EN
CONTENTS
I. Introduction
1. The Marcinkiewicz interpolation theorem..................................... 5
2. The classical results: Theorem B.......................................................... 11
3. The classical results: Theorem C......................................................... 13
4. Remarks..................................................................................................... 21
II. Preliminaries
5. Lorentz spaces; operators of strong and weak types (p, g)...... 22
6. Variations on Hardy's inequalities......................................................... 24
7. Averaging operators $A_p$, $B_p$, $G_p$ and $D_p$................... 28
III. The Lorentz-Zygmund spaces
8. Lorentz-Zygmund spaces Lila{logL)a.......................................... 29
9. Inclusion relations.................................................................................... 30
10. The classical function spaces............................................................. 34
11. The auxiliary spaces $ℒ^{pa}(logℒ)^a$ and $M^{pa}(logM)^a$... 38
12. The embedding theorem....................................................................... 42
IV. Operators of weak type (p, q; r, s)
13. Definition and elementary properties.......................................... 43
14. The Fourier transform............................................................................. 49
15. The Hardy-Littlewood maximal operator............................................. 49
16. The (maximal) Hilbert transform........................................................... 50
17. The fractional integrals........................................................................... 58
V. Proof of the main results
18. Proof of Theorems B and C........................................................... 59
19. Connections with interpolation space theory..................................... 62
References........................................................................................................... 65