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In my research, I need to evaluate an integral:

$\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}log(|\xi^{2}|+\frac{1}{4})d\xi$, here $x\in R^{3}$, $y\geq0$. Moreover, I want to see whether it is a biharmonic function on the up-half space or not. I also want to know will $log(|x|^{2}+(y+\frac{1}{2})^{2})-c\int_{R^{3}} \frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}log(|\xi^{2}|+\frac{1}{4})d\xi$ be bounded on the up half space? here c is a constant such that $c\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}d\xi=1$

In my research, I need to evaluate an integral: $$\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}\log(|\xi^{2}|+\frac{1}{4})d\xi$$ where $x\in R^{3}$, $y\geq0$. Moreover, I want to see whether it is a biharmonic function on the up-half space or not. I also want to know will $$ \log(|x|^{2}+(y+\frac{1}{2})^{2})-c\int_{R^{3}} \frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}} \log(|\xi^{2}|+\frac{1}{4})d\xi$$ be bounded on the up half space? Here $c$ is a constant such that $$c\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}d\xi=1$$

In my research, I need to evaluate an integral:

$\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}log(|\xi^{2}|+\frac{1}{4})d\xi$, here $x\in R^{3}$, $y\geq0$. Moreover, I want to see whether it is a biharmonic function on the up-half space or not. I also want to know will $log(|x|^{2}+(y+\frac{1}{2})^{2})-\int_{R^{3}} \frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}log(|\xi^{2}|+\frac{1}{4})d\xi$ be bounded on the up half space?

In my research, I need to evaluate an integral:

$\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}log(|\xi^{2}|+\frac{1}{4})d\xi$, here $x\in R^{3}$, $y\geq0$. Moreover, I want to see whether it is a biharmonic function on the up-half space or not. I also want to know will $log(|x|^{2}+(y+\frac{1}{2})^{2})-c\int_{R^{3}} \frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}log(|\xi^{2}|+\frac{1}{4})d\xi$ be bounded on the up half space? here c is a constant such that $c\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}d\xi=1$

In my research, I need to evaluate an integral:

$\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}log(|\xi^{2}|+\frac{1}{4})d\xi$, here $x\in R^{3}$, $y>0$. Moreover, I want to see whether it is a biharmonic function on the up-half space or not. Of course,

$$ \Delta^{2}_{x,y}(\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}})=0 $$

however in order to make the interchange between integral and derivative work, one needs kind fast decay of the function, I am not sure in this case. Also, I want to know if the function
$$ log(|x|^{2}+(y+\frac{1}{2})^{2})-\int_{R^{3}} \frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}log(|\xi^{2}|+\frac{1}{4})d\xi$$ is bounded?

Thanks a lot.

In my research, I need to evaluate an integral:

$\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}log(|\xi^{2}|+\frac{1}{4})d\xi$, here $x\in R^{3}$, $y\geq0$. Moreover, I want to see whether it is a biharmonic function on the up-half space or not. I also want to know will $log(|x|^{2}+(y+\frac{1}{2})^{2})-\int_{R^{3}} \frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}log(|\xi^{2}|+\frac{1}{4})d\xi$ be bounded on the up half space?