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Abstract
This thesis contains the results of the author from Roos (2021) (joint work with Joris Roos and Andreas Seeger), Rskor (2022), Srivastava (2020) and Srivastava (2020).
In Chapter 2, we prove Lp → Lq estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve bounds on sparse domination for global maximal operators. We also consider lacunary variants, and extensions to Métivier groups.
In Chapter 3, we prove Lp → Lq estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp Lp → Lq estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds in Ganguly (2021) for the associated lacunary maximal operator.
In Chapter 4, we exhibit the necessary range for which functions in the Sobolev spaces Lsp can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle-Lemarié wavelets. We also consider the natural extensions to Triebel-Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich (Seeger, 2017), where analogous results were established for the Haar wavelet system.
In Chapter 5, we prove the Lp boundedness of a Cotlar-type maximal operator associated with a dyadic frequency decomposition of a Fourier multiplier, under a weak regularity assumption.
