We develop a new technique for studying the boundary limiting behavior of a holomorphic function on a domain $\Omega$ -- both in one and several complex variables. The approach involves two new localized maximal functions.

As a result of this methodology, theorems of Calderón type about local boundary behavior on a set of positive measure may be proved in a new and more natural way.

We also study the question of nontangential boundedness (on a set of positive measure) versus admissible boundedness. Under suitable hypotheses, these two conditions are shown to be equivalent.

This is a survey paper on the topic of proving or disproving a priori $L^2$ estimates for non-selfadjoint operators. Our framework will be limited to the case of scalar semi-classical pseudodifferential operators of principal type. We start with recalling the simple conditions following from the sign of the first bracket of the real and imaginary part of the principal symbol. Then we introduce the geometric condition $(\psi)$ and show the necessity of that condition for obtaining a weak $L^2$ estimate. Considering that condition satisfied, we investigate the finite-type case, where one iterated bracket of the real and imaginary part does not vanish, a model of subelliptic operators. The last section is devoted partly to rather recent results, although we begin with a version of the 1973 theorem of R. Beals and C. Fefferman on solvability with loss of one derivative under condition $P$; next, we present a 1994 counterexample by N. L. establishing that $(\psi)$ does not ensure an estimate with loss of one derivative. Finally, we show that condition $(\psi)$ implies an estimate with loss of 3/2 derivatives, following the recent papers by N. Dencker and N. L. Our goal is to provide a general overview of the subject and of the methods; we do not enter in the details of the proofs, although we provide some key elements of the arguments, in particular in the last section.

We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as the results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to ‘heat spheres’ is proved for heat equation with respect to evolving Riemannian metrics via a spacetime consideration. Some new monotonicity formulae are derived. As applications of the new local monotonicity formulae, some local regularity theorems concerning Ricci flow are proved.

We consider a smooth CR mapping $f$ from a real-analytic generic submanifold $M$ in $\Bbb C^N$ into $\Bbb C^N$. For $M$ of finite type and essentially finite at a point $p\in\ M$, and $f$ formally finite at $p$, we give a necessary and sufficient condition for $f$ to extend as a holomorphic mapping in some neighborhood of $p$. In a similar vein, we consider a formal holomorphic mapping $H$ and give a necessary and sufficient condition for $H$ to be convergent.

We show that various notions of local homogeneity for CR-manifolds are equivalent. In particular, if germs at any two points of a CR-manifold are CR-equivalent, there exists a transitive local Lie group action by CR-automorphisms near every point.