In this paper we consider operators of the form$H$=λ(-$i$∇), with λ analytic in a strip and with some specific growth conditions at infinity, and prove Hardy type estimates in$L$^{2}(ℝ^{$n$}) with exponential weights. In fact we extend our previous results [19] from bounded analytic functions on a strip to analytic functions with polynomial growth in that strip.

The purpose of this note is to give an extension of the symbolic calculus of Melin for convolution operators on nilpotent Lie groups with dilations. Whereas the calculus of Melin is restricted to stratified nilpotent groups, the extension offered here is valid for general homogeneous groups. Another improvement concerns the$L$^{2}-boundedness theorem, where our assumptions on the symbol are relaxed. The zero-class conditions that we require are of the type $$|D^{\alpha}a(\xi)|\le C_{\alpha}\prod_{j=1}^R\rho_j(\xi)^{-|\alpha_j|},$$ where ρ_{$j$}are “partial homogeneous norms” depending on the variables ξ_{$k$}for$k$>$j$in the natural grading of the Lie algebra (and its dual) determined by dilations. Finally, the class of admissible weights for our calculus is substantially broader. Let us also emphasize the relative simplicity of our argument compared to that of Melin.

We give an elementary proof of the formula χ($K$^{$n$}A)=$n$^{3}σ($n$) for the Euler characteristic of the generalized Kummer variety$K$^{$n$}$A$, where σ($n$) denotes the sum of divisors function.

We consider subordination chains of simply connected domains with smooth boundaries in the complex plane. Such chains admit Hamiltonian and Lagrangian interpretations through the Löwner–Kufarev evolution equations. The action functional is constructed and its time variation is obtained. It represents the infinitesimal version of the action of the Virasoro–Bott group over the space of analytic univalent functions.

We describe the behavior of certain strictly plurisubharmonic functions near some real hypersurfaces in ℂ^{$n$},$n$≥3. Given a hypersurface we study continuous plurisubharmonic functions which are zero on the hypersurface and have Monge–Ampère mass greater than one in a one-sided neighborhood of the hypersurface. If we can find complex curves which have sufficiently high contact order with the hypersurface then the plurisubharmonic functions we study cannot be globally Lipschitz in the one-sided neighborhood.