Boundary Value Problems and Hardy Spaces for Elliptic Systems with Block Structure

✍ Scribed by Pascal Auscher; Moritz Egert

• Publisher: Birkhäuser
• Year: 2023
• Tongue: English
• Leaves: 310
• Series: Progress in Mathematics 346
• Category: Library

✦ Synopsis

In this monograph, for elliptic systems with block structure in the upper half-space and t-independent coefficients, the authors 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, and the authors also elucidate optimal ranges for problems with fractional regularity data. The first part of the monograph, which can be read independently, provides optimal ranges of exponents for functional calculus and adapted Hardy spaces for the associated boundary operator. Methods use and improve, with 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.

✦ Table of Contents

Preface
Acknowledgments
Contents
1 Introduction and Main Results
1.1 Objective of the Monograph
1.2 The Elliptic Equation
1.3 The Critical Numbers
1.4 Square Root Problem and Hardy Spaces
1.5 Main Results on Dirichlet Problems
1.6 Dirichlet Problems with Fractional Spaces of Data
1.7 Neumann Problems
1.8 Synthesis
1.9 Notation
2 Preliminaries on Function Spaces
2.1 Lebesgue Spaces and Distributions
2.2 Tent Spaces
2.3 Z-spaces
2.4 Hardy Spaces
2.5 Homogeneous Smoothness Spaces
2.6 Interpolation Functors
3 Preliminaries on Operator Theory
3.1 Definition of the Elliptic Operators
3.2 (Bi)sectorial Operators
3.3 Classes of Holomorphic Functions
3.4 Holomorphic Functional Calculi
3.5 Adjoints
3.6 Kato Problem and Riesz Transform
3.7 Off-Diagonal Estimates
4 Hp - Hq Bounded Families
4.1 Abstract Principles
4.2 Off-Diagonal Estimates
4.3 Interpolation Principles
4.4 Applications to the Functional Calculus
5 Conservation Properties
6 The Four Critical Numbers
6.1 General Facts on Critical Numbers
6.2 Worst-Case Estimates for the Critical Numbers
6.3 a-Independence of Critical Numbers
7 Riesz Transform Estimates: Part I
7.1 Sufficient Condition for 1<p<2
7.2 Sufficient Condition for p>2
7.3 Necessary Condition for 1<p<2
7.4 Necessary Condition for p>2
8 Operator-Adapted Spaces
8.1 Bisectorial Operators with First-Order Scaling
8.2 Sectorial Operators with Second-Order Scaling
8.3 Molecular Decomposition for Adapted Hardy Spaces
8.4 Connection with the Non-tangential Maximal Function
8.5 D-Adapted Spaces
8.6 Spaces Adapted to Perturbed Dirac Operators
9 Identification of Adapted Hardy Spaces
9.1 Identification Regions
9.2 The Identification Theorem
Part 1: p-(L) < h-(L) and p+(L) > h+(L)
Part 2: Inclusion Lp C IHp for 2<p<oo
Part 3: Injection of Classical Spaces into L-Adapted Spaces for 1<p<2
Part 4: Injection of L-Adapted Spaces into Classical Spaces for p<2
Part 5: Injection of Classical Spaces into L-Adapted Spaces for p<1
Part 6: h-(L) < p-(L)
Part 7: Upper bound for h1-
Part 8: h+(L) < p+(L)
Part 9: h1+(L) > q+(L)
Part 10: h1+(L) < q+(L)
9.3 Consequences for Square Functions
10 A Digression: H∞-Calculus and Analyticity
11 Riesz Transform Estimates: Part II
12 Critical Numbers for Poisson and Heat Semigroups
12.1 Identification of the Critical Heat Numbers
12.2 Identification of the Critical Poisson Numbers
12.3 More on Off-diagonal Decay for the Poisson Semigroup
13 Lp Boundedness of the Hodge Projector
13.1 Compatible Adapted Hodge Decompositions
13.2 Adapted Hodge Decompositions
13.3 Characterizations of P(L0)
14 Critical Numbers and Kernel Bounds
14.1 Consequences of p-(L)<1
14.2 Equivalence with Kernel Estimates
14.3 Dirichlet Property, Stability, and Examples
14.4 Remarks on Multiplicative Perturbations
14.5 Kernel estimates for L=-a-1x
15 Comparison with the Auscher–Stahlhut Interval
16 Basic Properties of Weak Solutions
16.1 Energy Solutions
16.2 Semigroup Solutions
16.3 Interior Estimates
17 Existence in Hp Dirichlet and Regularity Problems
17.1 Estimates Toward the Dirichlet Problem
17.2 Estimates Toward the Regularity Problem
17.3 Conclusion of the Existence Part
18 Existence in the Dirichlet Problems with α-Data
Part 1: Well-Definedness of the Solution
Part 2: Proof of (ii)
Part 3: The Upper Bound for the Carleson Functional
Part 4: Compatibility
Part 5: The Lower Bound for the Carleson Functional
Part 6: Proof of (iii)
19 Existence in Dirichlet Problems with Fractional Regularity Data
19.1 Fractional Identification Regions
19.2 Solvability for Fractional Regularity Data
20 Single Layer Operators for L and Estimates for L
21 Uniqueness in Regularity and Dirichlet Problems
21.1 The Strategy of Proof
21.2 Uniqueness for (R)pL—Conclusion of the Proofof Theorem 1.2
Case 1: p- v 1* < p < p-v1
Case 2: p- v 1 < p < p+
21.3 Uniqueness for (D)pL—Conclusion of the Proofof Theorem 1.1
Case 1: (p- v 1)< p < p+
Case 2: p- < 1=p
Case 3: p+ < p n and the Vertical Segment
Case 4: The Right-Hand Triangle
22 The Neumann Problem
A Non-tangential Maximal Functions and Traces
B The Lp-Realization of a Sectorial Operator in L2
References
Index