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Michael Greenblatt

Publications and source records attributed to Michael Greenblatt.

29 records · Page 2Linked to original sources
arXiv2014

Estimates for Fourier transforms of surface measures in R^3 and PDE applications

A local two-dimensional resolution of singularities theorem and arguments based on the Van der Corput lemma are used to give new estimates for the decay rate of the Fourier transform of a locally defined smooth hypersurface measure in R^3, as well as to provide new proofs of some known estimates. These are then used to give L^q bounds on solutions to certain PDE problems in terms of the L^p norms of their initial data for various values of p and q. Unlike much of the earlier work in this subject, no use is made of the adapted coordinate systems that have been often been used to study two-dimensional oscillatory integrals; all of the needed information is furnished by the resolution of singularities theorem.

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arXiv2014

Uniform bounds for Fourier transforms of surface measures in R^3 with nonsmooth density

We prove uniform estimates for the decay rate of the Fourier transform of measures supported on real-analytic hypersurfaces in R^3. If the surface contains the origin and is oriented such that its normal at the origin is in the direction of the z-axis and if dS denotes the surface measure for this surface, then the measures under consideration are of the form K(x,y)g(z) dS where K(x,y)g(z) is supported near the origin and both K(x,y) and g(z) are allowed to have singularities. The estimates here generalize the previously known sharp uniform estimates for when K(x,y)g(z) is smooth. The methods used in this paper involve an explicit two-dimensional resolution of singularities theorem, iterated twice, coupled with Van der Corput-type lemmas.

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arXiv2011

Stability of oscillatory integral asymptotics in two dimensions

The stability under phase perturbations of the decay rate of local scalar oscillatory integrals in two dimensions is analyzed. For a smooth phase S(x,y) and a smooth perturbation function f(x,y), the decay rate for phase S(x,y) + tf(x,y) is shown to be the same for all but finitely many t and given an explicit description. The decay rate of the generic S(x,y) + tf(x,y) is always at least as fast as that of S(x,y), and the "good" cases where it is the same as that of S(x,y) are explicitly described. Uniform stability of the decay rate is proven for S(x,y) + f(x,y) for small enough such good f(x,y), and the coefficient of the leading term of the asymptotics is shown to be Lipschitz of some order alpha, again for small enough good perturbations f(x,y).

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arXiv2010

L^p boundedness of maximal averages over hypersurfaces in R^3

Extending the methods developed in the author's previous paper and using adapted coordinate systems in two variables, an L^p boundedness theorem is proven for maximal operators over hypersurfaces in R^3 when p > 2. When the best possible p is greater than 2, the theorem typically provides sharp estimates. This gives another approach to the subject of recent work of Ikromov, Kempe, and Muller on this subject

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arXiv2010

Maximal averages over hypersurfaces and the Newton polyhedron

Using some resolution of singularities and oscillatory integral methods in conjunction with appropriate damping and interpolation techniques, L^p boundedness theorems for p > 2 are obtained for maximal operators over a wide range of hypersurfaces. These estimates are sharp in many situations, including the convex hypersurfaces of finite line type considered by Iosevich, Sawyer, and others. As a corollary, we also give a generalization of the result of Sogge and Stein that for some finite p the maximal operator corresponding to a hypersurface whose Gaussian curvature does not vanish to infinite order is bounded on L^p. Analogous estimates are proven for Fourier transforms of surface measures, and these are sharp for the same hypersurfaces as the maximal operators.

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arXiv2009

Oscillatory Integral Decay, Sublevel Set Growth, and the Newton Polyhedron

Using some resolution of singularities methods of the author, a generalization of a well-known theorem of Varchenko relating decay of oscillatory integrals to the Newton polyhedron is proven. They are derived from analogous results for sublevel integrals, proven here. Varchenko's theorem requires a certain nondegeneracy condition on the faces of the Newton polyhedron on the phase. In this paper, it is shown that the estimates of Varchenko's theorem also hold for a significant class of phase functions for which this nondegeneracy condition does not hold. Thus in problems where one wants to switch coordinates to a coordinate system where Varchenko's estimates are valid, one has greater flexibility. Some additional estimates are also proven for more degenerate situations, including some too degenerate for the Newton polyhedron to give the optimal decay in the sense of Varchenko.

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arXiv2009

The asymptotic behavior of degenerate oscillatory integrals in two dimensions

A theorem of Varchenko gives the order of decay of the leading term of the asymptotic expansion of a degenerate oscillatory integral with real-analytic phase in two dimensions. His theorem expresses this order of decay in a simple geometric way in terms of its Newton polygon once one is in certain coordinate systems called adapted coordinate systems. In this paper, we give explicit formulas that not only provide the order of decay of the leading term, but also the coefficient of this term. There are three rather different formulas corresponding to three different types of Newton polygon. Analogous results for sublevel integrals are proven, as are analogues for the more general case of smooth phase. The formulas require one to be in certain "superadapted" coordinates. These are a type of adapted coordinate system which we show exists for any smooth phase.

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arXiv2009

Resolution of Singularities in Two Dimensions and the Stability of Integrals

For a small disk D centered at the origin in R^2, a smooth real-valued function S(x,y) on D, and a positive epsilon, we consider the measure of the points in D where |S(x,y)| < epsilon, as well as oscillatory integral analogues. Specifically, we consider the effect of perturbing S(x,y) on these quantities. Besides being of intrinsic interest, these questions are important in the analysis of Fourier transforms of surface-supported measures. Complex and higher-dimensional analogues of these questions are also connected to various issues in algebraic and complex geometry. For real-analytic S(x,y), this question has been investigated for example by Karpushkin, using versal deformation theory, and by Phong-Stein-Sturm, who developed a method often referred to as the method of algebraic estimates. In this paper, we show how the use of resolution of singularities algorithms in two dimensions, along with some one-dimensional Van der Corput-type lemmas, provides another method for dealing with such questions. As a result, we prove new estimates and theorems for these and related quantities. Furthermore, since these algorithms apply to all smooth functions, the theorems will hold for all smooth functions as opposed to the earlier real-analytic results.

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arXiv2008

Resolution of singularities, asymptotic expansions of oscillatory integrals, and related phenomena

The elementary resolution of singularities algorithm of the author's earlier paper (math.CA/0609217) is developed further, replacing the quasibump functions in the blown up coordinates with the characteristic function of a rectangle times a smooth function. Such functions are easier to deal with, and as application the existence of asymptotic expansions for oscillatory integrals and related objects is given an elementary proof. In addition, some more detailed information about these expansions is given.

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arXiv2008

Adapted coordinates in two dimensions and a proof of Puiseux's theorem

A method for finding Puiseux series goes back to Isaac Newton, which gives the terms of Puiseux series through an infinite recursive process; an additional argument is then used to show that the resulting Puiseux series are convergent. This paper provides an argument based on Newton's method and some ideas from resolution of singularities that gives a quick proof of both the existence and convergence of Puiseux series. It is then shown that similar ideas can be used to give a short proof of the existence of smooth adapted coordinates for oscillatory integrals in two dimensions, a result first proved in the real-analytic case by Varchenko [V] and then recently for the general smooth case by Ikromov-Muller [IM]. The arguments of this paper are entirely elementary.

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