Trends in Mathematics Michael Ruzhansky Ville Turunen Editors Fourier Analysis Pseudo-differential Operators, Time-Frequency Analysis and Partial Differential Equations Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of T X is acceptable, but the entire collection of E files must be in one particular dialect of T X and unified according to simple E instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference. Fourier Analysis Pseudo-differential Operators, Time-Frequency Analysis and Partial Differential Equations Michael Ruzhansky Ville Turunen Editors Editors ISBN 978-3-319-02549-0 ISBN 978-3-319-02550-6 (eBook) DOI10.1007/978-3-319-02550-6 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014930298 © Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printedonacid-freepaper Contents Preface .................................................................. vii N. Bez and M. Sugimoto Optimal Constant for a Smoothing Estimate of Critical Index ....... 1 T.B.N. Bui and M. Reissig The Interplay Between Time-dependent Speed of Propagation and Dissipation in Wave Models .................................... 9 F. Colombini, D. Del Santo, F. Fanelli and G. M´etivier A Note on Complete Hyperbolic Operators with log-Zygmund Coefficients ........................................... 47 E. Cordero, F. Nicola and L. Rodino A Few Remarks on Time-frequency Analysis of Gevrey, Analytic and Ultra-analytic Functions ........................................ 73 S. Coriasco, K. Johansson and J. Toft Global Wave-front Sets of Intersection and Union Type ............. 91 V. Fischer and M. Ruzhansky A Pseudo-differential Calculus on Graded Nilpotent Lie Groups ..... 107 G. Garello and A. Morando Lp Microlocal Properties for Vector Weighted Pseudodifferential Operators with Smooth Symbols ................. 133 D. Grieser and E. Hunsicker A Parametrix Construction for the Laplacian on Q-rank 1 Locally Symmetric Spaces .............................. 149 N. Habal, W. Rungrottheera and B.-W. Schulze A Class of Elliptic Operators on a Manifold with Edge and Boundary ...................................................... 187 vi Contents C. Iwasaki A Representation of the Fundamental Solution for the Fokker–Planck Equation and Its Application ....................................... 211 B. Kanguzhin and N. Tokmagambetov The Fourier Transform and Convolutions Generated by a Differential Operator with Boundary Condition on a Segment ................... 235 S. Katayama and H. Kubo Global Existence for Quadratically Perturbed Massless Dirac Equations Under the Null Condition ................................ 253 M. Lassas and T. Zhou Singular Partial Differential Operators and Pseudo-differential Boundary Conditions in Invisibility Cloaking ........................ 263 T. Matsuyama Perturbed Besov Spaces by a Short-range Type Potential in an Exterior Domain .............................................. 285 T. Nishitani On the Cauchy Problem for Hyperbolic Operators with Double Characteristics, a Transition Case ................................... 311 S.Ya. Serovajsky Differentiation Functor and Its Application in the Optimization Control Theory ................................. 335 K. Shakenov The Solution of the Initial Mixed Boundary Value Problem for Hyperbolic Equations by Monte Carlo and Probability Difference Methods ................................................. 349 S. Tikhonov and M. Zeltser Weak Monotonicity Concept and Its Applications ................... 357 Y. Wakasugi Critical Exponent for the Semilinear Wave Equation with Scale Invariant Damping ............................................ 375 K. Yagdjian Semilinear Hyperbolic Equations in Curved Spacetime .............. 