MA8106 Harmonisk analyse, våren 2024

Final lecture on Tuesday 23 April. No lecture on Friday.

• Duality of Hardy spaces
• An introduction to bounded mean oscillation (BMO). See Garnett Ch. VI for further reading.

Lectures on Tuesday 16 April and Friday 19 April, based on Bounded Analytic Functions, by J. B. Garnett (G.), chapters G. II.4-II.5:

• Outer functions
• Inner functions
• The inner-outer factorization of the Hardy space
• Associated exercises from Garnett: Ch. II: 1, 2(b), 5(a), 5(b), 8

Lectures on Tuesday 9 April and Friday 12 April, based on Bounded Analytic Functions, by J. B. Garnett (G.), chapters G. II.1-II.2:

• Harnack's principle for harmonic functions
• Harmonic majorants
• Blaschke products
• Jensen's formula
• The zeros of Hardy space functions

Lectures on Tuesday 2 April and Friday 5 April:

• Exercises from Muscalu–Schlag: Ex. 3.4, Problems 3.1, 3.5*, 3.6, 3.10, 3.12.
• Two definitions of analytic Hardy spaces.
• The F. & M. Riesz theorem on H1-functions.
• Riesz projection
• The The F. & M. Riesz theorem on integrability of log|f|

No lectures in week 12 (18-22 March), due to the faculty allmøte and the department work environment study meeting.

Lectures on Tuesday 12 March and Friday 15 March:

• Kernel representation of the Hilbert transform.
• Endpoint estimates (i.e. L1 and Linfty).
• Subharmonic functions
• Nontangential maximal function
• The F. and M. Riesz theorem
• Associated exercises from Muscalu–Schlag: Ex. 3.4, Problems 3.1, 3.5*, 3.6, 3.10, 3.12.
• *This problem is very challenging. Hint: Calderon-Zygmund-type decomposition w.r.t. the measure. Use the Besicovitch covering lemma (problem 2.5).

Exercise Session on Friday 8th March 2024. 12:15–14:00. Room 656 Sentralbygg 2. Resolution of the exercises in the attached list Exercise4.pdf. Solution to Exercise 3 from List 4, and Exercise 7 from List 2 can be found in Blackboard: Other new content/Undervisningsmateriell.

Lecture on Tuesday 5th March 2024. 12:15–14:00: We have seen:

• Proof of the Calderón-Zygmund lemma. Lemma 2.17, Chapter 2, Muscalu-Schlag.
• Further properties for the dyadic and usual maximal function, see Exercises 2.10 and 2.12.
• The (p,p) strong weighted estimate for A_p. Theorem 2.19.

New exercise list, which we will discussed on Friday 8th March. Exercise4.pdf

Exercise Session Lecture on Friday 1st March 2024. 12:15–14:00. Room 656 Sentralbygg 2. We have seen:

• Further properties of A_p weights. Another characterization of A_p; properties (2.17) and (2.18).
• The (p,p) weak weighted estimate for A_p. Theorem 2.16.
• The Calderón-Zygmund decomposition of a function. Lemma 2.17.

Lecture on Tuesday 27th February 2024. 12:15–14:00: We have seen:

• End of the proof of L^p convergence of Fourier sums. Theorem 3.20 in Chapter 3, Muscalu-Schlag.
• Doubling measures and Muckenhoupt A_p weights. Section 2.5, Proposition 2.14, and p. 43.
• The problem of characterizing the maximal weighted inequalities. Subsection 2.5.2.
• Basic properties of A_p weights. Exercise 2.7, Proposition 2.15, Exercise 2.9.

Exercise Session Lecture on Friday 23rd February 2024. 12:15–14:00. Room 656 Sentralbygg 2. We have seen:

• Weak (1,1) estimate for the Hilbert transform. Corollary 3.16 in Chapter 3, Muscalu-Schlag.
• The (p,p) strong estimate for the Hilbert transform. Theorem 3.17.
• L^p convergence of the Fourier Series. Theorem 3.20.

Lecture on Tuesday 20th February 2024. 12:15–14:00: We have seen:

• Basic properties of harmonic conjugates. Lemma 3.11 in Chapter 3, Muscalu-Schlag.
• Fourier coefficients and "little-hardy"-properties of harmonic conjugates. Lemma 3.12, Proposition 3.13 and Corollary 3.14.
• The radial maximal function and the corresponding weak estimate for h^1 functions. Definition 3.6 and the subsequent comment.
• Weak type estimate for the radial maximal function of the harmonic conjugate. Theorem 3.15.
• The Hilbert transform. Beginning of Section 3.5.

Exercise Session Lecture on Friday 16th February 2024. 12:15–14:00. Room 656 Sentralbygg 2. We have seen:

• Almost everywhere convergence for convolution of functions. Theorem 2.12 from Muscalu-Schlag.
• Almost everywhere convergence for convolution of measures. Lemma 2.13 and Exercise 2.6.
• The harmonic conjugate: definition and basic properties. Definition 3.10 and Lemma 3.11 in Chapter 3.

Lecture on Tuesday 13th February 2024. 12:15–14:00: We have seen:

• Maximal function inequalities for functions and measures. Section 2.3, Chapter 2, from Muscalu-Schlag.
• Radially bounded approximate identities. Definition 2.10.
• Pointwise estimates via the maximal function. Lemma 2.11.

