A Few Remarks about the Hilbert Transform

A generalized integral similar to integrability B is used to study the Hilbert transform onX=∑1⩽p<∞ Lp(R), with a view to obtaining (i) a mollifier which commutes with the Hilbert transform onXand coincides with the Friedrichs mollifier onLloc1(R); (ii) estimates for nonlinear equations; (iii) an integral representation for the Hilbert transform of a regular Schwartz distribution; (iv) a generalized multiplier representation of the Hilbert transform onL1(R); (v) an elementary proof of the injectivity of H onX.