Complex centered Poisson contractivity at exponent three

Prove that the centered Poisson operator \widetilde P_r f=P_r f-\widehat f(0) is Schwarz-contractive from complex-valued L^\infty(T) to L^3(T), namely that \int_T|\widetilde P_r f|^3\le r^3 for every 0<r<1 whenever \|f\|_\infty\le1.

Background

The paper proves the corresponding statement for real-valued functions but notes that the proof does not extend to complex-valued functions. Numerical evidence is reported in support of both the centered L^3 contractivity statement and the key arc-average estimate used in the real-valued proof.

The centered operator removes the constant Fourier component, making this conjecture distinct from the zero-mean complex conjecture for the uncentered Poisson operator.