arXiv:2611.13285v1 Announce Type: new Abstract: For $1<p<2$ and $c>0$, we determine the exact norm of the weighted analytic--coanalytic projection operator $F\mapsto(P_+F,\sqrt c\,P_-F)$ from $L^p(\T)$ to $L^p(\T;\ell_2^2)$. The sharp constant is \[ a_{p,c}=\left(\frac{1+c+\sqrt{1+c^2+2c\cos(2\pi/p)}}{2\sin^2(\pi/p)}\right)^{1/2}. \] We construct an explicit plurisubharmonic minorant of the corresponding pointwise obstacle and obtain an origin-corrected inequality for arbitrary harmonic decompositions. The uncorrected inequality holds under an optimal angular condition at the origin. The scalar estimate underlying the construction is proved by angular minimization and a two-sided comparison with solutions of $y''=y$. An explicit family establishes sharpness.
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