Green’s function of poisson’s equation in general dimensions

Question

Green’s function of the Helmholtz equation, in the general dimension D, can be calculated as follows: \begin{align} G(\mathbf{r})&=\int \frac{d^D{k}}{(2\pi)^D} \frac{e^{-i \mathbf{k}\mathbf{r}}}{\mathbf{k}^2+m^2}\\ &=\int \frac{d^D{k}}{(2\pi)^D} e^{-i \mathbf{k}\mathbf{r}} \int_0^\infty dt\ e^{-t(\mathbf{k}^2+m^2)}\\ &= \frac{1}{2\pi}\Big(\frac{m}{2\pi r} \Big)^{\frac{D}{2}-1} K_{\frac{D}{2}-1}(mr). \end{align} I’m looking for the same representation about Poisson’s equation, but it is difficult because the similar integral is divergent. In fact, \begin{align} G(\mathbf{r})&=\int \frac{d^D{k}}{(2\pi)^D} \frac{e^{-i \mathbf{k}\mathbf{r}}}{\mathbf{k}^2}\\ &=\int \frac{d^D{k}}{(2\pi)^D} e^{-i \mathbf{k}\mathbf{r}} \int_0^\infty dt\ e^{-t\mathbf{k}^2}\\ &=\frac{1}{(4\pi)^{\frac{D}{2}}}\int_0^\infty dt \frac{1}{t^{\frac{D}{2}}} e^{-\frac{r^2}{4t}}\\ \end{align} is divergent especially when D=1 or 2. Does anybody know about the closed expression or treatment of this divergent integral which can be used even in one or two dimension? Of course, I know the other way to calculate Green’s function in low dimensions. I want to know how to treat this integral. I guess this integral needs some regularization, but I don’t know.