Knowlpedia

Theorem

Heat smoothing estimate

Gaussian convolution gains spatial derivatives and improves integrability with explicit powers of time.

Statement

For 1≤p≤q≤∞1\le p\le q\le\infty, a spatial multi-index α\alpha, and t>0t>0, the satisfies

∥∂xαTtf∥q≤C(νt)−∣α∣/2−n2(1/p−1/q)∥f∥p.\|\partial_x^\alpha T_tf\|_q \le C(\nu t)^{-|\alpha|/2-\frac n2(1/p-1/q)}\|f\|_p.

The constant depends on n,p,q,αn,p,q,\alpha, and is independent of ν,t,f\nu,t,f.

Kernel estimate

Every Gaussian derivative is a polynomial times a Gaussian. Scaling gives

∥∂αGν(t)∥r=Cα,n,r(νt)−∣α∣/2−n2(1−1/r).\|\partial^\alpha G_\nu(t)\|_r =C_{\alpha,n,r}(\nu t)^{-|\alpha|/2-\frac n2(1-1/r)}.

Choose rr with 1+1/q=1/p+1/r1+1/q=1/p+1/r and apply Young's convolution inequality. The negative powers as t↓0t\downarrow0 express the cost of recovering derivatives from rough initial data.