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 .