5 research outputs found
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A Nonlinear Plancherel Theorem with Applications to Global Well-Posedness for the Defocusing Davey-Stewartson Equation and to the Inverse Boundary Value Problem of Calderón
We prove a Plancherel theorem for a nonlinear Fourier transform in two
dimensions arising in the Inverse Scattering method for the defocusing
Davey-Stewartson II equation. We then use it to prove global well-posedness and
scattering in for defocusing DSII. This Plancherel theorem also implies
global uniqueness in the inverse boundary value problem of Calder\'on in
dimension , for conductivities \sigma>0 with .
The proof of the nonlinear Plancherel theorem includes new estimates on
classical fractional integrals, as well as a new result on -boundedness of
pseudo-differential operators with non-smooth symbols, valid in all dimensions
Strichartz estimates on Schwarzschild black hole backgrounds
We study dispersive properties for the wave equation in the Schwarzschild
space-time. The first result we obtain is a local energy estimate. This is then
used, following the spirit of earlier work of Metcalfe-Tataru, in order to
establish global-in-time Strichartz estimates. A considerable part of the paper
is devoted to a precise analysis of solutions near the trapping region, namely
the photon sphere.Comment: 44 pages; typos fixed, minor modifications in several place
Recommended from our members
A Nonlinear Plancherel Theorem with Applications to Global Well-Posedness for the Defocusing Davey-Stewartson Equation and to the Inverse Boundary Value Problem of Calderón
We prove a Plancherel theorem for a nonlinear Fourier transform in two
dimensions arising in the Inverse Scattering method for the defocusing
Davey-Stewartson II equation. We then use it to prove global well-posedness and
scattering in for defocusing DSII. This Plancherel theorem also implies
global uniqueness in the inverse boundary value problem of Calder\'on in
dimension , for conductivities \sigma>0 with .
The proof of the nonlinear Plancherel theorem includes new estimates on
classical fractional integrals, as well as a new result on -boundedness of
pseudo-differential operators with non-smooth symbols, valid in all dimensions