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

==== For every real sss and every real r≥0r\ge0r≥0 there exists C=C(ν,r)C=C(\nu,r)C=C(ν,r) such that for all t>0t>0t>0 and all fff with finite HsH^{s}Hs-norm, ====

∥Gtf∥Hs+r  ≤  C t−r/2 ∥f∥Hs.\boxed{\qquad
\|G_t f\|_{H^{s+r}} \;\le\; C\,t^{-r/2}\,\|f\|_{H^s}.
\qquad}∥Gtf∥Hs+r≤Ct−r/2∥f∥Hs.
Consequences:
* If f∈Hsf\in H^sf∈Hs (or L2L^2L2, or a distribution), then for every t>0t>0t>0 the function GtfG_t fGtf is C∞C^\inftyC∞ in xxx.
* More concretely, for each multiindex α\alphaα there is CαC_\alphaCα with ∥∂xαGtf∥L2≤Cα t−∣α∣/2 ∥f∥L2.\|\partial_x^\alpha G_t f\|_{L^2} \le C_\alpha\, t^{-|\alpha|/2}\,\|f\|_{L^2}.∥∂xαGtf∥L2≤Cαt−∣α∣/2∥f∥L2.
'' Kernel form: for each multiindex α\alphaα there is CαC_\alphaCα such that ∣∂xαKt(x)∣≤Cα t−(3+∣α∣)/2exp⁡ ⁣(−∣x∣2/(8νt)).|\partial_x^\alpha K_t(x)| \le C_\alpha\, t^{-(3+|\alpha|)/2}\exp\!\big(-|x|^2/(8\nu t)\big).∣∂xαKt(x)∣≤Cαt−(3+∣α∣)/2exp(−∣x∣2/(8νt)). Hence Gtf=Kt∗fG_t f=K_t''fGtf=Kt∗f is C∞C^\inftyC∞ for all t>0t>0t>0 even if fff is merely a tempered distribution.