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— The surjectivity of the operator D2 + ixfDi from the Gevrey space S w (R), s ̂ 1, onto itself and its non-surjectivity from S w (R) to S w (R) is proved.

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Global analytic and Gevrey surjectivity of the Mizohata operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msub><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:msubsup><mml:mi>x</mml:mi> <mml:mn>2</mml:mn> <mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow> </mml:msubsup><mml:msub><mml:mi>D</mml:mi> <mml:mn>1</mml:mn> </mml:msub></mml:mrow></mml:math>

Global analytic and Gevrey surjectivity of the Mizohata operator D 2 +ix 2 2k D 1