Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach

We study well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional Besov-Hardy-Sobolev (BHS) spaces. Our approach uses minimal assumptions on the coefficients, and in particular does not require De Giorgi-Nash-Moser estimates. Our results are completely new for the Hardy-Sobolev case, and in the Besov case they extend results recently obtained by Barton and Mayboroda. First we develop a theory of BHS spaces adapted to operators which are bisectorial on $L^2$, with bounded $H^\infty$ functional calculus on their ranges, and which satisfy $L^2$ off-diagonal estimates. In particular, this theory applies to perturbed Dirac operators $DB$. We then prove that for a nontrivial range of exponents (the identification region) the BHS spaces adapted to $DB$ are equal to those adapted to $D$ (which correspond to classical BHS spaces). Our main result is the classification of solutions of the elliptic system $\operatorname{div} A \nabla u = 0$ within a certain region of exponents. More precisely, we show that if the conormal gradient of a solution belongs to a weighted tent space (or one of their real interpolants) with exponent in the classification region, and in addition vanishes at infinity in a certain sense, then it has a trace in a BHS space, and can be represented as a semigroup evolution of this trace in the transversal direction. As a corollary, any such solution can be represented in terms of an abstract layer potential operator. Within the classification region, we show that well-posedness is equivalent to a certain boundary projection being an isomorphism. We derive various consequences of this characterisation, which are illustrated in various situations, including in particular that of the Regularity problem for real equations.Comment: Changed title and fixed some minor typos. To appear in the CRM Monograph Serie

Orthonormal bases of regular wavelets in spaces of homogeneous type

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 H\"older regularity. This is used to build an orthonormal basis of H\"older-continuous wavelets with exponential decay in any space of homogeneous type. As in the classical theory, wavelet bases provide a universal Calder\'on 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.Comment: We have made improvements to section 2 following the referees suggestions. In particular, it now contains full proof of formerly Theorem 2.7 instead of sending back to earlier works, which makes the construction of splines self-contained. One reference adde

Addendum to : Orthonormal bases of regular wavelets in spaces of homogeneous type

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.Comment: 1 page. we make a precision on the density lemma in arXiv:1110.5766. This reference had been published in ACH

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 {\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.Comment: This is an extended version of the paper, containing some new material and a road map to proofs on suggestion from the referee

Representation and uniqueness for boundary value elliptic problems via first order systems

Given any elliptic system with $t$-independent coefficients in the upper-half space, we obtain representation and trace for the conormal gradient of solutions in the natural classes for the boundary value problems of Dirichlet and Neumann types with area integral control or non-tangential maximal control. The trace spaces are obtained in a natural range of boundary spaces which is parametrized by properties of some Hardy spaces. This implies a complete picture of uniqueness vs solvability and well-posedness.Comment: submitted, 70 pages. A number of maths typos have been eliminate