Eine Erweiterung der Infinitesimalrechnung

Ohne Zusammenfassung     

Uniform Estimates for the Fourier Transform of Surface Carried Measures in ?3 and an Application to Fourier Restriction

Let S be a hypersurface in \BbbR³{\Bbb{R}}^{3} which is the graph of a smooth, finite type function φ, and let μ=ρ  dσ be a surface carried measure on S, where dσ denotes the surface element on S and ρ a smooth density with sufficiently small support. We derive uniform estimates for the Fourier transform [^(m)]\hat{\mu} of μ, which are sharp except for the case where the principal face of the Newton polyhedron of φ, when expressed in adapted coordinates, is unbounded. As an application, we prove a sharp L ^p -L ² Fourier restriction theorem for S in the case where the original coordinates are adapted to φ. This improves on earlier joint work with M. Kempe.     

Geometrically Convex Solutions of Certain Difference Equations and Generalized Bohr-Mollerup Type Theorems

Let G: (0, ∞) → (0, ∞) be logarithmically concave on a neighbourhood of ∞ and suppose lim_x→∞ G(x + δ)/G(x) = 1 for some δ > 0. Then, the functional equation $$g(x+1)=G(x)\cdot g(x),\ \ \ x\in (0,\infty),$$ admits, up to a multiplicative constant, at most one solution g: (0, ∞) → (0, ∞), geometrically convex on a neighbourhood of ∞. Sufficient conditions on G are given, for which also such a unique geometrically convex solution of (D) exists. This result improves the classical theorems of Bohr-Mollerup type and gives a new characterization of the gamma function and the q-gamma function for q ∈ (0, 1).     

An Asymptotic Formula for the Iterates of a Function

Let IK be either IR or ? and D an open set of IK containing 0 and starlike with respect to 0 (i.e. an open interval containig 0 in the case IK = IR). If f: D » IK is a continuous function with fixed point 0, then under certain conditions stated below we can prove for the kn- th iterates of f the following asymptotic formula: 1 $$f^{(kn)}\bigg({x \over n}\bigg )=\sum_{i-1}^r{1\over (nk)^i}\ f_i(kx)+o \bigg({1\over n^r}\bigg),$$ for n » ∞, k, n and r beeing positive integers and x close enough to 0. The functions f _i are continuous and uniquely determined by f. In particular (1) holds for any function holomorphic on a neighbourhood of zero, having a convergent power series expansion of the form $$f(z)=z+a_2z^2+\cdots=\sum_{j=1}^\infty\ a_jz^j,\ a_j\in {\cal C},a_1=1,$$ and for any integers k, r with r > 0.     

Local solvability for positive combinations of generalized sub-Laplacians on the Heisenberg group

As one step in a program to understand local solvability of complex coefficient second order differential operators on the Heisenberg group in a complete way, solvability of operators of the form , where the leading term is a ``positive combination of generalized and degenerate generalized sub-Laplacians', has been studied in a recent article by M. Peloso, F. Ricci and the first-named author (J. Reine Angew Math. 513 (1999)). It was shown that there exists a discrete set of ``critical' values , such that solvability holds for . The case remained open, and it is the purpose of this note to close this gap. Our results extend corresponding results in another article by the above-mentioned authors (J. Funct. Anal. 148 (1997)), by means of an even simplified approach which should allow for further generalizations.

     

Complete Minimal Kähler Surfaces in R6

We construct the first examples of minimal Kähler surfaces in R⁶ which are irreducible and neither holomorphic nor complex ruled.     

Self-diffusion in a non-uniform one dimensional system of point particles with collisions

Summary We generalize the results of Spitzer, Jepsen and others [1–4] on the motion of a tagged particle in a uniform one dimensional system of point particles undergoing elastic collisions to the case where there is also an external potential U(x). When U(x) is periodic or random (bounded and statistically translation invariant) then the scaled trajectory of a tagged particle converges, as A , to a Brownian motion W _D (t) with diffusion constant , where is the average density, is the mean absolute velocity and ^–1 the temperature of the system. When U(x) is itself changing on a macroscopic scale, i.e. , then the limiting process is a spatially dependent diffusion. The stochastic differential equation describing this process is now non-linear, and is particularly simple in Stratonovich form. This lends weight to the belief that heuristics are best done in that form.Dedicated to Frank Spitzer on the occasion of his 60th birthdayWork supported in part by NSF Grants No. PHY 8201708 and No. DMR 81-14726Heisenberg-fellowAlso Department of Physics     

Wave Equation and Multiplier Estimates on Damek–Ricci Spaces

Let S be a Damek–Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form ${\rm {e}}^{it\sqrt{ L}}\psi\big(\sqrt{ L}/{\lambda}\big)$ for arbitrary time t and arbitrary λ>0, where ψ is a smooth bump function supported in [?2,2] if λ<1 and supported in [1,2] if λ≥1. This generalizes previous results in Müller and Thiele (Studia Math. 179:117–148, 2007). We also prove pointwise estimates for the gradient of these convolution kernels. As a corollary, we reprove basic multiplier estimates from Hebish and Steger (Math. Z. 245:37–61, 2003) and Vallarino (J. Lie Theory 17:163–189, 2007) and derive Sobolev estimates for the solutions to the wave equation associated to L.