A 𝑇(1)-theorem in relation to a semigroup of operators and applications to new paraproducts

Bernicot, Frédéric

doi:10.1090/S0002-9947-2012-05609-1

A $T(1)$-theorem in relation to a semigroup of operators and applications to new paraproducts
HTML articles powered by AMS MathViewer

by Frédéric Bernicot PDF

Trans. Amer. Math. Soc. 364 (2012), 6071-6108 Request permission

Abstract:

In this work, we are interested in developing new directions of the famous $T(1)$-theorem. More precisely, we develop a general framework where we look to replace the John-Nirenberg space $BMO$ (in the classical result) by a new $BMO_{L}$, associated to a semigroup of operators $(e^{-tL})_{t>0}$. These new spaces $BMO_L$ (including $BMO$) have recently appeared in numerous works in order to extend the theory of Hardy and $BMO$ space to more general situations. Then we give applications by describing boundedness for a new kind of paraproduct, built on the considered semigroup. In addition we obtain a version of the classical $T(1)$-theorem for doubling Riemannian manifolds.

References

L. Ambrosio, M. Miranda Jr., and D. Pallara, Special functions of bounded variation in doubling metric measure spaces, Calculus of variations: topics from the mathematical heritage of E. De Giorgi, Quad. Mat., vol. 14, Dept. Math., Seconda Univ. Napoli, Caserta, 2004, pp. 1–45. MR 2118414
Pascal Auscher and Philippe Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249 (1998), viii+172 (English, with English and French summaries). MR 1651262
Pascal Auscher, On $L^p$ estimates for square roots of second order elliptic operators on $\Bbb R^n$, Publ. Mat. 48 (2004), no. 1, 159–186. MR 2044643, DOI 10.5565/PUBLMAT_{4}8104_{0}8
Pascal Auscher, On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\Bbb R^n$ and related estimates, Mem. Amer. Math. Soc. 186 (2007), no. 871, xviii+75. MR 2292385, DOI 10.1090/memo/0871
Pascal Auscher, Thierry Coulhon, Xuan Thinh Duong, and Steve Hofmann, Riesz transform on manifolds and heat kernel regularity, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 911–957 (English, with English and French summaries). MR 2119242, DOI 10.1016/j.ansens.2004.10.003
Pascal Auscher, Alan McIntosh, and Emmanuel Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008), no. 1, 192–248. MR 2365673, DOI 10.1007/s12220-007-9003-x
Pascal Auscher and José María Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights, Adv. Math. 212 (2007), no. 1, 225–276. MR 2319768, DOI 10.1016/j.aim.2006.10.002
Pascal Auscher and Emmanuel Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $\Bbb R^n$, J. Funct. Anal. 201 (2003), no. 1, 148–184. MR 1986158, DOI 10.1016/S0022-1236(03)00059-4
Nadine Badr and Frédéric Bernicot, Abstract Hardy-Sobolev spaces and interpolation, J. Funct. Anal. 259 (2010), no. 5, 1169–1208. MR 2652185, DOI 10.1016/j.jfa.2010.02.012
Frédéric Bernicot and Jiman Zhao, New abstract Hardy spaces, J. Funct. Anal. 255 (2008), no. 7, 1761–1796. MR 2442082, DOI 10.1016/j.jfa.2008.06.018
Frédéric Bernicot, Use of abstract Hardy spaces, real interpolation and applications to bilinear operators, Math. Z. 265 (2010), no. 2, 365–400. MR 2609316, DOI 10.1007/s00209-009-0520-0
F. Bernicot and J. Zhao, Abstract framework for John Nirenberg inequalities and applications to Hardy spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (2012) available at http://arxiv.org/abs/1003.0591.
Jean-Michel Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 209–246 (French). MR 631751, DOI 10.24033/asens.1404
Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI 10.4064/cm-60-61-2-601-628
Ronald R. Coifman and Yves Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Société Mathématique de France, Paris, 1978 (French). With an English summary. MR 518170
Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR 0499948, DOI 10.1007/BFb0058946
Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
Thierry Coulhon, Itération de Moser et estimation gaussienne du noyau de la chaleur, J. Operator Theory 29 (1993), no. 1, 157–165 (French). MR 1277971
Thierry Coulhon and Xuan Thinh Duong, Riesz transforms for $1\leq p\leq 2$, Trans. Amer. Math. Soc. 351 (1999), no. 3, 1151–1169. MR 1458299, DOI 10.1090/S0002-9947-99-02090-5
Guy David and Jean-Lin Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2) 120 (1984), no. 2, 371–397. MR 763911, DOI 10.2307/2006946
G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), no. 4, 1–56 (French). MR 850408, DOI 10.4171/RMI/17
E. B. Davies, Non-Gaussian aspects of heat kernel behaviour, J. London Math. Soc. (2) 55 (1997), no. 1, 105–125. MR 1423289, DOI 10.1112/S0024610796004607
Xuan Thinh Duong and Lixin Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), no. 4, 943–973. MR 2163867, DOI 10.1090/S0894-0347-05-00496-0
