17 research outputs found
Estimates for maximal functions associated to hypersurfaces in with height Part II -- A geometric conjecture and its proof for generic 2-surfaces
In this article, we continue the study of -boundedness of the maximal
operator associated to averages along isotropic dilates of a
given, smooth hypersurface in 3-dimensional Euclidean space. We focus here
on small surface-patches near a given point exhibiting singularities of
type in the sense of Arnol'd at this point; this is the situation
which had yet been left open. Denoting by the minimal Lebesgue exponent
such that is -bounded for we are able to identify
for all analytic surfaces of type (with the exception of a
small subclass), by means of quantities which can be determined from associated
Newton polyhedra. Besides the well-known notion of height at a new
quantity, which we call the effective multiplicity, turns out to play a crucial
role here. We also state a conjecture on how the critical exponent might
be determined by means of a geometric measure theoretic condition, which
measures in some way the order of contact of arbitrary ellipsoids with
even for hypersurfaces in arbitrary dimension, and show that this conjecture
holds indeed true for all classes of 2-hypersurfaces for which we have
gained an essentially complete understanding of so far. Our
results lead in particular to a proof of a conjecture by Iosevich-Sawyer-Seeger
for arbitrary analytic 2-surfaces.Comment: 101 pages, 8 figure
Sharp time decay estimates for the discrete Klein-Gordon equation
We establish sharp time decay estimates for the the Klein-Gordon equation on
the cubic lattice in dimensions . The
dispersive decay rate is for , for and
for . These decay rates are faster than conjectured by
Kevrekidis and Stefanov (2005). The proof relies on oscillatory integral
estimates and proceeds by a detailed analysis of the the singularities of the
associated phase function. We also prove new Strichartz estimates and discuss
applications to nonlinear PDEs and spectral theory
Sharp time decay estimates for the discrete Klein-Gordon equation
We establish sharp time decay estimates for the Klein–Gordon equation on the cubic lattice in dimensions d = 2, 3, 4. The ℓ1 → ℓ∞ dispersive decay rate is |t|−3/4 for d = 2, |t|−7/6 for d = 3 and |t|−3/2 log|t| for d = 4. These decay rates are faster than conjectured by Kevrekidis and Stefanov (2005). The proof relies on oscillatory integral estimates and proceeds by a detailed analysis of the singularities of the associated phase function. We also prove new Strichartz estimates and discuss applications to nonlinear PDEs and spectral theory.<br