Kato's square root problem in Banach spaces

Let $L$ be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces $L^{p}(R^{n};X)$ of $X$-valued functions on $R^n$. We characterize Kato's square root estimates $\|\sqrt{L}u\|_{p} \eqsim \|\nabla u\|_{p}$ and the $H^{\infty}$-functional calculus of $L$ in terms of R-boundedness properties of the resolvent of $L$, when $X$ is a Banach function lattice with the UMD property, or a noncommutative $L^{p}$ space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case $X=C$, we get a new approach to the $L^p$ theory of square roots of elliptic operators, as well as an $L^{p}$ version of Carleson's inequality.Comment: 44 page

Heat flow and quantitative differentiation

For every Banach space (Y, parallel to . parallel to(Y)) that admits an equivalent uniformly convex norm we prove that there exists c = c(Y) is an element of(0, infinity) with the following property. Suppose that n is an element of N and that X is an n-dimensional normed space with unit ball B-X. Then for every 1-Lipschitz function f : B-X -> Y and for every epsilon is an element of(0, 1/2] there exists a radius r >= exp (1/epsilon(cn)), a point x is an element of B-X with x + r B-X subset of B-X, and an affine mapping Lambda : X -> Y such that parallel to f (y) - Lambda (y)parallel to(Y)Peer reviewe

The L-p-to-L-q boundedness of commutators with applications to the Jacobian operator

Supplying the missing necessary conditions, we complete the characterisation of the L-p -> L-q boundedness of commutators [b, T] of pointwise multiplication and Calderon-Zygmund operators, for arbitrary pairs of 1 q, our results are new even for special classical operators with smooth kernels. As an application, we show that every f is an element of L-p(R-d) can be represented as a convergent series of normalised Jacobians J(u) = det del uof u is an element of (over dot(W))(1,dp)(R-d)(d). This extends, from p = 1 to p > 1, a result of Coifman, Lions, Meyer and Semmes about J:. (over dot(W))(1,d)(R-d)(d) -> H-1(R-d), and supports a conjecture of Iwaniec about the solvability of the equation Ju = f is an element of L-p(R-d). (C) 2021 The Author(s). Published by Elsevier Masson SAS.Peer reviewe

On Besov regularity of Brownian motions in infinite dimensions

We extend to the vector-valued situation some earlier work of Ciesielski and Roynette on the Besov regularity of the paths of the classical Brownian motion. We also consider a Brownian motion as a Besov space valued random variable. It turns out that a Brownian motion, in this interpretation, is a Gaussian random variable with some pathological properties. We prove estimates for the first moment of the Besov norm of a Brownian motion. To obtain such results we estimate expressions of the form \E \sup_{n\geq 1}\|\xi_n\|, where the $\xi_n$ are independent centered Gaussian random variables with values in a Banach space. Using isoperimetric inequalities we obtain two-sided inequalities in terms of the first moments and the weak variances of $\xi_n$.Comment: to appear in Probab. Math. Statist (2008

Vector-Valued Local Approximation Spaces

We prove that for every Banach space Y, the Besov spaces of functions from the n-dimensional Euclidean space to Y agree with suitable local approximation spaces with equivalent norms. In addition, we prove that the Sobolev spaces of type q are continuously embedded in the Besov spaces of the same type if and only if Y has martingale cotype q. We interpret this as an extension of earlier results of Xu (J Reine Angew Math 504:195-226, 1998), and Martinez et al. (Adv Math 203(2):430-475, 2006). These two results combined give the characterization that Y admits an equivalent norm with modulus of convexity of power type q if and only if weakly differentiable functions have good local approximations with polynomials.Peer reviewe

The local Tb theorem with rough test functions

We prove a local Tbtheorem under close to minimal (up to certain 'buffering') integrability assumptions, conjectured by S. Hofmann (El Escorial, 2008): Every cube is assumed to support two non-degenerate functions b(Q)(1) is an element of L-p and b(Q)(2) is an element of L-q such that 1(2Q)Tb(Q)(1) is an element of L-q' and 1(2Q)T*b(Q)(2) is an element of L-p', with appropriate uniformity and scaling of the norms. This is sufficient for the L-2-boundedness of the Calderon-Zygmund operator T, for any p, q is an element of(1, infinity), a result previously unknown for simultaneously small values of pand q. We obtain this as a corollary of a local Tbtheorem for the maximal truncations T-# and (T*)(#): for the L-2-boundedness of T, it suffices that 1(Q)T#b(Q)(1) and 1Q(T*)# b(Q)(2) be uniformly in L-0. The proof builds on the technique of suppressed operators from the quantitative Vitushkin conjecture due to Nazarov-Treil-Volberg. (C) 2020 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer reviewe

R-boundedness of smooth operator-valued functions

In this paper we study $R$-boundedness of operator families \mathcal{T}\subset \calL(X,Y), where $X$ and $Y$ are Banach spaces. Under cotype and type assumptions on $X$ and $Y$ we give sufficient conditions for $R$-boundedness. In the first part we show that certain integral operator are $R$-bounded. This will be used to obtain $R$-boundedness in the case that $\mathcal{T}$ is the range of an operator-valued function T:\R^d\to \calL(X,Y) which is in a certain Besov space B^{d/r}_{r,1}(\R^d;\calL(X,Y)). The results will be applied to obtain $R$-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems.Comment: some typos correcte