Xuan Thinh Duong and Lixin Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005), no. 10, 1375–1420. MR 2162784, DOI 10.1002/cpa.20080
Jacek Dziubański, Atomic decomposition of $H^p$ spaces associated with some Schrödinger operators, Indiana Univ. Math. J. 47 (1998), no. 1, 75–98. MR 1631616, DOI 10.1512/iumj.1998.47.1479
Jacek Dziubański, Spectral multipliers for Hardy spaces associated with Schrödinger operators with polynomial potentials, Bull. London Math. Soc. 32 (2000), no. 5, 571–581. MR 1767710, DOI 10.1112/S0024609300007311
Jacek Dziubański, Note on $H^1$ spaces related to degenerate Schrödinger operators, Illinois J. Math. 49 (2005), no. 4, 1271–1297. MR 2210363
Jacek Dziubański and Jacek Zienkiewicz, Hardy spaces $H^1$ for Schrödinger operators with compactly supported potentials, Ann. Mat. Pura Appl. (4) 184 (2005), no. 3, 315–326. MR 2164260, DOI 10.1007/s10231-004-0116-6
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
Loukas Grafakos and José María Martell, Extrapolation of weighted norm inequalities for multivariable operators and applications, J. Geom. Anal. 14 (2004), no. 1, 19–46. MR 2030573, DOI 10.1007/BF02921864
Loukas Grafakos, Liguang Liu, Carlos Pérez, and Rodolfo H. Torres, The multilinear strong maximal function, J. Geom. Anal. 21 (2011), no. 1, 118–149. MR 2755679, DOI 10.1007/s12220-010-9174-8
Alexander Grigor′yan, Estimates of heat kernels on Riemannian manifolds, Spectral theory and geometry (Edinburgh, 1998) London Math. Soc. Lecture Note Ser., vol. 273, Cambridge Univ. Press, Cambridge, 1999, pp. 140–225. MR 1736868, DOI 10.1017/CBO9780511566165.008
Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. MR 1683160, DOI 10.1090/memo/0688
Steve Hofmann, An off-diagonal $T1$ theorem and applications, J. Funct. Anal. 160 (1998), no. 2, 581–622. With an appendix “The Mary Weiss lemma” by Loukas Grafakos and the author. MR 1665299, DOI 10.1006/jfan.1998.3323
Steve Hofmann, Local $Tb$ theorems and applications in PDE, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1375–1392. MR 2275650
Steve Hofmann and Svitlana Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), no. 1, 37–116. MR 2481054, DOI 10.1007/s00208-008-0295-3
Tuomas P. Hytönen, An operator-valued $Tb$ theorem, J. Funct. Anal. 234 (2006), no. 2, 420–463. MR 2216905, DOI 10.1016/j.jfa.2005.11.001
Tuomas Hytönen and Lutz Weis, A $T1$ theorem for integral transformations with operator-valued kernel, J. Reine Angew. Math. 599 (2006), 155–200. MR 2279101, DOI 10.1515/CRELLE.2006.081
F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317
Stephen Keith and Xiao Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. (2) 167 (2008), no. 2, 575–599. MR 2415381, DOI 10.4007/annals.2008.167.575
Andrei K. Lerner, Sheldy Ombrosi, Carlos Pérez, Rodolfo H. Torres, and Rodrigo Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222–1264. MR 2483720, DOI 10.1016/j.aim.2008.10.014
Alan McIntosh, Operators which have an $H_\infty$ functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210–231. MR 912940
Alan McIntosh and Yves Meyer, Algèbres d’opérateurs définis par des intégrales singulières, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 8, 395–397 (French, with English summary). MR 808636
José María Martell, Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications, Studia Math. 161 (2004), no. 2, 113–145. MR 2033231, DOI 10.4064/sm161-2-2
Y. Meyer, Régularité des solutions des équations aux dérivées partielles non linéaires, Sém. Bourbaki 22 no. 550, (1979).
Yves Meyer, Remarques sur un théorème de J.-M. Bony, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981, pp. 1–20 (French). MR 639462
F. Nazarov, S. Treil, and A. Volberg, The $Tb$-theorem on non-homogeneous spaces, Acta Math. 190 (2003), no. 2, 151–239. MR 1998349, DOI 10.1007/BF02392690
F. Nazarov, S. Treil, and A. Volberg, Accretive system $Tb$-theorems on nonhomogeneous spaces, Duke Math. J. 113 (2002), no. 2, 259–312. MR 1909219, DOI 10.1215/S0012-7094-02-11323-4
Laurent Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289, Cambridge University Press, Cambridge, 2002. MR 1872526
Adam Sikora, Sharp pointwise estimates on heat kernels, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 187, 371–382. MR 1412562, DOI 10.1093/qmath/47.3.371
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Xavier Tolsa, Littlewood-Paley theory and the $T(1)$ theorem with non-doubling measures, Adv. Math. 164 (2001), no. 1, 57–116. MR 1870513, DOI 10.1006/aima.2001.2011
A. V. Vähäkangas, A $T1$ theorem for weakly singular integral operators, submitted, available at http://arxiv.org/abs/1001.5072.
Dachun Yang, $T1$ theorems on Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and their applications, Z. Anal. Anwendungen 22 (2003), no. 1, 53–72. MR 1962076, DOI 10.4171/ZAA/1132