Positivity ( IF 1 ) Pub Date : 2020-06-05 , DOI: 10.1007/s11117-020-00763-9 Qianjun He , Dunyan Yan
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.
中文翻译：
Morrey空间上的双线性分数积分算子
我们证明了形式为双线性分数积分算子的亚当斯类型的有界性的过多
$$ \ begin {aligned} B _ {\ alpha}（f，g）（x）= \ int _ {{\ mathbb {R}} ^ {n}} \ frac {f（xy）g（x + y） } {| y | ^ {n- \ alpha}} dy，\ quad 0 <\ alpha <n。\ \ end {aligned} $$对于\（1 <t \ le s <\ infty \），我们通过已知的Adams类型结果证明了非加权情况。并且我们证明了Adams类型的这些结果是最优的。对于\（0 <t \ le s <\ infty \）和\（0 <t \ le 1 \），我们得到了加权理论的新结果，该理论描述了如果两个权重\（（v，\ vec {w}）\）满足
$$ \ begin {aligned}＆{[} v，\ vec {w}] _ {t，\ vec {q} / {a}} ^ {r，as} = \ mathop {\ sup _ {Q，Q ^ {\ prime} \ in {\ mathscr {D}}}} _ {Q \ subset Q ^ {\ prime}} \ left（\ frac {| Q |} {| Q ^ {\ prime} |} \ right ）^ {\ frac {1-s} {as}} | Q ^ {\ prime} | ^ {\ frac {1} {r}} \ left（\ fint _ {Q} v ^ {\ frac {t} {1-t}} \ right）^ {\ frac {1-t} {t}} \\＆\ quad \ prod _ {i = 1} ^ {2} \ left（\ fint _ {Q ^ {\质数}} w_ {i} ^ {-（q_ {i} / a）^ {\ prime}} \ right）^ {\ frac {1} {（q_ {i} / a）^ {\ prime}}} <\ infty，\ quad 0 <t <s <1 \ end {aligned} $$和
$$ \ begin {aligned}＆{[} v，\ vec {w}] _ {t，\ vec {q} / {a}} ^ {r，as}：= \ mathop {\ sup _ {Q， Q ^ {\ prime} \ in {\ mathscr {D}}}} _ {Q \ subset Q ^ {\ prime}} \ left（\ frac {| Q |} {| Q ^ {\ prime} |} \ right）^ {\ frac {1-as} {as}} | Q ^ {\ prime} | ^ {\ frac {1} {r}} \ left（\ fint _ {Q} v ^ {\ frac {t } {1-t}} \ right）^ {\ frac {1-t} {t}} \\＆\ quad \ prod _ {i = 1} ^ {2} \ left（\ fint _ {Q ^ { \ prime}} w_ {i} ^ {-（q_ {i} / a）^ {\ prime}} \ right）^ {\ frac {1} {{q_ {i} / a）^ {\ prime}} } <\ infty，\ quad s \ ge 1 \ end {aligned} $$其中\（\ Vert的v \ Vert的_ {L ^ {\ infty}（Q）} = \ SUP _ {Q}伏\）时\（T = 1 \） ，一个，- [R ，s ^，吨和\（\ vec {q} \）满足适当条件。在某些应用中，我们在分数积分的Morrey空间上建立了Olsen不等式，Fefferman-Stein型对偶不等式和Stein-Weiss不等式的双线性形式。
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