190 research outputs found
Addendum to "Maximal regularity and Hardy spaces"
No full textInternational audienceWe 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.
Real Harmonic Analysis Lectures by Pascal Auscher with the Assistance of Lashi Bandara
No full textReal Harmonic Analysis originates from the seminal works of Zygmund and Calderón, pursued by Stein, Weiss, Fefferman, Coifman, Meyer and others
Boundary value problems for elliptic systems
No full textDans cette thèse, nous étudions des problèmes aux limites pour les systèmes elliptiques sous forme divergence avec coefficients complexes dans L^{infty}. Nous prouvons des estimations a priori, discutons de la solvabilité et d'extrapolation de la solvabilité. Nous utilisons une transformation via des équations Cauchy-Riemann généralisées due à P. Auscher, A. Axelsson et A. McIntosh. On peut résoudre les équations Cauchy-Riemann généralisées via la semi-groupe engendré par un opérateur différentiel perturbé d'ordre un de type Dirac. A l'aide du semi-groupe, nous étudions la théorie L^{p} avec une discussion sur la bisectorialité, le calcul fonctionnel holomorphe et les estimations hors-diagonales pour des opérateurs dans le calcul fonctionnel. En particulier, nous développons une théorie L^{p}-L^{q} pour des opérateurs dans le calcul fonctionnel d'opérateur de type Dirac perturbé. Les problèmes de Neumann, Régularité et Dirichlet se formulent avec des estimations quadratiques et des estimations pour la fonction maximale nontangentielle. Cela conduit à à démontrer de telles estimations pour le semi-groupe d'opérateur de Dirac Pour cela, nous utilisons les espaces Hardy associés et les identifions dans certains cas avec des sous-espaces des espaces de Hardy et Lebesgue classiques. Nous obtenons enfin des estimations a priori pour les problème aux limites via une extension utilisant des espaces de Sobolev associés. Nous utilisons les estimations a priori pour une discussion sur la solvabilité des problèmes aux limites et montrer un théorème d'extrapolation de la solvabilité.In this this thesis we study boundary value problems for elliptic systems in divergence form with complex coefficients in L^{\infty}. We prove a priori estimates, discuss solvability and extrapolation of solvability. We use a transformation to generalized Cauchy-Riemann equations due to P. Auscher, A. Axelsson, and A. McIntosh. The generalized Cauchy-Riemann equations can be solved by the semi-group generated by a perturbed first order Dirac/differential operator. In relation to semi-group theory we setup the L^p theory by a discussion of bisectoriality, holomorphic functional calculus and off-diagonal estimates for operators in the functional calculus. In particular, we develop an L^p-L^q theory for operators in the functional calculus of the first order perturbed Dirac/differential operators. The formulation of Neumann, Regularity and Dirichlet problems involve square function estimates and nontangential maximal function estimates. This leads us to discuss square function estimates and nontangential maximal function estimates involving operators in the functional calculus of the perturbed first order Dirac/differential operator. We discuss the related Hardy spaces associated to operators and prove identifications by subspaces of classical Hardy and Lebesgue spaces. We obtain the a priori estimates by an extension of the square function estimates and nontangential maximal function estimates to Sobolev spaces associated to operators. We use the a priori estimates for a discussion of solvability and extrapolation of solvability
The maximal regularity operator on tent spaces
No full textInternational audienceRecently, Auscher and Axelsson gave a new approach to non-smooth boundary value problems with data, that relies on some appropriate weighted maximal regularity estimates. As part of the development of the corresponding theory, we prove here the relevant weighted maximal estimates in tent spaces for in a certain open range. We also study the case
On L 2 Solvability of BVPs for Elliptic Systems
No full textIn this article we prove solvability results for L2 boundary value problems of some elliptic systems Lu=0 on the upper half-space ℝn+1 + n≥1, with transversally independent coefficients. We use the first order formalism introduced by Auscher-Axelsson
Boundary value problems and Hardy spaces for elliptic systems with block structure
No full textFor elliptic systems with block structure in the upper half-space and
t-independent coefficients, we settle the study of boundary value problems by
proving compatible well-posedness of Dirichlet, regularity and Neumann problems
in optimal ranges of exponents. Prior to this work, only the two-dimensional
situation was fully understood. In higher dimensions, partial results for
existence in smaller ranges of exponents and for a subclass of such systems had
been established. The presented uniqueness results are completely new. We also
elucidate optimal ranges for problems with fractional regularity data. Methods
use and improve, with some new results, all the machinery developed over the
last two decades to study such problems: the Kato square root estimates and
Riesz transforms, Hardy spaces associated to operators, off-diagonal estimates,
non-tangential estimates and square functions and abstract layer potentials to
replace fundamental solutions in the absence of local regularity of solutions.
