65 research outputs found
On Muckenhoupt-Wheeden Conjecture
Let M denote the dyadic Maximal Function. We show that there is a weight w,
and Haar multiplier T for which the following weak-type inequality fails: (With T replaced by M, this is a well-known fact.) This shows
that a dyadic version of the so-called Muckenhoupt-Wheeden Conjecture is false.
This accomplished by using current techniques in weighted inequalities to show
that a particular consequence of the inequality above does not hold.Comment: 14 pages, 2 figures, corrected typo
Sharp Bekolle estimates for the Bergman projection
We prove sharp estimates for the Bergman projection in weighted Bergman
spaces in terms of the Bekolle constant. Our main tools are a dyadic model
dominating the operator and an adaptation of a method of Cruz-Uribe, Martell
and Perez.Comment: 12 pages, 1 figur
A theory for the Bergman projection
The Bergman space is the closed subspace of consisting of analytic functions, where denotes the unit disk. One considers the projection from into , such a projection can be written as a convolution operator with a singular kernel. In this talk, we will present the recent advances on the one weight theory for the Bergman projection that resulted from combining techniques from complex analysis and the theory of singular integrals in harmonic analysis. We will pay special attention to the development of a theory and its applications in Operator Theory. This is joint work with A. Aleman and S. Pott
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
Sparse Bounds for Bochner-Riesz Multipliers
The Bochner-Riesz multipliers on are shown
to satisfy a range of sparse bounds, for all . The
range of sparse bounds increases to the optimal range, as increases
to the critical value, , even assuming only partial
information on the Bochner-Riesz conjecture in dimensions . In
dimension , we prove a sharp range of sparse bounds. The method of proof
is based upon a `single scale' analysis, and yields the sharpest known weighted
estimates for the Bochner-Riesz multipliers in the category of Muckenhoupt
weights.Comment: 15 pages, 2 figure
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