arXiv subjects
Publications and source records attributed to Pascal Auscher.
We bring a precision to our cited work concerning the notion of "Borel measures", as the choice among different existing definitions impacts on the validity of the results.
We prove a number of \textit{a priori} estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value problems of Dirichlet and Neumann type in various topologies. We work in classes of solutions which include the energy solutions. For those solutions, we use a description using the first order systems satisfied by their conormal gradients and the theory of Hardy spaces associated with such systems but the method also allows us to design solutions which are not necessarily energy solutions. We obtain precise comparisons between square functions, non-tangential maximal functions and norms of boundary trace. The main thesis is that the range of exponents for such results is related to when those Hardy spaces (which could be abstract spaces) are identified to concrete spaces of tempered distributions. We consider some adapted non-tangential sharp functions and prove comparisons with square functions. We obtain boundedness results for layer potentials, boundary behavior, in particular strong limits, which is new, and jump relations. One application is an extrapolation for solvability ''á la {Š}ne{\uı}berg". Another one is stability of solvability in perturbing the coefficients in $L^\infty$ without further assumptions. We stress that our results do not require De Giorgi-Nash assumptions, and we improve the available ones when we do so.
We show that, under general conditions, the operator $\bigl (-\nabla \cdot μ\nabla +1\bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(Ω)$ and $L^p(Ω)$, for $p \in {]1,2[}$ if one presupposes that this isomorphism holds true for $p=2$. The domain $Ω$ is assumed to be bounded, the Dirichlet part $D$ of the boundary has to satisfy the well-known Ahlfors-David condition, whilst for the points from $\overline {\partial Ω\setminus D}$ the existence of bi-Lipschitzian boundary charts is required.
We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_{2}$ weight. We obtain representations and boundary traces for solutions in appropriate classes, perturbation results for solvability and solvability in some situations. The technology of earlier works of the first two authors can be adapted to the weighted setting once the needed quadratic estimate is established and we even improve some results in the unweighted setting. The proof of this quadratic estimate does not follow from earlier results on the topic and is the core of the article.
Let $A = -{\rm div} \,a(\cdot) \nabla$ be a second order divergence form elliptic operator on $\R^n$ 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 $$ u(t) = \int_0^t e^{-(t-s)A}g(s)\,dW(s), \quad t\ge 0,$$ satisfies, for all $1\le p<\infty$, a conical maximal $L^p$-regularity estimate $$\E \n \nabla u \n_{ T_2^{p,2}(\R_+\times\R^n)}^p \le C_p^p \E \n g \n_{ T_2^{p,2}(\R_+\times\R^n;H)}^p.$$ Here, $T_2^{p,2}(\R_+\times\R^n)$ and $T_2^{p,2}(\R_+\times\R^n;H)$ are the parabolic tent spaces of real-valued and $H$-valued functions, respectively. This contrasts with Krylov's maximal $L^p$-regularity estimate $$\E \n \nabla u \n_{L^p(\R_+;L^2(\R^n;\R^n))}^p \le C^p \E \n g \n_{L^p(\R_+;L^2(\R^n;H))}^p$$ which is known to hold only for $2\le p<\infty$, even when $A = -Δ$ and $H = \R$. 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.
We give a new proof of a well-known result of Koch and Tataru on the well-posedness of Navier-Stokes equations in $\R^n$ with small initial data in $BMO^{-1}(\R^n)$. The proof is formulated operator theoretically and does not make use of self-adjointness of the Laplacian.
The purpose of this article is to study extrapolation of solvability for boundary value problems of elliptic systems in divergence form on the upper half-space assuming De Giorgi type conditions. We develop a method allowing to treat each boundary value problem independently of the others. We shall base our study on solvability for energy solutions, estimates for boundary layers, equivalence of certain boundary estimates with interior control so that solvability reduces to a one-sided Rellich inequality. Our method then amounts to extrapolating this Rellich inequality using atomic Hardy spaces, interpolation and duality. In the way, we reprove the Regularity-Dirichlet duality principle between dual systems and extend it to $H^1-BMO$. We also exhibit and use a similar Neumann-Neumann duality principle.
