Estimates for Lusin-area and Littlewood-Paley g(lambda)* functions over spaces of homogeneous type

作者：Meng, Yan; Nakai, Eiichi; Yang, Dachun^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2010, 72(5): 2721-2736.

DOI：10.1016/j.na.2009.11.019

• Lusin-area function
• Littlewood-Paley g(lambda)* function
• L-p (X)
• BMO(X)
• Campanato spaces
• Space of homogeneous type

Let(X, d, mu) be a space of homogeneous type and mu(X) = infinity. Under the assumptions that the measure mu satisfies the Volume regularity property (P) and the Lusin-area function S is bounded on L-2(X), the authors prove, without invoking any regularity on the kernels considered, that if f belongs to BMO(X), S(f) is either infinite everywhere or finite almost everywhere, and in the latter case, [S(f)](2) is bounded from BMO(X) into its proper subspace BLO(X). As an application, the authors also obtain the boundedness on L-p(mu) with P is an element of (2, infinity) for the operator S. Furthermore, exploiting the L-p(X)-boundedness of S, the authors prove that if f belongs to a certain Campanato space epsilon(alpha-p)(X) with suitable indices, S(f) is either infinite everywhere or finite almost everywhere, and in the latter case, [S(f)](2) is bounded from epsilon(alpha.p)(X) into epsilon(2 alpha.p/2)(*) (X). Moreover, the authors establish corresponding results for the Littlewood-Paley g(lambda)* function without invoking any regularity of the kernels considered and the property (P) of X. The authors also show that epsilon(alpha,p)(*)(X) is a proper subspace of epsilon(alpha,p)(X) with suitable indices.

• 出版日期2010-3-1
• 单位中国人民大学; 北京师范大学