**Preprint series:**
97-08, Analysis

**The paper is published:**
J. London Math. Soc., 59 (1), 287-310, 1999

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

**Abstract:** 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

$\dot{{\cal H}}^{s_1}_{p_1} \times \dot{{\cal H}}^{s_2}_{p_2} \times \ldots \times \dot{{\cal H}}^{s_m}_{p_m}$ into ${\cal Z}'$.

Here all spaces are defined on ${Bbb R}$ and $2 \le m \le n$.

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

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

**Upload:** 1999-02-10

**Update:** 1999-06-30

The author(s) agree, that this abstract may be stored as full text and distributed as such by abstracting services.