Lemma 9.11.5199DF4FD89931

Proof. With the normalization of $\mathrm{dd}^{\mathrm{c}}$ fixed in the Conventions, $\mathrm{dd}^{\mathrm{c}}v=(\Delta v)/(4\pi )\, dA$ and $\mathrm{dd}^{\mathrm{c}}\log |z|^2=\delta _0$, as in lemma 5.37.18. Cutting off this logarithm in a coordinate disk, with a cutoff equal to one near zero, gives a function $L$ on $E$ with $\mathrm{dd}^{\mathrm{c}}L=\delta _0+q\beta _0$, where $q$ is smooth and $\int _ Eq\beta _0=-1$ by Stokes' theorem.

Let $e_\xi $ be the characters of the torus, with $-\Delta e_\xi =\lambda _\xi e_\xi $. The nonconstant characters have $\lambda _\xi >0$, growing quadratically on the dual lattice. Define the correction $S$, of mean zero, by its Fourier coefficients

where $(q+1)_\xi $ is the Fourier coefficient of $q+1$. Integration by parts shows that these coefficients decay faster than any power of $\lambda _\xi $. Hence the series and all its derivatives converge uniformly, and $\mathrm{dd}^{\mathrm{c}}S=-(q+1)\beta _0$. Conjugate symmetry of the coefficients makes $S$ real. Subtracting the mean of $L+S$ from $L+S$ proves (9.11.52). In particular, $G$ belongs to $L^ r$ for every finite $r$.

The convolution uses the group law and the Haar probability $\beta _0$. Write $K=G*(r\beta _0)$. Testing against smooth functions and using Fubini's theorem proves its distributional equation. Smooth input gives smooth output by the same Fourier argument. For $p'=p/(p-1)$, Hölder's inequality gives

Approximating $G$ by smooth functions shows that translation is continuous in $L^{p'}$, and hence that $K$ is continuous. $\square$