Boundedness of intrinsic Littlewood--Paley functions on Musielak--Orlicz Morrey and Campanato spaces - Liang, Yiyu; Nakai, Eiichi; Yang, Dachun; Zhang, Junqiang (Banach journal of mathematical analysis, 2014)
Let [theta]: Rnx[0,[infinity])[rightarrow][0,[infinity]) be such that [theta](x,[dot]) is nondecreasing, [theta](x,0) = 0,[theta](x,t)>0 when t>0, lim t[rightarrow][infinity][theta](x,t) = [infinity] and [theta]([dot],t) is a Muckenhoupt A[infinity](Rn) weight uniformly in t. Let [theta]: [0,[infinity])[rightarrow][0,[infinity]) be nondecreasing. In this article, the authors introduce the MusielakOrlicz Morrey space M[theta],[theta](Rn) and obtain the boundedness on M[theta],[theta](Rn) 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(Rn) functions, where [alpha](0,1], [lambda](min{max{3,p1}, 3 + 2[alpha]/n},[infinity]) and p1 denotes the uniformly upper type index of [theta]. Let [theta] : [0,[infinity])[rightarrow][0,[infinity]) be nondecreasing, [theta](0) = 0, [theta](t)>0 when t>0, and lim t[rightarrow][infinity][theta](t) = [infinity], wA[infinity](Rn) and [theta]: (0,[infinity])[rightarrow](0,[infinity]) be nonincreasing. The authors also introduce the weighted OrliczMorrey space M[theta],[theta]w(Rn) and obtain the boundedness on M[theta], [theta]w(Rn) of the aforementioned intrinsic LittlewoodPaley functions and their commutators with BMO(Rn) functions. Finally, for q[1,[infinity]), the boundedness of the aforementioned intrinsic LittlewoodPaley functions on the Musielak-Orlicz Campanato space L[theta],q(Rn) is also established.