271 research outputs found
Uniform resolvent estimates for Schr\"odinger operator with an inverse-square potential
We study the uniform resolvent estimates for Schr\"odinger operator with a
Hardy-type singular potential.
Let where is the usual Laplacian on
and where and
is a real function such that
the operator is a strictly positive
operator on . We prove some new uniform weighted
resolvent estimates and also obtain some uniform Sobolev estimates associated
with the operator .Comment: Comments are welcome.To appear in Journal of Functional Analysi
The Defocusing Energy-Critical Wave Equation with a Cubic Convolution
In this paper, we study the theory of the global well-posedness and
scattering for the energy-critical wave equation with a cubic convolution
nonlinearity in spatial dimension . The main difficulties are the absence of the classical finite speed of
propagation (i.e. the monotonic local energy estimate on the light cone), which
is a fundamental property to show the global well-posedness and then to obtain
scattering for the wave equations with the local nonlinearity . To compensate it, we resort to the extended causality
and utilize the strategy derived from concentration compactness ideas. Then,
the proof of the global well-posedness and scattering is reduced to show the
nonexistence of the three enemies: finite time blowup; soliton-like solutions
and low-to-high cascade. We will utilize the Morawetz estimate, the extended
causality and the potential energy concentration to preclude the above three
enemies.Comment: 19 pages, In this version, we prove the result in Proposition 4.1 in
an averaged-in-time sense, and we utilize the potential energy concentration
in an averaged-in-time sense and the Morawetz estimate to kill finite time
blow up solutions in Section
Linear adjoint restriction estimates for paraboloid
We prove a class of modified paraboloid restriction estimates with a loss of
angular derivatives for the full set of paraboloid restriction conjecture
indices. This result generalizes the paraboloid restriction estimate in radial
case from [Shao, Rev. Mat. Iberoam. 25(2009), 1127-1168], as well as the result
from [Miao et al. Proc. AMS 140(2012), 2091-2102]. As an application, we show a
local smoothing estimate for a solution of the linear Schr\"odinger equation
under the assumption that the initial datum has additional angular regularity.Comment: 24 page
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