# Analysis Seminar/Third Year Requirement: "Modulation Invariant operators near L^1"

*Abstract: It is a natural question whether the partial Fourier sums of a function in $ L^2(\mathbb{T}) $ converge almost everywhere to $ f $. In order to study this question one defines and studies the corresponding maximal operator $ C(f)(x)=\displaystyle{\sup_{N \in \N} | S_N(f)(x)|} $ which is called the Carleson operator.In this talk we will briefly talk about the seminar approach of Lacey and Thiele in the Walsh model.*
The Carleson operator is well known to be bounded on $L^p(\mathbb T)$ and a natural thing to ask whether we have $ \| Cf \|_{1,\infty} \lesssim \| f \|_{L^{\Phi(\mathbb T)}}$, where $\Phi$ is an Orlicz function so that $L^1(\mathbb T) \supsetneq L^{\Phi(\mathbb T)} \supsetneq L^p(\mathbb T)$. A standing conjecture,the so called " $L \log L$ conjecture", states that the above result holds for the space $ L \log L(\mathbb T)$. Weakened versions of this conjecture are estimates on the behavior of $ \displaystyle{\| C \|_{L^p \to L^{p,\infty}}} $ .

*Host: John McCarthy*