Singular Integrals and Hardy Type Spaces for the Inverse Gauss Measure

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