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

Abstract

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 ερα,p(χ) and Morrey-Campanato-BLO spaces ε̃ρα,p(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 ερa,p(χ) to ε̃ρa,p(χ) without invoking any regularity of considered kernels. The same is true for the gλ* 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 ℝd with the d-dimensional Lebesgue measure and have a wide applications.