Difficulties of convergence of the partial sums of the Fourier inversion formula

Question

Define the Fourier inversion formula over $\mathbb{R}^n$ by $$ f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\hat{f}d\xi $$ where $\hat{f}$ is the Fourier transform over $\mathbb{R}^n$ $$ \hat{f}(\xi)=\int_{\mathbb{R^n}}e^{-2\pi ix\cdot\xi}f(x)dx. $$ of $f$. Define the partial sums of $f$ as $$ S_Nf(x)=\int_{|\xi|\leq R}e^{2\pi ix\cdot \xi}\hat{f}(\xi)d\xi. $$ A famous question in harmonic analysis is the convergence of of the partial sums to $f$ in $L^p$ norm, specifically $$ S_Nf\to f~\text{in}~L^p~\text{norm}? $$ I have two questions:

$\textbf{Question 1}:$ Why is pointwise convergence so difficult especially in higher dimensions? I have studied papers whose lone purpose is to establish pointwise convergence using tools such as spherical means and spectral measure but I do not understand why pointwise convergence is so complicated.

$\textbf{Question 2}:$ Why is weak convergence trivial/no issue? What information is present that means we do not have to worry about weak convergence?

Thank you in advance for your comments and assistance.

Answer 1

Young's and Holder's inequalities make weak convergence easy to study, but the pointwise convergence depends on the behaviour of a maximal operator (the Carleson operator) for which it is not that easy to provide effective upper bounds. Carleson's greatest idea was probably to modify usual decomposition techniques in the Calderon-Zygmund theory in the following way:

1. Instead of using a spatial decomposition only, using a decomposition of the space-frequency plane in terms of dyadic tiles of unit area (that accounts for the Heisenberg indetermination principle);
2. Considering the translation, dilation and modulation operators;
3. Approximate the projection operator on the negative frequencies in terms of the previous operators and decomposition;
4. Provide effective bounds from combinatorial arguments.

To carry this plan on in full detail is not an easy task.

We may regard the Carleson's theorem as the Lebesgue differentiation theorem, in particular the weak $L^1$-estimates for the Hardy-Littlewood maximal operator, under steroids. Heavy steroids.

A recommended lecture is this brilliant piece of work of Lacey.