BOUNDEDNESS OF INTRINSIC LITTLEWOOD-PALEY FUNCTIONS ON MUSIELAK-ORLICZ MORREY AND CAMPANATO SPACES

Let phi : R-n x [0, infinity) -> [0, infinity) be such that phi(x, center dot) is nondecreasing, phi(x, 0) = 0, phi(x, t) > 0 when t > 0, lim(t ->infinity) phi(x, t) = 1 and phi(center dot, t) is a Muckenhoupt A(infinity)(R-n) weight uniformly in t. Let phi : [0, infinity) -> [0, infinity) be nondecreasing. In this article, the authors introduce the Musielak-Orlicz Morrey spaceM(pi,phi) (R-n) and obtain the boundedness on M-pi,M-phi (R-n) of the intrinsic Lusin area function S-alpha, the intrinsic g-function g(alpha), the intrinsic g(lambda)*-function g(lambda, alpha)* and their commutators with BMO(R-n) functions, where alpha is an element of (0, 1], lambda is an element of (min{max{3, p(1)}, 3 + 2 alpha/n}, 1 infinity) and p(1) denotes the uniformly upper type index of phi. Let Phi : [0, infinity) -> [0, infinity) be nondecreasing, Phi(0) = 0, Phi(t) > 0 when t > 0, and lim(t ->infinity) Phi(t) = infinity, w 2 A infinity (R-n) and phi : (0, infinity) -> (0, infinity) be nonincreasing. The authors also introduce the weighted Orlicz-Morrey space M-w(Phi,phi) (R-n) and obtain the boundedness on M-w(Phi,phi) (R-n) of the aforementioned intrinsic Littlewood-Paley functions and their commutators with BMO(R-n) functions. Finally, for q is an element of [1, infinity), the boundedness of the aforementioned intrinsic Littlewood-Paley functions on the Musielak-Orlicz Campanato space L-phi,L- q(R-n) is also established.