Morrey spaces : introduction and applications to integral operators and PDE's . Volume I / / Yoshihiro Sawano, Chuo University, Giuseppe Di Fazio, University of Catania, Denny Ivanal Hakim, Bandung Institute of Technology

• Autore: Sawano Yoshihiro
• Pubblicazione: Boca Raton : , : CRC Press, Taylor & Francis Group, , 2020
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource
• Disciplina: 515/.732
• Soggetto topico: Banach spaces; Harmonic analysis; Differential equations, Partial - Numerical solutions; Differential equations, Elliptic - Numerical solutions; Integral operators; MATHEMATICS / General; MATHEMATICS / Differential Equations; MATHEMATICS / Functional Analysis
• Persona (resp. second.): Di FazioGiuseppe <1963->; HakimDenny Ivanal
• Note generali: "A Chapman & Hall book."
• Nota di contenuto: Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Contents -- Preface -- Acknowledgement -- Notation in this book -- 1. Banach function lattices -- 1.1 Lp spaces -- 1.1.1 Measure space -- 1.1.2 Integration theorems -- 1.1.3 Fubini theorem and Lebesgue spaces -- 1.1.4 Exercises -- 1.2 Morrey spaces -- 1.2.1 Morrey norms -- 1.2.2 Examples of functions in Morrey spaces -- 1.2.3 The role of the parameters -- 1.2.4 Inclusions in Morrey spaces -- 1.2.5 Weak Morrey spaces -- 1.2.6 Morrey spaces and ball Banach function spaces -- 1.2.7 Exercises -- 1.3 Local Morrey spaces, Bσ-spaces, Herz spaces and Herz-Morrey spaces -- 1.3.1 Local Morrey spaces -- 1.3.2 Herz spaces and Herz-Morrey spaces -- 1.3.3 Exercises -- 1.4 Distributions and Lorentz spaces -- 1.4.1 Distribution function -- 1.4.2 Lorentz spaces -- 1.4.3 Hardy operators and Hardy's inequality -- 1.4.4 Inequalities for monotone functions and their applications to Lorentz norms -- 1.4.5 Exercises -- 1.5 Young functions and Orlicz spaces -- 1.5.1 Young functions -- 1.5.2 Orlicz spaces -- 1.5.3 Orlicz-averages -- 1.5.4 Lebesgue spaces with a variable exponent -- 1.5.5 Exercises -- 1.6 Smoothness function spaces -- 1.6.1 Sobolev spaces -- 1.6.2 Hölder-Zygmund spaces -- 1.7 Notes -- 2. Fundamental facts in functional analysis -- 2.1 Normed spaces and Banach spaces -- 2.1.1 Hahn-Banach theorem and Banach-Alaoglu theorem -- 2.1.2 Refinement of the triangle inequality -- 2.1.3 Sum and intersection of Banach spaces -- 2.1.4 Exercises -- 2.2 Hilbert spaces -- 2.2.1 Komlos theorem -- 2.2.2 Cotlar's lemma -- 2.2.3 Exercises -- 2.3 Bochner integral -- 2.3.1 Measurable functions -- 2.3.2 Convergence theorems -- 2.3.3 Fubini's theorem for Bochner integral -- 2.3.4 Exercises -- 2.4 Notes -- 3. Polynomials and harmonic functions -- 3.1 Preliminary facts on polynomials -- 3.1.1 The space Pk (Rn). ; 3.1.2 Moment inequalities -- 3.1.3 Control of derivatives by integrals -- 3.1.4 Best approximation -- 3.1.5 Exercises -- 3.2 Spherical harmonic functions -- 3.2.1 The spaces Hk (Rn) and Hk(Rn) -- 3.2.2 Norm estimates for spherical harmonics -- 3.2.3 Laplacian and integration by parts formula -- 3.2.4 Exercises -- 3.3 Notes -- 4. Various operators in Lebesgue spaces -- 4.1 Maximal operators -- 4.1.1 Hardy-Littlewood maximal operator -- 4.1.2 Hardy-Littlewood maximal inequality -- 4.1.3 Local estimates for the Hardy-Littlewood maximal operator -- 4.1.4 Fefferman-Stein vector-valued maximal inequality -- 4.1.5 Orlicz-maximal operators -- 4.1.6 Composition of the maximal operators -- 4.1.7 Local boundednss of the Φ-maximal operators -- 4.1.8 Estimates for convolutions -- 4.1.9 Exercises -- 4.2 Sharp maximal operators -- 4.2.1 Sharp-maximal inequalities -- 4.2.2 Distributional maximal function and median -- 4.2.3 Generalized dyadic grid and the Lerner-Hyt onen decomposition -- 4.2.4 Exercises -- 4.3 Fractional maximal operators -- 4.3.1 Fractional maximal operators -- 4.3.2 Local estimates for the maximal operators and the fractional maximal operators -- 4.3.3 Sparse estimate for fractional maximal operators -- 4.3.4 Exercises -- 4.4 Fractional integral operators -- 4.4.1 Fractional integral operators on Lebesgue spaces -- 4.4.2 Local estimates for the fractional integral operators -- 4.4.3 Sparse estimate of the fractional integral operators -- 4.4.4 Fundamental solution to the elliptic differential operators -- 4.4.5 The Bessel potential operator (1 - Δ)- 2, s > -- 0 -- 4.4.6 Morrey's lemma -- 4.4.7 Exercises -- 4.5 Singular integral operators -- 4.5.1 Riesz transform -- 4.5.2 Calderón-Zygmund operators -- 4.5.3 Calderón-Zgymund decomposition -- 4.5.4 Weak-(1, 1) boundedness and strong-(p, p) boundedness -- 4.5.5 Truncation and pointwise convergence. ; 4.5.6 Examples of singular integral operators -- 4.5.7 Sparse estimate of singular integral operators -- 4.5.8 Local estimates for singular integral operators -- 4.5.9 Exercises -- 