Global Hypoellipticity of a Class of Second Order Operators

Abstract

We show that almost all perturbations P — λ, λ € C, of an arbitrary constant coefficient partial differential operator P are globally hypoelliptic on the torus. We also give a characterization of the values λ € C for which the operator is globally hypoelliptic; in particular, we show that the addition of a term of order zero may destroy the property of global hypoellipticity of operators of principal type, contrary to that happens with the usual (local) hypoellipticity.

Global hypoellipticity for a class of periodic Cauchy operators

“…and thus L is not globally hypoelliptic if and only if one of the numbers α j is either a rational or Liouville. It can be noted here that this example recaptures results in [13].…”

On global hypoellipticity on the torus

“…EXAMPLE 2. Now we take $p=p(t)$ a real symbol that satisfies (1) and the additional condition: $|p(1)|>2$ or $|p(1)|=1$ . $(^{*})$ Then, we have the operators $p(D_{1}^{2})+e^{2ix_{1}}+ae^{-2ix_{1}},$ $a=\pm 1$ , which are (GH) on $T^{n}$ ; indeed, we can show that $0<|t_{1}|\leq 1$ and this implies $t_{1}^{2}+2a_{1,0}t_{1}+$ $a_{1,0}^{2}+1\neq 0$ , which, in tum, is shown to be equivalent to $t_{1}+\tilde{t}_{1}+a_{1,0}\neq 0$ .…”

“…Under hypothesis (1), we present a necessary and sufflcient condition for the operators in (1) to be (GH). Our examples show, in particular, that in the case when $p(t)=\lambda t^{2},1<\lambda<2$ , the situation $m>1$ is different from the case $m=1$ , (see [5]); namely, when $m>1$ , the operator may fail to be (GH).…”

“…Other related works dealing with global hypoellipticity are [6], [7], [1]. In [6] the operators $D_{1}^{2}+2cosx_{1}-\lambda,$ $\lambda\in C$ , are considered; in [7] this result is extended to cover more general operators with the same perturbation of order zero.…”

“…In [6] the operators $D_{1}^{2}+2cosx_{1}-\lambda,$ $\lambda\in C$ , are considered; in [7] this result is extended to cover more general operators with the same perturbation of order zero. In [1], the effect of perturbations by terms of order zero is considered only in the case of constant coefficients. Further related recent works are [2], [3].…”