Lecture notes

PDF Lecture 1: Maximal functions Locally integrable functions, the centred and uncentred Hardy–Littlewood maximal functions, and the comparison between them. Sep 2026 PDF Lecture 2: Marcinkiewicz interpolation Decomposing $L^p$ into $L^q + L^r$, linear and sublinear operators, weak and strong type, and the Marcinkiewicz interpolation theorem. Sep 2026 PDF Lecture 3: Lebesgue differentiation and approximate identities The Lebesgue differentiation theorem, the resulting $L^p$ bounds on the maximal function, and approximate identities. Sep 2026 PDF Lecture 4: Tempered distributions and Calderón–Zygmund The Schwartz class and tempered distributions, differentiation and convolution on $\mathcal{S}'$, the Fourier transform, translation-invariant operators, the Hilbert and Riesz transforms, and the Calderón–Zygmund decomposition. Sep 2026 PDF Lecture 5: Good and bad parts, and the weak (1,1) bound The Calderón–Zygmund decomposition proved by dyadic subdivision, splitting $f$ into a bounded good part and a mean-zero bad part, and the start of the weak $(1,1)$ estimate for convolution operators. Sep 2026 PDF Lecture 6: Weak (1,1), $L^p$ bounds, and Littlewood–Paley Finishing the weak $(1,1)$ estimate with the Hörmander condition, then Marcinkiewicz and duality for boundedness on $L^p$, $1<p<\infty$, and the Littlewood–Paley square function applied to lacunary Fourier series. Sep 2026 PDF Lecture 7: Littlewood–Paley theory and Khintchine A smooth dyadic partition of unity on the frequency side, Rademacher functions and Khintchine's inequalities, and randomising the pieces to turn the square function into a uniformly bounded family of Calderón–Zygmund operators. Sep 2026 PDF Lecture 8: The lower square function bound, and atoms Closing the Littlewood–Paley inequality from below by duality, using almost orthogonality of the dyadic blocks, then $L^2$ atoms and the atomic space $X$ on which a Calderón–Zygmund operator maps into $L^1$. Sep 2026 PDF Lecture 9: The Hardy space $H^1$ and BMO Four equivalent descriptions of $H^1$, by nontangential maximal function, atoms, Riesz transforms and the square function, then its dual: Fefferman and Stein's theorem that $(H^1)^* = \mathrm{BMO}$, with $\log|x|$ as the function that is of bounded mean oscillation without being bounded. Sep 2026 PDF Lecture 10: John–Nirenberg, and $H^1$–BMO duality The John–Nirenberg inequality proved by iterating the Calderón–Zygmund decomposition, the resulting equivalence of the $L^p$ and $L^1$ oscillation norms on BMO, and both halves of the pairing behind the Fefferman–Stein duality $(H^1)^* = \mathrm{BMO}$. Oct 2026 PDF Lecture 11: Carleson measures Measures on the upper half-space that charge a tent no more than its base, the Whitney decomposition extending the condition from cubes to open sets, and Carleson's theorem that this is exactly when $f \mapsto \Phi_t * f$ is bounded from $L^p(\mathbb{R}^d)$ into $L^p(\mu)$. Oct 2026 PDF Lecture 12: Carleson measures and BMO A second proof of the Carleson embedding theorem through the radially decreasing majorant, then the bridge to BMO: $|b*\psi_t|^2\,dx\,dt/t$ is a Carleson measure with constant $\|b\|_{\mathrm{BMO}}^2$, with the two lemmas that make $b*\psi_t$ converge at all. Oct 2026

PDF Seminar 1: Exercises Five exercises: the $M_c f \le Mf \le 2^d M_c f$ comparison, lower semicontinuity of $Mf$, $L^q + L^r \subseteq L^1_{loc}$, convergence of the averages, and the oscillation bound behind Lebesgue differentiation. Solutions on request. Sep 2026 PDF Seminar 2: Exercises Four exercises: the Minkowski integral inequality, radial decreasing majorants and a.e. convergence of approximate identities, harmonicity of the Poisson integral, and $\widehat{\mathrm{p.v.}(1/x)} = -i\pi\,\mathrm{sgn}$. Solutions on request. Sep 2026 PDF Seminar 3: Exercises Five exercises: pulling $T$ through the bad part, the enlarged cube inside the Hörmander region, derivative bounds on the Littlewood–Paley pieces, the Hörmander condition from kernel estimates, and completeness of the atomic space $X$. Solutions on request. Oct 2026