Analysis in Banach Spaces. Volume III: Harmonic Analysis and Spectral Theory

Preface
Contents
Symbols and notations
Standing assumptions
11 Singular integral operators
	11.1 Local oscillations of functions
		11.1.a Sparse collections and Lerner\'s formula
		11.1.b Almost orthogonality in Lp
		11.1.c Maximal oscillatory norms for Lp spaces
		11.1.d The dyadic Hardy space and BMO
	11.2 Singular integrals and extrapolation of Lp0 bounds
		11.2.a Calder on{Zygmund decomposition and case  (1, p0)
		11.2.b Local oscillations of p (p0, ∞)
		11.2.c The action of singular integrals on L∞
	11.3 Calder on{Zygmund operators and sparse bounds
		11.3.a An abstract domination theorem
		11.3.b Sparse operators and domination
		11.3.c Sparse domination of Calder on{Zygmund operators
		11.3.d Weighted norm inequalities and the A2 theorem
		11.3.e Sharpness of the A2 theorem
	11.4 Notes
		Section 11.1
		Section 11.2
		Section 11.3
		Further results
		A summary of sharp weighted bounds for classical operators
12 Dyadic operators and the T(1) theorem
	12.1 Dyadic singular integral operators
		12.1.a Haar multipliers
		12.1.b Nested collections of unions of dyadic cubes
		12.1.c The elementary operators of Figiel
	12.2 Paraproducts
		12.2.a Necessary conditions for boundedness
		12.2.b Su cient conditions for boundedness
		12.2.c Symmetric paraproducts
		12.2.d Mei\'s counterexample: no simple su cient conditions
	12.3 The The T(1) theorem for abstract bilinear forms
		12.3.a Weakly de ned bilinear forms
		12.3.b The BCR algorithm and Figiel\'s decomposition
		12.3.c Figiel\'s T(1) theorem
		12.3.d Improved estimates via random dyadic cubes
	12.4 The T(1) theorem for singular integrals
		12.4.a Consequences of the T(1) theorem
		12.4.b The dyadic representation theorem
	12.5 Notes
		Section 12.1
		Section 12.2
		Section 12.3
		Section 12.4
		(1) theorems on other function spaces
13 The Fourier transform and multipliers
	13.1 Bourgain\'s theorem on Fourier type
		13.1.a Hinrichs\'s inequality: breaking the trivial bound
		13.1.b The nite Fourier transform and sub-multiplicativity
		13.1.c Key lemmas for an initial uniform bound
		13.1.d Conclusion via duality and interpolation
	13.2 Fourier multipliers as singular integrals
		13.2.a Smooth multipliers have Calder on{Zygmund kernels
		13.2.b Mihlin multipliers have H ormander kernels
	13.3 Necessity of UMD for multiplier theorems
	13.4 Notes
		Section 13.1
		Section 13.2
		Section 13.3
14 Function spaces
	14.1 Summary of the main results
	14.2 Preliminaries
		14.2.a Notation
		14.2.b A density lemma and Young\'s inequality
		14.2.c Inhomogeneous Littlewood{Paley sequences
	14.3 Interpolation of Lp-spaces with change of weights
		14.3.a Complex interpolation
		14.3.b Real interpolation
	14.4 Besov spaces
		14.4.a De nitions and basic properties
		14.4.b Fourier multipliers
		14.4.c Embedding theorems
		14.4.d Di erence norms
		14.4.e Interpolation
		14.4.f Duality
	14.5 Besov spaces, random sums, and multipliers
		14.5.a The Fourier transform on Besov spaces
		14.5.b Smooth functions have R-bounded ranges
	14.6 Triebel–{Lizorkin spaces
		14.6.a The Peetre maximal function
		14.6.b De nitions and basic properties
		14.6.c Fourier multipliers
		14.6.d Embedding theorems
		14.6.e Di erence norms
		14.6.f Interpolation
		14.6.g Duality
		14.6.h Pointwise multiplication by 1
	14.7 Bessel potential spaces
		14.7.a General embedding theorems
		14.7.b Embedding theorems under geometric conditions
		14.7.c Interpolation
		14.7.d Pointwise multiplication by 1
	14.8 Notes
		Section 14.2
		Section 14.3
		Section 14.4
		Section 14.5
		Section 14.6
		Function spaces on domains and extension operators
		Weighted function spaces
		Lp–Lq-multipliers
15 Extended calculi and powers of operators
