Complex zero-mean Poisson contractivity at exponent four

Prove that the Poisson operator is Schwarz-contractive from the complex-valued zero-mean space L^\infty_0(T) to L^4(T), namely that \int_T |P_r f|^4\le r^4 for every 0<r<1 whenever \|f\|_\infty\le1 and \widehat f(0)=0.

Background

For complex-valued functions with zero mean, the paper proves Schwarz-contractivity of the Poisson operator into L^3(T), while a separate obstruction shows that contractivity fails for every exponent p>4. Thus, if p_C denotes the supremum of admissible exponents, the paper establishes 3\le p_C\le4. The conjecture asserts that the upper endpoint is attained, so that the critical complex exponent is exactly four.

The conjecture is motivated by the authors’ previously established limiting case as r\to0 and by the sharpness of the exponent-four upper bound in that infinitesimal regime.