Introduction. In this paper some problems of the teory of weighted Sobolev–

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1992

ON TWO CLASSES OF

WEIGHTED SOBOLEV–SLOBODETSKI˘ I SPACES IN A DIHEDRAL ANGLE

J. R O S S M A N N

Department of Mathematics, University of Rostock Universit¨ atsplatz 1, D-O-2500 Rostock, Germany

Introduction. In this paper some problems of the teory of weighted Sobolev–

Slobodetski˘ı spaces are dealt with. Weighted Sobolev spaces play an important role in the study of elliptic boundary value problems in non-smooth domains (see e.g. [2]–[4], [6]). Here two classes of weighted spaces are of special interest: the spaces V

_p,β^l

(G) with the homogeneous norms

(0.1) kuk

_V^l

p,β(G)

=  R

G

X

|α|≤l

r

^{p(β−l+|α|)}

|D

^α

u|

^p

dx 

1/p

(r = r(x) denotes the distance of x to the set of singularities on the boundary) and the spaces W

_p,β^l

(G) with the inhomogeneous norms

(0.2) kuk

_W^l

p,β(G)

=

 R

G

X

|α|≤l

r

^pβ

|D

^α

u|

^p

dx



1/p

.

We restrict ourselves to the case that G = D is a dihedral angle. This is the model case for domains with smooth non-intersecting edges. For domains with conical points the weighted Sobolev spaces V

_p,β^l

(G) and W

_p,β^l

(G) were investigated e.g.

in [2], [5], [8].

The present paper consists of two sections. In Section 1 the main properties of the weighted Sobolev spaces V

_p,β^l

(D), W

_p,β^l

(D) (l an integer, l ≥ 0, β ∈ R, 1 < p < ∞) and of the corresponding weighted Sobolev–Slobodetski˘ı spaces V

_p,β^s

(D), W

_p,β^s

(D) (s real, s ≥ 0, β ∈ R, 1 < p < ∞) will be investigated. It will be shown e.g. that the following imbeddings are valid:

V

_p,β^s

(D) ⊂ V

_p,β^s00

(D) if s ≥ s

⁰

, s − β = s

⁰

− β

⁰

,

W

_p,β^s

(D) ⊂ W

_p,β^s0 0

(D) if s ≥ s

⁰

, s − β = s

⁰

− β

⁰

, β

⁰

> −2/p .

[399]

Furthermore, we give a connection between the spaces V

_p,β^s

(D) and W

_p,β^s

(D). It will be proved that every function u ∈ W

_p,β^s

(D) can be written as the sum of a “quasi-polynomial” and a function from V

_p,β^s

(D) (see Theorem 5). This is a generalization of Lemma 1.3 of [6] (see also [7]) where such a representation has been shown if s is an integer and β + 2/p is not an integer. Analogous results are obtained for the space B

^◦s−1/p_p,β

(Γ

^±

) and B

_p,β^s−1/p

(Γ

^±

) of the traces of functions from V

_p,β^s

(D) and W

_p,β^s

(D) on the sides Γ

⁺

and Γ

⁻

of D, respectively.

Section 2 is concerned with two applications of the given results. In Section 2.1 we investigate conditions on g

^±_k

∈ B

^s−m

± k−1/p

p,β

(Γ

^±

) under which there exists u ∈ W

_p,β^s

(D) satisfying the boundary conditions

(0.3) B

_k^±

u = g

_k^±

on Γ

^±

(k = 1, . . . , p

^±

)

where {B

_k⁺

} and {B

_k⁻

} are normal systems of homogeneous boundary operators of order m

^±_k

with constant coefficients on Γ

⁺

and Γ

⁻

, respectively. In the spaces V

_p,β^s

the existence of u is always ensured (see [4]). We will show that u ∈ W

_p,β^s

(D) satisfying (0.3) exists if and only if the boundary data g

^±_k

satisfy some compati- bility conditions on the edge M (see Theorem 7).

In Section 2.2 the following regularity assertion for solutions of elliptic bound- ary value problems will be proved: If u ∈ W

_p,β^2m

(D) is a solution of an elliptic boundary value problem Lu = f in D, B

_k^±

u = g

_k^±

on Γ

^±

(k = 1, . . . , m) where f ∈ W

_p,β+s^s

(D), g

^±_k

∈ B

^s+2m−m

± k−1/p

p,β+s

(Γ

^±

) then u ∈ W

_p,β+s^s+2m

(D).

The corresponding result for the spaces V

_p,β^s

has been proved in [4].

1. Weighted Sobolev–Slobodetski˘ ı spaces in a dihedral angle

1.1. The spaces V

_p,β^s

(D). Let D = K × R

ⁿ⁻²

= {x = (y, z) ∈ R

ⁿ

: y ∈ K, z = (z

1

, . . . , z

n−2

) ∈ R

ⁿ⁻²

} be a dihedral angle in R

ⁿ

where K is a plane wedge which has the following representation in polar coordinates r, ω:

K = {y = (y

1

, y

2

) ∈ R

²

: 0 < r < ∞, ω ∈ Ω = (−

¹₂

ω

0

,

¹₂

ω

0

)}

(0 < ω

0

≤ 2π). The boundary of D consists of the (n − 1)-dimensional half-planes Γ

^±

= {x = (x, z) ∈ R

ⁿ

: 0 < r < ∞, ω = ±

¹₂

ω

0

, z ∈ R

ⁿ⁻²

}

and of the edge M = {(0, 0)} × R

ⁿ⁻²

. In the sequel we will denote the coordinates of a point x = (x

1

, . . . , x

n

) by y

1

, y

2

, z

1

, . . . , z

n−2

, i.e. y

1

= x

1

, y

2

= x

2

, z

j

= x

j+2

(j = 1, . . . , n − 2).

