Double Hilbert transform on [inline-graphic not available: see fulltext]

We introduce a space [graphic not available: see fulltext], where [graphic not available: see fulltext] is the testing function space whose functions are infinitely differentiable and have bounded support, and [graphic not available: see fulltext] is the space the double Hilbert transform acting on the testing function space. We prove that the double Hilbert transform is a homeomorphism from [graphic not available: see fulltext] onto itself.