Contents Multilinear Embedding and Hardy s Inequality Real-variable Theory of Orlicz-type Function Spaces Associated with Operators A Survey

Transcription

1 Contents Multilinear Embedding and Hardy s Inequality... 1 William Beckner 1 Multilinear convolution inequalities Diagonal trace restriction for Hardy s inequality Diagonal trace restriction for a multilinear fractional integral Multilinear integrals and rearrangement Acknowledgements References Real-variable Theory of Orlicz-type Function Spaces Associated with Operators A Survey Der-Chen Chang, Dachun Yang and Sibei Yang 1 Introduction Orlicz type spaces associated with operators satisfying Poisson estimates Musielak-Orlicz type spaces associated with nonnegative self-adjoint operators satisfying Davies-Gaffney estimates Musielak-Orlicz type spaces associated with operators satisfying bounded H -functional calculus Acknowledgements References Boundedness of Rough Strongly Singular Integral Operators Jiecheng Chen, Dashan Fan and Meng Wang 1 L p L q boundedness on rough operators The phase function is not radial The kernel satisfies a Lipschitz condition The kernel is C References

2 II Contents On the Dimension Dependence of Some Weighted Inequalities Alberto Criado and Fernando Soria 1 Introduction The maximal operator over radial functions Proofs of the main results Kakeya maximal operator Acknowledgements References L p Estimates for Multi-parameter and Multilinear Fourier Multipliers and Pseudo-differential Operators Wei Dai, Guozhen Lu and Lu Zhang 1 Introduction L p estimates for multi-parameter and multi-linear paraproducts, multipliers and pseudo-differential operators L p estimates for bilinear and multi-parameter Hilbert transforms L p estimates for bilinear operators given by non-smooth symbols with one-dimensional singularity set in the range 1/2 <p 2/ Acknowledgements References Existence and Uniqueness Theory for the Fractional Schrödinger Equation on the Torus Seckin Demirbas, M. Burak Erdoğan and Nikolaos Tzirakis 1 Introduction Notation and preliminaries Strichartz estimates Local well-posedness via the X s,b method A smoothing estimate Global well-posedness via high-low frequency decomposition References Compactness of Maximal Commutators of Bilinear Calderón-Zygmund Singular Integral Operators Yong Ding, Ting Mei and Qingying Xue 1 Introduction and main results

3 Contents III 2 The proof of Theorem The proof of Theorem References Weak Hardy Spaces Loukas Grafakos and Danqing He 1 Introduction Relevant background The proof of Theorem Properties of H p, Square function characterization of H p, References ALocalTb Theorem with Vector-valued Testing Functions AnaGraudelaHerrán and Steve Hofmann 1 Introduction, history, preliminaries A local Tb theorem with vector-valued testing functions Application of Theorem 2.13 to the theory of layer potentials Appendix: a generalized Christ-Journé T 1 Theorem for square functions References Non-homogeneous Local T 1 Theorem: Dual Exponents Michael T. Lacey and Antti V. Vähäkangas 1 Introduction Preliminaries Perturbations and a basic decomposition A stopping tree construction The inside-paraproduct term The inside-stopping/error term The separated term Preparations for the nearby term The nearby-non-boundary term The nearby-boundary term References

4 IV Contents The Dynamics of the NLS with the Combined Terms in Five and Higher Dimensions Changxing Miao, Guixiang Xu and Lifeng Zhao 1 Introduction Preliminaries Variational characterization Part I: blow up for K Profile decomposition Part II: GWP and scattering for K Acknowledgements References Sharp Estimates for Bilinear Fourier Multiplier Operators Akihiko Miyachi and Naohito Tomita 1 Introduction Product type Sobolev scale Estimates for L 2 L L Estimates for H 1 L L Estimates for L L BMO Estimates for H 1 H 1 L 1/ Estimates for H 1 L 2 L 2/ Proof of the only if part Isotropic Besov scale References Weighted Estimates for Fractional Type Marcinkiewicz Integral Operators Associated to Surfaces Yoshihiro Sawano and Kôzô Yabuta 1 Introduction Preparation for the proof of Theorem Proof of Theorem Proof of Proposition Appendix: complex interpolation of homogeneous weighted Triebel-Lizorkin spaces References

5 Contents V Commutator Estimates for the Dirichlet-to-Neumann Map in Lipschitz Domains Zhongwei Shen 1 Introduction Dahlberg s bilinear estimate, Part I Dahlberg s bilinear estimate, Part II Trilinear estimates and proof of Theorem Proof of Theorem References ANoteonL p -norms of Quasi-modes Christopher D. Sogge and Steve Zelditch 1 Introduction and main results Proof that Proposition 1.3 implies Theorems 1.1 and Proof of Proposition Applications to breaking convexity bounds References Astala s Conjecture from the Point of View of Singular Integrals on Metric Spaces Alexander Volberg 1 Introduction A simple proof of Theorem 1. The weighted estimate of Ahlfors-Beurling transform = unweighted estimate of a certain non-symmetric Calderón-Zygmund operator on a metric space T 1 theorem for non-homogeneous metric measure spaces Acknowledgements References p,q C.S.I. for Besov Spaces Λ α (Rn ) with ( α, (p, q) ) (0, 1) ( (0, 1] (0, 1] \{(1, 1)} ) Jie Xiao and Zhichun Zhai 1 Introduction C.S.I Applications Acknowledgements

6 VI Contents References A List of Ph.D. Students, Post-doctors and Visiting Scholars Supervised by Professor Shanzhen Lu and Foreign Collaborators Who Worked with Professor Shanzhen Lu