arXiv subjects
Publications and source records attributed to Pascal Auscher.
The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from $\RR^n$. The analysis interestingly combines use of Poincaré inequalities and of some Hardy type inequalities.
We prove weighted estimates for the maximal regularity operator. Such estimates were motivated by boundary value problems. We take this opportunity to study a class of weak solutions to the abstract Cauchy problem. We also give a new proof of maximal regularity for closed and maximal accretive operators following from Kato's inequality for fractional powers and almost orthogonality arguments.
We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms.
Using the framework of a previous article joint with Axelsson and McIntosh, we extend to systems two results of S. Hofmann for real symmetric equations and their perturbations going back to a work of B. Dahlberg for Laplace's equation on Lipschitz domains, The first one is a certain bilinear estimate for a class of weak solutions and the second is a criterion which allows to identify the domain of the generator of the semi-group yielding such solutions.
In this work, we exhibit abstract conditions on a functional space E who insure the existence of a global mild solution for small data in E or the existence of a local mild solution in absence of size constraints for a class of semi-linear parabolic equations, which contains the incompressible Navier-Stokes system as a fundamental example. We also give an abstract criterion toward regularity of the obtained solutions. These conditions, given in terms of Littlewood-Paley estimates for products of spectrally localized elements of $E$, are simple to check in all known cases: Lebesgue, Lorents, Besov, Morrey... spaces. These conditions also apply to non-invariant spaces E and we give full details in the case of some 2-microlocal spaces. The following comments did not show on the first version: This article was written around 1998-99 and never published, because at that time, Koch and Tataru announced their result on well-posedness of Navier-stokes equations with initial data in $BMO^{-1}$. We believe though that some results and counterexamples here are of independent interest and we make them available electronically.
We correct an inaccuracy in the original proof
We correct an inaccuracy in a previous article [Auscher, Pascal; Bernicot, Frédéric; Zhao, Jiman. Maximal regularity and Hardy spaces. Collect. Math. 59 (2008), no. 1, 103-127.]
We give an overview of the generalized Calderón-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. $L^p-L^q$ off-diagonal estimates when $p\le q$ play an important role and we present them. They are particularly well suited to the semigroups generated by second order elliptic operators and the range of exponents $(p,q)$ rules the $L^p$ theory for many operators constructed from the semigroup and its gradient. Such applications are summarized.
This paper has been withdrawn, and is replaced with paper "Solvability of elliptic systems with square integrable boundary data" by the same authors.
We consider second order elliptic divergence form systems with complex measurable coefficients $A$ that are independent of the transversal coordinate, and prove that the set of $A$ for which the boundary value problem with $L_2$ Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when $A$ is either Hermitean, block or constant. Our methods apply to more general systems of PDEs and as an example we prove perturbation results for boundary value problems for differential forms.
We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-$λ$ method that does not use any size or smoothness estimates for the kernels.
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.
In this work, we consider the Cauchy problem for $u' - Au = f$ with $A$ the Laplacian operator on some Riemannian manifolds or a sublapacian on some Lie groups or some second order elliptic operators on a domain. We show the boundedness of the operator of maximal regularity $f\mapsto Au$ and its adjoint on appropriate Hardy spaces which we define and study for this purpose. As a consequence we reobtain the maximal $L^q$ regularity on $L^p$ spaces for $p,q$ between 1 and $\infty$.
We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that any singular integral operator can be written as the sum of a bounded operator on $L^p$, $1<p<\infty$, and of a perfect dyadic singular integral operator. This allows to deduce a local $T(b)$ theorem for singular integral operators from the one for perfect dyadic singular integral operators obtained by Hofmann, Muscalu, Thiele, Tao and the first author.
We prove that the Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in $L_2$ for small complex $L_\infty$ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for $k$-forms are well posed for small perturbations of block matrices.
We show various $L^p$ estimates for Schrödinger operators $-Δ+V$ on $\RR^n$ and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen \cite{Sh1}. Our main tools are improved Fefferman-Phong inequalities and reverse Hölder estimates for weak solutions of $-Δ+V$ and their gradients.
Let $M$ be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces $H^p$ of differential forms on $M$ and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the $H^p$-boundedness for Riesz transforms on $M$, generalizing previously known results. Further applications, in particular to $H^{\infty}$ functional calculus and Hodge decomposition, are given.
This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a measure of decay. This substitute is that of off-diagonal estimates expressed in terms of local and scale invariant $L^p-L^q$ estimates. We propose a definition in spaces of homogeneous type that is stable under composition. It is particularly well suited to semigroups. We study the case of semigroups generated by elliptic operators.