Type: Article
Publication Date: 1983-09-01
Citations: 21
DOI: https://doi.org/10.2140/pjm.1983.108.221
For an appropriate surface o in R'\ we prove that the multiple Hilbert transform along a is a bounded operator on L p (R n ), for p sufficiently close to 2. Our analysis of this singular integral operator proceeds via Fourier transform techniques-that is, on the "multiplier side"-with applications of Stein's analytic interpolation theorem and the Marcinkiewicz multiplier theorem.At the heart of our argument we have estimates of certain trigonometric integrals.CONTENTS I. Introduction 221 II.Outline of the Argument 222 III.The Worsened Multipliers: m ENz for Re(z) > 0 223 IV.The Improved Multipliers: m eNz for Re(z) < 0 232 V. Conclusion 233 Appendix 235 References 241 222 JAMES T. VANCE, JR.sufficiently close to 2 proceeds under somewhat more stringent conditions on the exponents.What is the interest in the operators H and T1 They occur in the study of certain singular convolution operators Kf=%*f.If the kernel % is odd and satisfies a one-parameter homogeneity condition-the simplest being %(tx) = t~n%(x) (x E R n , t> 0)-then H arises when one decomposes K by an appropriate variant of the Calderon-Zygmund "method of rotations", and one sees that L p inequalities for H imply the same for K.In [6], Nagel and Wainger impose a multiple-parameter homogeneity condition upon % and are led to T via the method of rotations.Again, bounds on T imply bounds on K.Moreover, in this case the kernel % may fail to be locally integrable at a set of points of positive dimension-e.g.along a line in R n ; this stands in contrast to previously studied singular convolution operators in which the kernel could be non-integrable only at the origin and at infinity.For a more detailed discussion, one should see [6] and Part I of [12].