Pointwise multipliers on Orlicz-Campanato spaces

Abstract

Let ψ:(0,∞)→(0,∞) be a non-decreasing function with critical lower and upper type indices σ−⁎ and σ+⁎. Given a Campanato type space Cψ(Rn) that was defined associated to the function ψ, under σ−⁎>⌊σ+⁎⌋−1 when n≥2 or σ−⁎>σ+⁎−1 when n=1, the authors prove that any function is a pointwise multiplier on Cψ(Rn) if and only if it is bounded and belongs to a Musielak-Orlicz-Campanato space CΨ(Rn) that was associated toΨ(x,r):=|∫r1ψ(t)dtt|+∫1max⁡{r,e+|x|}(max⁡{r,e+|x|}t)⌊σ+⁎⌋ψ(t)dtt, where x∈Rn and r∈(0,∞). This generalizes the known characterizations of pointwise multipliers on the BMO(Rn) space and the Campanato space.