Global Solvability and Simultaneously Approximable Vectors

Abstract

We consider a class of sum of squares operators on a torus and we prove that global solvability is equivalent to an algebraic condition involving simultaneously approximable vectors. # 2002 Elsevier Science (USA)Key Words: global solvability; Fourier transform; simultaneous approximability. INTRODUCTIONIt is well known that the theory for the problem of global solvability of PDE is not well developed, in comparison with the local solvability case. One of the main obstacles is the appearance of diophantine phenomena. We hope that our result will provide some insight into this problem. In this paper, we prove global solvability on a torus for a class of operators in the form of a sum of squares of real vector fields. More precisely, we present a necessary and sufficient condition for global solvability on torus for the following class of operators:where ðt 1 ; . . . ; t m ; x 1 ; . . . ; x n Þ ¼ ðt; xÞ 2 T mþn and a j ; j ¼ 1; . . . ; n; are realvalued functions in C 1 ðT m Þ: Since there is an intimate connection between the global solvability of P and the global hypoellipticity of t P (a standard functional analysis argument applies: the global hypoellipticity of t P implies the global solvability of P ), this work can be regarded as a natural continuation of Himonas and Petronilho [10] where a study of the global hypoellipticity for this operator was presented. Since the class of operators that we are studying has the

Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions

“…, n, are realvalued functions defined on T m . This result extends to analytic and Gevrey classes the one of Petronilho [11] where the C ∞ -global solvability for such a class of operators was characterized and to any dimensions of one of [14]. On the other hand, since there is a connection between the global analytic and Gevrey solvability of any partial differential operator P and the global analytic and Gevrey hypoellipticity of t P as it was shown in [15] (or see [16]), this paper can be regarded as a continuation of Himonas [6] where a study of the global analytic and Gevrey hypoellipticity for this class of operators was presented.…”

“…, n, are real-valued functions defined on T m . We recall that such a class of operators was studied in [8,11] and in [6] in the frame of C ∞ -class and of G s -classes respectively. In [6,8] the authors investigated conditions which are necessary and sufficient for the global hypoellipticity; while, in [11] the global solvability in D (T m+n ) was characterized.…”

“…to [8,11] for the proper analogue definitions in the setting of C ∞ -class while for simultaneous G s type Diophantine conditions see [1] (s = 1) and [3,23] for s ≥ 1. In particular, the definition of simultaneously s-approximable is similar to the s-Siegel condition in [23] for commuting vector fields.…”

“…torus, in large scales of functional spaces (see e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references listed therein). It is well known that the theory of global properties of differential operators is not well developed in comparison with the one of local properties.…”

Перiодичнi розв’язки у просторi Соболєва для рiвняння iз частинними похiдними, коефiцiєнти якого залежать вiд параметра

“…[1,5,6,8,13,14] i посилання в них), де також виникає проблема малих зна-менникiв, якi є полiномами L(k), k ∈ Z p , зi сталими коефiцiєнтами. Зокрема, у роботi [8] показано, що диференцiальний оператор зi сталими коефiцiєнтами глобально гiпо-елiптичний тодi i тiльки тодi, коли його повний символ задовольняє дiофантовi умови типу Зiгеля.…”