Propagation of low regularity for solutions of nonlinear PDEs on a Riemannian manifold with a sub-Laplacian structure

Bernicot, Frédéric; Sire, Yannick

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 5, pp. 935-958.

Following Bernicot (2012) [7], 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.

Classification : 35S05, 58J47
Mots clés : Paralinearization, Sub-Laplacian operator, Riemannian manifold

[1] L. Ambrosio, M. Miranda, 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), 1-45 | Zbl

[2] P. Auscher, On necessary and sufficient conditions for $L^p$ estimates of Riesz transforms associated to elliptic operators on ${ℝ}^n$ and related estimates, Mem. Amer. Math. Soc. 186 (2007), 871

[3] N. Badr, F. Bernicot, E. Russ, Algebra properties for Sobolev spaces – Applications to semilinear PDEʼs on manifolds, J. Anal. Math. 118 (2012), 509-544, arXiv:1107.3826 | Zbl

[4] H. Bahouri, C. Fermanian-Kammerer, I. Gallagher, Phase Space Analysis and Pseudodifferential Calculus on the Heisenberg Group, Astérisque vol. 342, Société Math. de France (2012) | Zbl

[5] D. Bakry, M. Émery, Diffusions hypercontractives, Lecture Notes in Math. vol. 1123 (1985) | Numdam | Zbl

[6] F. Bernicot, J. Zhao, New Abstract Hardy Spaces, J. Funct. Anal. 255 (2008), 1761-1796 | Zbl

[7] F. Bernicot, A $T\left(1\right)$-Theorem in relation to a semigroup of operators and applications to new paraproducts, Trans. Amer. Math. Soc. 364 (2012), 6071-6108, arXiv:1005.5140 | Zbl

[8] J.M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Scient. E.N.S. 14 (1981), 209-246 | EuDML | Numdam | Zbl

[9] H.-Q. Bui, X.T. Duong, L. Yan, Calderón reproducing formulas and new Besov spaces associated with operators, Adv. Math. 229 no. 4 (2012), 2449-2502 | Zbl

[10] R.R. Coifman, Y. Meyer, Au-delà des opérateurs pseudo-diffeŕentiels, Astérisque vol. 57, Société Math. de France (1978) | Numdam | Zbl

[11] R. Coifman, G. Weiss, Analyse harmonique sur certains espaces homogènes, Lecture Notes in Math. vol. 242 (1971) | Zbl

[12] T. Coulhon, E. Russ, V. Tardivel-Nachef, Sobolev algebras on Lie groups and Riemannian manifolds, Amer. J. Math. 123 (2001), 283-342 | Zbl

[13] N. Dungey, A.F.M. Ter Elst, D.W. Robinson, Analysis on Lie groups with polynomial growth, Progress in Mathematics vol. 214, Birkäuser Boston Inc., Boston, MA (2003) | Zbl

[14] X.T. Duong, L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 no. 4 (2005), 943-973 | Zbl

[15] X.T. Duong, L. Yan, New function spaces of BMO type, the John–Nirenberg inequality, interplation and applications, Commun. Pure Appl. Math. 58 no. 10 (2005), 1375-1420 | Zbl

[16] C. Fefferman, A. Sánchez-Calle, Fundamental solutions for second order subelliptic operators, Ann. of Math. 124 no. 2 (1986), 247-272 | Zbl

[17] D. Frey, Paraproducts via $H^{\infty }$-functional calculus and a $T\left(1\right)$-Theorem for non-integral operators, Phd thesis, available at http://digbib.ubka.uni-karlsruhe.de/volltexte/documents/1687378.

[18] D. Frey, Paraproducts via $H^{\infty }$-functional calculus, arXiv:1107.4348 | Zbl

[19] G. Furioli, C. Melzi, A. Veneruso, Littlewood–Paley decompositions and Besov spaces on Lie groups of polynomial growth, Math. Nachrichten 279 (2006), 1028-1040 | Zbl

[20] I. Gallagher, Y. Sire, Besov algebras on Lie groups of polynomial growth and related results, arXiv:1010.0154

[21] Y. GuivarcʼH, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333-379 | EuDML | Numdam | Zbl

[22] L. Hörmander, Hypoelliptic differential operators, Ann. de lʼInst. Four. 11 (1961), 477-492 | EuDML | Numdam | Zbl

[23] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171 | Zbl

[24] O. Ivanovici, F. Planchon, On the energy critical Schrödinger equation in 3D non-trapping domains, Ann. I. H. Poincaré - A.N. 27 (2010), 1153-1177 | Numdam | Zbl

[25] D.S. Jerison, A. Sáncher-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J. 35 (1986), 835-854 | Zbl

[26] L. Kapitanskii, Some generalizations of the Strichartz–Brenner inequality, Leningrad Math. J. 1 (1990), 693-726 | Zbl

[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] Y. Meyer, Remarques sur un théorème de J.M. Bony, Suppl. Rendiconti del Circ. Mate. di Palermo 1 (1980), 8-17

[29] A. Nagel, E.M. Stein, S. Wainger, Balls and metrics defined by vector fields I: Basic properties, Acta Math. 155 (1985), 103-147 | Zbl

[30] D.W. Robinson, Elliptic Operators and Lie Groups, Oxford Univerisity Press (1991)

[31] A. Sánchez-Calle, Fundamental solutions and geometry of sum of squares of vector fields, Inv. Math. 78 (1984), 143-160 | EuDML | Zbl

[32] H. Triebel, Spaces of Besov–Hardy–Sobolev type on complete Riemannian manifolds, Ark. Mat. 24 (1986), 299-337 | Zbl

[33] N.Th. Varopoulos, Analysis and nilpotent groups, J. Funct. Anal. 66 (1986), 406-431 | Zbl