Recently the matrix A_2 conjecture was disproved. Indeed, the growth of the vector Hilbert transform in the matrix weighted L^2(W) space was shown to be at best a constant multiple of [W]_𝐀_2^3/2. This bound had previously been established and it was thus proved that it is sharp and the conjectured linear growth cannot be obtained. It is a natural question to see if the 3/2 power persists if we replace the classical matrix A_2 characteristic by the "fattened", larger, so-called matrix Poisson A_2 characteristic. We show that the 3/2 power, even in this case, cannot be improved.

We define a time faithful dyadic shift operator of complexity one, that is an antisymmetric antiinvolution. We show that the Hilbert transform with values in a Banach space is L^p bounded if and only if the dyadic shift is – with a linear two sided norm dependence. The results reduce the famous UMD conjecture to a pair of simple dyadic operators.

The main goal of this article is the celebration of Jill Pipher, who was postdoctoral mentor of the author and continued to be her mentor. We review the Fefferman-Kenig-Pipher $$A_{\infty }$$ A ∞ inequality for scalar weights using the heat extension. We transition from a Bellman function proof in the heat setting to its dyadic scalar version. From here, we transition to the matrix analog of this inequality. The matrix result itself is implicit in papers by Treil and Volberg. In this text we present a more classical looking Bellman proof with derivative estimates resulting in the dyadic convexity estimate. We also prove via Bellman function an unrelated general embedding sum whose scalar version has been used in sharp paraproduct estimates. In the matrix case this inequality holds in the sense of operators and it appears to be new.

We present a fundamentally new proof of the dimensionless Lp boundedness of the Bakry Riesz vector on manifolds with bounded geometry. Our proof has the significant advantage that it allows for a much stronger conclusion than previous arguments, namely that of some new dimensionless weighted estimates with optimal exponent. Part of the importance of this task lies in the novelty of the techniques: we develop the self similarity argument known as sparse domination in the setting of uniformly integrable cadlag Hilbert space valued martingales and extend the domination to a process with infinite memory. We provide a range of optimal weighted estimates and weak type estimates for these stochastic processes. Previous geometric Riesz transform estimates relied on Bellman functions and did not provide this range of weighted estimates. The development of sparse domination in this probabilistic setting and its use for high dimensional problems is new.

We show that the famous matrix $A_2$ conjecture is false: the norm of the Hilbert Transform in the space $L^2(W)$ with matrix weight $W$ is estimated below by $C[W]_{{A}_2}^{3/2}$.

We develop a biparameter theory for matrix weights and provide various biparameter matrix-weighted bounds for Journé operators as well as other central operators under the assumption of the product matrix Muckenhoupt condition. In particular, we provide a complete theory for biparameter Journé operator bounds on matrix-weighted L2 spaces. We also achieve bounds in the general case of matrix-weighted Lp spaces, for 1<p<∞ for paraproduct-free Journé operators. Finally, we expose an open problem involving a matrix-weighted Fefferman–Stein inequality, on which our methods rely in the general setting of matrix-weighted bounds for arbitrary Journé operators and p≠2.

We characterize dyadic little BMO via the boundedness of the tensor commutator with a single well chosen dyadic shift. It is shown that several proof strategies work for this problem, both in the unweighted case as well as with Bloom weights. Moreover, we address the flexibility of one of our methods.

We derive a dyadic model operator for the Riesz vector. We show linear upper $L^p$ bounds for $1 < p < \infty$ between this model operator and the Riesz vector, when applied to functions with values in Banach spaces. By an upper bound we mean that the boundedness of the dyadic Riesz vector implies the boundedness of the Riesz vector. The same holds for single dyadic Riesz transforms and their continuous counterparts. The linear dependence is with constant one.

The purpose of this text is both instructive and historic. We give a review of the classical Bellman technique mainly in the "weak" (dualized) form for dyadic martingales. From here, we approach techniques and novelties required to pass to their use for continuous time martingales with jumps. The historic part shows the development in dyadic analysis of a Bellman function for a specific problem. We then study this Bellman function and show it has some additional properties, useful for the analogous question in the continuous case.

We show that if the dyadic Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the Hilbert transform, with a linear relation of the bounds. This result is the counterpart of [arXiv:2212.00090] where the opposite bound was proven.

The convex body maximal operator is a natural generalization of the Hardy–Littlewood maximal operator. In this paper we are considering its dyadic version in the presence of a matrix weight. To our surprise it turns out that this operator is not bounded. This is in a sharp contrast to a Doob's inequality in this context. At first, we show that the convex body Carleson Embedding Theorem with matrix weight fails. We then deduce the unboundedness of the matrix-weighted convex body maximal operator.

This brief note describes an error in the following paper: Multiparameter Riesz commutators, Amer J. Math. 131 (2009), no. 3, 731-769. A correction of this error seems to require new ideas, and has not been produced as of this note.

