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