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

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For each 1≤p≤∞1\le p\le\infty1≤p≤∞ and each t>0t>0t>0,

Gtf=Kt∗f,Kt(x)=(4πνt)−3/2e−∣x∣2/(4νt),G_t f = K_t * f, \qquad K_t(x) = (4\pi\nu t)^{-3/2} e^{-|x|^2/(4\nu t)},Gtf=Kt∗f,Kt(x)=(4πνt)−3/2e−∣x∣2/(4νt),
and

∥Gtf∥Lp≤∥f∥Lp(in fact ∥Gt∥Lp→Lp=1).\|G_t f\|_{L^p} \le \|f\|_{L^p}\qquad(\text{in fact }\|G_t\|_{L^p\to L^p}=1).∥Gtf∥Lp≤∥f∥Lp(in fact ∥Gt∥Lp→Lp=1).
Proof (sketch). The heat kernel KtK_tKt is nonnegative and has unit mass ∫Kt=1\int K_t=1∫Kt=1. By Young’s convolution inequality,

∥Gtf∥Lp=∥Kt∗f∥Lp≤∥Kt∥L1∥f∥Lp=∥f∥Lp.\|G_t f\|_{L^p} = \|K_t * f\|_{L^p} \le \|K_t\|_{L^1}\|f\|_{L^p} = \|f\|_{L^p}.∥Gtf∥Lp=∥Kt∗f∥Lp≤∥Kt∥L1∥f∥Lp=∥f∥Lp.
For p=∞p=\inftyp=∞ the same follows from ∥Kt∥L1=1\|K_t\|_{L^1}=1∥Kt∥L1=1.
(Strict contractivity may hold on some spaces; the important fact is boundedness with norm ≤1\le1≤1.)

References. Standard: Evans, Partial Differential Equations, §2.3; Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations; also Henry, Geometric Theory of Semilinear Parabolic Equations.

===== For any s∈Rs\in\mathbb{R}s∈R and any r≥0r\ge0r≥0, =====

∥Gtf∥Hs+r≤C t−r/2 ∥f∥Hs,t>0,\|G_t f\|_{H^{s+r}} \le C\, t^{-r/2}\,\|f\|_{H^s},\qquad t>0,∥Gtf∥Hs+r≤Ct−r/2∥f∥Hs,t>0,
where C=C(ν,r)C=C(\nu,r)C=C(ν,r). Equivalently, for integer k≥0k\ge0k≥0,

∥∇kGtf∥L2≤C t−k/2 ∥f∥L2.\|\nabla^k G_t f\|_{L^2} \le C\, t^{-k/2}\,\|f\|_{L^2}.∥∇kGtf∥L2≤Ct−k/2∥f∥L2.
Proof (Fourier). Using the Fourier transform Gtf^(ξ)=e−νt∣ξ∣2f^(ξ)\widehat{G_t f}(\xi)=e^{-\nu t|\xi|^2}\widehat f(\xi)Gtf(ξ)=e−νt∣ξ∣2f(ξ),

∥Gtf∥Hs+r2=∫R3(1+∣ξ∣2)s+re−2νt∣ξ∣2∣f^(ξ)∣2 dξ.\|G_t f\|_{H^{s+r}}^2 = \int_{\mathbb{R}^3} (1+|\xi|^2)^{s+r} e^{-2\nu t|\xi|^2} |\widehat f(\xi)|^2\,d\xi.∥Gtf∥Hs+r2=∫R3(1+∣ξ∣2)s+re−2νt∣ξ∣2∣f(ξ)∣2dξ.
Factor (1+∣ξ∣2)re−2νt∣ξ∣2≤C t−r(1+|\xi|^2)^{r} e^{-2\nu t|\xi|^2} \le C\, t^{-r}(1+∣ξ∣2)re−2νt∣ξ∣2≤Ct−r (supremum over ξ\xiξ is O(t−r)O(t^{-r})O(t−r)). Thus the integral is ≤Ct−r∥f∥Hs2\le C t^{-r}\|f\|_{H^s}^2≤Ct−r∥f∥Hs2. Take square root to get the stated estimate.

References. Standard Fourier-semigroup calculation; see Pazy (semigroup theory), Evans §2.3, and Temam, Navier–Stokes Equations.

===== GtG_tGt is a strongly continuous semigroup on LpL^pLp and on HsH^sHs: for each fff fixed, =====

lim⁡t↓0∥Gtf−f∥X=0,X=Lp or Hs.\lim_{t\downarrow 0} \|G_t f - f\|_{X} = 0,\qquad X=L^p\ \text{or}\ H^s.t↓0lim∥Gtf−f∥X=0,X=Lp or Hs.
Proof (sketch). For LpL^pLp this follows from the approximation-identity property of KtK_tKt as t→0t\to0t→0. For HsH^sHs, use the Fourier representation: (1+∣ξ∣2)s/2(e−νt∣ξ∣2−1)f^(ξ)→0(1+|\xi|^2)^{s/2}(e^{-\nu t|\xi|^2}-1)\widehat f(\xi)\to0(1+∣ξ∣2)s/2(e−νt∣ξ∣2−1)f(ξ)→0 pointwise and dominated convergence given (1+∣ξ∣2)s∣f^∣2∈L1(1+|\xi|^2)^{s}|\widehat f|^2\in L^1(1+∣ξ∣2)s∣f∣2∈L1. Thus strong continuity holds.

References. Pazy, Semigroups; see also functional-analytic texts and Evans.

===== For 1≤p≤q≤∞1\le p\le q\le\infty1≤p≤q≤∞, =====

∥Gtf∥Lq≤Ct−32(1p−1q)∥f∥Lp,\|G_t f\|_{L^q} \le C t^{-\frac{3}{2}(\frac{1}{p}-\frac{1}{q})} \|f\|_{L^p},∥Gtf∥Lq≤Ct−23(p1−q1)∥f∥Lp,
with the usual Gaussian scaling. This provides short-time control near t=0t=0t=0.

Proof. Estimate KtK_tKt in LrL^rLr norms and apply Young inequality. Standard.

References. Beckner/Young/heat kernel estimates; see Stein, Harmonic Analysis or Davies, Heat Kernels and Spectral Theory.