Equivalence of estimates on domain and its boundary

Let Ω be a pseudoconvex domain in Cn with smooth boundary bΩ. We define general estimates (f-M)kΩ and (f-M)kbΩ on k-forms for the complex Laplacian □ on Ω and the Kohn–Laplacian □ b on bΩ. For 1 ≤ k ≤ n−2, we show that (f-M) kbΩ holds if and only if (f-M) kΩ and (f-M)n−k−1Ωhold. Our proof relies on Kohn’s method in Kohn (Ann. Math.156(2), 213–248, 2002).