Springer Proceedings in Mathematics & Statistics. Volume 206

Transcription

1 Springer Proceedings in Mathematics & Statistics Volume 206

2 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at

3 Pankaj Jain Hans-Jürgen Schmeisser Editors Function Spaces and Inequalities New Delhi, India, December

4 Editors Pankaj Jain Department of Mathematics South Asian University New Delhi, Delhi India Hans-Jürgen Schmeisser Faculty of Mathematics and Computer Science Friedrich Schiller University Jena Jena, Thuringia Germany ISSN ISSN (electronic) Springer Proceedings in Mathematics & Statistics ISBN ISBN (ebook) DOI / Library of Congress Control Number: Mathematics Subject Classification (2010): 26D10, 43B20, 42B25, 42B35, 46E30, 46E35, 47B38 Springer Nature Singapore Pte Ltd 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. 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore , Singapore

5 Preface With the development in the last few decades, the theory of Function Spaces has become a powerful tool in several areas of mathematical and physical sciences, engineering etc. In particular, the concept of generalized functions (distributions) enables to study Partial Differential Equations and Boundary Value Problems in a much wider perspective. In order to deal with such problems, quite often, function spaces and mapping properties of operators connected with corresponding norm inequalities come into picture. With this in mind, an International Conference on Function Spaces and Inequalities was organized under the leadership of Pankaj Jain at South Asian University, New Delhi during December 11 15, The aim of the conference was to bring together experts and young researchers working in the field of Function Spaces and Inequalities to share their latest interests/investigations. The topics covered in the conference include (Variable/Grand/Small) Lebesgue Spaces, Orlicz Spaces, Lorentz Spaces, Sobolev Spaces, Morrey Spaces, Sequence spaces, Weight Theory, Integral Operators of Hardy Type, Sobolev Type Imbeddings, Function Algebras, Banach Algebras, Spaces & Algebras of, Analytic Functions, Geometry of Banach Spaces, Isometries of Function Spaces, (Weighted) Integral and Discrete Inequalities, Convexity Theory, Harmonic Analysis. Simultaneously, it was proposed to bring out an edited volume based on the theme of the conference. This volume consists of original work as well as survey articles on topics of Function Spaces and Inequalities by distinguished mathematicians worldwide. The survey articles are self-contained and provide up-to-date knowledge in the respective area. Contributions are also from those who could not or did not attend the conference. All the articles are thoroughly refereed. We express our deep gratitude to all the authors for their valuable contribution. Also, we are thankful to the extremely talented mathematicians who contributed in the reviewing process. We owe a great debt of gratitude to Dr Kavita A Sharma, President, South Asian University for her encouragement and support towards the organization of the conference. We record our gratitude to National Board for Higher Mathematics v

6 vi Preface (NBHM), Department of Science and Technology (DST) and South Asian University for providing partial financial support for the conference. New Delhi, India Pankaj Jain Hans-Jürgen Schmeisser

7 Contents The Fundamental Function of Certain Interpolation Spaces Generated by N-Tuples of Rearrangement-Invariant Spaces... 1 Fernando Cobos and Luz M. Fernández-Cabrera Sobolev Embeddings for Herz-Type Triebel-Lizorkin Spaces Douadi Drihem Order Sharp Estimates for Monotone Operators on Orlicz Lorentz Classes Mikhail L. Goldman Complex Interpolation of Morrey Spaces Denny Ivanal Hakim and Yoshihiro Sawano Gagliardo-Nirenberg Inequalities for Spaces with Dominating Mixed Derivatives Dorothee D. Haroske and Hans-Jürgen Schmeisser Recent Trends in Grand Lebesgue Spaces Pankaj Jain, Monika Singh and Arun Pal Singh On Certain New Method to Construct Weighted Hardy-Type Inequalities and Its Application to the Sharp Hardy-Poincaré Inequalities Agnieszka Kałamajska and Iwona Skrzypczak Intrinsic Characterization and the Extension Operator in Variable Exponent Function Spaces on Special Lipschitz Domains Henning Kempka The Boundedness of Sublinear Operators in Weighted Morrey Spaces Defined on Spaces of Homogeneous Type Vakhtang Kokilashvili and Alexander Meskhi vii

8 viii Contents Essentially Algebraic Composition Operators on Lorentz Sequence Spaces with a Weight Romesh Kumar, Ajay K. Sharma, Sumit Dubey and Shagoon Wasir Higher Dimensional Hardy-Type Inequalities Santosh Kumari Recent Advances on Generalized Trigonometric Systems in Higher Dimensions Jan Lang and Osvaldo Méndez Pointwise Multipliers on Musielak-Orlicz-Morrey Spaces Eiichi Nakai The Fatou Property of Function Spaces, Heat Kernels, Admissible Norms and Mapping Properties Hans Triebel A Survey on Some Variable Function Spaces Dachun Yang, Wen Yuan and Ciqiang Zhuo