Cristina Benea

The problems presented in this thesis were motivated by the study of a Rubio de Francia operator for iterated Fourier integrals associated to arbitrary intervals. This further led to vector-valued estimates for the bilinear Hilbert transform $BHT$. The vector spaces can be iterated $l^p$ or $L^p$ spaces, and whenever all these are locally in $L^2$, we recover the $BHT$ range. This is illustrated in Chapter 4.

The methods of the proof apply for paraproducts as well, as seen in Chapter 5. We prove boundedness of vector-valued paraproducts, within the same range as scalar paraproducts.

In Chapter 6, we present a few consequences: the boundedness of the initial Rubio de Francia operator for iterated Fourier integrals, the boundedness of tensor products of $n$ paraproducts and one $BHT$ in $L^p$ spaces, and new estimates for tensor products of bilinear operators in $L^p$ spaces with mixed norms. Since paraproducts act as mollifiers for products of functions, possibly the most important application is a new Leibniz rule in mixed norm $L^p$ spaces.

A Rubio de Francia theorem for paraproducts is described in Chapter 3. The approach is completely different from the more abstract vector-valued method, and it is an instance where maximal paraproducts appear.

Finally, in Chapter 7 we employ our methods for re-proving vector-valued Carleson operator estimates, as well as estimates for the square function. As opposed to the Calderón–Zygmund decomposition which yields $L^1 \to L^{1,\infty}$ estimates, our ``localization" method is useful for proving $L^p \to L^p$, when $p \geq 2$. In both cases, the general result follows by duality.