Fourier transform inequalities with measure weights
The authors prove Fourier transform inequalities with measure weights on the left hand side, \[ \|\widehat f\|_{q,\mu}\leq C\| f\|_{p,v}. \] The inequality is first proved on moment subspaces of \(L^ p_ v(R^ n)\). Conditions are given on \(v\) that guarantee that the closure of the moment subspace is all of \(L^ p_ v\). The conditions on \(\mu\) and \(v\) are computable, but two points are worthy of mention. The results do not cover the usual restriction theorems (\(v=1\), \(\mu=\sigma_{n-1}\), the uniformly distributed unit mass on the sphere in \(n\)-space); indeed, with that \(\mu\) and \(v(t)=| t|^{\alpha}\), the conditions require \(n(p-1)<\alpha<(n+p')(p-1)\). Second, the result of the extension may be some generalized version of the Fourier transform. On page 207 it is pointed out that when \(\mu=\delta\), the extension derived is the zero function. Benedetto and Lakey will discuss this question. The results are obtained by using recent results of G. Sinnamon for Hardy inequalities with measure weights, combined with familiar splittings of the Fourier transform.
scientific article; zbMATH DE number 221235
scientific article