Boundary value problems and Hardy spaces for elliptic systems with block structure

Revised version of our monograph. Section 14 supersedes the treatment of multiplicative perturbations in v1. Section 10 on functional calculus is new. Finally, there is the proof for a-independence of critical numbers (Section 6.3). Many further improvements throughout the text.For elliptic systems with block structure in the upper half-space and t-independent coefficients, we settle the study of boundary value problems by proving compatible well-posedness of Dirichlet, regularity and Neumann problems in optimal ranges of exponents. Prior to this work, only the two-dimensional situation was fully understood. In higher dimensions, partial results for existence in smaller ranges of exponents and for a subclass of such systems had been established. The presented uniqueness results are completely new. We also elucidate optimal ranges for problems with fractional regularity data.The first part of the monograph, which can be read independently, provides optimal ranges of exponents for functional calculus and adapted Hardy spaces for the associated boundary operator.Methods use and improve, with new results, all the machinery developed over the last two decades to study such problems: the Kato square root estimates and Riesz transforms, Hardy spaces associated to operators, off-diagonal estimates, non-tangen\-tial estimates and square functions and abstract layer potentials to replace fundamental solutions in the absence of local regularity of solutions.This mostly self-contained monograph provides a comprehensive overview on the field and unifies many earlier results that have been obtained by a variety of methods