Exercise Session on Friday 9th February 2024. 12:15–14:00. Room 656 Sentralbygg 2: Resolution of the exercises in the attached list Exercise3.pdf.

Lecture on Tuesday 6th February 2024. 12:15–14:00: We have seen:

• Review basic properties of Poisson kernel (without proof). Pp. 30–33, Chapter 2, from Muscalu-Schlag.
• The little hardy spaces h^p of harmonic functions. See Definition 2.4.
• Isometries between h^1 and complex measures, and between h^p and L^p. Theorem 2.5, Lemma 2.6, and Lemma 2.7.

Exercise Session on Friday 2nd February 2024. 12:15–14:00. Room 656 Sentralbygg 2: Resolution of exercises 3,5,6,7 in the list ExerciseWeek2.pdf. See the updated hints in Blackboard: Other new content/Undervisningsmateriell.

Lecture on Tuesday 30th January 2024. 12:15–14:00: We have seen:

• Proof of Marcinkiewicz theorem. Theorem 1.17 in Muscalu-Schlag.
• The Riesz-Thorin interpolation theorem (without proof). See Theorem 1.6.
• Harmonic functions. Definition and basic properties: the mean value property and the maximum principle. Chapter 2, Section 2.1: Lemma 2.1 and Corollary 2.3.
• The Poisson kernel. Definition and motivation. Section 2.2.

Exercise Session on Friday 26th January 2024. 12:15–14:00. Room 656 Sentralbygg 2: Resolution of the exercises in the attached list ExerciseWeek2.pdf.

Lecture on Tuesday 23rd January 2024. 12:15–14:00: We have seen:

• The Wiener Algebra. Motivation and basic properties. pp. 17-19 in Muscalu-Schlag.
• Embedding of Hölder and H^s classes into the Wiener algebra. Theorem 1.13.
• Failure in the extremal case C^{1/2}. Proposition 1.14.
• Interpolation of operators between L^p spaces. Recalling the L^{p,weak} spaces. First example: the Hardy-Littlewood maximal function.
• The Marcinkiewicz interpolation theorem. Theorem 1.17.

PLEASE NOTE: From now on, we reschedule the Wednesday's sessions to Fridays 12:15–14:00. Room 656 Sentralbygg 2. The sessions on Tuesdays remain the same: Tuesday 12:15–14:00 . F3 Gamle fysikk.

Exercise Session on Wednesday 17th January 2024. 10:15–12:00: Resolution of the exercises in the attached list ExerciseWeek1.pdf.

Lecture on Tuesday 16th January 2024. 12:15–14:00: We have seen:

• Lack of convergence of Fourier series in L^1 and in the space of continuous functions. Corollary 1.10 in Muscalu-Schlag.
• A positive result: uniform convergence of Fourier series for Hölder-continuous functions. Statement in Theorem 1.2 (and proof from p.12, using approximation via Féjer kernel). We also note the rate of decay O(N^{-\alpha}) for \alpha-Hölder functions; see the beginning of Subsection 1.4.4 (pp. 17-18).
• The de la Vallée Poussin kernel. Definition and basic properties, in particular, a bound for the L^1 norm of the derivatives V_N' of V_N which depends linearly on N. See the proof of Proposition 1.11.
• The Bernstein Inequality for trigonometric polynomials. Proposition 1.11.

Lecture on Wednesday 10th January 2024. 10:15–12:00: Continuing on the previous lecture, we have seen:

• Properties of convergence for approximate identities (for continuous or L^p functions, and for measures). A bonus property that holds for the Féjer kernel (meaning that when the approximate identity is the Féjer kernel, and an L^1 function "f" is continuous at a point "t", then the convolution of this kernel at "t" converges to "f(t)", without assuming that "f" is bounded on the torus). Proposition 1.5 in Muscalu-Schlag.
• Corollaries: The Riemann-Lebesgue lemma; the identity theorem; density of trigonometric polynomials. Corollaries 1.8, 1.7, and 1.6(i).
• Convergence in L^2. Corollary 1.6.
• A criteria for convergence in L^p and uniform convergence, based on the Banach-Steinhaus theorem. Proposition 1.9.

Lecture on Tuesday 9th January 2024. 12:15–14:00: An introductory lecture, where we have seen:

• Definition of integral convolution and the related L^p-preserving properties. Also, that convolution "preserves" differentiability, provided those functions (and its derivatives) are bounded. Subsection 1.1.2 and 1.2.1 in Muscalu-Schlag.
• Fourier coefficients and Fourier sums. Subsection 1.1.1.
• Dirichlet and Féjer kernels: Definitions, main properties, trigonometric expressions, and pointwise and L^1 estimates. Subsection 1.1.2 and 1.2.1.
• Definition of an approximate identity. Definition 1.3.

References: Chapter 1 from the monograph: Muscalu C, Schlag W.; Classical and Multilinear Harmonic Analysis Volume 1, Cambridge University Press; 2013. For basic Fourier Analysis and convolution; see chapters 2.3 in Stein, E. M.; Shakarchi, R.; Fourier Analysis: An Introduction. Princeton University Press: 2003.