Weighted norm inequalities, off-diagonal estimates and elliptic operators

Pascal Auscher (1) , José Maria Martell (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 IMFF - Instituto de Matematicas y Fisica Fundamental

We give an overview of the generalized Calder\'{o}n-Zygmund theory for ``non-integral'' singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. $L^p-L^q$ off-diagonal estimates when $p\le q$ play an important role and we present them. They are particularly well suited to the semigroups generated by second order elliptic operators and the range of exponents $(p,q)$ rules the $L^p$ theory for many operators constructed from the semigroup and its gradient. Such applications are summarized.

Mots clés: Calder\'{o}n-Zygmund theory; spaces of homogeneous type; Muckenhoupt weights; singular non-integral operators; commutators with $\BMO$ functions; elliptic operators in divergence form; holomorphic calculi; Riesz transforms; square functions; Riemannian manifolds

Domaines: Analyse classique [math.CA]; Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]

Pascal Auscher, José Maria Martell. Weighted norm inequalities, off-diagonal estimates and elliptic operators. Patricio Cifuentes, José García-Cuerva, Gustavo Garrigós, Eugenio Hernández, José María Martell, Javier Parcet, Alberto Ruiz, Fernando Soria, José Luis Torrea and Ana Vargas. Harmonic Analysis and Partial Differential Equations,, pp.61-83, 2010, contemporary mathematics, 978-0-8218-4770-1. ⟨hal-00331495⟩