The question has been motivated by the fact that the $1+1$ massless bosonic free field suffers the infrared problem as a "tempered distribution".

The reason is essentially that $\int_{\mathbb{R}} \frac{dp}{\lvert p \rvert}$ is logarithmically divergent.

Since this is a infrared problem, I am curious whether the issue will be resolved by introducing a infrared cutoff, which is mathematically interpreted as compact supports in the spacetime variable.

In other words, the $1+1$ massless bosonic free field can be defined as a "just distribution" instead of being tempered?

Or more concretely, does the following integral converges for an arbitrary compactly supported smooth function $f(x,y)$ on $\mathbb{R}^2$?:

\begin{equation} \int_{\mathbb{R}^2} dxdy \int_{\mathbb{R}}\frac{dp}{\lvert p \rvert} f(x,y)e^{i(-\lvert p \rvert x+ py)} \end{equation}

It seems nontrivial to evaluate the above integral for me. Could anyone please help?