Conservation de certaines propriétés à travers un contrôle épars d’un opérateur et applications au projecteur de Leray–Hopf

Benea, Cristina; Bernicot, Frédéric

Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2329-2379.

Nous poursuivons l’étude d’un contrôle épars d’un opérateur singulier. Plus précisément nous expliquons comment on peut conserver certaines propriétés de l’opérateur initial à travers un tel contrôle et décrivons quelques applications : bornitude de l’adjoint de la transformée de Riesz et du projecteur de Leray. De plus, nous nous intéresserons à donner un regard nouveau sur les dominations éparses à travers les oscillations et les fonctions carrées localisées. Aussi, nous dévoilerons une connexion entre les bons intervalles de la décomposition éparse et une décomposition atomique.

We pursue the study of a sparse control for a singular operator. More precisely, we describe how one can track some properties of the initial operator, through such a control and describe also some applications: boundedness of the adjoint of a Riesz transform and of the Leray projector. Moreover, we will be interested in giving a new insight on the sparse domination through the oscillations and the localized square functions. Also, we will reveal a connection between the good intervals of the sparse domination and the atomic decomposition for a function in a Hardy space.

Classification : 42B15, 42B25, 42B35
Mot clés : Opérateurs épars, poids, espaces de Hardy et BMO
Keywords: Sparse operators, weights, Hardy and BMO spaces

[1] Auscher, Pascal On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on ${ℝ}^n$ and related estimates, Mem. Am. Math. Soc., Volume 186 (2007) no. 871, xviii+75 pages | DOI | MR | Zbl

[2] Auscher, Pascal On the Calderón-Zygmund lemma for Sobolev functions (2008) (http://arxiv.org/abs/0810.5029)

[3] Auscher, Pascal; Hofmann, Steve; Muscalu, Camil; Tao, Terence; Thiele, Christoph Carleson measures, trees, extrapolation, and $T\left(b\right)$ theorems, Publ. Mat., Volume 46 (2002) no. 2, pp. 257-325 | DOI | MR | Zbl

[4] Auscher, Pascal; McIntosh, Alan; Russ, Emmanuel Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal., Volume 18 (2008) no. 1, pp. 192-248 | DOI | MR | Zbl

[5] Auscher, Pascal; Russ, Emmanuel Hardy spaces and divergence operators on strongly Lipschitz domains of ${ℝ}^n$, J. Funct. Anal., Volume 201 (2003) no. 1, pp. 148-184 | DOI | MR | Zbl

[6] Barbatis, Gerassimos Stability of weighted Laplace-Beltrami operators under $L^p$-perturbation of the Riemannian metric, J. Anal. Math., Volume 68 (1996), pp. 253-276 | DOI | MR | Zbl

[7] Benea, Cristina; Bernicot, Frédéric; Luque, Teresa Sparse bilinear forms for Bochner Riesz multipliers and applications, Trans. Lond. Math. Soc., Volume 4 (2017) no. 1, pp. 110-128 | MR | Zbl

[8] Benea, Cristina; Muscalu, Camil Multiple vector-valued inequalities via the helicoidal method, Anal. PDE, Volume 9 (2016) no. 8, pp. 1931-1988 | DOI | MR | Zbl

[9] Bernicot, Frédéric; Frey, Dorothee; Petermichl, Stefanie Sharp weighted norm estimates beyond Calderón-Zygmund theory, Anal. PDE, Volume 9 (2016) no. 5, pp. 1079-1113 | DOI | MR | Zbl

[10] Bruna, Joaquim; Korenblum, Boris A note on Calderón-Zygmund singular integral convolution operators, Bull. Am. Math. Soc., Volume 16 (1987) no. 2, pp. 271-273 | DOI | MR | Zbl

[11] Carleson, Lennart Interpolations by bounded analytic functions and the corona problem, Ann. Math., Volume 76 (1962), pp. 547-559 | DOI | MR | Zbl

[12] Coifman, Ronald R.; Meyer, Yves Wavelets, Calderón-Zygmund Operators and Multilinear Operators, Cambridge Studies in Advanced Mathematics, 48, Cambridge University Press, 1997, xix+314 pages | Zbl

