Conditional hypoellipticity and Fourier multipliers for weighted L_p-spaces with an exponential weight

The following conditional hypoellipticity for a linear partial differential operator P(D) with constant coefficients is examined: uin W^1_{p,rho}(Omega), P(D)uin W^infty_{p,rho}(Omega)implies uin W^infty_{p,rho}(Omega). Here Omega is an open set in bold R^n of the form Omega_mtimesbold R^{n-m} and rho is a continuous positive function on Omega; the Sobolev spaces above are constructed from the following weighted L_p-spaces: L_{p,rho}(Omega)={u {rm measurable in} Omegacolon rho uin L_p(G_mtimesbold R^{n-m}) {rm for all rectangles} G_msubsetOmega_m}. Let rho(x)=exp{A_{m+1}|x_{m+1}|^{epsilon_{m+1}}+cdots+A_n| x_n|^{epsilon_n}}, where A_{m+1},cdots,A_nnot= 0 and epsilon_{m+1},cdots,epsilon_n>0; the author gives sufficient conditions on widetilde P(xi)=bigg(sumSbalphainbold N^n (alpha_1,cdots,alpha_m)not= 0endSb |P^{(alpha)}(xi)|^2bigg)^{1/2} for P(D) to be conditionally hypoelliptic. These results generalize both Hörmander's theorem (the case m=n) and previous work by the author (the case rhoequiv 1). The proofs, not given in the paper, use Fourier multiplier techniques developed by the author and M. Sh. Tuyakbaev [Dokl. Akad. Nauk SSSR {bf 320} (1991), no.~1, 11--14; [msn] MR1151506 (93e:42026) [/msn]].