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We investigate the bilinear Hilbert transform with oscillatory factors and the truncated bilinear Hilbert transform. The main result is that the Lp1×Lp2→Lq-boundedness of the two operators is equivalent with 1≤p1,p2<∞, and 1/q=1/p1+1/p2. In addition, we also discuss the boundedness of a variant operator of bilinear Hilbert transform with a nontrivial polynomial phase

abstract and applied analysis

Criterion on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mrow><mml:…

Criterion onLp1×Lp2→