This thesis deals with sharp weighted estimates for classical operators in harmonic analysis. In the first chapter we study sharp weighted estimates in terms of “fattened” A_p characteristics. We present a general method that allows one to get from counterexamples for dyadic (martingale) models of the Hilbert transform to counterexamples for this operator itself. We apply this method to prove sharpness of one-weight estimates for the Hilbert transform in terms of “fattened” A_p characteristics. Moreover, a simpler application of the same general method allows one to construct a counterexample to the L^p analog of Sarason’s conjecture for all 1 < p < ∞. While our constructions rely heavily on the machinery developed by F. Nazarov for disproving Sarason’s conjecture, they are explicit, and do not involve the Bellman function method, unlike Nazarov’s aforementioned work. The results of this chapter had originally appeared in joint work of the author with S. Treil (2018). In the second chapter we prove that the two-weight estimates for the dyadic square function due to M. Lacey and K. Li (2016) are sharp from the point of view of the Muckenhoupt A_∞ characteristics of the involved weights. We use a family of examples introduced by A. Lerner (2006) for one of the two weights, and we construct examples inspired by counterexamples in nonhomogeneous settings due to J. Lai and S. Treil (2015) for the other weight. The results of this chapter had originally appeared in work by the author (2017). Finally, in the third chapter we continue the line of study of the relation between two-weight estimates for sparse square functions and the separated bump conjecture initiated by S. Treil and A. Volberg (2016). We show that two-weight L^2 bounds for sparse square functions (uniform with respect to sparseness constants, and in both directions) do not imply a two-weight L^2 bound for the Hilbert transform. We present an explicit counterexample, making use of a construction due to M. C. Reguera and C. Thiele (2012). At the same time, we show that our method fails in an essential way to disprove the separated bump conjecture itself. The results of this chapter had originally appeared in work by the author (2019).
Kakaroumpas, Spyridon,
"Sharp Weighted Estimates in Harmonic Analysis"
(2020).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/f4jw-3w60
