We consider a domain Ω with Lipschitz boundary, which is relatively compact in an$n$-dimensional Kähler manifold and satisfies some “logδ-pseudoconvexity” condition. We show that the $$\bar \partial $$ -equation with exact support in ω admits a solution in bidegrees ($p, q$), 1≤$q$≤$n$−1. Moreover, the range of $$\bar \partial $$ acting on smooth ($p, n$−1)-forms with support in $$\bar \Omega $$ is closed. Applications are given to the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on boundaries of weakly pseudoconvex domains in Stein manifolds and to the solvability of the tangential Cauchy-Riemann equations for currents on Levi flat$CR$manifolds of arbitrary codimension.

We introduce a non-linear injective transformation τ from the set of non-vanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula$T$[($a$_{$n$})_{$n$=1}^{∞}]_{$n$}= 1/$a$_{1}...$a$_{$n$}. Special cases of this transformation have appeared in various papers on exponential functionals of Lévy processes, partly motivated by mathematical finance. We give several examples of moment sequences arising from the transformation and provide the corresponding measures, some of which are related to$q$-series.

We investigate local properties of the Green function of the complement of a compact set$E$υ[0,1] with respect to the extended complex plane. We demonstrate, that if the Green function satisfies the 1/2-Hölder condition locally at the origin, then the density of$E$at 0, in terms of logarithmic capacity, is the same as that of the whole interval [0, 1]..

Using a geometric method, we characterize all entire functions that transform the Bloch space into a Bergman space by superposition in terms of their order and type. We also prove that all superposition operators induced by such entire functions act boundedly. Similar results hold for superpositions from BMOA into Bergman spaces and from the Bloch space into certain weighted Hardy spaces.

The sine(hyperbolic)-Gordon hierarchy is shown to be the extension of the modified Korteweg-de Vries (MKdV) hierarchy in the integrodifferential algebra extending the standard differential algebra by means of one antiderivative. The characterization by vanishing residues of the MKdV hierarchy yields the same characterization of the sine(hyperbolic)-Gordon hierarchy in the integrodifferential algebra.