Invertibility of parabolic pseudo differential operators

The paper is devoted to the problem of invertibility of parabolic pseudodifferential operators in the spaces of distributions on the half-space R-+(n+1) = {x is an element of Rn+1, x(0) > 0) where x(0) is an element of R is the time variable and x' = (x(1), x(2), ..., x(n)) is an element of R-n are the spatial variables. We study pseudodifferential operators with symbols a(x, xi) satisfying the estimates |partial derivative(beta)(x)partial derivative(alpha)(xi)a(x,xi(0),xi')| <= C-alpha beta lambda(m-rho|alpha|)(xi), rho is an element of[0,1] for all multiindeces alpha, beta, where lambda is a base function. We introduce functional spaces H-s(lambda, Rn+1) of distributions u is an element of D'(Rn+1) such that the Fourier transform (u) over cap in the sense of distributions is a measurable function and parallel to u parallel to(Hs(lambda, Rn+1)) = (integral Rn+1 |(u) over cap(xi)lambda(s)(xi)|(2) d xi)(1/2) < infinity Let H-s(lambda, R-+(n+1)) be a subspace of H-s(lambda, Rn+1) contains the distributions with the supports in (R) over bar (n+1)(+). Pseudodifferential operator is called parabolic, in other terminology it is called causal or Volterra operators if it is bounded from H-s(lambda, R-+(n+1)) into Hs-m(lambda, R-+(n+1)). We give sufficient conditions provide parabolicity of a pseudodifferential operator. The main aim of the paper is to obtain sufficient conditions for parabolic pseudodifferential operators to be invertible in the spaces H-s(lambda, [0, T] x R-n) with arbitrary T > 0 and in the spaces H-s(lambda, R-+(n+1)) This problem is closely connected with the problem of reconstruction of input spatial-time signals from known output ones in causal space-time variable filters.