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Radial Fourier Multipliers

Connections for Women: Harmonic Analysis January 19, 2017 - January 20, 2017

January 20, 2017 (03:30 PM PST - 04:00 PM PST)
Speaker(s): Laura Cladek (University of California, Los Angeles)
Location: MSRI: Simons Auditorium
Tags/Keywords
  • classical analysis

  • Fourier multipliers

  • radial

  • bochner-riesz

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Radial Fourier Multipliers

Abstract

Let $m$ be a radial multiplier supported in a compact subset away from the origin. For dimensions $d\ge 2$, it is conjectured that the multiplier operator $T_m$ is bounded on $L^p(R^d)$ if and only if the kernel $K=\hat{m}$ is in $L^p(R^d)$, for the range $1<p<2d/(d+1)$. Note that there are no a priori assumptions on the regularity of the multiplier. This conjecture belongs near the top of the tree of a number of important related conjectures in harmonic analysis, including the Local Smoothing, Bochner-Riesz, Restriction, and Kakeya conjectures. We discuss new progress on this conjecture in dimensions $d=3$ and $d=4$. Our method of proof will rely on a geometric argument involving sizes of multiple intersections of three-dimensional annuli

Supplements
27805?type=thumb Cladek Notes 674 KB application/pdf Download
Video/Audio Files

Radial Fourier Multipliers

H.264 Video 10-Cladek.mp4 498 MB video/mp4 rtsp://videos.msri.org/data/000/027/650/original/10-Cladek.mp4 Download
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