Let ${K_t}_{t>0}$ be the semigroup of linear operators generated by a Schrödinger operator -L = Δ - V with V ≥ 0. We say that f belongs to $H{¹}_L$ if $\left|\right|su{p}_{t>0}|K_t{f\left(x\right)\left|\right||}_{L¹\left(dx\right)}<\infty$. We state conditions on V and $K_t$ which allow us to give an atomic characterization of the space $H{¹}_L$.

We define Hardy spaces of pairs of conjugate temperatures on ${ℝ}_{+}^2$ using the equations introduced by Kochneff and Sagher. As in the holomorphic case, the Hilbert transform relates both components. We demonstrate that the boundary distributions of our Hardy spaces of conjugate temperatures coincide with the boundary distributions of Hardy spaces of holomorphic functions.

We apply general Hardy type inequalities, recently obtained by the author. As a consequence we obtain a family of Hardy-Poincaré inequalities with certain constants, contributing to the question about precise constants in such inequalities posed in [3]. We confirm optimality of some constants obtained in [3] and [8]. Furthermore, we give constants for generalized inequalities with the proof of their optimality.

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy’s version of the uncertainty principle. We also obtain corresponding results for heat evolutions.

We prove some Hardy-type inequalities related to quasilinear second-order degenerate elliptic differential operators $L_{p}u:=-{\nabla }_L^{*}\left({\left|{\nabla }_{L}u\right|}^{p-2}{\nabla }_{L}u\right)$. If $\phi$ is a positive weight such that $-L_p\phi \ge 0$, then the Hardy-type inequalityholds. We find an explicit value of the constant involved, which, in most cases, results optimal. As particular case we derive Hardy inequalities for subelliptic operators on Carnot Groups.

A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.

We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface $X$ with the $L^2$ complex relative to a suitable metric on the bundle and a complete metric on the punctured Riemann surface. Applying results of C. Simpson, we show the existence of a harmonic metric on this vector bundle, giving the same $L^2$ complex.

Let $L$ be an elliptic linear operator in a domain in ${\mathbf{R}}^n$. We imposse only weak regularity conditions on the coefficients. Then the adjoint $L^{*}$ exists in the sense of distributions, and we start by deducing a regularity theorem for distribution solutions of equations of type $L^{*}u=$ given distribution. We then apply to $L^{*}$ R.M. Hervé’s theory of adjoint harmonic spaces. Some other properties of $L^{*}$ are also studied. The results generalize earlier work of the author.

An elliptic system in ${ℝ}^n$, which is invariant under the action of the group ${\mathbb{Z}}^n$ is considered. We construct a holomorphic family of finite-dimensional subrepresentations of the group in the space of solutions (Floquet solutions), such that any solution of the growth $O\left(\exp \right(a\left|x\right|\left)\right)$ at infinity can be rewritten in the form of an integral over the family.