The method of orbits in interpolation theory

Front cover......Page 1
Mathematical Reports......Page 2
Title......Page 3
Series......Page 4
Title page......Page 5
Contents......Page 7
Editor's Introduction......Page 11
Introduction......Page 13
1.1 Definition of a Banach couple......Page 19
1.2 Examples of Banach couples......Page 21
1.3 The sum and the intersection of the spaces of a couple......Page 24
1.4 Mappings of Banach couples......Page 25
1.5 Ideals of maps of Banach couples......Page 28
1.6 Orbits of elements. Orbits and coorbits of spaces......Page 32
1.7 Interpolation functors. The characteristic function of an interpolation functor......Page 34
1.8 Examples of interpolation functors......Page 36
§2 Duality for orbits and coorbits......Page 39
3.1 Elementary properties of the $K$-functional......Page 46
3.2 The $K$-functional and nonlinear bounded operators......Page 50
3.3 The functors of orbits and the $K$-functional......Page 53
3.4 The computation of the $K$-functional in special Banach couples......Page 54
4.1 The notion of a generating element of a function......Page 60
4.2 The classification of weight sequences......Page 63
5.1 The classical $K$-monotone couples......Page 71
5.2 Interpolation from the couple $L_1(\bar{w})$ into an arbitrary Banach couple......Page 73
5.3 Interpolation from the couple $\{L_{p_0}(w_0), L_{p_1}(w_1)\}$ into $\{L_{q_0}(\sigma_0), L_{q_1}(\sigma_1)\}$......Page 80
5.4 $K$-monotone couples and the Fourier transform......Page 81
§6 Orbital equivalence......Page 83
7.1 The $K$-method......Page 85
7.2 The $J$-method......Page 94
7.3 The connection between $K$- and $J$-methods......Page 96
7.4 Reiteration for the $K$- and $J$-methods......Page 99
7.5 Duality between $K$- and $J$-methods......Page 102
7.6 Special cases of $K$- and $J$-methods......Page 105
7.7 A reiteration theorem connected with functions of the type $P^{+-}$......Page 111
8.1 Definition of functors......Page 113
8.2 The Calderon construction of an intermediate space for a couple of Banach lattices......Page 116
8.3 Special properties of the functors $G_1$, $G_2$, $G_3$......Page 128
8.4 Dual constructions......Page 134
8.5 Computation of the spaces of the dual constructions......Page 138
8.6 Reiteration theorems for functors $G$ and $H$......Page 146
8.7 Functors commuting with infinite direct products......Page 149
8.8 Duality between the functors $G$ and $H$......Page 151
§9 Orbital description of the complex method......Page 153
10.1 The orbits of coherently nuclear operators......Page 160
10.2 The interpolation orbits of the ideal $AS_\ast$......Page 162
10.3 Orbits of ideals connected with the couple $\{L_1, L_\infty\}$......Page 164
11.1 Interpolation of operators in the ideals $\mathfrak{G}_p$......Page 169
11.2 Interpolation of nuclear operators connected with Hilbert couples......Page 172
11.3 Some applications......Page 174
11.4 The description of all Hilbert interpolation spaces for Hilbert couples......Page 175
References......Page 176
Copyright......Page 180
Back cover......Page 181