Let $(T_t)_{t \geq 0}$ be a markovian (resp. submarkovian) semigroup on some $\sigma$-finite measure space $(\Omega,\mu)$. We prove that its negative generator $A$ has a bounded $H^\infty(\Sigma_\theta)$ calculus on the weighted space $L^2(\Omega,wd\mu)$ as long as the weight $w : \Omega \to (0,\infty)$ has finite characteristic defined by $Q^A_2(w) = \sup_{t > 0} \left\| T_t(w) T_t \left(w^{-1} \right) \right\|_{L^\infty(\Omega)}$ (resp. by a variant for submarkovian semigroups). Some additional technical conditions on the semigroup have to be imposed and their validity in examples is discussed. Any angle $\theta > \frac{\pi}{2}$ is admissible in the above $H^\infty$ calculus, and for some semigroups also certain $\theta = \theta_w < \frac{\pi}{2}$ depending on the size of $Q^A_2(w)$. The norm of the $H^\infty(\Sigma_\theta)$ calculus is linear in the $Q^A_2$ characteristic for $\theta > \frac{\pi}{2}$. We also discuss negative results on angles $\theta < \frac{\pi}{2}$. Namely we show that there is a markovian semigroup on a probability space and a $Q^A_2$ weight $w$ without H\"ormander functional calculus on $L^2(\Omega,w d\mu)$.

We show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The stochastic representation of the function and that of its Hilbert transform are under differential subordination and orthogonality relation with respect to the sharp bracket of quadratic covariation. This illustrates the Cauchy Riemann relations of analytic functions in this setting. This result is inspired by the seminal work of Gundy and Varopoulos on stochastic representation of the Hilbert transform in the continuous setting.

We prove the matrix $A_2$ conjecture for the dyadic square function, that is, a norm estimate of the matrix weighted square function, where the focus is on the sharp linear dependence on the matrix $A_2$ constant in the estimate. Moreover, we give a mixed estimate in terms of $A_2$ and $A_{\infty}$ constants. Key is a sparse domination of a process inspired by the integrated form of the matrix--weighted square function.

We prove failure of the natural formulation of a matrix weighted bilinear Carleson embedding theorem, featuring a matrix valued Carleson sequence as well as products of norms for the embedding. We show that assuming an A2 weight is also not sufficient. Indeed, a uniform bound on the conditioning number of the matrix weight is necessary and sufficient to get the bilinear embedding. We show that any improvement of a recent matrix weighted bilinear embedding, featuring a scalar Carleson sequence and inner products instead of norms must fail. In particular, replacing the scalar sequence by a matrix sequence results in failure even when maintaining the formulation using inner products. Any formulation using norms, even in the presence of a scalar Carleson sequence must fail. As a positive result, we prove the so called matrix weighted redundancy condition in full generality.

In this article we highlight the interplay ofmulti-parameter BMO spaces and boundedness of corresponding commutators. In a variety of settings, we discuss two-sided norm estimates for commutators of classical singular operators with a symbol function. In its classical form, this concerns a theorem by Nehari, factorisation of Hardy space, Hankel and Toeplitz forms. We highlight recent results in which a characterization of Lp boundedness of iterated commutators of multiplication by a symbol function and tensor products of Riesz and Hilbert transforms is obtained, completing a theory on characterisation of BMO spaces begun by Cotlar, Ferguson and Sadosky. In the light of real analysis, we discuss results in a more intricate situation; commutators of multiplication by a symbol function and Calderón-Zygmund or Journé operators. We show that the boundedness of these commutators is also determined by the inclusion of their symbol function in the same multi-parameter BMO class. In this sense the Hilbert or Riesz transforms or their tensor products are a representative testing class for Calderón-Zygmund or Journé operators.

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.

We show that the classical A_∞ condition is not sufficient for a lower square function estimate in the non-homogeneous weighted L^2 space. We also show that under the martingale A_2 condition, an estimate holds true, but the optimal power of the characteristic jumps from 1 / 2 to 1 even when considering the classical A_2 characteristic. This is in a sharp contrast to known estimates in the dyadic homogeneous setting as well as the recent positive results in this direction on the discrete time non-homogeneous martingale transforms. Last, we give a sharp A_∞ estimate for the n -adic homogeneous case, growing with n .

We prove optimal ${L}^2$ bounds for a pair of Hilbert space valued differentially subordinate martingales under a change of law. The change of law is given by a process called a weight and sharpness in this context refers to the optimal growth with respect to the characteristic of the weight. The pair of martingales are adapted, uniformly integrable, and cadlag. Differential subordination is in the sense of Burkholder, defined through the use of the square bracket. In the scalar dyadic setting with underlying Lebesgue measure, this was proved by Wittwer, where homogeneity was heavily used. Recent progress by Thiele-Treil-Volberg and Lacey, independently, resloved the so-called non-homogenous case of discrete in time filtrations with two completely different proofs. The general case for continuous-in-time filtrations remained open and is adressed here. As a by-product, we give the needed explicit expression of a Bellman function of four variables for the weighted estimate of subordinate martingales with jumps.