  We study well-posedness of boundary value problems of Dirichlet and Neumann
type for elliptic systems on the upper half-space with coefficients independent
of the transversal variable, and with boundary data in fractional
Besov-Hardy-Sobolev (BHS) spaces. Our approach uses minimal assumptions on the
coefficients, and in particular does not require De Giorgi-Nash-Moser
estimates. Our results are completely new for the Hardy-Sobolev case, and in
the Besov case they extend results recently obtained by Barton and Mayboroda.
  First we develop a theory of BHS spaces adapted to operators which are
bisectorial on $L^2$, with bounded $H^\infty$ functional calculus on their
ranges, and which satisfy $L^2$ off-diagonal estimates. In particular, this
theory applies to perturbed Dirac operators $DB$. We then prove that for a
nontrivial range of exponents (the identification region) the BHS spaces
adapted to $DB$ are equal to those adapted to $D$ (which correspond to
classical BHS spaces).
  Our main result is the classification of solutions of the elliptic system
$\operatorname{div} A \nabla u = 0$ within a certain region of exponents. More
precisely, we show that if the conormal gradient of a solution belongs to a
weighted tent space (or one of their real interpolants) with exponent in the
classification region, and in addition vanishes at infinity in a certain sense,
then it has a trace in a BHS space, and can be represented as a semigroup
evolution of this trace in the transversal direction. As a corollary, any such
solution can be represented in terms of an abstract layer potential operator.
Within the classification region, we show that well-posedness is equivalent to
a certain boundary projection being an isomorphism. We derive various
consequences of this characterisation, which are illustrated in various
situations, including in particular that of the Regularity problem for real
equations.
