Weight-k hyperbolic Laplacian (${\Delta }_k$)

The Weight-k hyperbolic Laplacian is a differential operator adapted to weight-$k$ transformation laws on the upper half-plane $z=x+iy$, $y>0$. Using the sign consistent with the nonnegative weight-zero hyperbolic Laplacian, and the holomorphic-weight convention, defineHere ${\partial }_z=\frac{1}{2}({\partial }_x-i{\partial }_y)$ and ${\partial }_{\bar{z}}=\frac{1}{2}({\partial }_x+i{\partial }_y)$ are Wirtinger derivatives. The cited paper uses the opposite overall sign, so its eigenvalues must also have their signs reversed.^[ref]

The correction terms let the operator preserve the transformation law $f((az+b)/(cz+d))=(cz+d{)}^{k}f(z)$ for determinant-one integer matrices, as used for modular forms and Maass forms. At $k=0$ it reduces to the ordinary hyperbolic Laplacian. Every holomorphic function is locally annihilated because ${\partial }_{\bar{z}}f=0$; being holomorphic alone does not supply modularity or growth conditions.

For a direct calculation, take $f(z)=y^s$. Its $x$ derivatives vanish, givingThus both $1$ and $y^{1-k}$ are annihilated; when $k=1$ these coincide and a second independent solution depending only on $y$ is $\log y$. These are local operator examples, not claims that all satisfy a modular transformation law. Conventions using a unit-modulus weight multiplier have a different first-derivative correction, so the displayed formula matters as much as the name.