We establish new Calderón reproducing formulas for self-adjoint operators $D$ that generate strongly continuous groups with finite propagation speed. These formulas allow the analysing function to interact with $D$ through holomorphic functional calculus whilst the synthesising function interacts with $D$ through functional calculus based on the Fourier transform. We apply these to prove the embedding $H^p_D(\wedge T^*M) \subseteq L^p(\wedge T^*M)$, $1\leq p\leq 2$, for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ, where $D=d+d^*$ is the Hodge--Dirac operator on a complete Riemannian manifold $M$ that has polynomial volume growth. This fills a gap in that work. The new reproducing formulas also allow us to obtain an atomic characterisation of $H^1_D(\wedge T^*M)$. The embedding $H^p_L \subseteq L^p$, $1\leq p\leq 2$, where $L$ is either a divergence form elliptic operator on $\R^n$, or a nonnegative self-adjoint operator that satisfies Davies--Gaffney estimates on a doubling metric measure space, is also established in the case when the semigroup generated by the adjoint $-L^*$ is ultracontractive.
We make some remarks on earlier works on $R-$bisectoriality in $L^p$ of perturbed first order differential operators by Hytönen, McIntosh and Portal. They have shown that this is equivalent to bounded holomorphic functional calculus in $L^p$ for $p$ in any open interval when suitable hypotheses are made. Hytönen and McIntosh then showed that $R$-bisectoriality in $L^p$ at one value of $p$ can be extrapolated in a neighborhood of $p$. We give a different proof of this extrapolation and observe that the first proof has impact on the splitting of the space by the kernel and range.
In this article we prove solvability results for $L^2$ boundary value problems of some elliptic systems $Lu=0$ on the upper half-space $\R^{n+1}_{+}, n\ge 1$, with transversally independent coefficients. We use the first order formalism introduced by Auscher-Axelsson-McIntosh and further developed with a better understanding of the classes of solutions in the subsequent work of Auscher-Axelsson. The interesting fact is that we prove only half of the Rellich boundary inequality without knowing the other half.
Adapting the recently developed randomized dyadic structures, we introduce the notion of spline function in geometrically doubling quasi-metric spaces. Such functions have interpolation and reproducing properties as the linear splines in Euclidean spaces. They also have Hölder regularity. This is used to build an orthonormal basis of Hölder-continuous wavelets with exponential decay in any space of homogeneous type. As in the classical theory, wavelet bases provide a universal Calderón reproducing formula to study and develop function space theory and singular integrals. We discuss the examples of $L^p$ spaces, BMO and apply this to a proof of the T(1) theorem. As no extra condition {(like 'reverse doubling', 'small boundary' of balls, etc.)} on the space of homogeneous type is required, our results extend a long line of works on the subject.
We extend the recent results concerning boundedness of the maximal regularity operator on tent spaces. This leads us to develop a singular integral operator theory on tent spaces. Such operators have operator-valued kernels. A seemingly appropriate condition on the kernel is time-space decay measured by off-diagonal estimates with various exponents.
We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second order, complex, elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning \textit{a priori} almost everywhere non-tangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying \textit{a posteriori} a separate work on bounded domains.
We prove sharp bounds for the equivalence of norms in tent spaces with respect to changes of angles. Some applications are given.
In the setting of spaces of homogeneous type, we give a direct proof of the local Tb theorem for singular integral operators. Motivated by questions of S. Hofmann, we extend it to the case when the integrability conditions are lower than 2, with an additional weak boundedness type hypothesis, which incorporates some Hardy type inequalities. The latter can be obtained from some geometric conditions on the homogeneous space. For example, we prove that the monotone geodesic property of Tessera suffices.
We study the difference between vertical and conical square functions in the abstract and also in the specific case where the square functions come from an elliptic operator.
Recently, Auscher and Axelsson gave a new approach to non-smooth boundary value problems with $L^{2}$ data, that relies on some appropriate weighted maximal regularity estimates. As part of the development of the corresponding $L^{p}$ theory, we prove here the relevant weighted maximal estimates in tent spaces $T^{p,2}$ for $p$ in a certain open range. We also study the case $p=\infty$.
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 {\em 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 {\em a priori} to well-posedness, is also a new fact. As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of $\|A-A_0\|_C$ and well-posedness for $A_0$, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients $A_0$ by the two authors and A. McIntosh in order to use functional calculus related to the Kato conjecture solution, and the main analytic tool for coefficients $A$ is an operational calculus to prove weighted maximal regularity estimates.