Boundary value problems and hardy spaces for elliptic systems with block structure / Pascal Auscher, Moritz Egert.
Auscher, Pascal, author.; Egert, Moritz, author.
2023
QA377
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Title Boundary value problems and hardy spaces for elliptic systems with block structure / Pascal Auscher, Moritz Egert.
Author Auscher, Pascal, author.
ISBN 9783031299735 (electronic bk.)
3031299736 (electronic bk.)
9783031299728
3031299728
DOI https://doi.org/10.1007/978-3-031-29973-5
Published Cham : Springer, 2023.
Language English
Description 1 online resource (314 pages) : illustrations (black and white, and color).
Item Number 10.1007/978-3-031-29973-5 doi
Call Number QA377
Dewey Decimal Classification 515/.3533
Summary 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.
Access Note Access limited to authorized users.
Source of Description Description based on print version record.
Added Author Egert, Moritz, author.
Series Progress in mathematics (Boston, Mass.) ; v. 346.
Available in Other Form Boundary value problems and hardy spaces for elliptic systems with block structure.
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Chapter. 1. Introduction and main results
Chapter. 2. Preliminaries on function spaces
Chapter. 3. Preliminaries on operator theory
Chapter. 4. Hp - Hq bounded families
Chapter. 5. Conservation properties
Chapter. 6. The four critical numbers
Chapter. 7. Riesz transform estimates: Part I
Chapter. 8. Operator-adapted spaces
Chapter. 9. Identification of adapted Hardy spaces
Chapter. 10. A digression: H -calculus and analyticity
Chapter. 11. Riesz transform estimates: Part II
Chapter. 12. Critical numbers for Poisson and heat semigroups
Chapter. 13. Lp boundedness of the Hodge projector
Chapter. 14. Critical numbers and kernel bounds
Chapter. 15. Comparison with the AuscherStahlhut interval
Chapter. 16. Basic properties of weak solutions
Chapter. 17. Existence in Hp Dirichlet and Regularity problems
Chapter. 18. Existence in the Dirichlet problems with data
Chapter. 19. Existence in Dirichlet problems with fractional regularity data
Chapter. 20. Single layer operators for L and estimates for L-1
Chapter. 21. Uniqueness in regularity and Dirichlet problems
Chapter. 22. The Neumann problem
Appendix A. Non-tangential maximal functions and traces
Appendix B. The Lp-realization of a sectorial operator in L2
References
Index.

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