Auscher P., Egert M. Boundary Value Problems and Hardy Spaces for Elliptic Systems with Block Structure

Basel: Birkhäuser, 2023. — 310 p.In this monograph, for elliptic systems with block structure in the upper half-space and t-independent coefficients, the authors 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, and the authors 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-tangential estimates and square functions, and abstract layer potentials to replace fundamental solutions in the absence of local regularity of solutions.Introduction and Main Results
Preliminaries on Function Spaces
Preliminaries on Operator Theory
Bounded Families
Conservation Properties
The Four Critical Numbers
Riesz Transform Estimates: Part I
Operator-Adapted Spaces
Identification of Adapted Hardy Spaces
A Digression: -Calculus and Analyticity
Riesz Transform Estimates: Part II
Critical Numbers for Poisson and Heat Semigroups
Boundedness of the Hodge Projector
Critical Numbers and Kernel Bounds
Comparison with the Auscher–Stahlhut Interval
Basic Properties of Weak Solutions
Existence in Dirichlet and Regularity Problems
Existence in the Dirichlet Problems with -Data
Existence in Dirichlet Problems with Fractional Regularity Data