174 research outputs found
Orthonormal bases of regular wavelets in spaces of homogeneous type
Adapting the recently developed randomized dyadic structures, we introduce
the notion of spline function in geometrically doubling quasi-metric spaces.
Such functions have interpolation and reproducing properties as the linear
splines in Euclidean spaces. They also have H\"older regularity. This is used
to build an orthonormal basis of H\"older-continuous wavelets with exponential
decay in any space of homogeneous type. As in the classical theory, wavelet
bases provide a universal Calder\'on reproducing formula to study and develop
function space theory and singular integrals. We discuss the examples of
spaces, BMO and apply this to a proof of the T(1) theorem. As no extra
condition {(like 'reverse doubling', 'small boundary' of balls, etc.)} on the
space of homogeneous type is required, our results extend a long line of works
on the subject.Comment: We have made improvements to section 2 following the referees
suggestions. In particular, it now contains full proof of formerly Theorem
2.7 instead of sending back to earlier works, which makes the construction of
splines self-contained. One reference adde
On the Carleson duality
As a tool for solving the Neumann problem for divergence form equations,
Kenig and Pipher introduced the space X of functions on the half space, such
that the non-tangential maximal function of their L_2-Whitney averages belongs
to L_2 on the boundary. In this paper, answering questions which arose from
recent studies of boundary value problems by Auscher and the second author, we
find the pre-dual of X, and characterize the pointwise multipliers from X to
L_2 on the half space as the well-known Carleson-type space of functions
introduced by Dahlberg. We also extend these results to L_p generalizations of
the space X. Our results elaborate on the well-known duality between Carleson
measures and non-tangential maximal functions.Comment: The second author has recently changed surname from previous name
Andreas Axelsso
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