Chapter 1 Introduction ......................................... 1
Chapter 2 Notation and preliminaries ........................... 5
2.1 Spaces of homogeneous type ................................. 5
2.2 Assumptions ................................................ 6
2.3 The classical Hardy space H1( n) ........................... 6
2.4 Hardy spaces via atoms ..................................... 7
2.5 Hardy spaces via molecules ................................. 7
2.6 Hardy spaces via square and maximal functions .............. 8
2.7 BMO spaces associated to operators ......................... 9
2.8 Historical notes .......................................... 11
Chapter 3 Davies-Gaffney estimates ............................ 13
3.1 Self-improving properties of Davies-Gaffney estimates ..... 13
3.2 Finite speed propagation for the wave equation and
Davies-Gaffney estimates .................................. 15
Chapter 4 The decomposition into atoms ........................ 17
4.1 Strategy of the proof of Theorem 4.1 ...................... 17
4.2  .......................... 19
4.3 The inclusion  ............... 21
4.4 Equivalence of  ................ 25
4.5 Inclusion among the spaces .... 27
Chapter 5 Relations between atoms and molecules ............... 31
Chapter 6 BMOL,m(X): Duality with Hardy spaces ................. 41
Chapter 7 Hardy spaces and Gaussian estimates ................. 45
7.1 Hardy spaces H1L,at,M(X), H1L,Sh(X) and H1L,Sp(X) and
Gaussian estimates ........................................ 45
7.2 Hardy spaces via maximal functions ........................ 47
7.3 The spaces BMOL(X) under Gaussian bounds .................. 50
Chapter 8 Hardy spaces associated to Schrodinger operators .... 53
8.1 Equivalences among H1L,at,M(n), H1L,Sh(n) and H1L,Sp(n) .... 53
8.2 Maximal characterization of H1L,at,M(n) .................... 54
8.3 H1L,at,M → H1 bounds for Riesz transforms of
Schrodinger operators ..................................... 61
Chapter 9 Further properties of Hardy spaces associated to
operators ........................................... 65
9.1 The semigroup with the conservation property .............. 65
9.2 Hardy spaces HPL(X) for 1 ≤ p < ∞ ......................... 66
Bibliography ................................................... 75
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