Double Hilbert transform on D\left({ℝ}^2\right)

Abstract

We introduce a space D_{HH}=D\left({ℝ}^2\right)\mathbf{\oplus }{H}_2{H}_{1}D\left({ℝ}^2\right), where D\left({ℝ}^2\right) is the testing function space whose functions are infinitely differentiable and have bounded support, and H_2{H}_{1}D\left({ℝ}^2\right) 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.

Key words： Double Hilbert transform; vanishing moments; homeomorphism

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