# Necessary and Sufficient conditions for integrable function [closed]

Suppose that $$a, b$$ and $$c$$ are constant.

Is there the necessary and sufficient conditions of $$a ,b, c$$ for the following integration is integrable? i.e. $$\int_0^\infty \int_0^\infty \int_0^\infty \frac{1}{(1+x)^a (1+y)^b (1+x+t)^c (1+y+t)^c} t^{-\frac{1}{2}}e^{-\frac{1}{t}} \, dx \, dy \, dt < \infty.$$

Let us assume that $$a,b,c$$ are real numbers. The integral in question equals $$$$I=I_1+I_2,$$$$ where $$$$I_1:=\int_0^2 dt\; t^{-1/2}e^{-1/t} J_a(t)J_b(t),\quad I_2:=\int_2^\infty dt\; t^{-1/2}e^{-1/t} J_a(t)J_b(t),$$$$ $$$$J_a(t):=\int_0^\infty\frac{dx}{(1+x)^a(1+x+t)^c}.$$$$ Clearly, if $$I<\infty$$, then $$J_a(t)<\infty$$ for almost all $$t>0$$, whence $$a+c>1$$. Similarly, it is necessary that $$b+c>1$$. So, without loss of generality $$$$a+c>1,\quad b+c>1. \tag{1}$$$$
So, for $$t\in(0,2]$$, $$$$J_a(t)\asymp\int_0^\infty\frac{dx}{(1+x)^a(1+x)^c}\asymp1,$$$$ whence one always has $$$$I_1\asymp1<\infty,$$$$ given the necessary condition (1).
For $$t>2$$, \begin{align} J_a(t)&\asymp\int_0^1\frac{dx}{t^c}+\int_1^t\frac{dx}{x^a t^c}+\int_t^\infty\frac{dx}{x^a x^c} \\ &\asymp t^{-c}+t^{-c}\int_1^t\frac{dx}{x^a}+t^{-c+1-a}, \end{align} whence $$$$J_a(t)\asymp \begin{cases} t^{(1-a)_+-c}&\text{ if } a\ne1,\\ t^{-c}\ln t&\text{ if } a=1. \end{cases}$$$$ Since $$e^{-1/t}\asymp1$$ for $$t>2$$, it follows that $$$$I_2<\infty\iff (1-a)_+-c+(1-b)_+-c-1/2<-1 \iff c>\tfrac14+(1-a)_+ +(1-b)_+.$$$$ In particular, the latter condition implies $$c>(1-a)_+\ge1-a$$ and similarly $$c>1-b$$, so that the necessary condition (1) holds.
Thus, for $$I<\infty$$ it is necessary and sufficient that $$$$c>\tfrac14+(1-a)_+ +(1-b)_+.$$$$