We give upper bounds for volume of sublevel sets of real polynomials. Our method is to combine a version of global Lojasiewicz inequality with some well known estimates on volume of tubes around real algebraic sets. Some applications to oscillatory integrals and integration indices of real polynomial are also given.

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Abstract. We study the B-regularity of some classes of domains in C n . The results include a complete characterization of B-regularity in the class of Reinhardt domains, we also give some sufficient conditions for Hartogs domains to be B-regular. The last result yields sufficient conditions for preservation of B-regularity under holomorphic mappings.

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