28 I. Theory of Distributions

Proof: Let ϕ0(xn) ∈ C0

∞(R1),

∞

−∞

ϕ0(xn)dxn = 1. For an arbi-

trary ϕ(x , xn) ∈ C0

∞(Rn)

set ϕ1(x ) =

∞

−∞

ϕ(x , xn)dxn. Let ψ(x , xn) =

ϕ(x , xn)−ϕ1(x )ϕ0(xn). Then

∞

−∞

ψ(x , xn)dxn = 0 and therefore ψ =

∂ψ1

∂xn

,

where ψ1(x , xn) =

xn

−∞

ϕ(x , t)dt − ϕ1(x )

xn

−∞

ϕ0(t)dt ∈ C0

∞(Rn).

We have

ϕ(x , xn) =

∂ψ1

∂xn

+ ϕ1(x )ϕ0(xn).

Therefore

(7.13) f(ϕ) = f(ϕ1(x )ϕ0(xn)),

since f(

∂ψ1

∂n

) = 0. Denote by f1 the following distribution in D

(Rn−1):

(7.14) f1(a(x )) = f(a(x )ϕ0(xn)), ∀a(x ) ∈ C0

∞(Rn−1).

Then

f(ϕ) = f1(ϕ1(x )).

Since ϕ1(x ) =

∞

−∞

ϕ(x , xn)dxn, we have f1(ϕ1(x )) =

∞

−∞

f1(ϕ(x , xn))dxn

(cf. the proof of Proposition 7.2). Therefore

(7.15) f(ϕ) =

∞

−∞

f1(ϕ(x , xn))dxn,

i.e., f = f1 × 1.

7.4. Partial hypoellipticity.

Let V × (a, b) ⊂ Ω, where Ω is a domain in

Rn,

V is a domain in

Rn−1,

x = (x , xn), x ∈ V, xn ∈ (a, b). We assume that V × [a, b] is compact

and V × [a, b] ⊂ Ω. Let u ∈ D (Ω),f ∈

C∞(V

× [a, b]). Suppose u is a

distribution solution of P (x,

∂

∂x

)u = f in V × (a, b), i.e., u(P (x, −

∂

∂x

)ϕ) =

f(ϕ), ∀ϕ ∈ C0

∞(V

× (a, b)), where P (x,

∂

∂x

) is a differential operator of the

form

(7.16) P x,

∂

∂x

=

∂ru

∂xn

r

+

r−1

k=0

pk x,

∂

∂x

∂k

∂xnk

.

Here pk(x,

∂

∂x

) are differential operators in

∂

∂x

with

C∞(V

× [a, b]) coeﬃ-

cients, x = (x1,...,xn−1).

Proposition 7.6. Let P (x,

∂

∂x

) has the form ( 7.16), u ∈ D (Ω), and Pu = f

in V × (a, b), where f ∈

C∞(V

× [a, b]). Then u is a

C∞

function of

xn ∈ (a, b) with values in D (V ).

This property of solutions of P (x,

∂

∂x

)u = f is called the partial hypoel-

lipticity property.

Proof: We rewrite Pu = f as a system of first order differential op-

erators in

∂

∂xn

. Let w1 = u, w2 =

∂u

∂xn

, . . . , wr =

∂r−1u

∂xn

. Then we have