Parabolic oblique derivative problem in generalized Morrey spaces

0.17 MB·English

by  Lubomira Softova

PARABOLIC OBLIQUE DERIVATIVE PROBLEM IN
GENERALIZED MORREY SPACES
3 LUBOMIRA G. SOFTOVA
1
0
2

n
a Abstract. Westudytheregularityofthesolutionsoftheoblique
J derivative problem for linear uniformly parabolic equations with

3 VMO coefficients. We show that if the right-hand side of the
parabolic equation belongs to certain generalized Morrey space

] Mp,ϕ(Q)thanthestrongsolutionbelongstothegeneralizedSobolev-
P
◦
A Morrey space Wp,ϕ(Q).
2,1
.
h
t 1. Introduction
a
m
In the present work we consider the regular oblique derivative prob-
[
lem for linear non-divergence form parabolic equations in a cylinder

1
v u aij(x)D u = f(x) a.e. in Q,
t ij
8 −
8 u(x′,0) = 0 on Ω,
3 ∂u/∂ℓ = ℓi(x)D u = 0 on S.
0 i
1. The unique strong solvability of this problem was proved in [23]. In
0 [24] we study the regularity of the solution in the Morrey spaces Lp,λ
3
1 with p (1, ), λ (0,n+2) and also its H¨older regularity. In [26] we
∈ ∞ ∈
: extend these studies on generalized Morrey spaces Lp,ω with a weight
v
i ω satisfying the doubling and integral conditions introduced in [18, 20].
X
The approach associated to the names of Caldero´n and Zygmund and
r
a developed by Chiarenza, Frasca and Longo in [7, 8] consists of obtain-
ing of explicit representation formula for the higher order derivatives
of the solution by singular and nonsingular integrals. Further the regu-
larity properties of the solution follows by the continuity properties of
these integrals in the corresponding spaces. In [24] and [25] we study
the regularity of the corresponding operators in the Morrey and gen-
eralized Morrey spaces while in [23] we dispose of the corresponding
results obtained in Lp by [9] and [5]. In recent works we study the reg-
ularity of the solutions of elliptic andparabolic problems with Dirichlet
1991 Mathematics Subject Classification. 35K20,35D35,35B45, 35R05.
Keywords andphrases. Uniformlyparabolicoperator,regularobliquederivative
problem, VMO, generalized Morrey spaces, unique solvability.
1
2 L.G. SOFTOVA
data on the boundary in Morrey-type spaces Mp,ϕ with a weight ϕ sat-
isfying (2.4) (cf. [12, 13]). Precisely, we obtain boundedness in Mp,ϕ
for sub-linear integrals generated by singular integrals as the Caldero´n-
Zygmund. More results concerning sub-linear integrals in generalized
Morreyspacescanbefoundin[3,11,25], seealsothereferencestherein.
Throughout this paper the following notations are to be used, x =
(x′,t) = (x′′,x ,t) Rn+1, Rn+1 = x′ Rn,t > 0 and Dn+1 =
x′′ Rn−1,xn > ∈0,t > 0 ,+D u ={∂u/∈∂x , D u =} ∂2u/∂+x ∂x ,
n i i ij i j
{ ∈ }
D u = u = ∂u/∂t stand for the corresponding derivatives while Du =
t t
(D u,... ,D u) and D2u = D u n mean the spatial gradient and
1 n { ij }i,j=1
the Hessian matrix of u. For any measurable function f and A Rn+1
⊂
we write
1/p 1
f = f(y) pdy , f = f(y)dy
p,A A
k k | | A
(cid:18)ZA (cid:19) | | ZA
where A is the Lebesgue measure of A. Through all the paper the
| |
standard summation convention on repeated upper and lower indexes
is adopted. The letter C is used for various constants and may change
from one occurrence to another.
