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Propagation of low regularity for solutions of nonlinear PDEs on a riemannian manifold with a sub-Laplacian structure. (English) Zbl 06295447
Summary: Following F. Bernicot [Trans. Am. Math. Soc. 364, No. 11, 6071–6108 (2012; Zbl 1281.46028)], we introduce a notion of paraproducts associated to a semigroup. We do not use Fourier transform arguments and the background manifold is doubling, endowed with a sub-Laplacian structure. Our main result is a paralinearization theorem in a non-Euclidean framework, with an application to the propagation of regularity for some nonlinear PDEs.

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
58J47 Propagation of singularities; initial value problems on manifolds
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[1] Ambrosio, L.; Miranda, M.; Pallara, D., 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, (2004), Dept. Math., Seconda Univ. Napoli Caserta), 1-45 · Zbl 1089.49039
[2] Auscher, P., On necessary and sufficient conditions for \(L^p\) estimates of Riesz transforms associated to elliptic operators on \(\mathbb{R}^n\) and related estimates, Mem. Amer. Math. Soc., 186, 871, (2007)
[3] Badr, N.; Bernicot, F.; Russ, E., Algebra properties for Sobolev spaces - applications to semilinear PDEʼs on manifolds, J. Anal. Math., 118, 509-544, (2012) · Zbl 1286.46033
[4] Bahouri, H.; Fermanian-Kammerer, C.; Gallagher, I., Phase space analysis and pseudodifferential calculus on the Heisenberg group, Astérisque, vol. 342, (2012), Société Math. de France · Zbl 1246.35003
[5] Bakry, D.; Émery, M., Diffusions hypercontractives, (Lecture Notes in Math., vol. 1123, (1985)), XIX: 12, pp. 177-206 · Zbl 0561.60080
[6] Bernicot, F.; Zhao, J., New abstract Hardy spaces, J. Funct. Anal., 255, 1761-1796, (2008) · Zbl 1171.42012
[7] Bernicot, F., A \(T(1)\)-theorem in relation to a semigroup of operators and applications to new paraproducts, Trans. Amer. Math. Soc., 364, 6071-6108, (2012), available at · Zbl 1281.46028
[8] Bony, J. M., Calcul symbolique et propagation des singularités pour LES équations aux dérivées partielles non linéaires, Ann. Scient. E.N.S., 14, 209-246, (1981) · Zbl 0495.35024
[9] Bui, H.-Q.; Duong, X. T.; Yan, L., Calderón reproducing formulas and new Besov spaces associated with operators, Adv. Math., 229, 4, 2449-2502, (2012) · Zbl 1241.46020
[10] Coifman, R. R.; Meyer, Y., Au-delà des opérateurs pseudo-diffeŕentiels, Astérisque, vol. 57, (1978), Société Math. de France · Zbl 0483.35082
[11] Coifman, R.; Weiss, G., Analyse harmonique sur certains espaces homogènes, Lecture Notes in Math., vol. 242, (1971) · Zbl 0224.43006
[12] Coulhon, T.; Russ, E.; Tardivel-Nachef, V., Sobolev algebras on Lie groups and Riemannian manifolds, Amer. J. Math., 123, 283-342, (2001) · Zbl 0990.43003
[13] Dungey, N.; ter Elst, A. F.M.; Robinson, D. W., Analysis on Lie groups with polynomial growth, Progress in Mathematics, vol. 214, (2003), Birkäuser Boston Inc. Boston, MA · Zbl 1041.43003
[14] Duong, X. T.; Yan, L., Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18, 4, 943-973, (2005) · Zbl 1078.42013
[15] Duong, X. T.; Yan, L., New function spaces of BMO type, the John-Nirenberg inequality, interplation and applications, Commun. Pure Appl. Math., 58, 10, 1375-1420, (2005) · Zbl 1153.26305
[16] Fefferman, C.; Sánchez-Calle, A., Fundamental solutions for second order subelliptic operators, Ann. of Math., 124, 2, 247-272, (1986) · Zbl 0613.35002
[17] D. Frey, Paraproducts via \(H^\infty\)-functional calculus and a \(T(1)\)-Theorem for non-integral operators, Phd thesis, available at http://digbib.ubka.uni-karlsruhe.de/volltexte/documents/1687378.
[18] Frey, D., Paraproducts via \(H^\infty\)-functional calculus, available at · Zbl 1277.42020
[19] Furioli, G.; Melzi, C.; Veneruso, A., Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth, Math. Nachrichten, 279, 1028-1040, (2006) · Zbl 1101.22006
[20] Gallagher, I.; Sire, Y., Besov algebras on Lie groups of polynomial growth and related results, available at
[21] Guivarcʼh, Y., Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France, 101, 333-379, (1973) · Zbl 0294.43003
[22] Hörmander, L., Hypoelliptic differential operators, Ann. de lʼInst. Four., 11, 477-492, (1961) · Zbl 0099.30101
[23] Hörmander, L., Hypoelliptic second order differential equations, Acta Math., 119, 147-171, (1967) · Zbl 0156.10701
[24] Ivanovici, O.; Planchon, F., On the energy critical Schrödinger equation in 3D non-trapping domains, Ann. I. H. Poincaré - A.N., 27, 1153-1177, (2010) · Zbl 1200.35066
[25] Jerison, D. S.; Sáncher-Calle, A., Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J., 35, 835-854, (1986) · Zbl 0639.58026
[26] Kapitanskii, L., Some generalizations of the Strichartz-Brenner inequality, Leningrad Math. J., 1, 693-726, (1990) · Zbl 0732.35118
[27] A. McInstosh, Operators which have an \(H_\infty\)-calculus, in: Miniconference on Operator Theory and Partial Differential Equations, Proc. Centre Math. Analysis, ANU, Canberra, vol. 14, 1986, pp. 210-231.
[28] Meyer, Y., Remarques sur un théorème de J.M. bony, Suppl. Rendiconti del Circ. Mate. di Palermo, 1, 8-17, (1980)
[29] Nagel, A.; Stein, E. M.; Wainger, S., Balls and metrics defined by vector fields I: basic properties, Acta Math., 155, 103-147, (1985) · Zbl 0578.32044
[30] Robinson, D. W., Elliptic operators and Lie groups, (1991), Oxford Univerisity Press · Zbl 0747.47030
[31] Sánchez-Calle, A., Fundamental solutions and geometry of sum of squares of vector fields, Inv. Math., 78, 143-160, (1984) · Zbl 0582.58004
[32] Triebel, H., Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Ark. Mat., 24, 299-337, (1986) · Zbl 0664.46026
[33] Varopoulos, N. Th., Analysis and nilpotent groups, J. Funct. Anal., 66, 406-431, (1986) · Zbl 0595.22008
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