If s is a non-negative integer and p, β are real numbers, 1 < p < ∞, then V

_p,β^s

(D) denotes the closure of C

₀^∞

(D \ M ) = {u ∈ C

^∞

(D) : supp u ⊂ D \ M , supp u compact} with respect to the norm

(1.1) kuk

_V^s

p,β(D)

=  R

D

X

|α|≤s

r

^{p(β−s+|α|)}

|D

^α

u|

^p

dx 

1/p

(D

^α

= D

_x^α

= D

_x^α₁¹

. . . D

^α_xnⁿ

, D

xj

= (1/i)∂

xj

= (1/i)

_∂x^∂

j

, r = |y|).

For non-integer positive s = l + σ (l an integer, 0 < σ < 1) the space V

_p,β^s

(D) is defined as the closure of C

₀^∞

(D \ M ) with respect to the norm

(1.2) kuk

_V^s

p,β(D)

=



R

D

X

|α|≤l

|y|

^{p(β−s+|α|)}

|(D

^α

u)(x)|

^p

dx

+ R

D

R

D

X

|α|=l

| |y|

^β

(D

^α

u)(y, z) − |y

⁰

|

^β

(D

^α

u)(y

⁰

, z

⁰

)|

^p

dx dx

⁰

|x − x

⁰

|

^n+pσ



1/p

.

It can easily be shown that the norm (1.2) is equivalent to the norm (1.3) kuk =



R

D

X

|α|≤l

|y|

^{p(β−s+|α|)}

|(D

^α

u)(x)|

^p

dx

+ X

|α|=l

R R

D D

|x−x⁰|<|y|/2

|y|

^pβ

|(D

^α

u)(y, z) − (D

^α

u)(y

⁰

, z

⁰

)|

^p

dx dx

⁰

|x − x

⁰

|

^n+pσ



1/p

.

Theorem 1. If s

⁰

> s and β

⁰

− s

⁰

= β − s then V

_p,β^s00

(D) is continuously imbedded in V

_p,β^s

(D).

P r o o f. If l ≤ s < s

⁰

< l + 1 (l an integer) then the imbedding immediately follows from the inequality

|y|

^β−β0

= |y|

^s−s0

≤ 2

^s−s0

|x − x

⁰

|

^s−s0

for |x − x

⁰

| <

¹₂

|y| . In the case l < s < s

⁰

= l + 1 we can apply the equation

(D

^α

u)(x) − (D

^α

u)(x

⁰

) = −

1

R

0

d

dt (D

^α

u)(x + t(x

⁰

− x)) dt

= (x − x

⁰

)

1

R

0

(∇D

^α

u)(x + t(x

⁰

− x)) dt to obtain

R R

D D

|x−x⁰|<|y|/2

|y|

^pβ

|(D

^α

u)(x) − (D

^α

u)(x

⁰

)|

^p

dx dx

⁰

|x − x

⁰

|

^n+p(s−l)

≤

1

R

0

R R

D D

|x−x⁰|<|y|/2

|y|

^pβ

|x − x

⁰

|

−n+p(l+1−s)

|(∇D

^α

u)(x + t(x − x

⁰

))|

^p

dx dx

⁰

dt

≤ R

D

|y|

^pβ0

|(∇D

^α

u)(x)|

^p

dx for |α| = l, i.e. kuk

_V^s

p,β(D)

≤ ckuk

_Vs0 p,β0(D)

.

In the sequel let ζ

ν

(ν = . . . , −1, 0, +1, . . .) be smooth functions on R

+

with support in the interval [2

^ν−1

, 2

^ν+1

] such that

(1.4)

∞

X

ν=−∞

ζ

ν

(r) = 1 and |D

_r^j

ζ

ν

(r)| < c

j

2

^−νj

with constants c

j

independent of r and ν. If we set r = |y| = (y

₁²

+ y

²₂

)

^1/2

we can interpret ζ

ν

= ζ

ν

(r) as functions on D. Analogously to Lemma 1.1 in [4] the following assertion can be proved.

Lemma 1. The norm k · k

V_p,β^s (D)

is equivalent to the norm kuk =  X

^∞

ν=−∞

kζ

_ν

uk

^p_Vs p,β(D)



1/p

.

Let B

^◦s−1/p_p,β

(Γ

^±

) (s > 1/p) be the space of the traces of functions from V

_p,β^s

(D) on Γ

⁺

and Γ

⁻

, respectively, provided with the norm

(1.5) kuk

B^◦s−1/p_p,β (Γ^±)

= inf{kvk

_V^s

p,β(D)

: v ∈ V

_p,β^s

(D), v|

_Γ^±

= u} . Lemma 2. The norm (1.5) is equivalent to the norm

kuk =  X

^∞

ν=−∞

kζ

_ν

uk

^p

B^◦s−1/p_p,β (Γ^±)



1/p

.