	15.1 Extended calculi
		15.1.a The primary calculus
		15.1.b The extended Dunford calculus
		15.1.c Extended calculus via compensation
	15.2 Fractional powers
		15.2.a De nition and basic properties
		15.2.b Representation formulas
	15.3 Bounded imaginary powers
		15.3.a De nition and basic properties
		15.3.b Identi cation of fractional domain spaces
		15.3.c Connections with sectoriality
		15.3.d Connections with almost γ-sectoriality
		15.3.e Connections with γ-sectoriality
		15.3.f Connections with boundedness of the H1-calculus
		15.3.g The Hilbert space case
		15.3.h Examples
	15.4 Strip type operators
		15.4.a Nollau\'s theorem
		15.4.b Monniaux\'s theorem
		15.4.c The Dore–Venni theorem
	15.5 The bisectorial H∞-calculus revisited
		15.5.a Spectral projections
		15.5.b Sectoriality versus bisectoriality
	15.6 Notes
		Section 15.1
		Section 15.2
		Section 15.3
		Section 15.4
		Section 15.5
		The Kato square root problem
16 Perturbations and sums of operators
	16.1 Sums of unbounded operators
	16.2 Perturbation theorems
		16.2.a Perturbations of sectorial operators
		16.2.b Perturbations of the H∞-calculus
	16.3 Sum-of-operator theorems
		16.3.a The sum of two sectorial operators
		16.3.b Operator-valued H∞-calculus and closed sums
		16.3.c The joint H∞-calculus
		16.3.d The absolute calculus and closed sums
		16.3.e The absolute calculus and real interpolation
	16.4 Notes
		Section 16.1
		Section 16.2
		Scales of fractional domain spaces and interpolation
		Section 16.3
		Sums of non-commuting operators
17 Maximal regularity
	17.1 The abstract Cauchy problem
	17.2 Maximal Lp-regularity
		17.2.a De nition and basic properties
		17.2.b The initial value problem
		17.2.c The role of semigroups
		17.2.d Uniformly exponentially stable semigroups
		17.2.e Permanence properties
		17.2.f Maximal continuous regularity
		17.2.g Perturbation and time-dependent problems
	17.3 Characterisations of maximal Lp-regularity
		17.3.a Fourier multiplier approach
		17.3.b The end-point cases p = 1 and p = ∞
		17.3.c Sum-of-operators approach
		17.3.d Maximal Lp-regularity on the real line
	17.4 Examples and counterexamples
		17.4.a The heat semigroup and the Poisson semigroup
		17.4.b End-point maximal regularity versus containment of c0
		17.4.c Analytic semigroups may fail maximal regularity
	17.5 Notes
		Section 17.1
		Section 17.2
		Section 17.3
		Section 17.4
		Maximal Lp-regularity for non-autonomous equations
		Miscellaneous topics
18 Nonlinear parabolic evolution equations in critical spaces
	18.1 Semi-linear evolution equations with F = FTr
	18.2 Local well-posedness for quasi-linear evolution equations
		18.2.a Setting
		Assumption 18.2.2.
		De nition 18.2.3 (Criticality).
		De nition 18.2.5.
		18.2.b Main local well-posedness result
		Theorem 18.2.6 (Local well-posedness for quasi-linear problems).
		18.2.c Proof of the main result
		Lemma 18.2.7.
		Lemma 18.2.8.
		Lemma 18.2.9.
		Lemma 18.2.10 (Smallness).
		Lemma 18.2.12 (Lipschitz estimates).
		18.2.d Maximal solutions
		De nition 18.2.13.
		Theorem 18.2.14 (Maximal solutions).
		Theorem 18.2.15 (Global well-posedness for quasi-linear equations).
		Theorem 18.2.17 (Global well-posedness for semi-linear equations).
	18.3 Examples and comparison
	18.4 Long-time existence for small initial data and F = Fc
	18.5 Notes
Q Questions
K Measurable semigroups
	K.1 Measurable semigroups
	K.2 Uniform exponential stability
	K.3 Analytic semigroups
	K.4 An interpolation result
	K.5 Notes
L The trace method for real interpolation
	L.1 Preliminaries
	L.2 The trace method
	L.3 Reiteration
	L.4 Mixed derivatives and Sobolev embedding
		L.4.a Results for the half-line
		L.4.b Extension operators
		L.4.c Results for bounded intervals
	L.5 Notes
References
Index