Scales of Banach Spaces, Theory of Interpolation and their Applications

Contents Preface / 7 Chapter 1 Fractional powers of operators / 11 Chapter 2 Interpolation spaces / 17 2.1. Spaces Dσp / 18 2.2. Definition of interpolation spaces S(p; θ;X; p; θ – 1; Y ) / 23 2.3. Complex interpolation space / 25 2.4. Another definition of interpolation spaces; Real interpolation space / 27 2.4.1. The K-method / 28 2.4.2. The trace method / 30 2.4.3. The Reiteration Theorem / 32 2.4.4. Some examples / 33 Chapter 3 Infinitesimal generators of semi-groups / 35 3.1. Infinitesimal generators of bounded semi-groups / 35 3.2. Infinitesimal generators of bounded analytic semi-groups / 36 Chapter 4 Scales of Banach Spaces / 39 4.1. Inductive Limits and Projective Limits of Sequences of Banach Spaces / 40 4.2. Regular Spaces and Hyper-spaces / 51 Chapter 5 Examples of scales of Banach spaces / 63 Chapter 6 Sectorial Operators / 71 6.1. Examples of Sectorial Operators / 73 Chapter 7 Applications / 79 Chapter 8 The abstract Cauchy problem / 89 8.1. Examples and applications / 90 Appendix A Theory of distributions and the Fourier transform / 99 A.1. Theory of distributions / 99 A.2. The Fourier transform of rapidly decreasing functions / 100 A.3. The Fourier transform of tempered distributions / 101 Bibliography / 103 Streszczenie / 107 Резюме / 107