Vector-Valued Extensions For Singular Bilinear Operators And Applications
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 L2 , 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 Calderon-Zygmund decomposition which yields L1 [RIGHTWARDS ARROW] L1,[INFINITY] esti- mates, our "localization" method is useful for proving L p [RIGHTWARDS ARROW] L p , when p [GREATER-THAN OR EQUAL TO] 2. In both cases, the general result follows by duality.
vector-valued paraproducts; vector-valued bilinear Hilbert transform; Leibniz rule
Strichartz,Robert Stephen; Berest,Yuri
Ph. D., Mathematics
Doctor of Philosophy
dissertation or thesis