2. Definitions and statement of the problem
Let Ω Rn, n 1 be a bounded C1,1-domain, Q = Ω (0,T) be a
cylinder i⊂n Rn+1, ≥and S = ∂Ω (0,T) stands for the late×ral boundary
+ ×
of Q. We consider the problem
Pu := u aij(x)D u = f(x) a.e. in Q,
t ij
−
(2.1) Iu := u(x′,0) = 0 on Ω,
Bu := ∂u/∂ℓ = ℓi(x)D u = 0 on S.
i
under the following conditions:
(i) The operator P is supposed to beuniformly parabolic, i.e. there
exists a constant Λ > 0 such that for almost all (a.a.) x Q
∈
Λ−1 ξ 2 aij(x)ξ ξ Λ ξ 2 ξ Rn,
i j
(2.2) | | ≤ ≤ | | ∀ ∈
 aij(x) = aji(x), i,j = 1,... ,n.

Thesymmetryofthecoefficient matrixa = aij n implieses-
{ }i,j=1
sentialboundednessofaij’sandweset a = n aij .
k k∞,Q i,j=1k k∞,Q
(ii) The boundary operator B is prescribed in terms of a direc-
P
tional derivative with respect to the unit vector field ℓ(x) =
(ℓ1(x),...,ℓn(x)) x S. We suppose that Bis a regular oblique
∈
OBLIQUE DERIVATIVE PROBLEM IN GENERALIZED MORREY SPACES 3
derivative operator, i.e., the field ℓ is never tangential to S:
(2.3) ℓ(x) n(x) = ℓi(x)n (x) > 0 on S, ℓi Lip(S¯).
i
h · i ∈
Here Lip(S¯) is the class of uniformly Lipschitz continuous func-
¯
tions on S and n(x) stands for the unit outward normal to ∂Ω.
In the following, besides the parabolic metric ̺(x) = max( x′ , t 1/2)
| | | |
and the defined by it parabolic cylinders
(x) = y Rn+1 : x′ y′ < r, t τ < r2 = Crn+2.
r r
I ∈ | − | | − | |I |
n o
|x′|2+√|x′|4+4t2 1/2
we use the equivalent one ρ(x) = (see [9]). The
2
(cid:18) (cid:19)
balls with respect to this metric are ellipsoids
x′ y′ 2 t τ 2
(x) = y Rn+1 : | − | + | − | < 1 = Crn+2.
Er ∈ r2 r4 |Er|
n o
Because of the equivalence of the metrics all estimates obtained over
ellipsoids hold true also over parabolic cylinders and in the following
we shall use this without explicit references.
Definition 2.1. ([14, 22]) Let a L1 (Rn+1), denote by
∈ loc
1
η (R) = sup f(y) f dy for every R > 0
a | − Er|
Er,r≤R |Er| ZEr
where ranges over all ellipsoids in Rn+1. The Banach space BMO
r
E
(bounded mean oscillation) consists of functions for which the following
norm is finite
a = supη (R) < .
∗ a
k k ∞
R>0
A function a belongs to VMO (vanishing mean oscillation) with VMO-
modulus η (R) provided
a
lim η (R) = 0.
a
R→0
For any bounded cylinder Q we define BMO(Q) and VMO(Q) taking
a L1(Q) and Q = Q instead of in the definition above.
r r r
∈ ∩I E
According to [1, 15] having a function a BMO/VMO(Q) it is
possible to extend it in the whole Rn+1 prese∈rving its BMO-norm or
VMO-modulus, respectively. In the following we use this property
without explicit references.
Definition 2.2. Let ϕ : Rn+1 R R be a measurable function
+ +
and p (1, ). A function f ×Lp (→Rn+1) belongs to the generalized
∈ ∞ ∈ loc
4 L.G. SOFTOVA
parabolic Morrey space Mp,ϕ(Rn+1) provided
1/p
f = sup ϕ(x,r)−1 r−(n+2) f(y) pdy < .
p,ϕ;Rn+1
k k (x,r)∈Rn+1×R+ ZEr(x)| | ! ∞
The space Mp,ϕ(Q) consists of Lp(Q) functions provided the following
norm is finite
1/p
f = sup ϕ(x,r)−1 r−(n+2) f(y) pdy .
p,ϕ;Q
k k (x,r)∈Q×R+ ZQr(x)| | !
