Report 30/2005: Real Analysis, Harmonic Analysis and Applications to PDE (July 3rd – July 9th, 2005 (0527)).

*(English)*Zbl 1106.42300Summary: There have been important developments in the last few years in the point-of-view and methods of harmonic analysis, and at the same time significant concurrent progress in the application of these to partial differential equations and related subjects.

The conference brought together experts and young scientists working in these two directions, with the objective of furthering these important interactions.

Introduction: Major areas and results represented at the workshop are:

I. Methods in harmonic analysis

(a) Multilinear analysis: this is an outgrowth of the method of ‘tile decomposition’ which has been so successful in solving the problems of the bilinear Hilbert transform. Recent progress involves the control of maximal trilinear operators and an extension of the Carleson-Hunt theorem.

(b) Geometry of sets in \(\mathbb{R}^d\): This includes recent progress on the interaction of Fourier analysis and geometric combinatorics related to the Falconer distance problem.

(c) Singular integrals: A break trough has been obtained on singular integrals on solvable Iwasawa AN- groups, which require a new type of Calderón-Zygmund decomposition, since the underlying spaces have exponential volume growth. Further significant progress involves the theory of operator-valued Calderón-Zygmund operators and their connection to maximal regularity of evolution equations.

(d) Oscillatory integrals, Fourier integral operators and Maximal operators: This includes estimates of maximal operators related to polynomial polyhedra and their relations with higher dimensional complex analysis, sharp estimates for maximal operators associated to hypersurfaces in \(\mathbb{R}^3\), estimates for degenerate Radon transforms and linear and bilinear estimates for oscillatory integral operators, as well as optimal Sobolev regularity for Fourier integral operators.

II. Applications to P.D.E.

The conference brought together experts and young scientists working in these two directions, with the objective of furthering these important interactions.

Introduction: Major areas and results represented at the workshop are:

I. Methods in harmonic analysis

(a) Multilinear analysis: this is an outgrowth of the method of ‘tile decomposition’ which has been so successful in solving the problems of the bilinear Hilbert transform. Recent progress involves the control of maximal trilinear operators and an extension of the Carleson-Hunt theorem.

(b) Geometry of sets in \(\mathbb{R}^d\): This includes recent progress on the interaction of Fourier analysis and geometric combinatorics related to the Falconer distance problem.

(c) Singular integrals: A break trough has been obtained on singular integrals on solvable Iwasawa AN- groups, which require a new type of Calderón-Zygmund decomposition, since the underlying spaces have exponential volume growth. Further significant progress involves the theory of operator-valued Calderón-Zygmund operators and their connection to maximal regularity of evolution equations.

(d) Oscillatory integrals, Fourier integral operators and Maximal operators: This includes estimates of maximal operators related to polynomial polyhedra and their relations with higher dimensional complex analysis, sharp estimates for maximal operators associated to hypersurfaces in \(\mathbb{R}^3\), estimates for degenerate Radon transforms and linear and bilinear estimates for oscillatory integral operators, as well as optimal Sobolev regularity for Fourier integral operators.

II. Applications to P.D.E.

- (a)
- Dispersive linear and nonlinear equations: Far reaching new approaches to dispersive estimates for Schrödinger equations via coherent state decompositions and a related new phase space transform adapted to the wave operator were introduced. Further significant progress includes \(L^p\)- estimates respectively blow up rates for eigenfunctions and quasimodes of elliptic operators on compact manifolds with and without boundary, and related problems for globally elliptic pseudodifferential operators, and well-posedness of the periodic KP-I equations. All these results are based in part on important ideas in harmonic analysis (such as I(d) above).
- (b)
- Schrödinger operators with rough potentials: This includes quantitative unique continuation theorems and their relations with spectral properties and the study of embedded eigenvalues of Schrödinger operators.

Contributions: - –
- Carlos E. Kenig, Quantitative unique continuation theorems (p. 1683)
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- Waldemar Hebisch, Singular integral of Iwasawa AN groups (p. 1683)
- –
- Malabika Pramanik (joint with Andreas Seeger), Optimal Sobolev regularity for Fourier integral operators on Rd (p. 1684)
- –
- Isroil A. Ikromov, Boundedness problem for maximal operators associated to non-convex hypersurfaces (p. 1686)
- –
- Alexandru Ionescu (joint with C. E. Kenig), Well-posedness theorems for the KP-I initial value problem on \(\mathbb T \times \mathbb T\) and \(\mathbb R \times \mathbb T\) (p. 1689)
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- Michael Cowling (joint with M. Sundari), Hardy’s Uncertainty Principle (p. 1691)
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- Daniel Tataru (joint with Dan Geba), A phase space transform adapted to the wave equation (p. 1693)
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- Hart F. Smith (joint with Chris D. Sogge), Wave packets and boundary value problems (p. 1695)
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- Svetlana Roudenko (joint with M. Frazier, F. Nazarov), Littlewood-Paley theory for matrix weights (p. 1697)
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- Alexander Nagel (joint with Malabika Pramanik), Two problems related to polynomial polyhedra (p. 1700)
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- D. H. Phong (joint with Jacob Sturm), Energy functionals and flows in Kähler geometry (p. 1703)
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- Jong-Guk Bak, Endpoint estimates for some degenerate Radon transforms in the plane (p. 1706)
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- Christoph Thiele (joint with Ciprian Demeter, Terence Tao), A maximal trilinear operator (p. 1708)
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- Camil Muscalu (joint with Xiaochun Li), A generalization of the Carleson-Hunt theorem (p. 1711)
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- Mihail N. Kolountzakis (joint with Evangelos Markakis, Aranyak Mehta), Fourier zeros of Boolean functions (p. 1713)
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- Lars Diening (joint with Peter Hästö), Traces of Sobolev Spaces with Variable Exponents (p. 1714)
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- Ralf Meyer, \(L^p\)-estimates for the wave equation associated to the Grusin operator (p. 1716)
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- Stefanie Petermichl (joint with Sergei Treil, Brett Wick), Analytic Embedding in the unit ball of \(\mathbb {C}^n\) (p. 1719)
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- Peer Christian Kunstmann, Generalized Gaussian Estimates and Applications to Calderon-Zygmund Theory (p. 1721)
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- Christopher Sogge (joint with John Toth and Steve Zelditch), Blowup rates of eigenfunctions and quasimodes (p. 1724)
- –
- H. Koch (joint with D. Tataru), Dispersive estimates and absence of positive eigenvalues (p. 1727)
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- Loredana Lanzani (joint with R. Brown, L. Capogna), On the Mixed Boundary Value Problem for Laplace’s Equation in Planar Lipschitz Domains (p. 1729)
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- Sanghyuk Lee, Linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces (p. 1731)
- –
- Lutz Weis, Calderón-Zygmund operators with operator-valued kernels and evolution equations (p. 1734).

##### MSC:

42-06 | Proceedings, conferences, collections, etc. pertaining to harmonic analysis on Euclidean spaces |

43-06 | Proceedings, conferences, collections, etc. pertaining to abstract harmonic analysis |

22-06 | Proceedings, conferences, collections, etc. pertaining to topological groups |

35-06 | Proceedings, conferences, collections, etc. pertaining to partial differential equations |

00B05 | Collections of abstracts of lectures |