27.

[16] Sehba, B. F., Sawyer-type characterizations and sharp weighted norm estimates for Bergman-type operators. Positivity 25 (2021), no. 1, 49–71.

[17] Sehba, B. F., Weighted norm inequalities for fractional Bergman operators. Constr. Approx. 51 (2020), no. 1, 225–245.

[18] Sehba, B. F., Maximal functions and measures on the upper-half plane. Anal. Math. 45 (2019), no. 1, 177–199.

[19] Sehba, B. F., On the weighted estimate of the Bergman projection. Czechoslovak Math. J. 68 (143) (2018), no. 2, 497–511.

[20] Sehba, B. F., Weighted boundedness of maximal functions and fractional Bergman operators. J. Geom. Anal. 28 (2018), no. 2, 1635–1664.

[21] Sehba, B. F., On two-weight norm estimates for multilinear fractional maximal function. J. Math. Soc. Japan 70 (2018), no. 1, 71–94.

[22] Sehba, B. F., Logarithmic mean oscillation on the polydisc, multi-parameter para- products and iterated commutators. J. Fourier Anal. Appl. 20 (2014), no. 3, 500–523. [23] Sehba, B. F.: Operators on some analytic function spaces and their dyadic counter-

parts. Ph. D thesis, University of Glasgow 2009.
[24] Stegenga, D. A:Bounded Toeplitz operators on H1H1 and applications of the du-

ality between H1 and the functions of bounded mean oscillation. Amer. J. Math. 98 (1976), no. 3, 57309.

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[25] Treil S.:H1 and dyadic H1. Linear and complex analysis, 17903, Amer. Math. Soc. Transl. Ser. 2, 226, Amer. Math. Soc., Providence, RI, 2009.

[26] Viviani, B. E.: An atomic decomposition of the predual of BMO(ρ). Rev. Mat. Iberoamericana 3 (1987), no. 3-4, 401 – 425.