Bernstein's theorem on weighted Besov spaces
Degree GrantorUniversity of Canterbury
Degree NameResearch report
It is a well-known fact that the integrability of the Fourier transform of a function (or a distribution) is intimately related to its smoothness. The result of this type is usually called Bernstein's theorem. A most important result in this area is probably the Herz-Beurling theorem, which characterizes the Fourier transform of a distribution in the homogeneous Besov space Bα 2,q (see ). The results of Herz  also improve the Hausdorff-Young theorem. Closely related to this latter theorem is the weighted estimate for the Fourier transform, which can be traced back to the theorem by Pitt and the uncertainty inequality. We refer to , , , , ,,  for various such weighted estimates. In this note, which is one in a series of papers begun with  and devoted to a study of weighted function spaces, we propose to give a weighted version of Herz's results. The theorems we shall prove sharpen a number of weighted estimates mentioned above in the same manner as Herz's results did for the HausdorffYoung theorem. As an application of these theorems, we derive sufficient conditions for a function to be a multiplier on weighted Besov spaces with power weights. The results in this paper were announced in .
SubjectsField of Research::01 - Mathematical Sciences::0101 - Pure Mathematics
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