We prove $L^p$ estimates, $1\le p\le \infty$, for solutions to the tangential Cauchy–Riemann equations ${\overline{\partial }}_{b}u=\varphi$ on a class of infinite type domains $\mathrm{\Omega }\subset {\mathbb{C}}^2$. The domains under consideration are a class of convex ellipsoids, and we show that if $\varphi$ is a ${\overline{\partial }}_b$‐closed (0,1)‐form with coefficients in $L^p$, then there exists an explicit solution u satisfying ${\parallel u\parallel }_{L^p(b\mathrm{\Omega })}\le C{\parallel \varphi \parallel }_{L^p(b\mathrm{\Omega })}$. Moreover, when $p=\infty$, we show that there is a gain in regularity to an f‐Hölder space.

中文翻译：

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一类无限类型域上的∂é方程的Lp估计

我们证明 ${\mathrm{大号}}^p$ 估计， $\mathrm{1个}\le p\le \infty$，用于切向Cauchy–Riemann方程的解 ${\overline{\partial }}_bü=\varphi$ 在一类无限类型域上 $\mathrm{\Omega }\subset {\mathbb{C}}^2$。所考虑的域是一类凸椭圆体，我们证明了如果$\varphi$ 是一个 ${\overline{\partial }}_b$-（0,1）形式-系数为 ${\mathrm{大号}}^p$，则存在一个明确的解决方案ü满足${\parallel ü\parallel }_{{\mathrm{大号}}^p（b\mathrm{\Omega }）}\le C{\parallel \varphi \parallel }_{{\mathrm{大号}}^p（b\mathrm{\Omega }）}$。而且，什么时候$p=\infty$，我们证明f- Hölder空间的规则性有所提高。