Integrable function [closed]

Question

Suppose that $a, b, c_1$ and $c_2$ are real constant.

Is there the necessary and sufficient conditions of $a ,b, c_1,c_2 $ for the following integration is integrable? i.e. $$\int_1^{\infty}\int_1^{\infty}\int_1^{\infty}\frac{1}{~x^{a}~y^{b}~(x+y)^{c_1}~(x+y+t)^{c_2}}~t^{-\frac{1}{2}}e^{-\frac{1}{t}} dx dy dt < \infty.$$

#

# I know the necessary and sufficient conditions of $a ,b, c_1$ for

$$\int_1^{\infty}\int_1^{\infty}\frac{1}{~x^{a}~y^{b}~(x+y)^{c_1}}dx dy < \infty$$

is $a+c_1>1, b+c_1>1$ and $a+b+c_1>2.$

Answer 1

The factor $e^{-1/t}$ is $\asymp1$ for $t>1$, and so, it may be dropped. So, the integral in question is finite iff $I_j<\infty$ for all $j=1,\dots,6$, where \begin{equation} I_j:=\iiint\limits_{R_j}\frac{dx\,dy\,dt}{x^a\,y^b\,(x+y)^{c_1}\,(x+y+t)^{c_2}\,t^{1/2}}, \end{equation} \begin{align} R_1&:=\{(x,y,t)\colon 1<x<y<t\}, \\ R_2&:=\{(x,y,t)\colon 1<y<x<t\}, \\ R_3&:=\{(x,y,t)\colon 1<x<t<y\}, \\ R_4&:=\{(x,y,t)\colon 1<y<t<x\}, \\ R_5&:=\{(x,y,t)\colon 1<t<x<y\}, \\ R_6&:=\{(x,y,t)\colon 1<t<y<x\}. \end{align}

We have \begin{equation} I_1\asymp\int_1^\infty\frac{dx}{x^a}\int_x^\infty\frac{dy}{y^{b+c_1}} \int_y^\infty\frac{dt}{t^{c_2+1/2}}, \end{equation} so that $I_1<\infty$ iff \begin{equation} c_2+1/2>1,\quad b+c_1+c_2>3/2,\quad a+b+c_1+c_2>5/2. \end{equation} Similarly, $I_2<\infty$ iff \begin{equation} c_2+1/2>1,\quad a+c_1+c_2>3/2,\quad a+b+c_1+c_2>5/2. \end{equation}

The integrals $I_3,\dots,I_6$ are treated similarly. As the result, the integral in question is finite iff \begin{equation} c_2+1/2>1,\quad a+c_1+c_2>3/2,\quad b+c_1+c_2>3/2,\quad a+b+c_1+c_2>5/2. \end{equation}