V4B5: Real and harmonic analysis
Summer Semester 2018
- Dr. Pavel Zorin-Kranich
- Dr. Olli Saari
- Tu 14-16, 1.008
- Th 14-16, 1.008
- Fr 8-10, 1.008
- Fr 12-14, 0.011
We have covered the following topics
- (weak) Lp spaces
- Hölder, Minkowski, and Young convolution inequalities
- Banach spaces, duality, Hahn-Banach theorem (in extension form and in separation form)
- Dual spaces of Lp spaces
- Hardy-Littlewood maximal operator (weak type and Lp estimates), Lebesgue differentiation
- Real interpolation
- Complex interpolation (Riesz, Stein, iterated Lp)
- Hardy-Littlewood-Sobolev inequality, Sobolev embedding
- Fourier transform
- Distributions, Schwartz space, tempered distributions
- Action on Gaussians, multiplication formula, inversion formula, Plancherel theorem, Hausdorff-Young inequality
- Poisson summation formula
- Heisenberg uncertainty principle
- Hilbert transform
- Calderón-Zygmund theory
- CZ decomposition
- Cotlar-Stein lemma, cancellative CZ kernels
- Calderón-Vaillancourt theorem on pseudodifferential operators
- Norm convergence of Fourier integrals in dimension 1
- Cotlar's inequality
- Mihlin-Hörmander multipliers
- Littlewood-Paley theory
- Cauchy integral on Lipschitz curves
- Adapted Haar basis, almost orthogonal expansion in L2
- Analytic capacity
- Bounded functions that are mapped to bounded functions by CZO
- Oscillatory integrals
- Fourier transform of surface-carried measures
- Lacunary spherical maximal function
- Hardy and BMO spaces
- Atomic decomposition
- Complex interpolation between H1 and Lp
- Spherical maximal theorem for d≥3
- H1-BMO duality
- John-Nirenberg inequality
- Div-curl lemma
- Sharp maximal function
- Perturbation of constant coefficient elliptic PDE
- Ball multiplier
- Sequence valued extensions of linear operators
- Perron tree
These notes provide a record of which topics we discuss.
Many details are omitted, but references are often provided.
Lebesgue measure and integration, functional analysis (Banach spaces and operators).
- Exercises 1, due on 17 April.
- Exercises 2, due on 24 April.
- Exercises 3, due on 3 May.
- Exercises 4, due on 8 May.
- Exercises 5, due on 15 May.
- Exercises 6, due on 29 May.
- Exercises 7, due on 5 June.
- Exercises 8, due on 12 June.
- Exercises 9, due on 19 June.
- Exercises 10, due on 26 June.
- Exercises 11, due on 3 July.
- Exercises 12, due on 10 July.
- Exercises 13, due on 17 July.
- E. M. Stein and R. Shakarchi,
Introduction to further topics in analysis.
- E. M. Stein,
Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals.
- Muscalu and Schlag,
Classical and multilinear harmonic analysis, Vol 1.
Classical Fourier Analysis.
- M. Christ,
Lectures on singular integral operators.
Lectures on harmonic analysis
- T. W. Körner,
Fourier analysis (interesting historical notes)