Let$f=g$_{t}+$h$_{t}be the optimal decomposition for calculating the exact value of the$K$-functional$K(t, f$; $$\bar X$$ ) of an element$f$with respect to a couple $$\bar X$$ =($X$_{0},$X$_{1}) of Banach lattices of measurable functions. It is shown that this decomposition has a rather simple form in many cases where one of the spaces$X$_{0}and$X$_{1}is either$L$^{∞}or$L$^{1}. Many examples are given of couples of lattices $$\bar X$$ for which |$g$_{t}| increases monotonically a.e. with respect to$t$. It is shown that this property implies a sharpened estimate from above for the Brudnyi-Krugljak$K$-divisibility constant γ( $$\bar X$$ ) for the couple. But it is also shown that certain couples $$\bar X$$ do not have this property. These also provide examples of couples of lattices for which γ( $$\bar X$$ ).

Let Ω be a bounded convex domain in C_{$n$}, with smooth boundary of finite type$m$.

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The subject of this paper is a Jacobian, introduced by F. Lazzeri (unpublished), associated with every compact oriented Riemannian manifold whose dimension is twice an odd number. We study the Torelli and Schottky problem for Lazzeri's Jacobian of flat tori and we compare Lazzeri's Jacobian of Kähler manifolds with other Jacobians.

In this paper we give an explicit formula for the solution of the non-homogeneous complex Cauchy problem with Cauchy data given on a bounded smooth strictly convex domain in a non-characteristic hyperplane. These formulas are obtained using the explicit version of the fundamental principle given in terms of residue currents; moreover, we characterize the domain of definition of the solution and we generalize these techniques to the non-homogeneous Goursat problem.