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We prove sharp anisotropic H\"older estimates for the local solutions of the tangential Cauchy-Riemann equation in q-concave CR manifolds and we derive the same kind of estimates for global solutions when the manifold is compact.

We consider perturbations of CR structures which preserve the complex tangent bundle. For a compact generic CR manifold its concavity properties and hence the finiteness of some $\overline\partial_b$-cohomology groups are also preserved by such perturbations of the CR structure. Here we study the stability of the vanishing of these groups.

We have obtained the explicit versions and precisions for the Hodge-Riemann decomposition of formes on affine algebraic curve V. The main application consists in the construction of Faddeev-Green function for Laplacian on V. Basing on this [HM](arXiv:0804.3951 and J.Geom.Anal., 2008,18), we extended from the case X in C to the case of bordered Riemann surface X in V the R.Novikov (1988) scheme for the effective reconstruction of conductivity function sigma on X through Dirichlet-to-Neumann mapping on bX for solutions of d(sigma d^cU)=0. In Sec.4 we give a correction of the paper [HM].

Let $\g\_2$ be the Hochschild complex of cochains on $C^\infty(\RM^n)$ and $\g\_1$ be the space of multivector fields on $\RM^n$. In this paper we prove that given any $G\_\infty$-structure ({\rm i.e.} Gerstenhaber algebra up to homotopy structure) on $\g\_2$, and any $C\_\infty$-morphism $\phi$ ({\rm i.e.} morphism of commutative, associative algebra up to homotopy) between $\g\_1$ and $\g\_2$, there exists a $G\_\infty$-morphism $\Phi$ between $\g\_1$ and $\g\_2$ that restricts to $\phi$. We also show that any $L\_\infty$-morphism ({\rm i.e.} morphism of Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a $G\_\infty$-morphism, using Tamarkin's method for any $G\_\infty$-structure on $\g\_2$. We also show that any two of such $G\_\infty$-morphisms are homotopic., Comment: 10 pages, case of $C\_\infty$-morphisms is studied, existence of lift is proved in that case, final version, to appear in Proc. of the A.M.S

Let M be a differential manifold. Using different methods, Kontsevich and Tamarkin have proved a formality theorem, which states the existence of a Lie homomorphism 'up to homotopy' between the Lie algebra of Hochschild cochains on C infin(M) and its cohomology (Gamma(M, LambdaTM), [–, –] S ). Suppose M is a Poisson manifold equipped with a Poisson tensor pgr; then one can deduce from this theorem the existence of a star product $$\star$$ on C infin(M). In this Letter we prove that the formality theorem can be extended to a Lie (and even Gerstenhaber) homomorphism 'up to homotopy' between the Lie (resp. Gerstenhaber 'up to homtoptopy') algebra of Hochschild cochains on the deformed algebra (C infin(M), *) and the Poisson complex (Gamma(M, LambdaTM), [pgr, –] S ). We will first recall Tamarkin's proof and see how the formality maps can be deduced from Etingof and Kazhdan's theorem using only homotopies formulas. The formality theorem for Poisson manifolds will then follow.