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Let Ω⊂$R$^{$n$}be an arbitrary open set. In this paper it is shown that if a Sobolev function$f$∈$W$^{1,$p$}(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, then$f$is weakly zero on ϖΩ in the sense that$f$∈$W$_{0}^{1,$p$}(Ω).

We prove the following Hartogs-Bochner type theorem: Let M be a connected C^{2}hypersurface of P_{n}(C) (n≥2) which divides P_{n}(C) in two connected open sets Ω_{1}and Ω_{2}. Suppose that M has at most one open CR orbit. Then there exists i∈{1,2} such that C^{1}CR functions defined on M extends holomorphically to Ω_{$i$}.

The cosmological observations provide a strong evidence that there is a positive cosmological constant in our universe and thus the spacetime is asymptotical de Sitter space. The conjecture of gravity as the weakest force in the asymptotical dS space leads to a lower bound on the U(1) gauge coupling $g$, or equivalently, the positive cosmological constant gets an upper bound $\rho_V \leq g^2 M_p^4$ in order that the U(1) gauge theory can survive in four dimensions. This result has a simple explanation in string theory, i.e. the string scale $\sqrt{\alpha '}$ should not be greater than the size of the cosmic horizon. Our proposal in string theory can be generalized to U(N) gauge theory and gives a guideline to the microscopic explanation of the de Sitter entropy. The similar results are also obtained in the asymptotical anti-de Sitter space.