P r o o f. Let v ∈ V

_p,β^s

(D) be an extension of u ∈ B

^◦s−1/p_p,β

(Γ

^±

) such that kvk

_V^s

p,β(D)

≤ 2kuk

B^◦s−1/p_p,β (Γ^±)

. Since ζ

ν

v is an extension of ζ

ν

u we get

 X

^∞

ν=−∞

kζ

ν

uk

^p

B^◦s−1/p_p,β (Γ^±)



1/p

≤  X

^∞

ν=−∞

kζ

ν

vk

^p_Vs p,β(D)



1/p

≤ ckvk

_V^s

p,β(D)

≤ 2ckuk

B^◦s−1/p_p,β (Γ^±)

. Furthermore, there exist extensions v

ν

of ζ

ν

u such that kv

ν

k

_V^s

p,β(D)

≤ 2kζ

ν

uk

B^◦s−1/p_p,β (Γ^±)

(ν = . . . , −1, 0, +1, . . .). Since w

ν

= (ζ

ν−1

+ ζ

ν

+ ζ

ν+1

)v

ν

= ζ

ν

u on Γ

^±

and w = P

∞

ν=−∞

w

ν

= u on Γ

^±

we obtain kuk

^p

B^◦s−1/p_p,β (Γ^±)

≤ kwk

^p_Vs

p,β(D)

≤ c

∞

X

ν=−∞

kζ

_ν

wk

^p_Vs

p,β(D)

= c

∞

X

ν=−∞

ζ

ν

∞

X

k=−∞

w

k

p V_p,β^s (D)

= c

∞

X

ν=−∞

ζ

ν

ν+2

X

k=ν−2

(ζ

k−1

+ ζ

k

+ ζ

k+1

)v

k

p V_p,β^s (D)

≤ c

⁰

∞

X

ν=−∞

kv

_ν

k

^p_Vs

p,β(D)

≤ c

⁰

2

^p

∞

X

ν=−∞

kζ

_ν

uk

^p

B^◦s−1/p_p,β (Γ^±)

.

In the half-planes Γ

⁺

, Γ

⁻

we introduce the Cartesian coordinates ξ

1

= r =

|y| = dist(x, M ), ξ

₂

= z

1

, . . . , ξ

n−1

= z

n−2

. Using Lemmas 1 and 2 we can give the following norm equivalent to (1.5).

Theorem 2. Let p, β and s (1 < p < ∞, s > 1/p) be real numbers and k an arbitrary integer such that k > s − 1/p. Then the norm (1.5) is equivalent to the norm

(1.6) kuk =



R

R+×Rⁿ⁻²

ξ

₁^p(β−s)+1

|u(ξ)|

^p

dξ

+ R R

R+×Rⁿ⁻²R+×Rⁿ⁻²

|ξ−ξ⁰|<ξ1/2

ξ

^pβ₁

   

k

X

j=0

(−1)

^j

k j



u  jξ + (k − j)ξ

⁰

k

   

p

dξ dξ

⁰

|ξ − ξ

⁰

|

^n−2+ps



1/p

.

P r o o f. Let u∈B

^◦s−1/p_p,β

(Γ

^±

). Define u

ν

(ξ) = ζ

ν

(2

^ν

ξ)u(2

^ν

ξ) (ν = 0, ±1, ±2, . . .).

Since suppu

ν

⊂ {ξ ∈ R

+

× R

ⁿ⁻²

: 1/2 < ξ

1

< 2} the norm of u can be estimated by the usual Besov space norm (see [11]), i.e.

c

1

ku

ν

k

^p

B^◦s−1/p_p,β (Γ^±)

≤ R

R+×Rⁿ⁻²

|u

ν

(ξ)|

^p

dξ

+ R R

R+×Rⁿ⁻² R+×Rⁿ⁻²

|ξ−ξ⁰|<1/2

   

k

X

j=0

(−1)

^j

k j

 u

ν

 jξ + (k − 1)ξ

⁰

k



p

dξ dξ

⁰

|ξ − ξ

⁰

|

^n−2+ps

   

≤ c

₂

ku

_ν

k

^p

B^◦s−1/p_p,β (Γ^±)

where the constants c

1

, c

2

are independent of u and ν. It can be easily verified that ku

ν

k

B^◦s−1/p_p,β (Γ^±)

= 2

^{ν(s−β−n/p)}

kζ

ν

uk

B^◦s−1/p_p,β (Γ^±)

. Hence, c

1

2

^{ν(ps−pβ−n)}

kζ

_ν

uk

^p

B^◦s−1/p_p,β (Γ^±)

≤ 2

^−ν(n−1)

R

R+×Rⁿ⁻²

|ζ

_ν

(ξ)u(ξ)|

^p

dξ

+ 2

^{−ν(n−ps)}

R R

R⁺×Rⁿ⁻² R⁺×Rⁿ⁻²

|ξ−ξ⁰|<ξ1/2

   

k

X

j=0

(−1)

^j

k j



× ζ

_ν

 jξ + (k − j)ξ

⁰

k



u  jξ + (k − j)ξ

⁰

k

   

p

dξ dξ

⁰

|ξ − ξ

⁰

|

^n−2+ps

≤ c

₂

2

^{ν(ps−pβ−n)}

kζ

_ν

uk

^p

B^◦s−1/p_p,β (Γ^±)

.

Applying Lemma 2 we obtain the assertion of the theorem.

R e m a r k 1. By means of other equivalent norms in the usual Besov space (see [11], Section 4.4) analogously to Theorem 2 equivalence of other weighted norms to the norm (1.5) can be proved. In particular, if s − 1/p is not an integer, s − 1/p = l + σ (l an integer, l ≥ 0, 0 < σ < 1) then (1.5) is equivalent to the norm

kuk

V_p,β^s−1/p(R+×Rⁿ⁻²)

=

 X

|α|≤l

R

R+×Rⁿ⁻²

ξ

p(β−s+|α|)+1

1

|(D

^α

u)(ξ)|

^p

dξ

+ X

|α|=l

R R

R+×Rⁿ⁻² R+×Rⁿ⁻²

|ξ−ξ⁰|<ξ1/2

ξ

₁^pβ

|(D

^α

u)(ξ) − (D

^α

u)(ξ

⁰

)|

^p

dξ dξ

⁰

|ξ − ξ

⁰

|

^n−1+pσ



1/p

or to the norm kuk =



R

R⁺×Rⁿ⁻²

ξ

₁^p(β−s)+1

|u(ξ)|

^p

dξ

+ X

|α|=l

R R

R+×Rⁿ⁻² R+×Rⁿ⁻²

|ξ−ξ⁰|<ξ1/2

ξ

₁^pβ

|(D

^α

u)(ξ) − (D

^α

u)(ξ

⁰

)|

^p

dξ dξ

⁰

|ξ − ξ

⁰

|

^n−1+pσ



1/p

.

