Sobolev and Hölder estimates for $\bar{\partial}$ on bounded convex domains in $\mathbb{C}^{2}$
Deyun Wu · Illinois Journal of Mathematics · 1998
The regularity of the Cauchy-Riemann operator 0 is a very important problem in both PDE's and Several Complex Variables.Numerous results have been proved by many mathematicians.Here we only list two recent results concerning 0 on convex domains.In 1991, Polking [12] proved the following L p estimates.THEOREM 1.Let D {z E C 2 p(z) O.Without loss of generality, we will assume that f is a (0,1) form f fd + .f2d-2.A (0, 1) form is in W'P(D), AP(D), if its coefficients are in W'P(D), p (D), respectively.For definitions and proporties of the Sobolev and HOlder spaces W'P(D) and AP(D), see [1], [17].