Analysis in Banach Spaces

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