1,071 research outputs found
A variational nonlinear Hausdorff-Young inequality in the discrete setting
Following the works of Lyons and Oberlin, Seeger, Tao, Thiele and Wright, we
relate the variation of certain discrete curves on the Lie group
to the corresponding variation of their linearized versions on
the Lie algebra. Combining this with a discrete variational
Menshov-Paley-Zygmund theorem, we establish a variational Hausdorff-Young
inequality for a discrete version of the nonlinear Fourier transform on
.Comment: 16 page
A variational restriction theorem
We establish variational estimates related to the problem of restricting the
Fourier transform of a three-dimensional function to the two-dimensional
Euclidean sphere. At the same time, we give a short survey of the recent field
of maximal Fourier restriction theory.Comment: 10 pages, v2: bibliography is updated, a short survey of the maximal
Fourier restriction is include
A sharp nonlinear Hausdorff-Young inequality for small potentials
The nonlinear Hausdorff-Young inequality follows from the work of Christ and
Kiselev. Later Muscalu, Tao, and Thiele asked if the constants can be chosen
independently of the exponent. We show that the nonlinear Hausdorff-Young
quotient admits an even better upper bound than the linear one, provided that
the function is sufficiently small in the norm. The proof combines
perturbative techniques with the sharpened version of the linear
Hausdorff-Young inequality due to Christ.Comment: 14 pages, v2: referee's suggestions incorporate
Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots
We study a recent result of Bourgain, Clozel and Kahane, a version of which
states that a sufficiently nice function
that coincides with its Fourier transform and vanishes at the origin has a root
in the interval , where the optimal satisfies . A similar result holds in higher dimensions. We improve the
one-dimensional result to , and the lower bound in
higher dimensions. We also prove that extremizers exist, and have infinitely
many double roots. With this purpose in mind, we establish a new structure
statement about Hermite polynomials which relates their pointwise evaluation to
linear flows on the torus, and applies to other families of orthogonal
polynomials as well.Comment: 26 pages, 4 figure
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