## Analysis of the $$\overline\partial$$-Neumann problem along a straight edge.(English)Zbl 1057.32015

In a previous article [Indiana Univ. Math. J. 52, No. 3, 629–666 (2003; Zbl 1056.32024)], the author studied singularities of solutions of the $$\overline{\partial}$$-Neumann problem (for data in the Schwartz space) on the product of two half-planes. Now the author gives a generalization to the case of domains of the form $$\{(z_1,z_2)\in {\mathbb C}^2: \operatorname{Im} z_1 > \alpha \operatorname{Im} z_2,$$ $$\operatorname{Im} z_2 >0\}$$, where $$\alpha\geq0$$. The main results are that there is a solution that belongs to the space $$L^p$$ when $$p>2$$, the solution is smooth away from the edge, and the singularities that arise at the edge have logarithmic and arctangent forms.

### MSC:

 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators 35B65 Smoothness and regularity of solutions to PDEs 35N15 $$\overline\partial$$-Neumann problems and formal complexes in context of PDEs

### Keywords:

singularity; logarithm; arctangent; Fourier transform

Zbl 1056.32024
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