RG Analysis and Partial Differential Equations

V4B5: Real and harmonic analysis

Summer Semester 2018

Dr. Pavel Zorin-Kranich
Dr. Olli Saari


Exercise classes


We have covered the following topics
  1. (weak) Lp spaces
    1. Hölder, Minkowski, and Young convolution inequalities
    2. Banach spaces, duality, Hahn-Banach theorem (in extension form and in separation form)
    3. Dual spaces of Lp spaces
    4. Hardy-Littlewood maximal operator (weak type and Lp estimates), Lebesgue differentiation
    5. Real interpolation
    6. Complex interpolation (Riesz, Stein, iterated Lp)
    7. Hardy-Littlewood-Sobolev inequality, Sobolev embedding
  2. Fourier transform
    1. Distributions, Schwartz space, tempered distributions
    2. Action on Gaussians, multiplication formula, inversion formula, Plancherel theorem, Hausdorff-Young inequality
    3. Poisson summation formula
    4. Heisenberg uncertainty principle
    5. Hilbert transform
  3. Calderón-Zygmund theory
    1. CZ decomposition
    2. Cotlar-Stein lemma, cancellative CZ kernels
    3. Calderón-Vaillancourt theorem on pseudodifferential operators
    4. Norm convergence of Fourier integrals in dimension 1
    5. Cotlar's inequality
    6. Mihlin-Hörmander multipliers
  4. Littlewood-Paley theory
  5. Cauchy integral on Lipschitz curves
    1. Adapted Haar basis, almost orthogonal expansion in L2
    2. Analytic capacity
    3. Bounded functions that are mapped to bounded functions by CZO
  6. Oscillatory integrals
    1. Fourier transform of surface-carried measures
    2. Lacunary spherical maximal function
  7. Hardy and BMO spaces
    1. Atomic decomposition
    2. Complex interpolation between H1 and Lp
    3. Spherical maximal theorem for d≥3
    4. H1-BMO duality
    5. John-Nirenberg inequality
    6. Div-curl lemma
    7. Sharp maximal function
    8. Perturbation of constant coefficient elliptic PDE
  8. Ball multiplier
    1. Sequence valued extensions of linear operators
    2. Perron tree

Lecture notes

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).

Problem sets