Low-dimensional maximal restriction principles for the Fourier transform

Following the ideas from a paper by the same author, we prove abstract maximal restriction results for the Fourier transform. Our results deal mainly with maximal operators of convolution-type and $r-$average maximal functions. As a by-product of our techniques we obtain spherical maximal restriction estimates, as well as restriction estimates for $2-$average maximal functions, answering thus points left open by V. Kova\v{c} and M\"uller, Ricci and Wright.