**Preprint series:**
97-09, Analysis

**The paper is published:**
J. London Math. Soc., 60 (2), 561-580, 1999.

**MSC:**- 46E35 Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

**Abstract:** Let $2 \leq m \leq n$. We give necessary and sufficient conditions on the parameters $s_1, s_2, \ldots, s_m, p_1, p_2, \ldots, p_m$ such that the Jacobian determinant extends to a bounded operator from ${\cal H}^{s_1}_{p_1}\times {\cal H}^{s_2}_{p_2} \times \cdots \times {\cal H}^{s_m}_{p_m}$ into ${\cal S}'$.

Here all spaces are defined on ${Bbb R}^n$ or on domains $\Omega\subset {Bbb R}^n$.

In almost all cases the regularity of the Jacobian determinant is calculated exactly.

**Keywords:** *Bessel potential spaces, Jacobian determinant, paraproducts*

**Upload:** 1999-02-10

**Update:** 2000-02-01

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