Analogously, the norm (1.2) is equivalent to the norm kuk =

 R

D

|y|

^p(β−s)

|u(x)|

^p

dx

+ R R

D D

|x−x⁰|<|y|/2

|y|

^pβ

|(D

^α

u)(y, z) − (D

^α

u)(y

⁰

, z

⁰

)|

^p

|x − x

⁰

|

^n+pσ

dx dx

⁰



1/p

.

1.2. The spaces W

_p,β^s

(D). Let p, β, s be real numbers, 1 < p < ∞, β > −2/p, s ≥ 0. If s is an integer then we define the space W

_p,β^s

(D) as the closure of C

₀^∞

(D) with respect to the norm

(1.7) kuk

_W^s

p,β(D)

=  X

|α|≤s

R

D

r

^pβ

|D

^α

u|

^p

dx 

1/p

.

If s = l + σ (l an integer, 0 < σ < 1) then the space W

_p,β^s

(D) will be defined as the closure of C

₀^∞

(D) with respect to the norm

(1.8) kuk

_W^s

p,β(D)

=

 X

|α|≤l

R

D

r

^pβ

|D

^α

u|

^p

dx

+ X

|α|=l

R R

D D

|x−x⁰|<|y|/2

r

^pβ

|(D

^α

u)(x) − (D

^α

u)(x

⁰

)|

^p

|x − x

⁰

|

^n+pσ

dx dx

⁰



1/p

.

Furthermore, we define W

_p,β^s

(R

+

× R

ⁿ⁻²

) (s ≥ 0, 1 < p < ∞, β > −1/p) as the closure of C

₀^∞

(R

+

× R

ⁿ⁻²

) with respect to the norm

(1.9) kuk

_W^s

p,β(R+×Rⁿ⁻²)

=

 R

R⁺×Rⁿ⁻²

r

^pβ

X

|α|≤s

|(D

^α

u)(r, z)|

^p

dr dz



1/p

if s is an integer, and (1.10) kuk

_W^s

p,β(R⁺×Rⁿ⁻²)

=



R

R⁺×Rⁿ⁻²

r

^pβ

X

|α|≤l

|(D

^α

u)(r, z)|

^p

dr dz

+ X

|α|=l

R R

R+×Rⁿ⁻² R+×Rⁿ⁻²

|(r−r⁰,z−z⁰)|<r/2

r

^pβ

|(D

^α

u)(r, z) − (D

^α

u)(r

⁰

, z

⁰

)|

^p

|(r, z) − (r

⁰

, z

⁰

)|

^n−1+pσ

dr dz dr

⁰

dz

⁰



1/p

if s = l + σ (l an integer, l ≥ 0, 0 < σ < 1).

Let u = u(y

1

, y

2

, z) be an arbitrary function on D. Then we denote by u

^◦

the function

(1.11) u

^◦

(r, z) = 1 ω

0

ω0/2

R

−ω0/2

u(r cos ω, r sin ω, z) dω .

Lemma 3. If u ∈ W

p,β^s

(D) then u

^◦

∈ W

_p,β+1/p^s

(R

+

× R

ⁿ⁻²

) and ku

^◦

k

_W^s

p,β+1/p(R⁺×Rⁿ⁻²)

≤ ckuk

_W^s

p,β(D)

.

If s is an integer then this assertion immediately follows from the definition of the spaces W

_p,β^s

(D), W

_p,β^s

(R

+

× R

ⁿ⁻²

). For s not an integer it is proved in [9].

We introduce the operator (Kg)(r, z) = χ(r) R

Rⁿ⁻²

R

R+

g(tr, z + τ r)K(t, τ ) dt dτ

where χ is a smooth cut-off function on R

+

, equal to unity in [0, 1] and to zero in (2, ∞) and K(t, τ ) = ϕ(t)ψ(τ

1

) . . . ψ(τ

n−2

) is a product of smooth functions ϕ ∈ C

₀^∞

(R

+

), ψ ∈ C

₀^∞

(R) satisfying

(1.12)

supp ϕ ⊂ (3/4, 5/4),

∞

R

0

t

^j

ϕ(t) dt = δ

0,j

,

supp ψ ⊂ (−1/4, 1/4),

∞

R

−∞

t

^j

ψ(t) dt = δ

0,j

(j = 0, 1, . . . , k). Here δ

0,j

denotes the Kronecker symbol.

Lemma 4. If g ∈ W

p,β^s

(R

+

×R

ⁿ⁻²

) then Kg ∈ T

∞

ν=1

W

p,β+hsi−s+ν^hsi+ν

(R

+

×R

ⁿ⁻²

).

