Related papers: Sharp rate of average decay of the Fourier transfo…
We consider spherical averages of the Fourier transform of fractal measures and improve both the upper and lower bounds on the rate of decay. Maximal estimates with respect to fractal measures are deduced for the Schr\"odinger and wave…
For spherical and parabolic averages of the Fourier transform of fractal measures, we obtain new upper bounds on rates of decay by an "intermediate dimension" trick.
The main result of this paper is, that if we suppose that a function is absolutely continuous and uniformly H\"older continuous and that its finite difference function does not oscillate infinitely often on a bounded interval, then the…
This work studies the Fourier transform of the characteristic function of planar convex bodies averaged over affine transformations. We establish lower and upper bounds on the latter quantities in terms of the geometric properties of the…
In a well-known paper by Bruna, Nagel and Wainger [BNW], Fourier transform decay estimates were proved for smooth hypersurfaces of finite line type bounding a convex domain. In this paper, we generalize their results in the following ways.…
We study $L^p$ boundedness of the maximal average over dilations of a smooth hypersurface $S$. When the decay rate of the Fourier transform of a measure on $S$ is $1/2$, we establish the optimal maximal bound, which settles the conjecture…
We consider Fourier transforms of densities supported on curves in R^d. We obtain sharp lower and close to sharp upper bounds for the L^q decay rates.
An improved lower bound is given for the decay of conical averages of Fourier transforms of measures, for cones of dimension $d \geq 4$. The proof uses a weighted version of the broad restriction inequality, a narrow decoupling inequality…
We study averaged decay estimates for Fourier transforms of measures when the averages are taken over space curves with non-vanishing torsion. We extend the previously known results to higher dimensions and discuss sharpness of the…
The study of Fourier transforms of probability measures on fractal sets plays an important role in recent research. Faster decay rates are known to yield enhanced results in areas such as metric number theory. This paper focuses on…
We study an aspect of the following general question: which properties of a signal can be characterized by its scattering transform? We show that the energy contained in high order scattering coefficients is upper bounded by the energy…
We study the properties of the set where a generalized function of bounded variation has infinite approximate limit, highlighting in this way the main geometric difference with functions of bounded variation. To this aim we prove a new…
We consider the volume preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long time asymptotics…
We prove uniform estimates for the decay rate of the Fourier transform of measures supported on real-analytic hypersurfaces in R^3. If the surface contains the origin and is oriented such that its normal at the origin is in the direction of…
We consider measures supported on sets of irrational numbers possessing many consecutive partial quotients satisfying a condition based on the previous partial quotients. We show that under mild assumptions, such sets will always support…
Description of linear continuous functionals on a space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in $\mathbb R^n$ in terms of their Fourier-Laplace transform is obtained.
We provide necessary and sufficient conditions on the characteristics of an infinitely divisible distribution under which its characteristic function $\phi$ decays polynomially. Under a mild regularity condition this polynomial decay is…
The existence of a smooth, nonnegative, compactly supported function with monotone (on the half-line) Fourier transform satisfying two-sided decay bounds is demonstrated.
We show that the Fourier transform of Patterson-Sullivan measures associated to convex cocompact groups of isometries of real hyperbolic space decays polynomially quickly at infinity. The proof is based on the $L^2$-flattening theorem…
We prove $L^p$-bounds on the Fourier transform of measures $\mu$ supported on two dimensional surfaces. Our method allows to consider surfaces whose Gauss curvature vanishes on a one-dimensional submanifold. Under a certain non-degeneracy…