Sparse technology in weighted harmonic analysis
This dissertation is the union of two major results that we discovered during my graduate time. Both of the results involve what are called sparse families of cubes. There will also be some partial results included though they were not submitted to any peer-reviewed journal. In this work, we first provide an example that directly disproves the Muckenhoupt-Wheeden conjectures for sparse operators. More specifically, we will construct a pair of weights $(u,v)$ for which the Hardy-Littlewood maximal function is bounded from $L^p(v)$ to $L^p(u)$ and from $L^{p'}(u^{1-p'})$ to $L^{p'}(v^{1-p'})$ while a dyadic sparse operator is not bounded on the same domain and range. Our construction also provides an example of a single weight for which the weak-type endpoint does not hold for sparse operators. Secondly, we prove several weighted estimates for bilinear fractional integral operators and their commutators with BMO functions. We also prove a maximal function control theorem for these operators, that is, we prove that the weighted $L^p$ norm is bounded by the weighted $L^p$ norm of a natural maximal operator when the weight belongs to $A_\infty$. The heart of the proofs is centered around a technique that transfers the considered operator to its associated sparse operator. As a corollary we are able to obtain new weighted estimates for the bilinear maximal function associated to the bilinear Hilbert transform.
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