Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols
The book systematically presents the theories of pseudo-differential operators with symbols singular in dual variables, fractional order derivatives, distributed and variable order fractional derivatives, random walk approximants, and applications of the
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Sabir Umarov
Introduction to Fractional and PseudoDifferential Equations with Singular Symbols
Developments in Mathematics VOLUME 41 Series Editors: Krishnaswami Alladi, University of Florida, Gainesville, FL, USA Hershel M. Farkas, Hebrew University of Jerusalem, Jerusalem, Israel
More information about this series at http://www.springer.com/series/5834
Sabir Umarov
Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols
123
Sabir Umarov Department of Mathematics University of New Haven West Haven, CT, USA
ISSN 1389-2177 Developments in Mathematics ISBN 978-3-319-20770-4 DOI 10.1007/978-3-319-20771-1
ISSN 2197-795X (electronic) ISBN 978-3-319-20771-1 (eBook)
Library of Congress Control Number: 2015945587 Mathematics Subject Classification (2010): 46Bxx, 46Sxx, 35Sxx, 35Gxx, 32W25, 47G30, 26A33, 60H10, 60G50, 35Q84 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 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. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)
To my parents, to my family
Preface
The 20th century was rich with great scientific and mathematical discoveries. One of the most influential events in mathematics was the introduction of the Lebesgue integral (Lebesgue, 1901) followed soon after by Borel’s development of measure theory (Borel, 1903). Combined with Cantor’s set theory, whose axiomatic form appeared in the beginning of the century (Zermelo, 1907), these sources with their deep and remarkable ideas made possible the development of a large number of function spaces that are extremely important in modern analysis, such as L p -spaces (F. Riesz, 1910), Sobolev spaces (Sobolev, 1938), Schwatz’s distributions (Schwartz, 1951), and many more. Undoubtedly, modern functional analysis, including measure theory, the theory of topological vector spaces, and operator theory, with t
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