Estimates for Lusin-area and Littlewood–Paley gλ∗ functions over spaces of homogeneous type

Let (X,d,μ)(X,d,μ) be a space of homogeneous type and μ(X)=∞μ(X)=∞. Under the assumptions that the measure μμ satisfies the volume regularity property (P)(P) and the Lusin-area function SS is bounded on L2(X)L2(X), the authors prove, without invoking any regularity on the kernels considered, that if ff belongs to BMO(X), S(f)S(f) is either infinite everywhere or finite almost everywhere, and in the latter case, [S(f)]2[S(f)]2 is bounded from BMO(X) into its proper subspace BLO(X). As an application, the authors also obtain the boundedness on Lp(μ)Lp(μ) with p∈(2,∞)p∈(2,∞) for the operator SS. Furthermore, exploiting the Lp(X)Lp(X)-boundedness of SS, the authors prove that if ff belongs to a certain Campanato space Eα,p(X)Eα,p(X) with suitable indices, S(f)S(f) is either infinite everywhere or finite almost everywhere, and in the latter case, [S(f)]2[S(f)]2 is bounded from Eα,p(X)Eα,p(X) into E∗2α,p/2(X). Moreover, the authors establish corresponding results for the Littlewood–Paley gλ∗ function without invoking any regularity of the kernels considered and the property (P)(P) of XX. The authors also show that E∗α,p(X) is a proper subspace of Eα,p(X)Eα,p(X) with suitable indices.