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DCDS

We consider semilinear equations of the form $p(D)u=F(u)$, with a locally bounded nonlinearity $F(u)$, and a linear part $p(D)$ given by a Fourier multiplier. The multiplier $p(\xi)$ is the sum of positively homogeneous terms, with at least one of them non smooth. This general class of equations includes most physical models for traveling waves in hydrodynamics, the Benjamin-Ono equation being a basic example.

We prove sharp pointwise decay estimates for the solutions to such equations, depending on the degree of the non smooth terms in $p(\xi)$. When the nonlinearity is smooth we prove similar estimates for the derivatives of the solution, as well as holomorphic extension to a strip, for analytic nonlinearity.

We prove sharp pointwise decay estimates for the solutions to such equations, depending on the degree of the non smooth terms in $p(\xi)$. When the nonlinearity is smooth we prove similar estimates for the derivatives of the solution, as well as holomorphic extension to a strip, for analytic nonlinearity.

CPAA

We investigate connections between certain dispersive estimates of
a (pseudo) differential operator of real principal type and the
number of non-vanishing curvatures of its characteristic manifold.
More precisely, we obtain sharp thresholds for the range of
Lebesgue exponents depending on the specific geometry.

CPAA

Time-frequency methods are used to study a class of Fourier Integral
Operators (FIOs) whose representation using Gabor frames is proved
to be very efficient. Indeed, similarly to the case of shearlets and curvelets
frames [10, 35], the matrix representation of a Fourier Integral Operator with
respect to a Gabor frame is well-organized. This is used as a powerful tool
to study the boundedness of FIOs on modulation spaces. As special cases,
we recapture boundedness results on modulation spaces for pseudo-differential
operators with symbols in $M^{\infty, 1}$ [33], for some Fourier multipliers [6] and
metaplectic operators [14, 31]. Moreover, this paper provides the mathematical
tools to numerically solving the Cauchy problem for Schr¨odinger equations
using Gabor frames [17]. Finally, similar arguments can be employed to study
other classes of FIOs [16].

DCDS

We consider a class of linear Schrödinger equations in $\mathbb{R}^d$ with rough Hamiltonian, namely with certain derivatives in the Sjöostrand class $M^{\infty,1}$. We prove that the corresponding propagator is bounded on modulation spaces. The present results improve several contributions recently appeared in the literature and can be regarded as the evolution counterpart of the fundamental result of Sjöstrand about the boundedness of pseudodifferential operators with symbols in that class.

Finally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the corresponding initial value problem in certain modulation spaces.

Finally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the corresponding initial value problem in certain modulation spaces.

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