4.6 Notes -- 5. BMO spaces and Morrey-Campanato spaces -- 5.1 The space BMO(Rn) and commutators -- 5.1.1 The space BMO -- 5.1.2 John-Nirenberg inequality -- 5.1.3 Exercises -- 5.2 Commutators -- 5.2.1 Commutators generated by BMO and singular integral operators -- 5.2.2 Commutators generated by BMO and fractional integral operators -- 5.2.3 Exercises -- 5.3 Morrey-Campanato spaces -- 5.3.1 Morrey-Campanato spaces -- 5.3.2 Morrey-Campanato spaces and Hölder-Zygmund spaces -- 5.3.3 Exercises -- 5.4 Notes -- 6. General metric measure spaces -- 6.1 Maximal operators on Euclidean spaces with general Radon measures -- 6.1.1 Covering lemmas on Euclidean spaces -- 6.1.2 Maximal operators on Euclidean spaces with general Radon measures -- 6.1.3 Differentiation theorem -- 6.1.4 Universal estimates -- 6.1.5 Examples of metric measure spaces -- 6.1.6 Exercises -- 6.2 Maximal operators on metric measure spaces with general Radon measures -- 6.2.1 Weak-(1, 1) estimate and strong-(p, p) estimate -- 6.2.2 Vector-valued boundedness of the Hardy-Littlewood maximal operators -- 6.2.3 Examples of metric measure spaces which require modification -- 6.2.4 Exercises -- 6.3 Notes -- 7. Weighted Lebesgue spaces -- 7.1 One-weighted norm inequality -- 7.1.1 The class A1 -- 7.1.2 The class Ap -- 7.1.3 The class A∞ -- 7.1.4 The class Ap,q -- 7.1.5 Extrapolation -- 7.1.6 A2-theorem -- 7.1.7 Exercises -- 7.2 Two-weight norm inequality -- 7.2.1 Weighted estimates for the Hardy operator -- 7.2.2 Two-weight norm inequality for fractional maximal operators -- 7.2.3 Two-weight norm inequality for singular integral operators -- 7.2.4 Exercises -- 7.3 Notes -- 8. Approximations in Morrey spaces. ; 8.1 Various closed subspaces of Morrey spaces -- 8.1.1 Closed subspaces generated by linear lattices -- 8.1.2 Closed subspaces generated by the translation -- 8.1.3 Inclusions in closed subspaces of Morrey spaces -- 8.1.4 Exercises -- 8.2 Approximation in Morrey spaces -- 8.2.1 Characterization of Mp(Rn) -- 8.2.2 Approximations and closed subspaces -- 8.2.3 Examples of functions in closed subspaces -- 8.2.4 Exercises -- 8.3 Notes -- 9. Predual of Morrey spaces -- 9.1 Predual of Morrey spaces -- 9.1.1 Definition of block spaces and examples -- 9.1.2 Finite decomposition and a dense subspace -- 9.1.3 Duality-block spaces and Morrey spaces -- 9.1.4 Fatou property of block spaces -- 9.1.5 Köthe dual of Morrey spaces -- 9.1.6 Decomposition and averaging technique in Morrey spaces -- 9.1.7 Exercises -- 9.2 Choquet integral and predual spaces -- 9.2.1 Hausdorff capacity -- 9.2.2 Choquet integral -- 9.2.3 Predual spaces of Morrey spaces by way of the Choquet integral -- 9.2.4 Exercises -- 9.3 Notes -- 10. Linear and sublinear operators in Morrey spaces -- 10.1 Maximal operators in Morrey spaces -- 10.1.1 Maximal operator in Morrey spaces -- 10.1.2 Maximal operator in local Morrey spaces -- 10.1.3 Exercises -- 10.2 Sharp maximal operators in Morrey spaces -- 10.2.1 Sharp maximal inequalities for Morrey spaces -- 10.2.2 Sharp maximal inequalities for local Morrey spaces -- 10.2.3 Exercises -- 10.3 Fractional integral operators in Morrey spaces -- 10.3.1 Fractional integral operators in Morrey spaces -- 10.3.2 Fractional integral operators in local Morrey spaces -- 10.3.3 Exercises -- 10.4 Singular integral operators in Morrey spaces -- 10.4.1 Singular integral operators in Morrey spaces -- 10.4.2 Singular integral operators in local Morrey spaces -- 10.4.3 Exercises -- 10.5 Commutators in Morrey spaces -- 10.5.1 Commutators in Morrey spaces. ; 10.5.2 Commutators in local Morrey spaces -- 10.5.3 Exercises -- 10.6 Notes -- Bibliography -- Index.
• Sommario/riassunto: Morrey spaces were introduced by Charles Morrey to investigate the local behaviour of solutions to second order elliptic partial dierential equations. The technique is very useful in many areas in mathematics, in particular in harmonic analysis, potential theory, partial dierential equations and mathematical physics. Across two volumes, the authors of Morrey Spaces: Introduction and Applications to Integral Operators and PDE's discuss the current state of art and perspectives of developments of this theory of Morrey spaces, with the emphasis in Volume I focused mainly on harmonic analysis. Features Provides a from-scratch' overview of the topic readable by anyone with an understanding of integration theory Suitable for graduate students, masters course students, and researchers in PDE's or Geometry Replete with exercises and examples to aid the reader's understanding
• Titolo autorizzato: Morrey spaces
• ISBN: 0-429-53202-4; 0-429-08592-3; 1-4987-6552-1
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910987815103321
• Lo trovi qui: Univ. Federico II