The generalized Sobolev-Morrey space Wp,ϕ(Q), p (1, ) consist
2,1 ∈ ∞
of all Sobolev functions u Wp (Q) with distributional derivatives
∈ 2,1
DlDsu Mp,ϕ(Q), 0 2l+ s 2, endowed by the norm
t x ∈ ≤ | | ≤
u Wp,ϕ(Q) = ut p,ϕ;Q + Dsu p,ϕ;Q.
k k 2,1 k k k k
|sX|≤2
◦
Wp,ϕ(Q) = u Wp,ϕ(Q) : u(x) = 0 x ∂Q ,
2,1 ∈ 2,1 ∈
kukW◦p,ϕ(Q) =nkukW2p,,1ϕ(Q) o
2,1
where ∂Q means the parabolic boundary Ω ∂Ω (0,T) .
∪{ × }
Theorem 2.3. (Main result) Let (i) and (ii) hold, a VMO(Q) and
∈
◦
u Wp (Q), p (1, ) be a strong solution of (2.1). If f Mp,ϕ(Q)
∈ 2,1 ∈ ∞ ∈
with ϕ(x,r) being measurable positive function satisfying
n+2
essinf ϕ(x,ζ)ζ p
∞ s
s<ζ<∞
(2.4) 1+ln ds C
Zr (cid:18) r(cid:19) sn+p2+1 ≤
◦
for each (x,r) Q R then u Wp,ϕ(Q) and
∈ × + ∈ 2,1
(2.5) kukW◦p,ϕ(Q) ≤ Ckfkp,ϕ;Q
2,1
with C = C(n,p,Λ,∂Ω,T,kak∞;Q,ηa) and ηa = ni,j=1ηaij.
Ifϕ(x,r) = r(λ−n−2)/p thenMp,ϕ Lp,λ andthPecondition(2.4)holds
≡
withaconstant depending onn,pandλ.Ifϕ(x,r) = ω(x,r)1/pr−(n+2)/p
with ω : Rn+1 R R satisfying the conditions
+ +
× →
ω(x ,s)
k 0 k x Rn+1, r s 2r
1 2 0
≤ ω(x ,r) ≤ ∀ ∈ ≤ ≤
0
∞ ω(x ,s)
0
ds k ω(x ,r) k > 0, i = 1,2,3
3 0 i
s ≤
Zr
OBLIQUE DERIVATIVE PROBLEM IN GENERALIZED MORREY SPACES 5
thanwe obtainthe spaces Lp,ω studied in[18, 20]. The following results
areobtained in [13] and treat continuity in Mp,ϕ of certain singular and
nonsingular integrals.
Definition 2.4. A measurable function K(x;ξ) : Rn+1 Rn+1 0 R
× \{ } →
is called variable parabolic Calder´on-Zygmund kernel (PCZK) if:
i) K(x; ) is a PCZK for a.a. x Rn+1 :
a) K· (x; ) C∞(Rn+1 0 )∈,
· ∈ \{ }
b) K(x;µξ) = µ−(n+2)K(x;ξ) µ > 0,
∀
c) K(x;ξ)dσ = 0, K(x;ξ) dσ < + .
ξ ξ
Sn Sn | | ∞
Z Z
ii) DβK M(β) < for each multi-index β.
ξ ∞;Rn+1×Sn ≤ ∞
(cid:13) (cid:13)
Consi(cid:13)der th(cid:13)e singular integrals
(cid:13) (cid:13)
f(x) = P.V. K(x;x y)f(y)dy
K Rn+1 −
Z
(2.6)
C[a,f](x) = P.V. K(x;x y)[a(y) a(x)]f(y)dy.
