Compactness Estimate for the $\overline{\partial}$-Neumann Problem on a $Q$-Pseudoconvex Domain in a Stein Manifold

Download PDF


DOI: 10.46793/KgJMat2304.627S


We consider a smoothly bounded q-pseudoconvex domain Ω in an n-dimensional Stein manifold X and suppose that the boundary bΩ of Ω satisfies (q P) property, which is the natural variant of the classical P property. Then, one prove the compactness estimate for the -Neumann operator Nr,s in the Sobolev k-space. Applications to the boundary global regularity for the -Neumann operator Nr,s in the Sobolev k-space are given. Moreover, we prove the boundary global regularity of the -operator on Ω.


Stein manifold, q-pseudoconvex domain, compactness estimate, -operator, -Neumann operator.


[1]   L. Baracco and G. Zampieri, Regularity at the boundary for on Q-pseudoconvex domains, J. Anal. Math. 95 (2005), 45–61.

[2]   L. Baracco and G. Zampieri, Boundary regularity for on q-pseudoconvex wedges of CN, J. Math. Anal. Appl. 313(1) (2006), 262–272.

[3]   D. Catlin, Global regularity of the -Neumann problem, Proc. Sympos. Pure Math. 41 (1984), 39–49.

[4]   S. C. Chen and M. C. Shaw, Partial Differential Equations in Several Complex Variables, American Mathematical Society, Providence, 2001.

[5]   G. B. Folland and J. J. Kohn, The Neumann Problem for the Cauchy-Riemann Complex, Princeton Univ. Press, Princeton, New Jersey, 1972.

[6]   H. Grauert and O. Riemenschneider, Kählersche mannigfaltigkeiten mit hyper-q-konvexem rand, Problems in Analysis: A Symposium in Honor of Salomon Bochner, Univ. Press Princeton, New Jersey, 1970, 61–79.

[7]   H. Grauert and I. Lieb, Das ramirezsche integral und die losung der gleichung f = α im bereich der beschrankten formen, in: Proceeding of Conference Complex Analysis, Rice University Studies 56 (1970), 29–50.

[8]   G. M. Henkin, Integral representation of functions in strictly pseudoconvex domains and applications to the -problem, Mathematics of the USSR-Sbornik 7 (1969), 579–616.

[9]   G. M. Henkin and A. Iordan, Regularity of on pseudococave compacts and applications, Asian J. Math. 4 (2000) 855–884.

[10]   A. Heungju, Global boundary regularity for the -equation on q-pseudoconvex domains, Mathematische Nachrichten 16 (2005), 5–9.

[11]   J. J. Kohn, Global regularity for on weakly pseudo-convex manifolds, Trans. Amer. Math. Soc. 181 (1973), 273–292.

[12]   J. J. Kohn, Methods of partial differential equations in complex analysis, Proc. Sympos. Pure Math. 30 (1977), 215–237.

[13]   J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492.

[14]   J. D. McNeal, A sufficient condition for compactness of the -Neumann operator, J. Funct. Anal. 195 (2002), 190–205.

[15]   S. Saber, Global boundary regularity for the -problem on strictly q-convex and q-concave domains, Complex Anal. Oper. Theory 6 (2012), 1157–1165.

[16]   S. Saber, The problem on q-pseudoconvex domains with applications, Math. Slovaca 63 (3) (2013), 521–530.

[17]   T. Vu Khanh and G. Zampieri, Compactness estimate for the -Neumann problem on a Q-pseudoconvex domain, Complex Var. Elliptic Equ. 57 (12) (2012), 1325–1337.

[18]   G. Zampieri, q-pseudoconvexity and regularity at the boundary for solutions of the -problem, Compos. Math. 121 (2000), 155–162.