We extend the known result that the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class \(BS^{-n/2}_{0,0}(\mathbb {R}^n)\) are bounded from \(L^2 \times L^2\) to \(h^1\). We show that those operators are also bounded from \(L^2 \times L^2\) to \(L^r \) for every \(1< r\le 2\). Moreover we give similar results for symbol classes wider than \(BS^{-n/2}_{0,0}(\mathbb {R}^n)\). We also give results for symbols of limited smoothness.

中文翻译：

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$$ S_ {0,0} $$ S 0，0型双线性伪微分算子在$$ L ^ 2 \ times L ^ 2 $$ L 2×L 2上的有界性

我们扩展了已知结果是双线性伪微分算子与双线性Hörmander类符号\（BS ^ { - N / 2} _ {0,0}（\ mathbb {R} ^ N）\）是从有界\ （L ^ 2 \ times L ^ 2 \）到\（h ^ 1 \）。我们证明了对于每个\（1 <r \ le 2 \），这些算子也从\（L ^ 2 \ times L ^ 2 \）到\（L ^ r \）有界。此外，对于大于\（BS ^ {-n / 2} _ {0,0}（\ mathbb {R} ^ n）\）的符号类，我们给出相似的结果。我们还给出了平滑度有限的符号的结果。