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

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
$$\mathcal{I}_{\alpha}(\overrightarrow{f})(x)=\int\limits_{(\mathbb{R}^{n})^{m}}\frac{f_1(y_1)⋯f_m(y_m)}{(|x−y_1|+⋯+|x−y_m|)^{mn-\alpha}}d}(\overrightarrow{y}, x∈R^n,$$ $0 < \alpha < mn$, where $L_r^q, L_l^q, L_{s_j}^{p_j}, j = 1,…,m,$ 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 Bα is the bilinear fractional integral operator given by the formula
$Bα(f_1,f_2)(x)=\int\limits_{R^n}\frac{f_1(x+t)f_2(x−t)}{|t|^{n−α}}dt,0<α<n$, Some of our results are new even in the linear case, i.e. when $m = 1$.