44 research outputs found
On localization of the Schr\"odinger maximal operator
In \cite{Lee:2006:schrod-converg}, when the spatial variable is
localized, Lee observed that the Schr\"odinger maximal operator
enjoys certain localization property in for frequency
localized functions. In this note, we give an alternative proof of this
observation by using the method of stationary phase, and then include two
applications: the first is on is on the equivalence of the local and the global
Schr\"odinger maximal inequalities; secondly the local Schr\"odinger maximal
inequality holds for , which implies that
converges to almost everywhere if . These results are not
new. In this note we would like to explore them from a slightly different
perspective, where the analysis of the stationary phase plays an important
role.Comment: 14 pages, no figure. Note
Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schr\"odinger equation
In this paper, we first show that there exists a maximizer for the
non-endpoint Strichartz inequalities for the Schr\"odinger equation in all
dimensions based on the recent linear profile decomposition results. We then
present a new proof of the linear profile decomposition for the Schr\"oindger
equation with initial data in the homogeneous Sobolev space; as a consequence,
there exists a maximizer for the Sobolev-Strichartz inequality.Comment: 14 pages; Various corrections, references update
On extremisers to a bilinear Strichartz inequality
In this note, we show that a pair of Gaussian functions are extremisers to a
bilinear Strichartz inequality, and unique up to the symmetry group of the
inequality.Comment: 6 pages. The constant in defining the inverse Fourier transform is
corrected;the expression of convolution of measures is correcte
A note on the cone restriction conjecture in the cylindrically symmetric case
In this note, we present two arguments showing that the classical
\textit{linear adjoint cone restriction conjecture} holds for the class of
functions supported on the cone and invariant under the spatial rotation in all
dimensions. The first is based on a dyadic restriction estimate, while the
second follows from a strengthening version of the Hausdorff-Young inequality
and the H\"older inequality in the Lorentz spaces.Comment: 9 pages, no figures. Referee's suggestions and comments incorporated;
to appear the Proceedings of the AM
Analyticity of extremisers to the Airy Strichartz inequality
We prove that there exists an extremal function to the Airy Strichartz
inequality, by
using the linear profile decomposition. Furthermore we show that, if is an
extremiser, then is extremely fast decaying in Fourier space and so can
be extended to be an entire function on the whole complex domain. The rapid
decay of the Fourier transform of extremisers is established with a bootstrap
argument which relies on a refined bilinear Airy Strichartz estimate and a
weighted Strichartz inequality.Comment: 18 page