Furthermore, (1.13) R

Rⁿ⁻²

R

R+

r

p(β−s+j+|γ|)

|D

^j_r

D

^γ_z

(Kg)(r, z)|

^p

dr dz ≤ ckgk

^p_Ws

p,β(R+×Rⁿ⁻²)

for j ≥ 1 or |γ| ≥ hsi + 1, and

(1.14) R

Rⁿ⁻²

R

R+

r

p(β−s+hsi)+1

|D

^γ_z

(Kg − g)|

^p

dr dz ≤ ckgk

^p_Ws

p,β(R⁺×Rⁿ⁻²)

for |γ| = hsi. Here hsi denotes the largest integer less than s.

P r o o f. For simplicity we restrict ourselves to the case n = 3. For n > 3 the lemma can be proved analogously.

If j + |γ| = hsi and r < 1 then D

^j_r

D

^γ_z

(Kg)(r, z) is a linear combination of terms of the form

T = R

R

R

R+

(D

^α

g)(tr, z + τ r)t

^µ

ϕ(t)τ

^ν

ψ(τ ) dt dτ

= r

⁻²

R

R

R

R⁺

(D

^α

g)(t, τ )  t r



µ

ϕ  t r

 τ − z r



ν

ψ  τ − z r

 dt dτ where |α| = j + |γ|, µ + ν = j.

Let j + |γ| ≥ hsi + 1, r < 1. Since

R

R

R

R⁺

r

⁻²

 t r



µ

ϕ  t r

 τ − z r



ν

ψ  τ − z r



dt dτ = R

R

R

R⁺

t

^µ

ϕ(t)τ

^ν

ψ(τ ) dt dτ is a constant, the function D

^j_r

D

^γ_z

Kg can be written as a finite sum of expressions of the form

T

⁰

= cr

−2+hsi−j−|γ|

R

R

R

R+

((D

^α

g)(t, τ ) − (D

^α

g)(r, z))

×  t r



µ1

ϕ

^(µ2)

 t r

 τ − z r



ν1

ψ

^(ν2)

 τ − z r

 dt dτ

= cr

^{hsi−j−|γ|}

R

R

R

R⁺

((D

^α

g)(tr, z + τ r) − (D

^α

g)(r, z))

× t

^µ1

ϕ

^(µ2)

(t)τ

^ν1

ψ

^(ν2)

(τ ) dt dτ where |α| = hsi. Consequently, for j + |γ| ≥ hsi + 1 we obtain

R

R 1

R

0

r

p(β−s+j+|γ|)

|D

_r^j

D

^γ_z

(Kg)(r, z)|

^p

dr dz

≤ c X

|α|=s

R

R

R

R+

r

^p(β−σ)

1/4

R

−1/4 5/4

R

3/4

|(D

^α

g)(tr, z + τ r) − (D

^α

g)(r, z)|

^p

dt dτ dr dz

= c X

|α|=s

R

R

R

R⁺

r

^{p(β−σ)−2}



^z+r/4

R

z−r/4 5r/4

R

3r/4

|(D

^α

g)(r

⁰

, z

⁰

) − (D

^α

g)(r, z)|

^p

dr

⁰

dz

⁰

 drdz

≤ c X

|α|=s

R

R

R

R+

r

^pβ



R R

R R⁺

|(r⁰−r,z⁰−z)|<r/2

|(D

^α

g)(r

⁰

, z

⁰

) − (D

^α

g)(r, z)|

^p

(|r − r

⁰

|

²

+ |z − z

⁰

|

²

)

^1+pσ/2

dr

⁰

dz

⁰

 drdz

≤ ckgk

^p_Ws

p,β(R+×Rⁿ⁻²)

.

Analogously, the inequality (1.14) can be proved. Furthermore, it can be shown that r

^β

D

_z^α

Kg ∈ L

_p

(R

+

× R

ⁿ⁻²

). Together with (1.13) and (1.14) this implies Kg ∈ T

∞

ν=1

W

p,β+hsi−s+ν^hsi+ν

(R

+

× R

ⁿ⁻²

).

R e m a r k 2. If we interpret Kg as a function on D (i.e. we define v(y, z) = (Kg)(|y|, z) = (Kg)(r, z)) then

Kg ∈

∞

\

ν=1

W

p,β+hsi−s+ν−1/p^hsi+ν

(D) for g ∈ W

_p,β^s

(R

+

× R

ⁿ⁻²

) . The following lemma is a consequence of the Hardy inequality

∞

R

0

r

^β−p

|f (r)|

^p

dr ≤

 p

|β − p + 1|



p ∞

R

0

r

^β

|f

⁰

(r)|

^p

dr , which is satisfied if f (0) = 0, β < p − 1 or if f (∞) = 0, β > p − 1.

Lemma 5. Let u ∈ W

p,β⁰

(D) (β > −2/p) be such that ∇u ∈ W

_p,β⁰ 0

(D) where β

⁰

> 1 − 2/p. Then u ∈ V

_p,β¹ 0

(D).

Now we can prove an imbedding analogous to Theorem 1 for the spaces W

_p,β^s

(D).

Theorem 3. Let s

⁰

≥ s, β

⁰

− s

⁰

= β − s and β > −2/p. Then W

_p,β^s0 0

(D) is continuously imbedded in W

_p,β^s

(D).

P r o o f. Without loss of generality we can assume that l ≤ s < s

⁰

≤ l + 1 where l is a non-negative integer. We consider the following cases: (a) s = l, s

⁰

= l + σ (0 < σ < 1), (b) s = l + σ, s

⁰

= l + σ

⁰

(0 < σ < σ

⁰

< 1), (c) s = l + σ, s

⁰

= l + 1 (0 < σ < 1).

(a) Let u ∈ W

_p,β^s0 0

(D) and v

α

= D

^α

u (|α| = l). By Lemma 5 and (1.14) we have Kv

^◦α

∈ W

_p,β+1−σ¹

(D) = W

_p,β+1¹

(D) ⊂ W

_p,β⁰

(D) and v

α

− Kv

^◦_α

∈ W

_p,β⁰

(D).

