Une note sur les lemmes div–curl

Let $\text{Ω}$ be a strongly Lipschitz domain of $\text{R}{}^{\mathrm{n}}$ (n?2). We give endpoint versions of div–curl lemmata on $\text{Ω}$, for a given function f on $\text{Ω}$ whose gradient belongs to a Hardy space on $\text{Ω}$. To cite this article: P. Auscher et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Weighted norm inequalities,off-diagonal estimates and elliptic operators. Part IV: Riesz transforms on manifolds and weights

This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace–Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds.      

Change of angle in tent spaces

We prove sharp bounds for the equivalence of norms in tent spaces with respect to changes of angles. Some applications are given.     

Conical stochastic maximal L^p-regularity for 1\leqslant p<\infty

Let \(A = -\mathrm{div} \,a(\cdot ) \nabla \) be a second order divergence form elliptic operator on \({\mathbb R}^n\) with bounded measurable real-valued coefficients and let \(W\) be a cylindrical Brownian motion in a Hilbert space \(H\) . Our main result implies that the stochastic convolution process $$\begin{aligned} u(t) = \int _0^t e^{-(t-s)A}g(s)\,dW(s), \quad t\geqslant 0, \end{aligned}$$ satisfies, for all \(1\leqslant p<\infty \) , a conical maximal \(L^p\) -regularity estimate $$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)}^p \leqslant C_p^p {\mathbb E}\Vert g \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)}^p. \end{aligned}$$ Here, \(T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)\) and \(T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)\) are the parabolic tent spaces of real-valued and \(H\) -valued functions, respectively. This contrasts with Krylov’s maximal \(L^p\) -regularity estimate $$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;{\mathbb R}^n))}^p \leqslant C^p {\mathbb E}\Vert g \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;H))}^p \end{aligned}$$ which is known to hold only for \(2\leqslant p<\infty \) , even when \(A = -\Delta \) and \(H = {\mathbb R}\) . The proof is based on an \(L^2\) -estimate and extrapolation arguments which use the fact that \(A\) satisfies suitable off-diagonal bounds. Our results are applied to obtain conical stochastic maximal \(L^p\) -regularity for a class of nonlinear SPDEs with rough initial data.     

Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I

We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with L ₂ boundary data. The coefficients A may depend on all variables, but are assumed to be close to coefficients A ₀ that are independent of the coordinate transversal to the boundary, in the Carleson sense ‖A−A ₀‖ _C defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour results under finiteness of ‖A−A ₀‖ _C . Our methods yield full characterization of weak solutions, whose gradients have L ₂ estimates of a non-tangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operator-valued kernel. Also, the non-tangential maximal function of a weak solution is controlled in L ₂ by the square function of its gradient. This estimate is new for systems in such generality, and even for real non-symmetric equations in dimension 3 or higher. The existence of a proof a priori to well-posedness, is also a new fact.     

On non-autonomous maximal regularity for elliptic operators in divergence form

We consider the Cauchy problem for non-autonomous forms inducing elliptic operators in divergence form with Dirichlet, Neumann, or mixed boundary conditions on an open subset \({\Omega \subseteq \mathbb{R}^n}\). We obtain maximal regularity in \({L^2(\Omega)}\) if the coefficients are bounded, uniformly elliptic, and satisfy a scale invariant bound on their fractional time-derivative of order one-half. Previous results even for such forms required control on a time-derivative of order larger than one-half.     

Square roots of elliptic second order divergence operators on strongly lipschitz domains:<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

We prove the Kato conjecture for square roots of elliptic second order non-self-adjoint operators in divergence formL = -div(A∇) on strongly Lipschitz domains in ℝⁿ, n≥2, subject to Dirichlet or to Neumann boundary conditions. The method relies on a transference procedure from the recent positive result on ℝⁿ in [2].     

Solution of two problems on wavelets

We solve two problems on wavelets. The first is the nonexistence of a regular wavelet that generates a wavelet basis for the Hardy space ?²(?). The second is the existence, given any regular wavelet basis for $\mathbb{H}^2 (\mathbb{R})$ , of aMulti-Resolution Analysis generating the wavelet. Moreover, we construct a regular scaling function for this Multi-Resolution Analysis. The needed regularity conditions are very mild and our proofs apply to both the orthonormal and biorthogonal situations. Extensions to more general cases in dimension 1 and higher are given. In particular, we show in dimension larger than 2 that a regular wavelet basis for $\mathbb{L}^2 (\mathbb{R}^n )$ arises from a Multi-Resolution Analysis that is regular modulo the action of a unitary operator, which is whenn = 2 a Calderón-Zygmund operator of convolution type.