150 research outputs found
Every P-convex subset of is already strongly P-convex
A classical result of Malgrange says that for a polynomial P and an open
subset of the differential operator is surjective on
if and only if is P-convex. H\"ormander showed that
is surjective as an operator on if and only if
is strongly P-convex. It is well known that the natural question
whether these two notions coincide has to be answered in the negative in
general. However, Tr\`eves conjectured that in the case of d=2 P-convexity and
strong P-convexity are equivalent. A proof of this conjecture is given in this
note
A Fourier integral formula for logarithmic energy
A formula which expresses logarithmic energy of Borel measures on R^n in
terms of the Fourier transforms of the measures is established and some
applications are given. In addition, using similar techniques a (known) formula
for Riesz energy is reinvented.Comment: 14 page
Mixing operators with non-perfectly spanning unimodular eigenvectors
For arbitrary closed countable subsets of the unit circle examples of
topologically mixing operators on Hilbert spaces are given which have a densely
spanning set of unimodular eigenvectors with eigenvalues restricted to . In
particular, these operators cannot be ergodic in the Gaussian sense
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