Hence, v

α

= D

^α

u ∈ W

_p,β⁰

(D) and by Lemma 5 we obtain D

^α

u ∈ W

_p,β⁰

(D) for

|α| ≤ l.

(b) Analogously to (a) we obtain Kv

^◦α

∈ W

_p,β¹ 0+1−σ⁰

(D) = W

_p,β+1−σ¹

(D) for

|α| = l. Since Kv

^◦_α

= 0 for r = |y| > 2 this implies Kv

^◦α

∈ W

_p,β+1¹

(D) ⊂ W

_p,β⁰

(D) for |α| = l. Furthermore, by (1.14), v

α

−Kv

^◦_α

∈ W

_p,β−σ⁰

(D)∩W

_p,β⁰ 0

(D) ⊂ W

_p,β⁰

(D) and analogously to (a) it follows that D

^α

u ∈ W

_p,β⁰

(D) for |α| ≤ l. Using the fact that β − β

⁰

= σ − σ

⁰

we get

R R

D D

|x−x⁰|<|y|/2

|y|

^pβ

|(D

^α

u)(x) − (D

^α

u)(x

⁰

)|

^p

dx dx

⁰

|x − x

⁰

|

^n+pσ

≤ c R R

D D

|x−x⁰|<|y|/2

|y|

^pβ0

|(D

^α

u)(x) − (D

^α

u)(x

⁰

)|

^p

dx dx

⁰

|x − x

⁰

|

^n+pσ0

for |α| = l. Hence, u ∈ W

_p,β^s

(D).

(c) By Lemma 5 the space W

_p,β^l+10

(D) is imbedded in W

_p,β^l

(D). Furthermore, the equation

(D

^α

u)(x) − (D

^α

u)(x

⁰

) = (x − x

⁰

)

1

R

0

(∇D

^α

u)(x + t(x

⁰

− x)) dt yields

R R

D D

|x−x⁰|<|y|/2

|y|

^pβ

|(D

^α

u)(x) − (D

^α

u)(x

⁰

)|

^p

dx dx

⁰

|x − x

⁰

|

^n+pσ

≤ c R

D

|y|

^pβ0

X

|α⁰|=l+1

|(D

^α0

u)(x)|

^p

dx for |α| = l. This implies W

_p,β^l+10

(D) ⊂ W

_p,β^s

(D).

Corollary 1. If β > s − 2/p then

W

_p,β^s

(D) ⊂ W

_p,β−s+hsi^hsi

(D) ⊂ W

p,β−s+hsi−1^hsi−1

(D) ⊂ . . . ⊂ W

_p,β−s⁰

(D) , i.e. W

_p,β^s

(D) ⊂ V

_p,β^s

(D).

R e m a r k 3. Analogously to Theorem 3 it can be proved that W

_p,β^s0 0

(R

⁺

× R

ⁿ⁻²

) is continuously imbedded in W

_p,β^s

(D) if s

⁰

≥ s, β

⁰

− s

⁰

= β − s, β > −1/p.

1.3. Traces of functions from W

_p,β^s

(D) on M. In [6] (Lemma 1.1) it has been proved that the trace of a function u ∈ W

_p,β^l

(D) belongs to the Besov space B

^l−β−2/pp

(M ) for l an integer, l > β + 2/p > 0. Here the norm in B

_p^κ

(M ) is defined by

(1.15) kuk

_Bκ p(M )

=

 kuk

^p_L

p(M )

+ R

M

R

M

|∆

^k_ζ

u(z)|

^p

dz dζ

|ζ|

^n−2+pκ



1/p

where ∆

^k_ζ

u(z) = P

k

ν=0

(−1)

^{ν k}_ν

u(z + νζ).

Theorem 4. Let u ∈ W

p,β^s

(D), s > β + 2/p > 0. Then the trace of u on M exists and belongs to the Besov space B

p^s−β−2/p

(M ).

P r o o f. Let {u

n

} be a sequence of functions from C

₀^∞

(D) which converges to u in W

_p,β^s

(D). Furthermore, let {f

n

} be the sequence of the traces of u

_n

on M , i.e. f

n

(z) = u

n

(0, 0, z) = u

^◦n

(0, z) (the function u

^◦n

is defined by (1.11)). Since the functions u

^◦n

can be interpreted as functions from W

_p,−1/p+1/2¹

(R

+

× R

ⁿ⁻²

) Lemma 4 yields

R

R⁺×Rⁿ⁻²

r

^−1−p/2

|u

^◦_n

(r, z) − (Ku

^◦n

)(r, z)|

^p

dr dz < ∞ .

Consequently, u

^◦n

(0, z) − (Ku

^◦n

)(0, z), i.e. f

n

is the trace of Ku

^◦n

on M . We denote the trace of Ku

^◦

on M by f . Then by Lemma 1.1 of [6] and Lemmas 3 and 4 we get

kf − f

_n

k

B^s−β−2/p_p (M )

≤ ckKu

^◦

− Ku

^◦_n

k

Wp,β−s+hsi+1^hsi+1 (D)

≤ c

⁰

ku − u

_n

k

_W^s

p,β(D)

. Hence, {f

n

} converges to f in B

p^s−β−2/p

(M ).

