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We give an overview of the generalized Calder\'on-Zygmund theory for"non-integral" singular operators, that is, operators without kernels boundsbut appropriate off-diagonal estimates. This theory is powerful enough toobtain weighted estimates for such operators and their commutators with $\BMO$functions. $L^p-L^q$ off-diagonal estimates when $p\le q$ play an importantrole and we present them. They are particularly well suited to the semigroupsgenerated by second order elliptic operators and the range of exponents $(p,q)$rules the $L^p$ theory for many operators constructed from the semigroup andits gradient. Such applications are summarized.

Weighted norm inequalities, off-diagonal estimates and elliptic operators