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In this paper we deal with several characterizations of the Hardy-Sobolev spaces in the unit ball of C, that is, spaces of holomorphic functions in the ball whose derivatives up to a certain order belong to the classical Hardy spaces. Some of our characterizations are in terms of maximal functions, area functions or Littlewood-Paley functions involving only complex-tangential derivatives. A special case of our results is a characterization of H itself involving only complex-tangential derivatives....

We will classify $n$-dimensional real submanifolds in ${ℂ}^n$ which have a set of parabolic complex tangents of real dimension $n-1$. All such submanifolds are equivalent under formal biholomorphisms. We will show that the equivalence classes under convergent local biholomorphisms form a moduli space of infinite dimension. We will also show that there exists an $n$-dimensional submanifold $M$ in ${ℂ}^n$ such that its images under biholomorphisms $(z_1,\cdots ,z_n)\mapsto (r{z}_1,\cdots ,r{z}_{n-1},r^2{z}_n)$, $r\>1$, are not equivalent to $M$ via any local volume-preserving holomorphic...