Memoirs of the American Mathematical Society 2007; 75 pp; softcover Volume: 186 ISBN10: 0821839411 ISBN13: 9780821839416 List Price: US$66 Individual Members: US$39.60 Institutional Members: US$52.80 Order Code: MEMO/186/871
 This memoir focuses on \(L^p\) estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. The author introduces four critical numbers associated to the semigroup and its gradient that completely rule the ranges of exponents for the \(L^p\) estimates. It appears that the case \(p<2\) already treated earlier is radically different from the case \(p>2\) which is new. The author thus recovers in a unified and coherent way many \(L^p\) estimates and gives further applications. The key tools from harmonic analysis are two criteria for \(L^p\) boundedness, one for \(p<2\) and the other for \(p>2\) but in ranges different from the usual intervals \((1,2)\) and \((2,\infty)\). Table of Contents  Beyond CalderónZygmund operators
 Basic \(L^2\) theory for elliptic operators
 \(L^p\) theory for the semigroup
 \(L^p\) theory for square roots
 Riesz transforms and functional calculi
 Square function estimates
 Miscellani
 Appendix A. CalderónZygmund decomposition for Sobolev functions
 Appendix. Bibliography
