ltivariate Hausdorff operators on the spaces H¹(Rⁿ) and BMO(Rⁿ)

Summary

A multivariate Hausdorff operator H = H(μ, c, A) is defined in terms of a σ-finite Borel measure μ on Rⁿ, a Borel measurable function c on Rⁿ, and an n  × n matrix A whose entries are Borel measurable functions on rⁿ and such that A is nonsingular μ-a.e. The operator H*:= H (μ, c | det A^-1|, A^-1) is the adjoint to H in a well-defined sense. Our goal is to prove sufficient conditions for the boundedness of these operators on the real Hardy space H¹(Rⁿ) and BMO (Rⁿ). Our main tool is proving commuting relations among H, H*, and the Riesz transforms R _j . We also prove commuting relations among H, H*, and the Fourier transform.