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
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