154 research outputs found
Second order Riesz transforms on multiply-connected Lie groups and processes with jumps
We study a class of combinations of second order Riesz transforms on Lie
groups that are multiply connected, composed of a discrete abelian component
and a compact connected component. We prove sharp estimates for these
operators, therefore generalising previous results.
We construct stochastic integrals with jump components adapted to functions
defined on our semi-discrete set. We show that these second order Riesz
transforms applied to a function may be written as conditional expectation of a
simple transformation of a stochastic integral associated with the function.
The analysis shows that Ito integrals for the discrete component must be
written in an augmented discrete tangent plane of dimension twice larger than
expected, and in a suitably chosen discrete coordinate system. Those artifacts
are related to the difficulties that arise due to the discrete component, where
derivatives of functions are no longer local. Previous representations of Riesz
transforms through stochastic integrals in this direction do not consider
discrete components and jump processes
A matrix weighted bilinear Carleson Lemma and Maximal Function
We prove a bilinear Carleson embedding theorem with matrix weight and scalar
measure. In the scalar case, this becomes exactly the well known weighted
bilinear Carleson embedding theorem. Although only allowing scalar Carleson
measures, it is to date the only extension to the bilinear setting of the
recent Carleson embedding theorem by Culiuc and Treil that features a matrix
Carleson measure and a matrix weight. It is well known that a Carleson
embedding theorem implies a Doob's maximal inequality and this holds true in
the matrix weighted setting with an appropriately defined maximal operator. It
is also known that a dimensional growth must occur in the Carleson embedding
theorem with matrix Carleson measure, even with trivial weight. We give a
definition of a maximal type function whose norm in the matrix weighted setting
does not grow with dimension.Comment: 15 pages, for proceeding
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