Pointwise Multipliers of Zygmund Classes on"Equation missing"

Abstract

It is well known that Lipschitz spaces on the torus are an algebra. It is no more the case in the non compact situation because of the behavior at infinity. This is a companion article to Bonami et al. (J Math Pures Appl (9) 131:130–170, 2019), where pointwise multipliers on Lipschitz spaces on $${\mathbb {R}}^n$$ are characterized for non-integer values of the parameter. In this article, the authors first establish two equivalent characterizations of a modified Zygmund space, and then characterize the pointwise multipliers on Lipschitz spaces on $${\mathbb {R}}^n$$ for the integer values of the parameter, in particular, for the Zygmund class, via the intersection of the Lebesgue space $$L^\infty ({\mathbb {R}}^n)$$ and the modified Zygmund space. This result can be used to show that the bilinear decomposition of the pointwise product of the Hardy space and its dual, in the integer values of the parameter, obtained in the aforementioned reference is sharp in the dual space sense.