A new way to express boundary values in terms of holomorphic functions on Lipschitz planar domains

Wedecompose p-integrablefunctionsontheboundaryofasimplyconnectedLipschitz domain  ⊂ C into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is holomorphic in  while the other is holomorphic in C \  and vanishes at infinity. This decomposition has been described previously for smooth functions on the boundary of a smooth domain (Bell, The Cauchy transform, potential theory, and conformal mapping, CRC Press, Boca Raton, 2016). Uniqueness of thedecompositionis elementaryinthesmoothcase, but extendingit tothe L p setting reliesuponaclassicalalbeitlittle-knownregularitytheoremfortheholomorphicHardy space h p (b) of planar domains for which we provide a new proof that is valid also in higher dimensions. An immediate consequence of our result will be a new characterization of the kernel of the Cauchy transform acting on L p (b). These results give a new perspective on the classical Dirichlet problem for harmonic functions and the Poisson formula even in the case of the disc. Further applications are presented along with directions for future work.