Rn+1 − −
Z
Theorem 2.5. For any f Mp,ϕ(Rn+1) with (p,ϕ) as in Theorem 2.3
∈
and a BMO there exist constants depending on n,p and the kernel
∈
such that
f C f ,
p,ϕ;Rn+1 p,ϕ;Rn+1
kK k ≤ k k
(2.7) C[a,f] C a f .
p,ϕ;Rn+1 ∗ p,ϕ;Rn+1
k k ≤ k k k k
Corollary 2.6. Let Q be a cylinder in Rn+1, f Mp,ϕ(Q), a
BMO(Q) and K(x,ξ) : Q Rn+1 0 + R. Th∈en the operator∈s
× + \ { } →
(2.6) are bounded in Mp,ϕ(Q) and
f C f ,
p,ϕ;Q p,ϕ;Q
kK k ≤ k k
(2.8) C[a,f] C a f
p,ϕ;Q ∗ p,ϕ;Q
k k ≤ k k k k
with C independent of a and f.
Corollary 2.7. Let a VMO and (p,ϕ) be as in Theorem 2.3. Then
∈
for any ε > 0 there exists a positive number r0 = r0(ε,ηa) such that for
any (x ) with a radius r (0,r ) and all f Mp,ϕ( (x ))
r 0 0 r 0
E ∈ ∈ E
(2.9) C[a,f] Cε f
k kp,ϕ;Er(x0) ≤ k kp,ϕ;Er(x0)
where C is independent of ε, f, r and x .
0
For any x′ Rn and any fixed t > 0 define the generalized reflection
∈ +
an(x′,t)
(2.10) (x) = ( ′(x),t), ′(x) = x′ 2x
n
T T T − ann(x′,t)
6 L.G. SOFTOVA
where an(x) is the last row of the coefficients matrix a(x) of (2.1).
The function ′(x) maps Rn into Rn and the kernel K(x; (x) y) =
K(x; ′(x) yT′,t τ) is a n+onsingula−r one for any x,y DTn+1. T−aking
T − − ∈ +
x = (x′′, x ,t) there exist positive constants κ and κ such that
n 1 2
−
(2.11) κ ρ(x y) ρ( (x) y) κ ρ(x y).
1 2
e − ≤ T − ≤ −
For any f Mp,ϕ(Dn+1) with a norm
∈ e+ e
1/p
f = sup ϕ(x,r)−1 r−(n+2) f(y) pdy
k kp,ϕ;Dn++1 (x,r)∈Dn++1×R+ ZEr(x)| | !
and a BMO(Dn+1) define the nonsingular integral operators
∈ +
f(x) = K(x; (x) y)f(y)dy
K Dn+1 T −
Z +
(2.12) e
C[a,f](x) = K(x; (x) y)[a(y) a(x)]f(y)dy.
Dn+1 T − −
Z +
Theoreme 2.8. Let a BMO(Dn+1) and f Mp,ϕ(Dn+1) with (p,ϕ)
∈ + ∈ +
as in Theorem 2.3. Then the operators f and C[a,f] are continuous
in Mp,ϕ(Dn+1) and K
+
e e
f C f ,
kK kp,ϕ;Dn++1 ≤ k kp,ϕ;Dn++1
(2.13) kCe[a,f]kp,ϕ;Dn++1 ≤ Ckak∗kfkp,ϕ;Dn++1
with a constant indeependend of a and f.
Corollary 2.9. Let a VMO, then for any ε > 0 there exists a
positive number r0 = r0(ε∈,ηa) such that for any Er+(x0) = Er(x0)∩Dn++1
with a radius r (0,r ) and center x0 = (x′′,0,0) and for all f
0
∈ ∈
Mp,ϕ( +(x0)) holds
Er
(2.14) C[a,f] Cε f ,
k kp,ϕ;Er+(x0) ≤ k kp,ϕ;Er+(x0)
where C is independent of ε, f, r and x0.
e
3. Proof of the main result
◦
Asitfollowsby[24],theproblem(2.1)isuniquelysolvableinWp (Q).
