Maximal operators associated with Fourier multipliers

A Fourier multiplier operator is a linear operator given by multiplication with a bounded function on the Fourier transform side. This bounded function is called a symbol or a Fourier multiplier. A fundamental question in Harmonic Analysis is which bounded functions give rise to bounded Fourier multiplier operators on Euclidean spaces. The purpose of this dissertation is to study a related question, namely the boundedness of a maximal operator formed by the supremum of a family of Fourier multiplier operators.; We show that maximal operators formed by dilations of Mikhlin-Hormander multipliers need not to be bounded on a Lebesgue space with power integrability. Given a finite family of Mikhlin-Hormander multipliers, with uniform estimates, we prove an optimal norm estimate for the associated maximal operator and related bounds for maximal functions generated by dilations. We prove similar results for bilinear operators. We establish transference for maximal operators of multilinear symbols and use it to obtain some results on convergence of bilinear Fourier series.