The extension of 𝐻^{𝑝}-functions from certain hypersurfaces of a polydisc

Abstract

Let E be a subvariety of the open unit polydisc U n , n ⩾ 2 {U^n},n \geqslant 2 , of pure dimension n − 1 n - 1 , satisfying the following conditions. There exists an annular domain Q n = { ( z 1 , … , z n ) ∈ C n : r > | z i | > 1 , 1 ⩽ i ⩽ n } {Q^n} = \{ ({z_1}, \ldots ,{z_n}) \in {{\mathbf {C}}^n}:r > |{z_i}| > 1,1 \leqslant i \leqslant n\} , a continuous function η : [ r , 1 ) → [ r , 1 ) \eta :[r,1) \to [r,1) , and a δ > 0 \delta > 0 , such that (i) | z n | ⩽ η ( ( | z 1 | + ⋯ + | z n − 1 | ) / ( n − 1 ) ) |{z_n}| \leqslant \eta ((|{z_1}| + \cdots + |{z_{n - 1}}|)/(n - 1)) whenever ( z 1 , … , z n ) ∈ E ∩ Q n ({z_1}, \ldots ,{z_n}) \in E \cap {Q^n} , (ii) | α − β | ⩾ δ |\alpha - \beta | \geqslant \delta whenever 1 ⩽ j ⩽ n 1 \leqslant j \leqslant n and ( ζ ′ , α , ζ ) ≠ ( ζ ′ , β , ζ ) (\zeta ’,\alpha ,\zeta ) \ne (\zeta ’,\beta ,\zeta ) are both in ( Q j − 1 × U × Q n − j ) ∩ E ({Q^{j - 1}} \times U \times {Q^{n - j}}) \cap E . Theorem. Let 0 > p > ∞ 0 > p > \infty , let g be holomorphic on E and let u be the real part of a holomorphic function on E. If | g ( z ) | p ⩽ u ( z ) |g(z){|^p} \leqslant u(z) for all z ∈ E z \in E , then g can be extended to a function in the Hardy space H p ( U n ) {H^p}({U^n}) .