2.3. HEAT EQUATION 61 Since |v|, |˜| v ≤ A for some constant A, Theorem 7 implies v ≡ ˜ v that is, (36) v(x, t) = t 0 Rn Φ(x − y, t − s)˜(y, f s) dyds. Now suppose (x, t) ∈ C . As ζ ≡ 0 off the cylinder C, (34) and (36) imply u(x, t) = C Φ(x − y, t − s)[(ζs(y, s) − Δζ(y, s))u(y, s) − 2Dζ(y, s) · Du(y, s)] dyds. Note in this equation that the expression in the square brackets vanishes in some region near the singularity of Φ. Integrate the last term by parts: (37) u(x, t) = C [Φ(x − y, t − s)(ζs(y, s) + Δζ(y, s)) + 2DyΦ(x − y, t − s) · Dζ(y, s)]u(y, s) dyds. We have proved this formula assuming u ∈ C∞. If u satisfies only the hypotheses of the theorem, we derive (37) with uε = ηε ∗ u replacing u, ηε being the standard mollifier in the variables x and t, and let ε → 0. 3. Formula (37) has the form (38) u(x, t) = C K(x, t, y, s)u(y, s) dyds ((x, t) ∈ C ), where K(x, t, y, s) = 0 for all points (y, s) ∈ C , since ζ ≡ 1 on C . Note also K is smooth on C − C . In view of expression (38), we see u is C∞ within C = C(x0,t0 1 2 r). c. Local estimates for solutions of the heat equation. Let us now record some estimates on the derivatives of solutions to the heat equa- tion, paying attention to the differences between derivatives with respect to xi (i = 1,...,n) and with respect to t. THEOREM 9 (Estimates on derivatives). There exists for each pair of integers k, l = 0, 1,... a constant Ck,l such that max C(x,t r/2) |DxDtu| k l ≤ Ckl rk+2l+n+2 u L1(C(x,t r)) for all cylinders C(x, t r/2) ⊂ C(x, t r) ⊂ UT and all solutions u of the heat equation in UT .

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