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~~With what classes of functions the equality $\int_0^\infty f(x)dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) dx$ leads to paradoxes?~~

With what classes of functions does the equality $\int_0^\infty f(x)dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) dx$ lead to paradoxes?

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:
1. 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.
2. 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 that lead to such *paradoxes* so to avoid them?

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 Laplace 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:
1. 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.
2. 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 that lead to such *paradoxes* so to avoid them?