Multivariate Hausdorff operators on the spaces Lp(Rn) ✩

The operators H indicated in the title are characterized by a σ -finite Borel measure μ on Rn, a Borel measurable function c on Rn, and an n ×n μ-a.e. nonsingular matrix A whose entries are also Borel measurable functions on Rn; and Hf is defined by means of a Lebesgue–Stieltjes integral with respect to μ.We give simple sufficient conditions in order that these operators be bounded on the Lebesgue spaces Lp(Rn) for some 1 p ∞. These sufficient conditions are exact even in the well-known special cases of the Cesàro and Copson operators.We also determine the Hausdorff operator H∗ which is adjoint to H in a certain sense.We reveal interrelations among these operators and the Fourier transform of a function f in L1(Rn). On closing, we present further special Hausdorff operators.  2002 Elsevier Science (USA). All rights reserved