A sharp estimate for multilinear Marcinkiewicz integral operator is obtained. By using this estimate, we obtain the weighted norm inequalities and $L\log L$ type estimate for the multilinear operator.

Functions which are moment maps of Hamiltonian actions on symplectic manifolds have the property that their stationary phase expansions have only finitely many nonzero terms and are therefore precise rather than asymptotic. In this paper, we exhibit another type of function which has this property and explain why in terms of equivariant cohomology and the geometry of the spaces involved.

We give an Euler-Maclaurin formula with remainder for the weighted sum of the values of a smooth function on the integral points in a simple integral polytope. Our work generalizes the formula obtained in [KSW2].

In a recent article, Cattaneo, Felder and Tomassini explained how the notion of formality can be used to construct flat Fedosov connections on formal vector bundles on a Poisson manifold $M$ and thus a star product on $M$ through the original Fedosov method for symplectic manifolds. In this paper, we suppose that $M$ is a fibre bundle manifold equipped with a Poisson tensor tangential to the fibers. We show that in this case the construction of Cattaneo-Felder- Tomassini gives tangential (to the fibers) star products.

In this paper we study the absolute continuity of self-similar measures defined by iterated function systems (IFS) whose contraction ratios are not uniform. We introduce a transversality condition for a multi-parameter family of IFS and study the absolute continuity of the corresponding self-similar measures. Our study is a natural extension of the study of Bernoulli convolutions by Solomyak, Peres, et al.

Examples of projective homogeneous varieties over the field of Laurent series over $p$-adic fields which admit zero-cycles of degree one and which do not have rational points are constructed.