We prove a plethora of the boundedness property of Adams type for bilinear fractional integral operators of the form Bα(f,g)(x)=∫Rnf(x-y)g(x+y)|y|n-αdy,0<α<n.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} B_{\alpha }(f,g)(x)=\int _{{\mathbb {R}}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-\alpha }}dy,\quad 0<\alpha <n.\ \end{aligned}$$\end{document}For 1<t≤s<∞\documentclass[12pt]{minimal}
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\begin{document}$$1<t\le s<\infty $$\end{document}, we prove the non-weighted case through the known Adams type result. And we show that these results of Adams type is optimal. For 0<t≤s<∞\documentclass[12pt]{minimal}
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\begin{document}$$0<t\le s<\infty $$\end{document} and 0<t≤1\documentclass[12pt]{minimal}
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\begin{document}$$0<t\le 1$$\end{document}, we obtain new result of a weighted theory describing Morrey boundedness of above form operators if two weights (v,w→)\documentclass[12pt]{minimal}
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\begin{document}$$(v,\vec {w})$$\end{document} satisfy [v,w→]t,q→/ar,as=supQ,Q′∈DQ⊂Q′|Q||Q′|1-sas|Q′|1r⨍Qvt1-t1-tt∏i=12⨍Q′wi-(qi/a)′1(qi/a)′<∞,0<t<s<1\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&{[}v,\vec {w}]_{t,\vec {q}/{a}}^{r,as}=\mathop {\sup _{Q,Q^{\prime }\in {\mathscr {D}}}}_{Q\subset Q^{\prime }}\left( \frac{|Q|}{|Q^{\prime }|}\right) ^{\frac{1-s}{as}}|Q^{\prime }|^{\frac{1}{r}}\left( \fint _{Q}v^{\frac{t}{1-t}}\right) ^{\frac{1-t}{t}}\\&\quad \prod _{i=1}^{2}\left( \fint _{Q^{\prime }}w_{i}^{-(q_{i}/a)^{\prime }}\right) ^{\frac{1}{(q_{i}/a)^{\prime }}}<\infty ,\quad 0<t<s<1 \end{aligned}$$\end{document}and [v,w→]t,q→/ar,as:=supQ,Q′∈DQ⊂Q′|Q||Q′|1-asas|Q′|1r⨍Qvt1-t1-tt∏i=12⨍Q′wi-(qi/a)′1(qi/a)′<∞,s≥1\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&{[}v,\vec {w}]_{t,\vec {q}/{a}}^{r,as}:=\mathop {\sup _{Q,Q^{\prime }\in {\mathscr {D}}}}_{Q\subset Q^{\prime }}\left( \frac{|Q|}{|Q^{\prime }|}\right) ^{\frac{1-as}{as}}|Q^{\prime }|^{\frac{1}{r}}\left( \fint _{Q}v^{\frac{t}{1-t}}\right) ^{\frac{1-t}{t}}\\&\quad \prod _{i=1}^{2}\left( \fint _{Q^{\prime }}w_{i}^{-(q_{i}/a)^{\prime }}\right) ^{\frac{1}{(q_{i}/a)^{\prime }}}<\infty , \quad s\ge 1 \end{aligned}$$\end{document}where ‖v‖L∞(Q)=supQv\documentclass[12pt]{minimal}
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\begin{document}$$\Vert v\Vert _{L^{\infty }(Q)}=\sup _{Q}v$$\end{document} when t=1\documentclass[12pt]{minimal}
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\begin{document}$$t=1$$\end{document}, a, r, s, t and q→\documentclass[12pt]{minimal}
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\begin{document}$$\vec {q}$$\end{document} satisfy proper conditions. As some applications we formulate a bilinear version of the Olsen inequality, the Fefferman–Stein type dual inequality and the Stein–Weiss inequality on Morrey spaces for fractional integrals.
