Triebel-Lizorkin spaces provide a framework that unifies classical function spaces such as Lp spaces, Hardy spaces, Lipschitz spaces, and BMO. In this thesis we will discuss boundedness of various linear operators on Triebel-Lizorkin spaces. This thesis contains results of the author from the papers [52] - [56].

In Chapter 1 we introduce the definition of Besov and Triebel-Lizorkin spaces and then study maximal inequalities of Fefferman-Stein type and Peetre type. These are key tools to study the boundedness of operators on the spaces. Particular emphasis is given to BMO-like spaces such as Fs,qinfinity, for which the classical definitions need to be changed.

In Chapter 2 we discuss characterizations of Besov and Triebel-Lizorkin spaces via differences and via oscillations. Our new results concern the case p = infinity for the Triebel- Lizorkin scale.

In Chapter 3 and Chapter 4 we deal with the boundedness of pseudo-differential operators of type (p,p), 0 < p ≤ 1 on (inhomogeneous) Besov spaces and Triebel- Lizorkin spaces. We first briefly explain preliminary results and then prove new sharp estimates of the operators. When p = infinity our new maximal inequality will be used.

In Chapter 5 we study sharp estimates of the classical Mikhlin-Hormander multiplier theorem on Fs,qp. In particular, we extend and improve results of Baernsten and Sawyer in [1].

Most of our new boundedness results are sharp. In the last part of the thesis we will discuss the sharpness of our several results. One of the key technique is a random construction in Christ and Seeger's paper [11].