Let \(I_ \alpha (f_ 1, \dots, f_ k) (x) = \int_{R^ n} f_ 1 (x - \theta_ 1y) \dots f_ k(x -\theta_ ky) | y |^{\alpha - n} dy\) be a multilinear fractional integration operator with \(\theta_ j \in \mathbb{R} \backslash 0\), \(1 \leq j \leq k\). The following theorem which generalizes the results of Adams and Grafakos is proved.
Theorem. Let \(k \geq 1\), \(p \in (1,\infty)\), \(\alpha = n/p\), \({1 \over p} = \sum^ k_{j = 1} {1 \over p_ j}\), \(p_ j \in (1,\infty]\). Then there exists a constant \(c_ 0 = c_ 0(p)\) such that for all \(f_ j \in L^{p_ j} (\mathbb{R}^ n)\) supported by \(\Omega \subset \mathbb{R}^ n\), \(| \Omega | < \infty\), \({1 \over | \Omega |} \int_ \Omega \exp ({n \over \omega_{n - 1}} \bigl| {LI_ \alpha (f_ 1, \dots, f_ k) (x) \over \| f_ 1 \|_{p_ 1} \dots \| f_ k \|_{p_ k}}\bigr|^{p'}) dx \leq c_ 0\), where \(L = \prod^ k_{j = 1} | \theta_ j |^{n/p_ j}\), \(\omega_{n-1}\) is a surface area of the unit sphere in \(\mathbb{R}^ n\), \({1\over p'}+ {1 \over p} = 1\). Furthermore, this inequality fails if \(n/ \omega_{n - 1}\) is replaced by a larger constant.
The result above is obtained by making use of a sharp interpolation theorem for Orlicz spaces with the Luxemburg norm.