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Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$ where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+\frac{\theta_{i}}{p}<1$ but $a_{i}q>1$ for any $q>1$.

Consider the quadrilinear form \begin{multline*} Q(f_{1},f_{2},f_{3},f_{4})\\:=\int_{\substack{0<x<\frac{1}{2}<y<1\\ x^2+y^2<1\\ 0<z<\frac{1}{2}<t<1}}\frac{f_{1}(x)f_{2}(y)f_{3}(z)f_{4}(t)}{|x|^{a_{1}-\delta(p)}(1-x^2-y^2)^{a_{2}}|x-z|^{a_{3}}|y-t|^{a_{4}}}dxdydzdt, \end{multline*} where $0<\delta(p)<a_{1}$ is such that $(a_{1}-\delta(p))p<1$ for all $1<p<2$.

Is there a way to show that $Q$ is bounded on $L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}$ for some such exponents $a_{i}$ ?

Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$ where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+\frac{\theta_{i}}{p}<1$ but $a_{i}p>1$ for any $p>1$.

Consider the quadrilinear form \begin{multline*} Q(f_{1},f_{2},f_{3},f_{4})\\:=\int_{\substack{0<x<\frac{1}{2}<y<1\\ x^2+y^2<1\\ 0<z<\frac{1}{2}<t<1}}\frac{f_{1}(x)f_{2}(y)f_{3}(z)f_{4}(t)}{|x|^{a_{1}-\delta(p)}(1-x^2-y^2)^{a_{2}}|x-z|^{a_{3}}|y-t|^{a_{4}}}dxdydzdt, \end{multline*} where $0<\delta(p)<a_{1}$ is such that $(a_{1}-\delta(p))p<1$ for all $1<p<2$.

Is there a way to show that $Q$ is bounded on $L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}$ for some such exponents $a_{i}$ ?

Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$ where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}<1$ but $a_{i}q>1$ for any $q>1$.

Consider the quadrilinear form \begin{multline*} Q(f_{1},f_{2},f_{3},f_{4})\\:=\int_{\substack{0<x<\frac{1}{2}<y<1\\ x^2+y^2<1\\ 0<z<\frac{1}{2}<t<1}}\frac{f_{1}(x)f_{2}(y)f_{3}(z)f_{4}(t)}{|x|^{a_{1}-\delta(p)}(1-x^2-y^2)^{a_{2}}|x-z|^{a_{3}}|y-t|^{a_{4}}}dxdydzdt, \end{multline*} where $0<\delta(p)<a_{1}$ is such that $(a_{1}-\delta(p))p<1$ for all $1<p<2$.

Is there a way to show that $Q$ is bounded on $L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}$ for some such exponents $a_{i}$ ?

Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$ where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+\frac{\theta_{i}}{p}<1$ but $a_{i}q>1$ for any $q>1$.

Consider the quadrilinear form \begin{multline*} Q(f_{1},f_{2},f_{3},f_{4})\\:=\int_{\substack{0<x<\frac{1}{2}<y<1\\ x^2+y^2<1\\ 0<z<\frac{1}{2}<t<1}}\frac{f_{1}(x)f_{2}(y)f_{3}(z)f_{4}(t)}{|x|^{a_{1}-\delta(p)}(1-x^2-y^2)^{a_{2}}|x-z|^{a_{3}}|y-t|^{a_{4}}}dxdydzdt, \end{multline*} where $0<\delta(p)<a_{1}$ is such that $(a_{1}-\delta(p))p<1$ for all $1<p<2$.

Is there a way to show that $Q$ is bounded on $L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}$ for some such exponents $a_{i}$ ?

Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$ where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}<1$ but $a_{i}q>1$ for any $q>1$.

Consider the quadrilinear form \begin{multline*} Q(f_{1},f_{2},f_{3},f_{4})\\:=\int_{\substack{0<x<\frac{1}{2}<y<1\\ x^2+y^2<1\\ 0<z<\frac{1}{2}<t<1}}\frac{f_{1}(x)f_{2}(y)f_{3}(z)f_{4}(t)}{|x|^{a_{1}-\delta(p)}(1-x^2-y^2)^{a_{2}}|x-z|^{a_{3}}|y-t|^{a_{4}}}dxdydzdt, \end{multline*} where $0<\delta(p)<a_{1}$ is such that $a_{1}p<1$ for all $1<p<2$.

Is there a way to show that $Q$ is bounded on $L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}$ for some such exponents $a_{i}$ ?

Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$ where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}<1$ but $a_{i}q>1$ for any $q>1$.

Consider the quadrilinear form \begin{multline*} Q(f_{1},f_{2},f_{3},f_{4})\\:=\int_{\substack{0<x<\frac{1}{2}<y<1\\ x^2+y^2<1\\ 0<z<\frac{1}{2}<t<1}}\frac{f_{1}(x)f_{2}(y)f_{3}(z)f_{4}(t)}{|x|^{a_{1}-\delta(p)}(1-x^2-y^2)^{a_{2}}|x-z|^{a_{3}}|y-t|^{a_{4}}}dxdydzdt, \end{multline*} where $0<\delta(p)<a_{1}$ is such that $(a_{1}-\delta(p))p<1$ for all $1<p<2$.

Is there a way to show that $Q$ is bounded on $L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}\times L^{p^{\prime}}$ for some such exponents $a_{i}$ ?