Bilinear fractional integral operators on Morrey spaces

Abstract

We prove a plethora of the boundedness property of Adams type for bilinear fractional integral operators of the form

$$\begin{aligned} B_{\alpha }(f,g)(x)=\int _{{\mathbb {R}}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-\alpha }}dy,\quad 0<\alpha <n.\ \end{aligned}$$

For $1<t\le s<\infty$, we prove the non-weighted case through the known Adams type result. And we show that these results of Adams type is optimal. For $0<t\le s<\infty$ and $0<t\le 1$, we obtain new result of a weighted theory describing Morrey boundedness of above form operators if two weights $(v,\vec {w})$ satisfy

$$\begin{aligned}&{[}v,\vec {w}]_{t,\vec {q}/{a}}^{r,as}=\mathop {\sup _{Q,Q^{\prime }\in {\mathscr {D}}}}_{Q\subset Q^{\prime }}\left( \frac{|Q|}{|Q^{\prime }|}\right) ^{\frac{1-s}{as}}|Q^{\prime }|^{\frac{1}{r}}\left( \fint _{Q}v^{\frac{t}{1-t}}\right) ^{\frac{1-t}{t}}\\&\quad \prod _{i=1}^{2}\left( \fint _{Q^{\prime }}w_{i}^{-(q_{i}/a)^{\prime }}\right) ^{\frac{1}{(q_{i}/a)^{\prime }}}<\infty ,\quad 0<t<s<1 \end{aligned}$$

and

$$\begin{aligned}&{[}v,\vec {w}]_{t,\vec {q}/{a}}^{r,as}:=\mathop {\sup _{Q,Q^{\prime }\in {\mathscr {D}}}}_{Q\subset Q^{\prime }}\left( \frac{|Q|}{|Q^{\prime }|}\right) ^{\frac{1-as}{as}}|Q^{\prime }|^{\frac{1}{r}}\left( \fint _{Q}v^{\frac{t}{1-t}}\right) ^{\frac{1-t}{t}}\\&\quad \prod _{i=1}^{2}\left( \fint _{Q^{\prime }}w_{i}^{-(q_{i}/a)^{\prime }}\right) ^{\frac{1}{(q_{i}/a)^{\prime }}}<\infty , \quad s\ge 1 \end{aligned}$$

where $\Vert v\Vert _{L^{\infty }(Q)}=\sup _{Q}v$ when $t=1$, a, r, s, t and $\vec {q}$ 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.