Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

===== The Leray projector PPP extends to a bounded linear operator on Lp(R3;R3)L^p(\mathbb{R}^3;\mathbb{R}^3)Lp(R3;R3) for every 1<p<∞1<p<\infty1<p<∞. More precisely, the components of PPP are combinations of singular integral operators (Riesz transforms) and the identity; hence Calderón–Zygmund theory applies. =====

Proof (sketch). Write I−ξ⊗ξ/∣ξ∣2=I−ξiξj∣ξ∣2I - \xi\otimes\xi/|\xi|^2 = I - \frac{\xi_i\xi_j}{|\xi|^2}I−ξ⊗ξ/∣ξ∣2=I−∣ξ∣2ξiξj. The scalar multipliers ξk/∣ξ∣\xi_k/|\xi|ξk/∣ξ∣ correspond to Riesz transforms Rk\mathcal{R}_kRk whose kernels are homogeneous of degree −3-3−3 and satisfy Calderón–Zygmund kernel bounds. The entries of P\mathbb{P}P are linear combinations of products of Riesz transforms (order-zero multipliers). By Calderón–Zygmund theory (or the Mikhlin multiplier theorem), these are bounded on LpL^pLp for 1<p<∞1<p<\infty1<p<∞.

References. Stein, Singular Integrals and Differentiability Properties of Functions (Riesz transforms & CZ theory); Grafakos, Classical Fourier Analysis (Mikhlin theorem, multipliers); also Temam or Constantin–Foias for Navier–Stokes.