391 Preface The interactions between the theory of pseudo-differential operators, the time- frequencyanalysis,andthetheoryofpartialdifferentialequationshavecontributed toprogressinalltheseareasandareanactivefieldofcurrentresearch.Tofacilitate further developments and links between these fields, the international conference “FourierAnalysisandPseudo-DifferentialOperators”,withapplicationstopartial differentialequations,washeld atAalto Universitynear Helsinki,Finland, on25– 29 June 2012. It was organised as a satellite meeting to the European Congress of Mathe- maticians that took place in Krakow the following week, and as the 6th meeting in the series “Fourier Analysis and Partial Differential Equations”, with previous meetings taking place at University of Osaka (2008), Imperial College London (2008),NagoyaUniversity(2009),UniversityofG¨ottingen(2010)and,finally,Im- perial College London (2011). The conference attracted around 90 participants presenting recent results of their work, with a total of around 75 sectional and plenary talks. The papers collected in this volume are authored by participants of that meeting. They focus on different aspects of current research in the above-mentioned subjects and are, in particular, centred around the following topics: • pseudo-differential operators in different settings; • microlocal analysis and Fourier integral operators; • pseudo-differential operators and noncommutative harmonic analysis; • time-frequency analysis and its applications; • linear and nonlinear evolution equations; • hyperbolic equations and systems; • dispersive, smoothing and Strichartz estimates; • applications: wave models, control theory, stochastic analysis. On one hand, the volume is aimed at being a rigorous presentation of recent re- searchdevelopmentsintheseareas,aswellasatemphasisinginteractionsbetween them. As such, all the contributions are full research papers presenting new re- sults. This allows experts in the field to describe the recent developments in their subjects, to present new results, and will hopefully lead to further collaborative work in the area. On the other hand, the volume gives an overview on the great variety of ongoing current research in several broad fields and, therefore, allows viii Preface researchers as well as students grasping new aspects and broadening their under- standing of these areas. Therefore, the papers provide a wide scope of ideas and detailed proofs of results. It is our pleasure to acknowledge the sponsorship of the conference and con- tributions by the following organisations: • Aalto University andAalto University Departmentof Mathematics andSys- tems Analysis; • Science Factories programme by Aalto Science Institute at Aalto University School of Science; • the Finnish National Graduate School in Mathematics and its Applications; • ISAAC (International Society for Analysis, its Applications and Computa- tion); • MagnusEhrnroothFoundationoftheFinnishSocietyofSciencesandLetters. Finally,wewouldalsoliketothankothermembersoftheorganisingcommittee of theconference,inparticular,JensWirthandMitsuruSugimoto,fortheirvaluable contributions in different ways. Michael Ruzhansky and Ville Turunen FourierAnalysis TrendsinMathematics,1–7 (cid:2)c 2014SpringerInternational Publishing Switzerland Optimal Constant for a Smoothing Estimate of Critical Index Neal Bez and Mitsuru Sugimoto Abstract. We generalise a result by Hoshiro [3] which considered a critical case of Kato–Yajima’s smoothing estimate (cid:2) (cid:2) (cid:2)|x|a−1|∇|aexp(−itΔ)f(cid:2)L2 (R×Rd) ≤C(cid:4)f(cid:4)L2(Rd) t,x for the Schor¨odinger propagator exp(−itΔ). An expression for the optimal constant is also given. MathematicsSubjectClassification(2010). Primary 35B45; Secondary35P10, 35B65. Keywords. Smoothing estimates, optimal constants, extremisers. 1. Introduction We consider the Cauchy problem for the Schr¨odinger equation (cid:2) (i∂ −Δ) u(t,x)=0, t u(0,x)=f(x)∈L2(Rd) which has solution u(t,x) = exp(−itΔ)f(x) for t∈ R and x ∈Rd. For each fixed time t∈R we have (cid:3)u(t,·)(cid:3) =(cid:3)f(cid:3) , L2x(Rd) L2(Rd) so that the L2-norm of the initial data f is preserved. However, it is well known that if we integrate the solution in time, we get improved regularity in space, and as an illustration of such a smoothing effect, Kato–Yajima[4] established the smoothing estimate (cid:3) (cid:3) (cid:3)|x|a−1|∇|aexp(−itΔ)f(cid:3) ≤C(cid:3)f(cid:3) (1.1) L2t,x(R×Rd) L2(Rd) for a ∈ [0,1) and d ≥ 3. Watanabe [15] implicitly pointed out that this estimate 2 in the critical case a= 1 is not true. Nevertheless, Hoshiro [3] gave an interesting 2