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

==== Recall the matrix-valued Fourier multiplier for PPP: ====

Pf^(ξ)=(I−ξ⊗ξ∣ξ∣2)f^(ξ)=:P(ξ)f^(ξ).\widehat{P f}(\xi) = \Big(I - \frac{\xi\otimes\xi}{|\xi|^2}\Big)\widehat f(\xi) =:\mathbb{P}(\xi)\widehat f(\xi).Pf(ξ)=(I−∣ξ∣2ξ⊗ξ)f(ξ)=:P(ξ)f(ξ).
Each component of P(ξ)\mathbb{P}(\xi)P(ξ) is a homogeneous rational function of degree 000 in ξ\xiξ (symbol of order 000).

===== 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.

===== PPP is bounded on Hs(R3)H^s(\mathbb{R}^3)Hs(R3) for every real sss; i.e. =====

∥Pf∥Hs≤Cs∥f∥Hs.\|P f\|_{H^s} \le C_s \|f\|_{H^s}.∥Pf∥Hs≤Cs∥f∥Hs.
Proof (Fourier / multiplier). PPP is a Fourier multiplier with symbol P(ξ)\mathbb{P}(\xi)P(ξ) that is smooth away from ξ=0\xi=0ξ=0 and homogeneous of degree 000. For an HsH^sHs-norm,

∥Pf∥Hs2=∫(1+∣ξ∣2)s∣P(ξ)f^(ξ)∣2 dξ≤sup⁡ξ∥P(ξ)∥2(1+∣ξ∣2)s∣f^(ξ)∣2 dξ=C∥f∥Hs2,\|P f\|_{H^s}^2 = \int (1+|\xi|^2)^s |\mathbb{P}(\xi)\widehat f(\xi)|^2\,d\xi
\le \sup_\xi \|\mathbb{P}(\xi)\|^2 (1+|\xi|^2)^s |\widehat f(\xi)|^2\,d\xi
= C \|f\|_{H^s}^2,∥Pf∥Hs2=∫(1+∣ξ∣2)s∣P(ξ)f(ξ)∣2dξ≤ξsup∥P(ξ)∥2(1+∣ξ∣2)s∣f(ξ)∣2dξ=C∥f∥Hs2,
since P(ξ)\mathbb{P}(\xi)P(ξ) is bounded (matrix entries are bounded by 1). More refined multiplier theorems (Mikhlin) give boundedness on Ws,pW^{s,p}Ws,p and Besov spaces.

References. Mikhlin multiplier theorem (Grafakos), Stein; for direct PDE context see Temam, Navier–Stokes Equations or Lions, Quelques méthodes.

===== By Littlewood–Paley theory and multiplier theorems, PPP is bounded on Besov spaces Bp,qsB^s_{p,q}Bp,qs and Hölder spaces CαC^\alphaCα (appropriate ranges) — essentially because PPP is an order-zero Fourier multiplier satisfying the standard symbol estimates. =====

References. Bahouri–Chemin–Danchin, Fourier Analysis and Nonlinear Partial Differential Equations; Triebel, Theory of Function Spaces.