2,1
◦
We are going to show that f Mp,ϕ(Q) implies u Wp,ϕ(Q). For this
∈ ∈ 2,1
goal we obtain an a priori estimate of u. The proof is divided in two
steps.
Interior estimate. For any x Rn+1 consider the parabolic semi-
0 ∈ +
cylinders (x ) = (x′) (t r2,t ). Let v C∞( ) and suppose
Cr 0 Br 0 × 0 − 0 ∈ 0 Cr
OBLIQUE DERIVATIVE PROBLEM IN GENERALIZED MORREY SPACES 7
that v(x,t) = 0 for t 0. According to [5, Theorem 1.4] for any x
≤ ∈
suppv the following representation formula for the second derivatives
of v holds true
D v(x) = P.V. Γ (x;x y)[ahk(y) ahk(x)]D v(y)dy
ij ij hk
Rn+1 − −
Z
(3.15)
+P.V. Γ (x;x y)Pv(y)dy+Pv(x) Γ (x;y)ν dσ ,
ij j i y
Rn+1 − Sn
Z Z
where ν(ν ,... ,ν ) is the outward normal to Sn. Here Γ(x;ξ) is the
1 n+1
fundamentalsolutionoftheoperatorPandΓ (x;ξ) = ∂2Γ(x;ξ)/∂ξ ∂ξ .
ij i j
Because of density arguments the representation formula (3.15) still
holds for any v Wp ( (x )). The properties of the fundamental
∈ 2,1 Cr 0
solution (cf. [5, 16, 23]) imply Γ are Caldero´n-Zygmund kernels in
ij
the sense of Definition 2.4. We denote by and C the singular
ij ij
K
integrals defined in (2.6) with kernels K(x;x y) = Γ (x;x y). Then
ij
− −
we can write that
D v(x) =C [ahk,D v](x)
ij ij hk
(3.16) + (Pv)(x)+Pv(x) Γ (x;y)ν dσ .
ij j i y
K Sn
Z
Because of Corollaries 2.6 and 2.7 and the equivalence of the metrics
we get
D2v C(ε D2v + Pu )
k kp,ϕ;Cr(x0) ≤ k kp,ϕ;Cr(x0) k kp,ϕ;Cr(x0)
for some r small enough. Moving the norm of D2v on the left-hand
side we get
D2v C Pv
k kp,ϕ;Cr(x0) ≤ k kp,ϕ;Cr(x0)
with a constant depending on n,p,ηa(r), a ∞,Q and DΓ ∞,Q. Define
k k k k
a cut-off function φ(x) = φ (x′)φ (t), with φ C∞( (x′)), φ
C∞(R) such that 1 2 1 ∈ 0 Br 0 2 ∈
0
1 x′ (x′) 1 t (t (θr)2,t ]
φ (x′) = ∈ Bθr 0 , φ (t) = ∈ 0 − 0
1 0 x′ 6∈ Bθ′r(x′0) 2 0 t < t0 −(θ′r)2
with θ (0,1), θ′ = θ(3 θ)/2 > θ andDsφ C[θ(1 θ)r]−s,
s = 0,1,∈2, φ D2φ . Fo−r any solution u | Wp| (≤Q) of (2.−1) define
| t| ∼ | | ∈ 2,1
v(x) = φ(x)u(x) Wp ( ). Then we get
∈ 2,1 Cr
D2u D2v C Pv
k kp,ϕ;Cθr(x0) ≤ k kp,ϕ;Cθ′r(x0) ≤ k kp,ϕ;Cθ′r(x0)
Du u
C f + k kp,ϕ;Cθ′r(x0) + k kp,ϕ;Cθ′r(x0) .
≤ k kp,ϕ;Cθ′r(x0) θ(1 θ)r [θ(1 θ)r]2 !
− −
8 L.G. SOFTOVA
By the choice of θ′ it holds θ(1 θ) 2θ′(1 θ′) which leads to
− ≤ −
2
θ(1 θ)r D2u
− k kp,ϕ;Cθr(x0)
h i
C r2 f +θ′(1 θ′)r Du + u .
