In this paper we consider operators of the form H=λ(-i∇), with λ analytic in a strip and with some specific growth conditions at infinity, and prove Hardy type estimates in L2(ℝn) with exponential weights. In fact we extend our previous results  from bounded analytic functions on a strip to analytic functions with polynomial growth in that strip.
"Hardy type inequalities with exponential weights for a class of convolution operators." Ark. Mat. 45 (1) 83 - 103, April 2007. https://doi.org/10.1007/s11512-006-0035-4