Complex and Real Hausdorff Operators

Hausdorff operators (Hausdorff summability methods) appeared long ago aiming to solve certain classical problems in analysis. Modern theory of Hausdorff operators started with the work of Siskakis in complex analysis setting and with the work of Georgakis and Liflyand-Móricz in the Fourier transform setting. While Hausdorff operators for power series are still studied mostly in dimension one, the center of attraction of interesting problems for the Hausdorff operators of Fourier integrals lies in the multi-variate setting. One of the most general definitions of the Hausdorff operator reads as (Hf)(x) = (HΦf)(x) = (HΦ,Af)(x) = ∫ Rn Φ(u)f ( xA(u) ) du, where A = A(u) = (aij)i,j=1 = ( aij(u) )n i,j=1 is the n×n matrix with the entries aij(u) being measurable functions of u. This matrix may be singular on a set of measure zero at most; xA(u) is the row n-vector obtained by multiplying the row n-vector x by the matrix A. However, we first give a brief overview of Hausdorff operators in other settings. For Fourier transforms, many details are given in dimension one then. Recent results in which conditions on the couple (Φ, A) are found to provide the boundedness of the operator in the real Hardy space are discussed. There now exist two proofs, one based on the H-BMO duality while the other on atomic decomposition. The case of product Hardy spaces is also studied. Many open problems in the subject are formulated.