Harmonic Analysis - 2016

重要課程資料:課程大綱
先備知識:實分析、富分析、廣義函數、基本偏微分方程。


Week 1: 1. Lebesgue’s differentiation theorem; 2. The Hardy-Littlewood maximal function; 3. Weak type (1,1) of the Hardy-Littlewood operator; 4. Covering lemma; 5. Strong type (p,p) of the Hardy-Littlewood maximal operator; 6. Proof of the Lebesgue’s differentiation theorem.
Week 2: 1. Marcinkiewicz interpolation theorem; 2. Weak type (p,q); 3. The Lebesgue set; 4. The regular family; 5. Point of density; 6. Approximation of the identity; 7. Pointwise convergence.
Week 3: 1. Fourier transform in L^p; 2. Plancherel theorem; 3. Riesz-Thorin interpolation theorem; 4. Young’s inequality; 5. Hausdorff-Young’s inequality; 6. Hadamard-Phragmen-Lindelof 3-line theorem; 7. Proof of Riesz-Thorin theorem.
Week 4: 1. Summability method of Fourier integrals; 2. Cesaro summability; 3. Abel-Poisson summability; 4. Gauss-Weierstrass summability; 5. Calderon-Zygmund decomposition; 6. Decomposition of open sets into cubes.
Week 5: 1. Dyadic maximal operators; 2. Dyadic Calderon-Zygmund decomposition; 3. The Hilbert transform; 4. Connection to the complex analysis; 5. Poisson and Conjugate Poisson kernels.
Week 6: 1. Characterization of L^2 bounded linear operators which commute with translations; 2. Fourier multipliers; 3. Characterization of the Hilbert transform; 4. Kolmogorov-Riesz Theorem; 5. Calderon-Zygmund decomposition.
Week 7: 1. Singular integral operators in higher dimensions; 2. Riesz transforms; 3. Directional Hardy-Littlewood maximal operators; 4. Directional Hilbert transforms; 5. Method of rotation.
Week 8: 1. Singular integrals with even kernels; 2. Connection with the partial differential operators; 3. Calderon-Zygmund singular integral operators (convolution type).
Week 9: 1. Calderon-Zygmund convolution operators; 2. Hormander’s condition; 3. Strong (p,p), weak (1,1) (proved using Calderon-Zygmund decomposition); 4. Dini’s type condition implies Hormander’s condition; 5. Truncated kernel.
Week 10: No classes.
Week 11: 1. Truncated kernel; 2. Singular integral operators defined by the limit the truncated kernels; 3. Generalized Calderon-Zygmund singular integral operators; 4. Standard kernels; 5. Schwartz kernel theorem.
Week 12: 1. Cauchy integral along a Lipschitz curve; 2. Plemelj’s formulas; 3. Calderon commutators; 4. Solving a differential equation; 5. Truncated kernels; 6. Calderon-Zygmund singular integrals; 7. Cotlar’s inequality.
Week 13: 1. Cotlar’s inequality; 2. Pointwise convergence of the truncated operator; 3. H^1 and atomic H^1; 4. Calderon-Zygmund operators acting on H^1; 5. BMO (bounds mean oscillation); 6. Equivalent characterization of BMO; 7. log|x| belongs to BMO.
Week 14: 1. Calderon-Zygmund operators acting on bounded functions; 2. An interpolation result involving BMO; 3. Good-lambda inequality.
Homework due on May 31.
Week 15: 1. John-Nirenberg inequality; 2. Almost orthogonality; 3. Cotlar’s lemma; 4. Schur’s lemma; 5. Application of Cotlar’s lemma.
Week 16: David-Journe T1 theorem.