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.

We give a Bellman-function proof of the dimension-free estimate R⃗ f _L^p(Ω; ℓ^2)≲ (p-1) f_L^p(Ω), 2≤ p<∞, for the vector of Riesz transforms associated with the Walsh number operator on the Hamming cube Ω={-1,1}^n, as well as for locally compact abelian groups, in particular Ω=ℤ^n. The argument is based on a Poisson semigroup representation, symmetrized estimates along edges of Ω, and a two-point inequality. This is the first non noncommutative proof of this result, after the seminal papers of Lust-Piquard and later Junge-Mei-Parcet. According to an example of Lamberton, for 1<p<2 such a dimension-free bound is known to be false.

The main result of this paper are dimension-free Lp inequalities, 10, and theta=theta(epsilon,p)is an element of(0,1) satisfying 1/p=theta/p+epsilon+1-theta/2 we obtain, for any function f:{-1,1}n -> C whose spectrum is bounded from above by d, the Bernstein-Markov type inequalities parallel to Delta(k)f parallel to p <= C(p,epsilon)(k)d(k)parallel to f parallel to(1-theta)(2)parallel to f parallel to(theta)(p+epsilon),k is an element of N. Analogous inequalities are also proved for p is an element of(1,2) with p-epsilon replacing p+epsilon. As a corollary, if f is Boolean-valued or f:{-1,1}n ->{-1,0,1}, we obtain the bounds parallel to Delta(k)f parallel to p <= C(p)(k)d(k)parallel to f parallel to p,k is an element of N. At the endpoint p=infinity we provide counterexamples for which a linear growth in d does not suffice when k=1. We also obtain a counterpart of this result on tail spaces. Namely, for p>2 we prove that any function f:{-1,1}n -> C whose spectrum is bounded from below by d satisfies the following upper bound on the decay of the heat semigroup: parallel to e(-t Delta)f parallel to(p)<= exp(-c(p,epsilon)td)parallel to f parallel to(1-theta)(2)parallel to f parallel to(theta)(p+epsilon),t>0, and an analogous estimate for p is an element of(1,2). The constants c(p,epsilon) and C(p,epsilon) depend only on p and epsilon; crucially, they are independent of the dimension n.

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.

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 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.

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.

We present a new proof of the dimensionless $L^p$ boundedness of the Riesz vector on manifolds with bounded geometry. Our proof has the significant advantage that it allows for a much stronger conclusion, namely that of a new dimensionless weighted $L^p$ estimate with optimal exponent. Other than previous arguments, only a small part of our proof is based on special auxiliary functions, the core of the argument is a weak type estimate and a sparse decomposition of the stochastic process by X.D. Li, whose projection is the Riesz vector.

We give several sharp estimates for a class of combinations of second order Riesz transforms on Lie groups G=G_x×G_y that are multiply connected, composed of a discrete abelian component G_x and a connected component G_y endowed with a biinvariant measure. These estimates include new sharp L^p estimates via Choi type constants, depending upon the multipliers of the operator. They also include weak-type, logarithmic and exponential estimates. We give an optimal L^q → L^p estimate as well. It was shown recently by Arcozzi, Domelevo and Petermichl that such 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 proofs of our theorems combine this stochastic integral representation with a number of deep estimates for pairs of martingales under strong differential subordination by Choi, Banuelos and Osekowski. When two continuous directions are available, sharpness is shown via the laminates technique. We show that sharpness is preserved in the discrete case using Lax-Richtmyer theorem.

We provide a mathematical analysis of a thermo-diffusive combustion model of lean spray flames, for which we prove the existence of travelling waves. In the high activation energy singular limit we show the existence of two distinct combustion regimes with a sharp transition – the diffusion limited regime and the vaporisation controlled regime. The latter is specific to spray flames with slow enough vaporisation. We give a complete characterisation of these regimes, including explicit velocities, profiles, and upper estimate of the size of the internal combustion layer. Our model is on the one hand simple enough to allow for explicit asymptotic limits and on the other hand rich enough to capture some particular aspects of spray combustion. Finally, we briefly discuss the cases where the vaporisation is infinitely fast, or where the spray is polydisperse.

We show that the norm of the vector of Riesz transforms as operator in the weighted Lebesgue space L^2(w) is bounded by a constant multiple of the first power of the Poisson-A_2 characteristic of w. The bound is free of dimension. Our argument requires an extension of Wittwer's linear estimate for martingale transforms to the vector valued setting with scalar weights, for which we indicate a proof. Extensions to L^p(w) for 1 1, the Poisson-A_2 class is properly included in the classical A_2 class.