Some toy Furstenberg sets and projections of the four-corner Cantor set
We give lower bounds for the Hausdorff dimensions of some model Furstenberg sets.
math.CA↗
arXiv subjects
Daniel M. Oberlin
Publications and source records attributed to Daniel M. Oberlin.
Restriction of Fourier transforms to curves: An endpoint estimate with affine arclength measure
Consider the Fourier restriction operator associated to a curve in $R^d$, $d\ge 3$. We prove for various classes of curves the endpoint restricted strong type estimate with respect to affine arclength measure on the curve. An essential ingredient is an interpolation result for multilinear operators with symmetries acting on sequences of vector-valued functions.
math.CA↗
Exceptional sets of projections, unions of k-planes, and associated transforms
We prove a generalization of a result of Peres and Schlag on the dimensions of certain exceptional sets of projections and then apply it to a geometric problem.
math.CA↗
A uniform Fourier restriction theorem for surfaces in R^d
We prove a Fourier restriction result, uniform over a certain collection of reference measures, for some indices in the Stein-Tomas range.
math.CA↗
Restricting Fourier transforms of measures to curves in R^2
We establish estimates for restrictions to certain curves in R^2 of the Fourier transforms of some fractal measures.
math.CA↗
Convolution with measures on flat curves in low dimensions
We prove convolution estimates for affine arclength measure on certain flat curves in dimensions 2, 3, and 4.
math.CA↗
A Uniform Estimate for Fourier Restriction to Simple Curves
We prove a uniform Fourier extension-restriction estimate for a certain class of curves in d-dimensional Euclidean space.
math.CA↗
Lower Bounds for Dimensions of Sums of Sets
We study lower bounds for the Minkowski and Hausdorff dimensions of the algebraic sum E+K of two subsets E and K of d-dimensional Euclidean space.
math.CA↗
Restriction of Fourier transforms to curves, II: Some classes with vanishing torsion
We consider the Fourier restriction operators associated to certain degenerate curves in R^d for which the highest torsion vanishes. We prove estimates with respect to affine arclength and with respect to the Euclidean arclength measure on the curve. The estimates have certain uniform features, and the affine arclength results cover families of flat curves.
math.CA↗
Restriction of Fourier transforms to curves and related oscillatory integrals
We prove sharp endpoint results for the Fourier restriction operator associated to nondegenerate curves in $\Bbb R^d$, $d\ge 3$, and related estimates for oscillatory integral operators. Moreover, for some larger classes of curves in $\Bbb R^d$ we obtain sharp uniform $L^p\to L^q$ bounds with respect to affine arclength measure, thereby resolving a problem of Drury and Marshall.
math.CA↗
Restricted Radon transforms and projections of planar sets
We establish a mixed norm estimate for the Radon transform in the plane when the set of directions has fractional dimension. This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to a conjecture analogous to a well-known conjecture of Furstenberg.
math.CA↗