Boundedness of Lusin-area and g* λ functions on localized Morrey-Campanato spaces over doubling metric measure spaces

Let χ be a doubling metric measure space and ρ an admissible function on χ. in this paper,the authors establish some equivalent characterizations for the localized morrey-campanato spaces ε α,p ρ(χ) and morrey-campanato-blo spaces ε̃ α,p ρ(χ) when α ∈ (-∞,0) and p ∈ [1,∞). if χ has the volume regularity property (p),the authors then establish the boundedness of the lusin-area function,which is defined via kernels modeled on the semigroup generated by the schrödinger operator,from ε α,p ρ(χ) to ε̃ α,p ρ(χ) without invoking any regularity of considered kernels. the same is true for the g*λ function and,unlike the lusin-area function,in this case,χ is even not necessary to have property (p). these results are also new even for ℝ d with the d-dimensional lebesgue measure and have a wide applications. copyright © 2011 hindawi publishing corporation.