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• W1781786423 abstract "We prove global analytic hypoellipticity on a product of tori for partial differential operators which are constructed as rigid (variable coefficient) quadratic polynomials in real vector fields satisfying the Hörmander condition and where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding=application/x-tex>P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies a “maximal” estimate. We also prove an analyticity result that is local in some variables and global in others for operators whose prototype is <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P equals left-parenthesis StartFraction partial-differential Over partial-differential x 1 EndFraction right-parenthesis squared plus left-parenthesis StartFraction partial-differential Over partial-differential x 2 EndFraction right-parenthesis squared plus left-parenthesis a left-parenthesis x 1 comma x 2 right-parenthesis StartFraction partial-differential Over partial-differential t EndFraction right-parenthesis squared> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mfrac> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>P=left ( frac partial {partial x_1}right ) ^2+left ( frac partial { partial x_2}right ) ^2+left ( a(x_1,x_2)frac partial {partial t}right )^2</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> (with analytic <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a left-parenthesis x right-parenthesis comma a left-parenthesis 0 right-parenthesis equals 0> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>a(x),a(0)=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, naturally, but not identically zero). The results, because of the flexibility of the methods, generalize recent work of Cordaro and Himonas in [<italic>Global analytic hypoellipticity of a class of degenerate elliptic operators on the torus</italic>, Math. Res. Lett. <bold>1</bold> (1994), 501–510] and Himonas in [<italic>On degenerate elliptic operators of infinite type</italic>, Math. Z. (to appear)] which showed that certain operators known not to be locally analytic hypoelliptic (those of Baouendi and Goulaouic [<italic>Analyticity for degenerate elliptic equations and applications</italic>, Proc. Sympos. Pure Math., vol. 23, Amer. Math. Soc., Providence, RI, 1971, pp. 79–84], Hanges and Himonas [<italic>Singular solutions for sums of squares of vector fields</italic>, Comm. Partial Differential Equations <bold>16</bold> (1991), 1503–1511], and Christ [<italic>Certain sums of squares of vector fields fail to be analytic hypoelliptic</italic>, Comm. Partial Differential Equations <bold>10</bold> (1991), 1695–1707]) were <italic>globally</italic> analytic hypoelliptic on products of tori." @default.
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• W1781786423 title "Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori" @default.
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