Global hypoellipticity for strongly invariant operators

Non-isotropic Gevrey Hypoellipticity for Grushin Operators

We shall determine non-isotropic Gevrey exponents for general Grushin operators based on the results given in the paper [26], where a method to determine isotropic (worst) Gevrey exponents was given. The ideas of the bracket calculus given in the paper [2] and FBI-transformation given in the paper [5] are also useful.

On global hypoellipticity

We consider a first order linear partial differential operator of principal type on a closed connected orientable two-dimensional manifold sending sections of one complex line bundle to sections of another. We prove that the assumption of global hypoellipticity of the operator implies a relation between the degrees of the line bundles and the Euler characteristic of the manifold.

Failure of Analytic Hypoellipticity in a Class of Differential Operators

For the hypoelliptic differential operators P = ∂ x + ( x∂y − x∂t 2 introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of k and l left open in the analysis, the operators P also fail to be analytic hypoelliptic (except for (k, l) = (0, 1)), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of e...

Φ-strongly Accretive Operators

Suppose that X is an arbitrary real Banach space and T : X → X is a Lipschitz continuous φ-strongly accretive operator or uniformly continuous φ-strongly accretive operator. We prove that under different conditions the three-step iteration methods with errors converge strongly to the solution of the equation Tx = f for a given f ∈ X.

Inequalities for Strongly Singular Convolution Operators