Approximate formulas for canonical homotopy operators for the $\bar \partial$ complex in strictly pseudoconvex domains

Abstract

Let $D=\{ \rho <0 \}$ be a smoothly bounded strictly pseudoconvex domain in $\boldsymbol C^n$ and $\rho$ a strictly plurisubharmonic smooth defining function. We construct explicit homotopy operators for the $\bar \partial$ complex, which are approximately equal to the homotopy operators that are canonical with respect to the metric $\Omega = i\varphi(-\rho)\partial \bar \partial \log(1/-\rho)$ and weights $(-\rho)^\alpha$, where $\varphi$ is a strictly positive smooth function. We also obtain an explicit operator which is approximately equal to the canonical homotopy operator for $\bar \partial_b$ on $\partial D$. From the explicit operators we obtain regularity results for these canonical operators, including $C^\infty$ regularity and $L^p$-boundedness for the orthogonal projections onto Ker $\bar \partial$ and Ker $\bar \partial_b$. Previously it has been proved, in the ball case and $\varphi \equiv 1$, that the boundary values of the canonical operators coincide with the values of well-known explicit operators due to Henkin and Skoda et al. Previously Lieb and Range have constructed an explicit homotopy operator which is approximately equal to the canonical operator with respect to the metric $i\varphi \partial \bar \partial_\rho$.