The following operators keep the area under the convergent integrals unchanged:
$$\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx=\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)](x)\,dx$$
But with divergent integrals applying the transform $\mathcal{L}_t[t f(t)](x)$ (I would denote it $\mathcal{T}f(x)$) to a function more than once may lead to a strictly greater or smaller function, which I refer to as an unacceptable paradox.
One such example is the function $f(x)=\frac1{\sqrt{x}}$. Applying the transform $\mathcal{T}$ to it twice, one arrives at function $\frac{\pi }{2 \sqrt{x}}$, which is strictly greater than the original function, and the problematic step seems to be the first one. It seems, one class of the problematic functions is $f(x)=\frac1{x^p}$, where $0<p<1$.
So, I wonder, what is the whole description of the class of the functions that lead to such paradoxes. I outline the two paradoxes I am seeking to avoid:
Functions such that $f(x)>\mathcal{T^n}[f(x)]$ or $f(x)<\mathcal{T^n}[f(x)]$ for all $x>0$, that is the function becomes strictly smaller or greater after the transform.
Functions such that the integral $\int_0^\infty (f(x)-\mathcal{T^n}[f(x)])dx$ is finite but nonzero.
Is it possible to clearly describe the class of functions than lead to such paradoxes so to avoid them?
Or should the assumed equivalence be modified to add some other terms?