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: $$\sup_{t>0}t w\left\{x\in\mathbb R \mid |Tf(x)|>t\right\}\le C \int_{\mathbb R}|f|Mw(x)dx.$$ (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 $L^2$ 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 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 $B_{\delta }$ on $\mathbb R ^{n}$ are shown to satisfy a range of sparse bounds, for all $0< \delta < \frac {n-1}2$. The range of sparse bounds increases to the optimal range, as $\delta$ increases to the critical value, $\delta =\frac {n-1}2$, even assuming only partial information on the Bochner-Riesz conjecture in dimensions $n \geq 3$. In dimension $n=2$, 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

A B∞B_\infty theory for the Bergman projection

The Bergman space $A^{2}(\mathbb D)$ is the closed subspace of $L^2(\mathbb D)$ consisting of analytic functions, where $\mathbb D$ denotes the unit disk. One considers the projection from $L^2(\mathbb D)$ into $A^{2}(\mathbb D)$, 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 $B_{\infty}$ theory and its applications in Operator Theory. This is joint work with A. Aleman and S. Pott