On Sharp Olsen’s and Trace Inequalities for Multilinear Fractional Integrals

Abstract

We establish a sharp Olsen type inequality

$ \big \| g {\mathcal {I}}_{\alpha }(f_{1}, {\dots } , f_{m}) \big \|_{{L^{q}_{r}} } \leq C \big \| g \big \|_{L^{q}_{\ell } } \prod\limits_{j=1}^{m} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}}} $

for multilinear fractional integrals ${\mathcal {I}}_{\alpha }(\vec {f} ) (x) = \int \limits _{({\Bbb {R}}^{n})^{m}}\frac {f_{1}(y_{1}){\cdots } f_{m}(y_{m})}{(|x-y_{1}|+ {\cdots } + |x-y_{m}|)^{mn-\alpha }} d\vec {y}, x\in {\Bbb {R}}^{n}$, 0 < α < mn, where ${L^{q}_{r}}$, $L^{q}_{\ell }$, $L^{p_{j}}_{s_{j}}$, j = 1,…,m, are Morrey space with indices satisfying certain homogeneity conditions. This inequality is sharp because it gives necessary and sufficient condition on a weight function V for which the inequality

$ \big \|{\mathcal {I}}_{\alpha }(f_{1}, {\dots } , f_{m}) \big \|_{{L^{q}_{r}}(V) } \leq C \prod\limits_{j=1}^{m} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}}} $

holds. Morrey spaces play an important role in relation to regularity problems of solutions of partial differential equations. They describe the integrability more precisely than Lebesgue spaces. We also derive a characterization of the trace inequality

$ \big \| B_{\alpha } (f_{1},f_{2})\big \|_{{L^{q}_{r}}(d\mu ) } \leq C \prod\limits_{j=1}^{2} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}} ({\Bbb {R}}^{n}) }, $

in terms of a Borel measure μ, where B_α is the bilinear fractional integral operator given by the formula $B_{\alpha }(f_{1},f_{2})(x) =\int \limits _{{\Bbb {R}}^{n}} \frac {f_{1}(x+t)f_{2}(x-t)}{|t|^{n-\alpha }} dt, 0< \alpha <n,$ Some of our results are new even in the linear case, i.e. when m = 1.