Charles Fefferman

A note on spherical summation multipliers

We give a new proof of a theorem of L. Carleson and P. Sjölin onLp-boundedness of spherical summation operators in two variables.

On Local Solvability of Linear Partial Differential Equations

The title indicates more or less what the talk is going to be about. I t is going to be about the problem which is probably the most primitive in partial differential equations theory, namely to know whether an equation does, or does not, have a solution. Even this is meant in the most primitive terms. I would like to begin by explaining what the terms are. As you all know, the really difficult analysis these days, and perhaps always, is the global analysis. Well, the problem that I am going to discuss is purely local—in the strictest possible sense: we would like to find out if a linear partial differential equation, with coefficients as smooth as you wish, admits locally a solution. Obviously, in this connection, negative results are very important: and negative results about local solvability have global implications. But of course positive results have also their importance. Let us state precisely what is the problem. The partial differential equation under study will be

Geometric constraints on potentially

We discuss necessary and sufficient conditions for the formation of finite time singularities (blow up) in the incompressible three dimensional Euler equations. The well-known result of Beale, Kato and Majda states that these equations have smooth solutions on the time interval (0,t) if, and only if lim/t{r_arrow}T {integral}{sup t}{sub 0} {parallel}{Omega}({center_dot},s){parallel}{sub L}{sup {infinity}} (dx)dx < {infinity} where {Omega} = {triangledown} X u is the vorticity of the fluid and u is its divergence=free velocity. In this paper we prove criteria in which the direction of vorticity {xi} = {Omega}/{vert_bar}{Omega}{vert_bar} plays an important role.

On the convergence of multiple Fourier series

Inequality (1) follows from the special case in which P is a triangle with a vertex at the origin; for any polygon breaks up into triangles, and the characteristic function of any triangle is a linear combination of characteristic functions of triangles with vertices at zero. Consequently, we can assume P has the form P= {(x, y)C£S\ (x> y)-t<a}, where 5 is a sector of angle <7r emanating from the origin, ££i? , and aÇ^R. Thus (1) is equivalent to

Ambient metric construction of Q-curvature in conformal and CR geometries

We give a geometric derivation of Bransons Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally invariant operators and can be applied to a large class of invariant operators. This procedure can be also applied to CR geometry and gives a CR analog of the Q-curvature; it then turns out that the Q-curvature gives the coefficient of the logarithmic singularity of the Szego kernel of 3-dimensional CR manifolds.

Relativistic Stability of Matter (I).

In this article, we study the quantum mechanics of N electrons and M nuclei interacting by Coulomb forces. Motivated by an important idea of Chandrasekhar and following Herbst [H], we modify the usual kinetic energy -? to take into account an effect from special relativity. As a result, the system can implode for unfavorable values of the nuclear charge Z and the fine structure constant a. This is analogous to the gravitational collapse of a heavy star. Our goal here is to find those values of a and Z for which the system is stable.

Lp bounds for pseudo-differential operators

SharpLp boundedness results are proven for pseudo-differential operators in the classSpδm.

Growth of solutions for QG and 2D Euler equations

The work of Constantin-Majda-Tabak [1] developed an analogy between the Quasi-geostrophic and 3D Euler equations. Constantin, Majda and Tabak proposed a candidate for a singularity for the Quasi-geostrophic equation. Their numerics showed evidence of a blow-up for a particular initial data, where the level sets of the temperature contain a hyperbolic saddle. The arms of the saddle tend to close in finite time, producing a sharp front. Numerics studies done later by Ohkitani-Yamada [8] and Constantin-Nie-Schorghofer [2], with the same initial data, suggested that instead of a singularity the derivatives of the temperature were increasing as double exponential in time. The study of collapse on a curve was first studied in [1] for the Quasi-geostrophic equation where they considered a simplified ansatz for classical frontogenesis with trivial topology. At the time of collapse, the scalar θ is discontinuous across the curve x2 = f(x1) with different limiting values for the temperature on each side of the front. They show that under this topology the directional field remains smooth up to the collapse, which contradicts the following theorem proven in [1]:

Honeycomb lattice potentials and Dirac points

We prove that the two-dimensional Schroedinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (Dirac points) at the vertices of its Brillouin zone. No assumptions are made on the size of the potential. We then prove the robustness of such conical singularities to a restrictive class of perturbations, which break the honeycomb lattice symmetry. General small perturbations of potentials with Dirac points do not have Dirac points; their dispersion surfaces are smooth. The presence of Dirac points in honeycomb structures is associated with many novel electronic and optical properties of materials such as graphene.

Can one learn history from the allelic spectrum

It is well known that the neutral allelic frequency spectrum of a population is affected by the history of population size. A number of authors have used this fact to infer history given observed allele frequency data. We ask whether perfect information concerning the spectrum allows precise recovery of the history, and with an explicit example show that the answer is in the negative. This implies some limitations on how informative allelic spectra can be.