Extensions of the theory of tent spaces and applications to boundary value problems
Date
2016
Authors
Amenta, Alex
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Abstract
We extend the theory of tent spaces from Euclidean spaces to
various types of metric measure spaces. For doubling spaces we
show that the usual 'global' theory remains valid, and for
'non-uniformly locally doubling' spaces (including R^n with the
Gaussian measure) we establish a satisfactory local theory. In
the doubling context we show that
Hardy–Littlewood–Sobolev-type embeddings hold in the scale of
weighted tent spaces, and in the special case of unbounded
AD-regular metric measure spaces we identify the real
interpolants (the 'Z-spaces') of weighted tent spaces.
Weighted tent spaces and Z-spaces on R^n are used to construct
Hardy–Sobolev and Besov spaces adapted to perturbed Dirac
operators. These spaces play a key role in the classification of
solutions to first-order Cauchy–Riemann systems (or
equivalently, the classification of conormal gradients of
solutions to second-order elliptic systems) within weighted tent
spaces and Z-spaces. We establish this classification, and as a
corollary we obtain a useful characterisation of well-posedness
of Regularity and Neumann problems for second-order
complex-coefficient elliptic systems with boundary data in
Hardy–Sobolev and Besov spaces of order s ∈ (−1, 0).
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tent spaces, metric measure spaces, interpolation, hardy-sobolev spaces, besov spaces, elliptic boundary value problems, functional calculus, off-diagonal estimates, cauchy-riemann systems, semigroups of operators
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