≤ k kp,ϕ;Q − k kp,ϕ;Cθ′r(x0) k kp,ϕ;Cθ′r(x0)
(cid:16) (cid:17)
Introducing the semi-norms
s
Θ = sup θ(1 θ)r Dsu s = 0,1,2
s − k kp,ϕ;Cθr(x0)
0<θ<1
h i
and taking the supremo with respect to θ and θ′ we get
(3.17) Θ C r2 f +Θ +Θ .
2 p,ϕ;Q 1 0
≤ k k
(cid:16) (cid:17)
The interpolation inequality [26, Lemma 4.2] gives that there exists a
positive constant C independent of r such that
C
Θ εΘ + Θ for any ε (0,2).
1 2 0
≤ ε ∈
Thus (3.17) becomes
[θ(1 θ)r]2 D2u Θ C r2 f +Θ
− k kp,ϕ;Cθr(x0) ≤ 2 ≤ k kp,ϕ;Q 0
(cid:16) (cid:17)
foreachθ (0,1).Takingθ = 1/2wegettheCaccioppoli-typeestimate
∈
1
D2u C f + u .
k kp,ϕ;Cr/2(x0) ≤ k kp,ϕ;Q r2k kp,ϕ;Cr(x0)
(cid:18) (cid:19)
To estimate u we exploit the parabolic structure of the equation and
t
the boundedness of the coefficients
u a D2u + f
k tkp,ϕ;Cr/2(x0) ≤ k k∞;Qk kp,ϕ;Cr/2(x0) k kp,ϕ;Cr/2(x0)
1
C f + u .
≤ k kp,ϕ;Q r2k kp,ϕ;Cr(x0)
(cid:16) (cid:17)
Consider cylinders Q′ = Ω′ (0,T) and Q′′ = Ω′′ (0,T) with Ω′ ⋐
× ×
Ω′′ ⋐ Ω, by standard covering procedure and partition of the unity we
get
(3.18) u Wp,ϕ(Q′) C f p,ϕ;Q+ u p,ϕ;Q′′
k k 2,1 ≤ k k k k
(cid:16) (cid:17)
whereC dependsonn,p,Λ,T, DΓ ∞;Q,ηa(r), a ∞,Q anddist(Ω′,∂Ω′′).
k k k k
Boundary estimates. For any fixed R > 0 and x0 = (x′′,0,0) define
the semi-cylinders
+(x0) = (x0) Dn+1.
CR CR ∩ +
Without lost of generality we can take x0 = (0,0,0). Define
+ = x′ < R,x > 0 , S+ = x′′ < R,x = 0,t (0,R2) and
BR {| | n } R {| | n ∈ }
OBLIQUE DERIVATIVE PROBLEM IN GENERALIZED MORREY SPACES 9
consider the problem
Pu := u aij(x)D u = f(x) a.e. in +,
t − ij CR
(3.19) Iu := u(x′,0) = 0 on +,
Bu := ℓi(x)D u = 0 on SBR+.
i R
Let u Wp(C+) with u = 0 for t 0 and x 0, then the following
∈ 2,1 R ≤ n ≤
representation formula holds (see [17, 23])
D u(x) = I (x) J (x)+H (x)
ij ij ij ij
−
where
I (x) =P.V. Γ (x;x y)F(x;y)dy
ij ij
ZCR+ −
+f(x) Γ (x;y)ν dσ , i,j = 1,...,n;
j i y
Sn
Z
J (x) = Γ (x; (x) y)F(x;y)dy;
ij ij
ZCR+ T −
l
∂ (x)
J (x) =J (x) = Γ (x; (x) y) T F(x;y)dy,
in ni il
ZCR+ T − ∂xn !
i,j = 1,... ,n 1
−
l s
∂ (x) ∂ (x)
J (x) = Γ (x; (x) y) T T F(x;y)dy;
nn ls
ZCR+ T − ∂xn ! ∂xn !