This self-contained monograph provides a comprehensive overview on the field
and unifies many earlier results that have been obtained by a variety of
methods.Comment: This is a preprint of the following work: P. Auscher and M. Egert,
Boundary value problems and Hardy spaces for elliptic systems with block
structure, 2023, Birkh\"auser reproduced with permission of Birkh\"auser. The
final authenticated version is available online at:
https://doi.org/10.1007/978-3-031-29973-5. We have corrected and clarified
the statements of Propositions 8.28 and 8.3
On the use of tent spaces for solving PDEs: A proof of the Koch-Tataru theorem
No full textLecture notes with complete detailsInternational audienceIn these notes we will present (a part of) the parabolic tent spaces theory and then apply it in solving some PDE's originated from the fluid mechanics. In more details, to our most interest are the incompressible homogeneous Navier-Stokes equations. These equations have been investigated mathematically for almost one century. Yet, the question of proving well-posedness (i.e. existence, uniqueness and regularity of solutions) lacks satisfactory answer. A large part of the known positive results in connection with Navier-Stokes equations are those in which the initial data is supposed to have a small norm in some critical or scaling invariant functional space. All those spaces are embedded in the homogeneous Besov space . A breakthrough was made in the paper [16] by Koch and Tataru, where the authors showed the existence and the uniqueness of solutions to the Navier-Stokes system in case when the norm is small enough. The principal goal of these notes is to present a new proof of the theorem by Koch and Tataru on the Navier-Stokes system, namely the one using the tent spaces theory. We also hope that after having read these notes, the reader will be convinced that the theory of tent spaces is highly likely to be useful in the study of other equations in fluid mechanics. These notes are mainly based on the content of the article [1] by P. Auscher and D. Frey. However, in [1] the authors deal with a slightly more general system of parabolic equations of Navier-Stokes type. Here we have chosen to write down a self-contained text treating only the relatively easier case of the classical incompressible homogeneous Navier-Stokes equations
On existence and uniqueness for non-autonomous parabolic Cauchy problems with rough coefficients
No full textThe proof of Proposition 2.12 has a gap. However it was recently proved in Pascal Auscher, Hedong Hou. On well-posedness for parabolic Cauchy problems of Lions type with rough initial data. 2024. ⟨hal-04620829⟩ Theorem 5.1, that this holds for . See Remark 5.4 there. It is also said that the use of this proposition in the article (Lemma 6.12, Thm 6.14, Thm 6.15, Prop 7.1, Thm 8.1) is always for (in which case ) when or for general . Hence all the results are valid. Moreover, this later reference improves some exponents. In particular, it confirms the validity of Remark 2.10 which says that the exponent of proposition 2.8 is . This improves the ranges of Lemma 6.7, Thm 6.14, Thm 6.15 to .International audienceWe consider existence and uniqueness issues for the initial value problem of parabolic equations ∂ t u = divA∇u on the upper half space, with initial data in L p spaces. The coefficient matrix A is assumed to be uniformly elliptic, but merely bounded measurable in space and time. For real coefficients and a single equation, this is an old topic for which a comprehensive theory is available, culminating in the work of Aronson. Much less is understood for complex coefficients or systems of equations except for the work of Lions, mainly because of the failure of maximum principles. In this paper, we come back to this topic with new methods that do not rely on maximum principles. This allows us to treat systems in this generality when p ≥ 2, or under certain assumptions such as bounded variation in the time variable (a much weaker assumption that the usual Hölder continuity assumption) when p < 2. We reobtain results for real coefficients, and also complement them. For instance, we obtain uniqueness for arbitrary L p data, 1 ≤ p ≤ ∞, in the class L ∞ (0, T ; L p (R n)). Our approach to the existence problem relies on a careful construction of propagators for an appropriate energy space, encompassing previous constructions. Our approach to the uniqueness problem, the most novel aspect here, relies on a parabolic version of the Kenig-Pipher maximal function, used in the context of elliptic equations on non-smooth domains. We also prove comparison estimates involving conical square functions of Lusin type and prove some Fatou type results about non-tangential convergence of solutions. Recent results on maximal regularity operators in tent spaces that do not require pointwise heat kernel bounds are key tools in this study
Calderón Reproducing formulas and applications to Hardy spaces
No full textWe establish new Calder\'{o}n reproducing formulas for self-adjoint operators that generate strongly continuous groups with finite propagation speed. These formulas allow the analysing function to interact with through holomorphic functional calculus whilst the synthesising function interacts with through functional calculus based on the Fourier transform. We apply these to prove the embedding , , for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ, where is the Hodge--Dirac operator on a complete Riemannian manifold that has doubling volume growth. This fills a gap in that work. The new reproducing formulas also allow us to obtain an atomic characterisation of . The embedding , , where is either a divergence form elliptic operator on~, 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 is ultracontractive
Calderon Reproducing Formulas and Applications to Hardy Spaces
No full textInternational audienceWe establish new Calder\'{o}n reproducing formulas for self-adjoint operators that generate strongly continuous groups with finite propagation speed. These formulas allow the analysing function to interact with through holomorphic functional calculus whilst the synthesising function interacts with through functional calculus based on the Fourier transform. We apply these to prove the embedding , , for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ, where is the Hodge--Dirac operator on a complete Riemannian manifold 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 . The embedding , , where is either a divergence form elliptic operator on~, 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 is ultracontractive
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