ANISOTROPIC SHUBIN OPERATORS AND EIGENFUNCTION EXPANSIONS IN GELFAND-SHILOV SPACES

摘要

We derive new results on the characterization of Gelfand-Shilov spaces S-nu(mu)(R-n), mu, nu > 0, mu + nu >= 1 by Gevrey estimates of the L-2 norms of iterates of (m, k) anisotropic globally elliptic Shubin (or Gamma) type operators, (- Delta)(m/2) + vertical bar x vertical bar(k) with m, k is an element of 2N being a model operator, and on the decay of the Fourier coefficients in the related eigenfunction expansions. Similar results are obtained for the spaces Sigma(mu)(nu)(R-n), mu, nu > 0, mu + nu > 1; cf. (1.2). In contrast to the symmetric case mu = nu and k = m (classical Shubin operators) we encounter resonance type phenomena involving the ratio kappa:= mu/nu; namely we obtain a characterization of S-nu(mu)(R-n) and Sigma(mu)(nu)(R-n) in the case mu = kt/(k + m), nu = mt/(k + m), t >= 1, that is, when kappa = k/m is an element of Q.