An elementary coordinate-dependent local resolution of singularities and applications

Abstract

An elementary classical analysis resolution of singularities method is developed, extensively using explicit coordinate systems. The algorithm is designed to be applicable to subjects such as oscillatory integrals and critical integrability exponents. As one might expect, the trade-off for such an elementary method is a weaker theorem than Hironaka's work [H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, Ann. of Math. (2) 79 (1964) 109–203; H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero II, Ann. of Math. (2) 79 (1964) 205–326] or its subsequent simplications and extensions such as [E. Bierstone, P. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (2) (1997) 207–302; S. Encinas, O. Villamayor, Good points and constructive resolution of singularities, Acta Math. 181 (1) (1998) 109–158; J. Kollar, Resolution of singularities—Seattle lectures, preprint; A.N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Funct. Anal. Appl. 18 (3) (1976) 175–196]. Nonetheless the methods of this paper can be used to prove a variety of theorems of interest in analysis. As illustration, two consequences are given. First and most notably, a general theorem regarding the existence of critical integrability exponents are established. Secondly, a quick proof of a well-known inequality of Lojasiewicz [S. Lojasiewicz, Ensembles semi-analytiques, Inst. Hautes Études Sci., Bures-sur-Yvette, 1964] is given. The arguments here are substantially different from the general algorithms such as [H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, Ann. of Math. (2) 79 (1964) 109–203; H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero II, Ann. of Math. (2) 79 (1964) 205–326], or the elementary arguments of [E. Bierstone, P. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988) 5–42] and [H. Sussman, Real analytic desingularization and subanalytic sets: an elementary approach, Trans. Amer. Math. Soc. 317 (2) (1990) 417–461]. The methods here have as antecedents the earlier work of the author [M. Greenblatt, A direct resolution of singularities for functions of two variables with applications to analysis, J. Anal. Math. 92 (2004) 233–257], Phong and Stein [D.H. Phong, E.M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997) 107–152], and Varchenko [A.N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Funct. Anal. Appl. 18 (3) (1976) 175–196].