The John-Nirenberg inequality characterizes functions in the space BMO in terms of the decay of the distribution function of their oscillations over a cube. In the first part of this thesis, separate necessary and sufficient John-Nirenberg type inequalities are proved for functions in the space [Special characters omitted.] . The results are a modified version of the conjecture made by Essén, Janson, Peng and Xiao, who introduce the space [Special characters omitted.] . The necessity for this modification is shown by two counterexamples.  The counterexamples provide us with a borderline case function for [Special characters omitted.] . In the second part, the discussion on the relation between the function and the space Q {460} in a wider range is presented. Moreover, the analytic and fractal properties of the function are studied and the fractal dimensions of the graph of the function are determined. These properties and dimensions illustrate some form of regularity for functions in [Special characters omitted.] .  Lastly, the relation between the tent spaces [Special characters omitted.] and L p , H p , and BMO for q ✹ 2 is discussed. By the theory of Triebel-Lizorkin spaces, a projection from [Special characters omitted.] to L p for 1 < p < {592}, to H p for p {600} 1, and to BMO for p = {592}, is shown to exist for 1 < q {600} 2.