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We study the Dirichlet and Neumann boundary value problems for the Laplacian in a Lipschitz domain Ω, with boundary data in the Besov space B p,p s (∂Ω). The novelty is to identify a way of measuring smoothness for the solution u which allows us to consider the case p < 1. This is accomplished by using a Besov-based non-tangential maximal function in place of the classical one. This builds on the works of D. Jerison and C. Kenig [JFA-1995] where the case p > 1 was treated.doi:10.1007/s10231-004-0125-5 fatcat:buinsaehi5gppesy3okocvm5xa