A note on the cone multiplier

Mockenhaupt, Gerd

doi:10.1090/S0002-9939-1993-1098404-6

A note on the cone multiplier
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by Gerd Mockenhaupt PDF

Proc. Amer. Math. Soc. 117 (1993), 145-152 Request permission

Abstract:

In this paper we study the convolution operator given on the Fourier transform side by multiplication by \[ {m_\alpha }(x,z) = \phi (z)(1 - |x|/z)_ + ^\alpha ,\qquad (x,z) \in {{\mathbf {R}}^2} \times {\mathbf {R}},\;\alpha > 0,\] where $\phi \in C_0^\infty (1,2)$. We will prove that ${m_\alpha }$ defines a bounded operator on ${L^4}({{\mathbf {R}}^3})$ if $\alpha > \tfrac {1} {8}$. Furthermore, as a generalization of a result of C. Fefferman (Acta Math. 124 (1970), 9-36), we will show that an $({L^2},{L^p})$ restriction theorem for compact ${C^\infty }$ submanifolds $M \subset {{\mathbf {R}}^n}$ of arbitrary codimension imply results for multipliers having a singularity of the form $\operatorname {dist} {(x,M)^\alpha }$ near $M$.

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