Schwarz-contractivity of the Poisson operator

We say that the Poisson operator $P_r$ is Schwarz-contractive from a space $X$ to a space $Y$ if $\mathrm{\parallel }{P}_r{\mathrm{\parallel }}_{X\to Y}\le r$ holds for all $0<r<1$. The classical Schwarz lemma is the case $H_0^{\mathrm{\infty }}\to H^{\mathrm{\infty }}$, the subscript indicating zero mean. We are concerned with the non-holomorphic case $L_0^{\mathrm{\infty }}\to L^p$. For real functions Schwarz-contractivity holds up to the sharp exponent $p_{\mathbb{R}}=4.109\dots$. For complex functions we prove it for $p\le 3$ and conjecture that the critical exponent is $4$.