Singular integrals and Hardy type spaces for the inverse Gauss measure

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Let γ₋₁ be the absolutely continuous measure on Rⁿ whose density is the reciprocal of a Gaussian and consider the natural weighted Laplacian A on L²(γ₋₁). In this paper, we prove boundedness and unboundedness results for the purely imaginary powers and the first order Riesz transforms associated with the translated operators A+λI, λ≥0, from certain new Hardy-type spaces adapted to γ₋₁ to L¹(γ₋₁). We also investigate the weak type (1,1) of these operators.