A Note on the Hilbert Transform
نویسنده
چکیده
If /(OG^P) £>!> then f(x)^Lpy and a considerable literature is devoted to studying the relationship of such pairs of "conjugate" functions to the theory of functions analytic in a half-plane. More to the point of the present note is a series of papers studying the Hubert transform along strictly real variable lines ([2, 3 ] ; further bibliography in [2]). Much less is known about f(x) when f(t) £ L i . Plessner found by applying complex variable methods to the theory of Fourier series that if / ( / ) £ J L I then f(x) exists almost everywhere (see [l, p. 145]). Besicovitch [4] proved Plessner's result using only the theory of sets, starting from his own previous real variable investigation of the L2 transform case. S. Pollard [5] showed how Besicovitch's proof could be extended to prove the existence a.e. of the principal value of the Stieltjes integral r°° dF(t) f{x)-P\ — l i , J _oo t — X
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