Optimal Hölder and $L^ p$ estimates for $\overline \partial _ b$ on the boundaries of real ellipsoids in $\textbf {C}^ n$

Abstract:

Let $D$ be a real ellipsoid in ${{\mathbf {C}}^n},n \geq 3$, with defining function $\rho (z) = \sum \nolimits _{k = 1}^n {(x_k^{2{n_k}} + y_k^{2{m_k}})} - 1$, ${z_k} = {x_k} + i{y_k}$, where ${n_k},{m_k} \in N$. In this paper we study the sharp Hàlder and ${L^p}$ estimates for the solutions of the tangential Cauchy-Riemann equations ${\overline \partial _b}$ on the boundary $\partial D$ of $D$ using the integral kernel method. In particular, we proved that if $\alpha \in L_{(0,1)}^\infty (\partial D)$ such that ${\overline \partial _b}\alpha = 0$ on $\partial D$ in the distribution sense, then there exists a $u \in {\Lambda _{1/2m}}(\partial D)$ satisfying ${\overline \partial _b}u = \alpha$ and ${\left \| u \right \|_{{\Lambda _{1/2m}}(\partial D)}} \leq c{\left \| \alpha \right \|_{{L^\infty }(\partial D)}}$ for some constant $c > 0$ independent of $\alpha$, where ${\Lambda _{1/2m}}(\partial D)$ is the Lipschitz space with exponent $\frac {1} {{2m}}$ and $2m = {\max _{1 \leq k \leq n}}\min (2{n_k},2{m_k})$ is the type of the domain $D$.