Conversely, it can be proved that every f ∈ B

p^s−β−2/p

(M ) (s > β + 2/p > 0) can be extended to a function v ∈ W

_p,β^s

(D). We define the extension operator K as follows:

(Kf )(r, z) = χ(r) R

Rⁿ⁻²

g(z + τ r)ψ(τ

1

) . . . ψ(τ

n−2

) dτ

where ψ ∈ C

₀^∞

(R) satisfies (1.12). By Lemma 1.2 of [6] the operator K defines a continuous map from B

_p^κ

(M ) into W

_{p,s−κ−2/p}^s

(D) for κ > 0, κ not an integer, s an integer, s + h−κ − 2/pi > −2. From Theorem 3 it follows that this is also true for real s, s + h−κ − 2/pi > −2.

1.4. Connection between V

_p,β^s

(D) and W

_p,β^s

(D). In [6], [7] it has been proved that every u ∈ W

_p,β^s

(D) is the sum of a “quasi-polynomial” and a function from V

_p,β^s

(D) if s is a non-negative integer and β + 2/p is not an integer. We will give a similar connection between V

_p,β^s

(D) and W

_p,β^s

(D) without any restrictions on s and β.

Let u ∈ W

_p,β^s

(D) (β > −2/p). We denote the derivatives ∂

_yⁱ₁

∂

_y^j₂

u (i + j ≤ hsi) by u

ij

. By using the properties of the operator K the following lemma can be easily proved (see [9]).

Lemma 6. If u∈W

p,β^s

(D) then the following inequality holds for i+j+|α|≤hsi:

R

D

r

p(β−s+i+j+|α|)

   

∂

ⁱ_y1

∂

_y^j₂

∂

_z^α

u −

hsi−i−j−|α|

X

µ+ν=0

(∂

_z^α

Ku

^◦_{i+µ, j+ν}

) y

₁^µ

y

^ν₂

µ!ν!

   

p

dx

≤ ckuk

^p_Ws p,β(D)

.

Lemma 6 implies the following corollary.

Corollary 2. If u ∈ W

p,β^s

(D) then u −

hsi

X

i+j=0

(Ku

^◦ij

)(r, z) y

ⁱ₁

y

₂^j

i!j! ∈ V

_p,β^s

(D) .

P r o o f. By Lemma 4 we have ∂

_y^µ₁

∂

_y^ν₂

∂

_z^α

(Ku

^◦ij

) ∈ V

p,β−s+i+j+µ+ν+|α|⁰

(D) if µ + ν ≥ 1 or |α| ≥ s − i − j. Therefore, from Lemma 6 it follows that

R

D

r

p(β−s+µ+ν+|α|)

   

∂

_y^µ₁

∂

^ν_y2

∂

_z^α

 u −

hsi

X

i+j=0

(Ku

^◦ij

) y

₁ⁱ

y

^j₂

i!j!

   

p

dx < ∞ for µ + ν + |α| ≤ hsi, which yields the desired conclusion.

By Corollary 2 for every u ∈ W

_p,β^s

(D) there exist v

ij

∈ W

_p,β+1/p^s

(R

+

× R

ⁿ⁻²

) such that

u −

hsi

X

i+j=0

(Kv

ij

)(r, z) y

₁ⁱ

y

₂^j

i!j! ∈ V

_p,β^s

(D) .

Now we investigate the question whether the v

ij

are (in some sense) uniquely determined by u. Similarly to [8] we introduce the following equivalence relation in W

_p,β+1/p^s

(R

+

× R

ⁿ⁻²

) (s > 0, β > −2/p):

(1.16) f

^β−s,p

∼ g ⇔ R

R+×Rⁿ⁻²

r

^p(β−s)+1

|(Kf )(r, z) − (Kg)(r, z)|

^p

dr dz < ∞ .

Another characterization is the following:

(1.17) f

^β−s,p

∼ g ⇔ K(f − g) ∈

∞

\

ν=0

V

p,β−s+ν+1/p^ν

(R

+

× R

ⁿ⁻²

) .

P r o o f. If f, g ∈ W

_p,β+1/p^s

(R

+

× R

ⁿ⁻²

) and f

^β−s,p

∼ g then by Lemma 4 K(f − g) ∈

∞

\

ν=hsi+1

W

p,β−s+ν+1/p^ν

(R

+

× R

ⁿ⁻²

) ∩ V

_p,β−s+1/p⁰

(R

+

× R

ⁿ⁻²

) , and (1.17) follows from W

_p,β^l

(R

+

×R

ⁿ⁻²

)∩V

_p,β−l⁰

(R

+

×R

ⁿ⁻²

) ⊂ V

_p,β^l

(R

+

×R

ⁿ⁻²

) (cf. Remark 1).

R e m a r k 4. Let f, g ∈ W

_p,β+1/p^s

(R

+

× R

ⁿ⁻²

).

(a) If β − s < −2/p then f

^β−s,p

∼ g ⇔ f |

M

= g|

M

. (b) If β − s = −2/p then

f

^β−s,p

∼ g ⇔ R

R⁺×Rⁿ⁻²

r

^p(β−s)+1

|f (r, z) − g(r, z)|

^p

dr dz < ∞ .

(c) If β − s > −2/p then f

^β−s,p

∼ g

^β−s,p

∼ 0.

P r o o f. (a) If β − s < −2/p then the trace of f − g on M exists and coincides with that of K(f − g). The Hardy inequality and Lemma 4 imply

(1.18) R

R+×Rⁿ⁻²

r

^p(β−s)+1

|K(f − g) − (f − g)|

_M

|

^p

dr dz

≤ c R

R⁺×Rⁿ⁻²

r

^{p(β−s+1)+1}

   

∂

∂r K(f − g)    

p

dr dz ≤ ckf − gk

^p_Ws

p,β+1/p

(R

+

× R

ⁿ⁻²

) . Hence, from (1.16) it follows that

R

R⁺×Rⁿ⁻²

r

^p(β−s)+1

|(f − g)|

_M

|

^p

dr dz < ∞ ,

i.e. f |

M

= g|

M

. Conversely, if f |

M

= g|

M

then (1.16) follows from (1.18).

