13,514 research outputs found
BCR algorithm and the theorem
We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that
any singular integral operator can be written as the sum of a bounded operator
on , , and of a perfect dyadic singular integral operator.
This allows to deduce a local theorem for singular integral operators
from the one for perfect dyadic singular integral operators obtained by
Hofmann, Muscalu, Thiele, Tao and the first author.Comment: Change of title. New abstract and new introductio
Regularity of the minimiser of one-dimensional interaction energies
We consider both the minimisation of a class of nonlocal interaction energies
over non-negative measures with unit mass and a class of singular integral
equations of the first kind of Fredholm type. Our setting covers applications
to dislocation pile-ups, contact problems, fracture mechanics and random matrix
theory. Our main result shows that both the minimisation problems and the
related singular integral equations have the same unique solution, which
provides new regularity results on the minimiser of the energy and new
positivity results on the solutions to singular integral equations.Comment: 46 page
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