$L^p$ Liouville theorems for pluriharmonic functions on gradient K\"ahler-Ricci solitons

We study Liouville-type theorems for real-valued pluriharmonic functions on complete gradient K"ahler-Ricci solitons under gradient integrability assumptions. For a complete gradient K"ahler-Ricci soliton $(M,g,J,f)$ and a real-valued pluriharmonic function $u$, we investigate conditions under which $u$ must be constant. By introducing a globally defined holomorphic quantity induced by the soliton potential, we obtain new Liouville-type results beyond the range available for harmonic functions. In the steady case, we prove that $u$ is constant whenever $${\int }_M\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p\mathrm{d}v<\mathrm{\infty }$$ for some $0<p<\mathrm{\infty }$. In the shrinking case, we prove the same conclusion for $0<p\le 2$. Finally, we construct a complete K"ahler example showing that the extension to the range $0<p<1$ relies essentially on the soliton structure and does not hold on general complete K"ahler manifolds.