Exponential estimates of solutions of parabolic pseudodifferential equations with discontinuous and growing symbols

Let Ω′⊂Rⁿ be an open set, and Ω₊ = R₊×Ω′ where R₊={fx₀:x₀} We consider pseudodifferential operators in domain Ω₊ with double symbols which have singularities near R₊×τΩ′ and super exponential growths at infinity. We suppose that symbols have analytic extension with respect to the variable dual to the time in the lower complex half-plane. We construct the theory of invertibility of such operators in weighted Sobolev spaces with weights connected with growths of symbols. We give applications to estimates of the fundamental solutions of such operators, in particular, to the heat equations with singular potentials of power, exponential and super exponential growths.