Markov's inequality on Koornwinder's domain in $L^p$ norms

Let $\mathrm{\Omega }=\{(x,y)\in {\mathbb{R}}^2:\mathrm{\mid }x\mathrm{\mid }<y+1,x^2>4y\}$. We prove that the optimal exponent in Markov’s inequality on $\mathrm{\Omega }$ in $L^p$ norms is 4.