F(x;y) =f(y)+[ahk(y) ahk(x)]D u(y)
hk
−
Hij(x) =(Gij 2 g)(x)+g(x′′,t) Gj(x;y′′,xn,τ)nidσ(y′′,τ),
∗ Sn
Z
i,j = 1,... ,n,
∂T(x) an1(x) ann−1(x)
= 2 ,... , 2 , 1 .
∂xn − ann(x) − ann(x) − !
Here the kernel G = Γ , is a byproduct of the fundamental solution
Q
and a bounded regular function . Hence its derivatives G behave as
ij
Q
Γ and the convolution that appears in H is defined as follows
ij ij
(G g)(x) = P.V. G (x;x′′ y′′,x ,t τ)g(y′′,0,τ)dy′′dτ,
ij 2 ij n
∗ ZSR+ − −
g(x′′,0,t) = ℓk(0) ℓk(x′′,0,t) D u ℓk(0)(Γ F) (x′′,0,t),
k k
− − ∗
h(cid:16) (cid:17) i(cid:12)(cid:12)xn=0
(cid:12)
(Γk F)(x) = Γk(x;x y)F(x;y)dy. (cid:12)
∗ ZCR+ −
10 L.G. SOFTOVA
Here I are a sum of singular integrals and bounded surface integrals
ij
hence the estimates obtained in Corollaries 2.6 and 2.7 hold true. On
the nonsingular integrals J we apply the estimates obtained in Theo-
ij
rem 2.8 and Corollary 2.9 that give
(3.20)
kIijkp,ϕ;CR+ +kJijkp,ϕ;CR+ ≤ C kfkp,ϕ;CR+ +ηa(R)kD2ukp,ϕ;CR+
(cid:16) (cid:17)
for all i,j = 1,... ,n. To estimate the norm of H we suppose that the
ij
vector field ℓ is extended in + preserving its Lipschitz regularity and
CR
the norm. This automatically leads to extension of the function g in
+ that is
CR
(3.21)
g(x) = ℓk(0) ℓk(x) D u(x) ℓk(0)(Γ F)(x).
k k
− − ∗
(cid:16) (cid:17)
Applying the estimates for the heat potentials [16, Chapter 4] and the
trace theorems in Lp [2, Theorems 7.48, 7.53] (see also [23, Theorem 1])
we get
(G g)(y) pdy C g(y) pdy + Dg(y) pdy .
ij 2
ZCR+| ∗ | ≤ ZCR+ | | ZCR+ | | !
Taking a parabolic cylinder (x) centered in some point x + we
Ir ∈ CR
have
rn+2 ϕ(x,r)−p
(G g)(y) pdy C g(y) pdy
ZCR+∩Ir(x)| ij ∗2 | ≤ ϕ(x,r)−p(cid:18) rn+2 ZCR+∩Ir(x)| |
ϕ(x,r)−p
+ Dg(y) pdy
rn+2 ZCR+∩Ir(x)| | (cid:19)
ϕ(x,r)−p
C g p + Dg p .
≤ rn+2 k kp,ϕ;C+ k kp,ϕ;C+
(cid:18) R R(cid:19)
Moving ϕ(x,r)−p on the left-hand side and taking the supremo with
rn+2
respect to (x,r) + R we get
∈ CR × +
G g p C g p + Dg p .
k ij ∗2 kp,ϕ;C+ ≤ k kp,ϕ;C+ k kp,ϕ;C+
R (cid:18) R R(cid:19)
An immediate consequence of (3.21) is the estimate
g [ℓk(0) ℓk( )]D u + ℓk(0)(Γ F)
k kp,ϕ;CR+ ≤ k − · k kp,ϕ;CR+ k k ∗ kp,ϕ;CR+
CR ℓ Du + Γ f
≤ k kLip(S¯)k kp,ϕ;CR+ k k ∗ kp,ϕ;CR+
+ Γ [ahk( ) ahk(x)]D u .
k k ∗ · − hk kp,ϕ;CR+