| Summary: | In this paper, we are devoted to establishing several necessary and su cient conditions for <em>f</em>∈<em>L</em><sup><em>p</em></sup>(R<sup><em>n</em></sup>); <em>g</em>∈<em>L</em><sup><em>q</em></sup>(R<sup><em>n</em></sup>) with (1/<em>p</em>) +(1/<em>q</em>)≤1 to satisfy the Bedrosian identity <em>H</em>(<em>fg</em>) =<em>fHg</em>, where <em>H</em> denotes the n-dimensional Hilbert transform. In addition, we also show that the distribution <em>f</em>∈<em>DL</em><sup><em>p</em></sup>' (R<sup><em>n</em></sup>) can be represented by functions in the Hardy space on tube.
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