We shall characterize the sets of locally uniform convergence of pointwise convergent sequences. Results obtained for sequences of holomorphic functions by Hartogs and Rosenthal in 1928 will be generalized for many other sheaves of functions. In particular, our Hartogs-Rosenthal type theorem holds for the sheaf of solutions to the second order elliptic PDE's as well as it has applications to the theory of harmonic spaces.

On the harmonic Bergman space of the ball, we give characterizations for an arbitrary positive Toeplitz operator to be a Schatten class operator in terms of averaging functions and Berezin transforms.

We study Toeplitz operators between the pluriharmonic Bergman spaces for positive symbols on the ball. We give characterizations of bounded and compact Toeplitz operators taking a pluriharmonic Bergman space $b^p$ into another $b^q$ for $1<p,q<\infty$ in terms of certain Carleson and vanishing Carleson measures.

Le faisceau des fonctions hyperharmoniques dans les ouverts de ${\mathbf{R}}^n$ vérifie le principe du minimum et est maximal parmi les faisceaux de cônes convexes de fonctions s.c.i. $\&gt;-\infty$ vérifiant ce principe du minimum.On se donne plus généralement un espace localement $\Omega$ dans lequel on définit différents principes du minimum, et on étudie la donnée d’un faisceau de cônes convexes de fonctions s.c.i. $\&gt;-\infty$ qui soit maximal par rapport à l’un de ces principes.On montre ainsi comment on peut caractériser certains de...

Cet article complète les résultats obtenus par J.-M. Bony et par l’auteur. On montre d’abord qu’on peut définir les fonctions harmoniques adjointes au faisceau donné, et qu’elles coïncident avec les solutions de l’équation adjointe. Puis, dans un ouvert assez régulier, la solution du problème de Dirichlet dans le cadre axiomatique est comparée à la solution au sens variationnel construite par M. Derridj.

We derive weighted rearrangement estimates for a large class of area integrals. The main approach used earlier to study these questions is based on distribution function inequalities.

We prove that the subharmonic envelope of a lower semicontinuous function on Omega is harmonic on a certain open subset of Omega, using a very classical method in potential theory. The result gives simple proofs of theorems on harmonic measures and Jensen measures obtained by Cole and Ransford.

Let ${\left(h_k\right)}_{k\ge 0}$ be the Haar system on [0,1]. We show that for any vectors $a_k$ from a separable Hilbert space and any ${\epsilon }_k\in [-1,1]$, k = 0,1,2,..., we have the sharp inequality $\left|\right|{\sum }_{k=0}^n{\epsilon }_k{a}_k{h}_k{\left|\right|}_{W\left(\right[0,1\left]\right)}\le 2\left|\right|{\sum }_{k=0}^n{a}_k{h}_k{\left|\right|}_{L^{\infty }\left([0,1]\right)}$, n = 0,1,2,..., where W([0,1]) is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ${\left|\right|Y\left|\right|}_{W\left(\Omega \right)}{\le 2\left|\right|X\left|\right|}_{L^{\infty }\left(\Omega \right)}$, where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic $\omega$-$\alpha$-Bloch space and characterize it in terms of $$\omega \left(\right(1-{\left|x\right|}^2{)}^{\beta }{(1-|y|}^2{)}^{\alpha -\beta })\left|\frac{f\left(x\right)-f\left(y\right)}{x-y}\right|$$ and $$\omega \left(\right(1-{\left|x\right|}^2{)}^{\beta }{(1-|y|}^2{)}^{\alpha -\beta })\left|\frac{f\left(x\right)-f\left(y\right)}{\left|x\right|y-x^{\text{'}}}\right|$$ where $\omega$ is a majorant. Similar results are extended to harmonic little $\omega$-$\alpha$-Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G. Ren, U. Kähler (2005).