Double Hilbert transform on D(R~2)

摘要

We introduce a space D_(HH) = D(R~2) ⊕ H_2H_1D(R~2), where D(R~2) is the testing function space whose functions are infinitely differentiable and have bounded support, and H_2H_1D(R~2) is the space the double Hilbert transform acting on the testing function space. We prove that the double Hilbert transform is a homeomorphism from D_(HH) onto itself.