This paper provides equivalence characterizations of homogeneous
Triebel-Lizorkin and Besov-Lipschitz spaces, denoted by
FΛp,qsβ(Rn) and BΛp,qsβ(Rn)
respectively, in terms of maximal functions of the mean values of iterated
difference. It also furnishes the reader with inequalities in
FΛp,qsβ(Rn) in terms of iterated difference and in terms of
iterated difference along coordinate axes. The corresponding inequalities in
BΛp,qsβ(Rn) in terms of iterated difference and in terms of
iterated difference along coordinate axes are also considered. The techniques
used in this paper are of Fourier analytic nature and the Hardy-Littlewood and
Peetre-Fefferman-Stein maximal functions