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Summary: A renormalized Riesz potential and applications
Mihai Putinar
Abstract. The convolution in Rn
with |x|-n
is a very singular
operator. Endowed with a proper normalization, and regarded as a
limit of Riesz potentials, it is equal to Dirac's distribution . However,
a different normalization turns the non-linear operator:
Ef = exp(
-2
|Sn-1|
|x|-n
f),
into a remarkable transformation. Its long history (in one dimension)
and some of its recent applications in higher dimensions make the
subject of this exposition. A classical extremal problem studied by A.
A. Markov is related to the operation E in one real variable. Later,
the theory of the spectral shift of self-adjoint perturbations was also
based on E. In the case of two real variables, the transform E has ap-
peared in operator theory, as a determinantal-characteristic function
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