Let's first consider the following Dirichlet problem on the upper half-space $\mathbb H^n=\{(x_1,\ldots, x_n)\in \mathbb R^n:x_n>0\}$. $$ \Delta u =0, u|_{x_n=0}=g(x). $$ Performing Fourier transform on $\Delta u=0$, changing the variables as $x_i\mapsto \xi_i, 1\leq i\leq n-1$, we get $$ i^{2} (\xi_1^2+ \xi_2^2+\ldots +\xi_{n-1}^2) (\mathcal F_{\partial\mathbb H_n} u)=-\partial^2_{x_{n}x_n}(\mathcal F_{\partial\mathbb H_n} u). $$ If we let $\xi=(\xi_1, \xi_2,\ldots ,\xi_{n-1}) ,|\xi|=\sqrt{\xi_1^2+ \xi_2^2+\ldots +\xi_{n-1}^2}$, then this is very easy to solve: $$ (\mathcal F_{\partial\mathbb H_n} u) (\xi,x_n)=A_+(\xi) \exp{(|\xi|x_n)}+A_-(\xi) \exp{(-|\xi|x_n)} $$ We now do the same transform on the boundary data. We have $$ \mathcal F_{\partial\mathbb H_n} u|_{x_n=0}=\mathcal F_{\partial\mathbb H_n} g(\xi_1,\ldots,\xi_{n-1}). $$ So, for even dimensions, the solution is unique if we limit ourselves to bounded solutions. In this case, $$ (\mathcal F_{\partial\mathbb H_n} u) (\xi,x_n)=(\mathcal F_{\partial\mathbb H_n} g)(\xi)\exp{(-|\xi|x_n)}. $$ To find an expression of $u$ in terms of an integral, we just evaluate the fourier transform of $\exp{(-|\xi|x_n)}$, and we get the solution $u$ in terms of a convolution: $$ u(x)=(g* P_{x_n}) (x_1\ldots, x_{n-1}), $$ where $P_{x_n}(x_1\ldots, x_{n-1})$ is the Poisson kernel.
Now, my question is, could I deduce in a similar fashion an explicit expression for the Poisson kernel for the unit ball? In other words, I am aiming to find $P_x(y)$ such that for all $y\in B(0,1)$, the function $$ u(y)=\int_{\partial B(0,1)} f(x) P_x(y) d\sigma $$ solves the Dirichlet problem $\Delta u =0, u|_{\partial B(0,1)}=f(x)$.
I have considered a conformal isomorphism $\phi:B(0,1)\to \mathbb H^n$, but the composition of a harmonic function with such a conformal map is not necessarily harmonic in higher dimensions.
Also, in odd dimensions, even the condition that $u$ is bounded could not gaurantee a unique solution, so I could use this idea in this case.
Edit: I already know other ways of reaching the Poisson Kernel; I just wish to see if we could use Fourier transform to derive it.
First, FT wrt variables $x_i$, $1\leq i\leq n-1 $, gives sum $-\xi_1^2-\ldots-\xi_{n-1}^2$, not a product. Second, For a ball usually one gets a formula for the Green function $G$ of the Dirichlet problem in a ball and then the integral representation of a harmonic function in a ball gives that the kernel you seek is $P_x(y)=\frac{\partial G(y,x)}{\partial r_x}$. All this is written in PDE textbooks.