[13] Conde-Alonso, José M.; Culiuc, Amalia; Di Plinio, Francesco; Ou, Yumeng A sparse domination principle for rough singular integrals, Anal. PDE, Volume 10 (2017) no. 5, pp. 1255-1284 | DOI | MR | Zbl

[14] Coulhon, Thierry; Dungey, Nicholas Riesz transform and perturbation, J. Geom. Anal., Volume 17 (2007) no. 2, pp. 213-226 | DOI | MR | Zbl

[15] Cruz-Uribe, David; Martell, José María; Pérez, Carlos Sharp weighted estimates for classical operators, Adv. Math., Volume 229 (2012) no. 1, pp. 408-441 | DOI | MR | Zbl

[16] Culiuc, Amalia; Di Plinio, Francesco; Ou, Yumeng Domination of multilinear singular integrals by positive sparse forms (2016) (http://arxiv.org/abs/1603.05317, to appear in J. Lond. Math. Soc.)

[17] Lacey, Michael An elementary proof of the $A_2$ Bound, Isr. J. Math., Volume 217 (2017), pp. 181-195 | Zbl

[18] Lee, Ming-Yi; Lin, Chin-Cheng The molecular characterization of weighted Hardy spaces, J. Funct. Anal., Volume 188 (2002) no. 2, pp. 442-460 | DOI | MR | Zbl

[19] Lee, Ming-Yi; Lin, Chin-Cheng; Lin, Ying-Chieh A wavelet characterization for the dual of weighted Hardy spaces, Proc. Am. Math. Soc., Volume 137 (2009) no. 12, pp. 4219-4225 | DOI | MR | Zbl

[20] Lee, Ming-Yi; Lin, Chin-Cheng; Yang, Wei-Chi $H_w^p$ boundedness of Riesz transforms, J. Math. Anal. Appl., Volume 301 (2005) no. 2, pp. 394-400 | DOI | MR | Zbl

[21] Lerner, Andrei K. A pointwise estimate for the local sharp maximal function with applications to singular integrals, Bull. Lond. Math. Soc., Volume 42 (2010) no. 5, pp. 843-856 | Zbl

[22] Lerner, Andrei K. Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Adv. Math., Volume 226 (2011) no. 5, pp. 3912-3926 | DOI | MR | Zbl

[23] Lerner, Andrei K. On an estimate of Calderón-Zygmund operators by dyadic positive operators, J. Anal. Math., Volume 121 (2013), pp. 141-161 | Zbl

[24] Lerner, Andrei K. A simple proof of the $A_2$ conjecture, Int. Math. Res. Not., Volume 14 (2013), pp. 3159-3170 | MR | Zbl

[25] Lerner, Andrei K. On pointwise estimates involving sparse operators, New York J. Math., Volume 22 (2016), pp. 341-349 http://nyjm.albany.edu:8000/j/2016/22_341.html | MR | Zbl

[26] Lerner, Andrei K.; Nazarov, Fedor Intuitive dyadic calculus$:$ the basics (2015) (http://arxiv.org/abs/1508.05639, to appear in Expo. Math.)

[27] Meyer, Yves Ondelettes et opérateurs. II, Actualités Mathématiques., Hermann, 1990, p. i-xii and 217–384 (Opérateurs de Calderón-Zygmund.) | MR | Zbl

[28] Muscalu, Camil; Schlag, Wilhem Classical and Multilinear Harmonic Analysis, Cambridge Studies in Advanced Mathematics, 137, Cambridge University Press, 2013, xviii+370 pages | Zbl

[29] Muscalu, Camil; Tao, Terence; Thiele, Christoph $L^p$ estimates for the biest. I. The Walsh case, Math. Ann., Volume 329 (2004) no. 3, pp. 401-426 | DOI | MR | Zbl

[30] Neugebauer, Christoph J. Iterations of Hardy-Littlewood maximal functions, Proc. Am. Math. Soc., Volume 101 (1987) no. 2, pp. 272-276 | DOI | MR | Zbl

[31] Stein, Elias M. Harmonic analysis$:$ real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993, xiv+695 pages (With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III) | MR | Zbl