Linear and Non-Linear Waveshttp://hdl.handle.net/20.500.11824/42024-03-02T03:33:57Z2024-03-02T03:33:57ZControl of pseudodifferential operators by maximal functions via weighted inequalitiesBeltran, D.http://hdl.handle.net/20.500.11824/17442024-02-09T23:18:04Z2019-01-01T00:00:00ZControl of pseudodifferential operators by maximal functions via weighted inequalities
Beltran, D.
We establish general weighted L 2 inequalities for pseudodifferential operators associated to the Hörmander symbol classes S ρ,δm . Such inequalities allow one to control these operators by fractional “non-tangential” maximal functions and subsume the optimal range of Lebesgue space bounds for pseudodifferential operators. As a corollary, several known Muckenhoupt-type bounds are recovered, and new bounds for weights lying in the intersection of the Muckenhoupt and reverse Hölder classes are obtained. The proof relies on a subdyadic decomposition of the frequency space, together with applications of the Cotlar–Stein almost orthogonality principle and a quantitative version of the symbolic calculus.
2019-01-01T00:00:00ZSubdyadic square functions and applications to weighted harmonic analysisBeltran, D.Bennett, J.http://hdl.handle.net/20.500.11824/17432024-02-09T23:18:03Z2017-02-05T00:00:00ZSubdyadic square functions and applications to weighted harmonic analysis
Beltran, D.; Bennett, J.
Through the study of novel variants of the classical Littlewood–Paley–Stein g-functions, we obtain pointwise estimates for broad classes of highly-singular Fourier multipliers on Rd satisfying regularity hypotheses adapted to fine (subdyadic) scales. In particular, this allows us to efficiently bound such multipliers by geometrically-defined maximal operators via general weighted L2 inequalities, in the spirit of a well-known conjecture of Stein. Our framework applies to solution operators for dispersive PDE, such as the time-dependent free Schrödinger equation, and other highly oscillatory convolution operators that fall well beyond the scope of the Calderón–Zygmund theory.
2017-02-05T00:00:00ZA Fefferman-Stein inequality for the Carleson operatorBeltran, D.http://hdl.handle.net/20.500.11824/17422024-02-09T23:18:01Z2018-01-01T00:00:00ZA Fefferman-Stein inequality for the Carleson operator
Beltran, D.
We provide a Fefferman-Stein type weighted inequality for maximally modulated Calderón-Zygmund operators that satisfy a priori weak type unweighted estimates. This inequality corresponds to a maximally modulated version of a result of Pérez. Applying it to the Hilbert transform we obtain the corresponding inequality for the Carleson operator C, that is C : Lp(Mp+1w) → Lp(w) for any 1 < p < ∞ and any weight function w, with bound independent of w. We also provide a maximalmultiplier weighted theorem, a vector-valued extension, and more general two-weighted inequalities. Our proof builds on a recent work of Di Plinio and Lerner combined with some results on Orlicz spaces developed by Pérez.
2018-01-01T00:00:00ZALMOST SURE POINTWISE CONVERGENCE OF THE CUBIC NONLINEAR SCHRODINGER EQUATION ON ̈ T 2Lucà, R.http://hdl.handle.net/20.500.11824/16852023-10-16T22:20:07Z2022-01-01T00:00:00ZALMOST SURE POINTWISE CONVERGENCE OF THE CUBIC NONLINEAR SCHRODINGER EQUATION ON ̈ T 2
Lucà, R.
We revisit a result from “Pointwise convergence of the Schr ̈odinger
flow, E. Compaan, R. Luc`a, G. Staffilani, International Mathematics Research
Notices, 2021 (1), 596-647” regarding the pointwise convergence of solutions
to the periodic cubic nonlinear Schr ̈odinger equation in dimension d = 2.
2022-01-01T00:00:00Z