(b) If β − s = −2/p then W

_p,β+1/p^s

(R

+

× R

ⁿ⁻²

) ⊂ W

p,β−hsi+1/p^s−hsi

(R

+

× R

ⁿ⁻²

) (see Remark 3) and Lemma 4 yields

R

R⁺×Rⁿ⁻²

r

^p(β−s)+1

|Kf − f |

^p

dr dz

≤ ckf k

^p

Wp,β−hsi+1/p^s−hsi (R+×Rⁿ⁻²)

≤ ckf k

^p_Ws

p,β+1/p(R+×Rⁿ⁻²)

. This proves the assertion.

(c) If β − s > −2/p then W

_p,β+1/p^s

(R

+

× R

ⁿ⁻²

) is imbedded in V

_p,β+1/p^s

(R

+

× R

ⁿ⁻²

) (cf. Corollary 1, Remark 3). Consequently, (1.16) is satisfied for all f, g ∈ W

_p,β+1/p^s

(R

+

× R

ⁿ⁻²

).

Lemma 7. Let f ∈ W

p,β+1/p^s

(R

+

× R

ⁿ⁻²

), β − s ≤ −1 − 2/p. Then f

^β−s,p

∼ 0 iff ∂f /∂z

i

β−s+1,p

∼ 0 for one i ∈ {1, . . . , n − 2}.

P r o o f. 1) If f

^β−s,p

∼ 0 then Kf ∈ V

_p,β+1/p^s

(R

+

× R

ⁿ⁻²

) and K ∂f

∂z

i

= ∂

∂z

i

Kf ∈ V

_p,β+1/p^s−1

(R

+

× R

ⁿ⁻²

) , i.e. ∂f /∂z

i

β−s+1,p

∼ 0.

2) Let ∂f /∂z

1

β−s+1,p

∼ 0. If β − s < −1 − 2/p this means

_∂z^∂

1

(f |

M

) =

_∂z^∂f

1

|

_M

= 0 where f |

M

∈ B

p^s−β−2/p

(M ). Hence, f |

M

= 0, i.e. f

^β−s,p

∼ 0. If β − s = −1 − 2/p we first assume that f (r, z) = 0 for r > 1 and |z

1

| > 1. Using the Hardy inequality we get

R

R⁺×Rⁿ⁻²

r

⁻¹

|f (0, z)|

^p

dr dz ≤ c R

R⁺×Rⁿ⁻²

(r

⁻¹

|f (r, z)|

^p

+r

⁻²

|f (r, z)−f (0, z)|

^p

) dr dz

≤ c R

R+×Rⁿ⁻²

 r

⁻¹

   

z1

R

−1

∂

∂t f (r, t, z

2

, . . . , z

n−2

) dt    

p

+    

∂f (r, z)

∂r    

p

 dr dz

≤ c

 R

R⁺×Rⁿ⁻²

r

⁻¹

   

∂f

∂z

1

   

p

dr dz +

∂f

∂r

p

Lp(R+×Rⁿ⁻²)

 .

Hence, f |

M

= 0, i.e. f

^β−s,p

∼ 0. If f has an arbitrary support then we show that (ϕf )|

M

= 0 for every cut-off function ϕ with compact support.

Theorem 5. Let u, v

ij

(i + j ≤ hsi) be arbitrary functions from W

_p,β^s

(D) and W

_p,β+1/p^s−i−j

(R

+

× R

ⁿ⁻²

), respectively. Then

(1.19) u −

hsi

X

i+j=0

1

i!j! (Kv

ij

)(r, z)y

₁ⁱ

y

^j₂

∈ V

_p,β^s

(D) if and only if v

ij

β−s+i+j,p

∼ u

^◦ij

for i + j ≤ hsi.

P r o o f. 1) Let v

ij

β−s+i+j,p

∼ u

^◦ij

. Then K(v

ij

− u

^◦_ij

) ∈ V

p,β−s+hsi+i+j+1+1/p^hsi+1

(R

+

× R

ⁿ⁻²

) for i + j ≤ hsi. Interpreting K(v

ij

− u

^◦_ij

) as functions on D we get K(v

ij

− u

^◦ij

) ∈ V

p,β−s+hsi+i+j+1^hsi+1

(D) and K(v

ij

− u

^◦ij

)y

₁ⁱ

y

₂^j

∈ V

p,β−s+hsi+1^hsi+1

(D) ⊂ V

_p,β^s

(D). Using Corollary 2 we obtain (1.19).

2) Assume that (1.19) is satisfied. Then by Corollary 2

(1.20) R

D

r

^{p(β−s+µ+ν)}

   

D

^µ_y1

D

_y^ν₂

hsi

X

i+j=0

(K(u

^◦ij

− v

_ij

)) y

ⁱ₁

y

₂^j

i!j!

   

p

dx < ∞ .

Since D

^γ_y

K(u

^◦_ij

− v

_ij

) ∈ V

p,β−s+i+j+|γ|⁰

(D) for |γ| ≥ 1 (see Lemma 4), (1.20) yields

(1.21) R

D

r

^{p(β−s+µ+ν)}

   

hsi−µ−ν

X

i+j=0

(K(u

^◦i+µ,j+ν

− v

_i+µ,j+ν

)) y

₁ⁱ

y

₂^j

i!j!

   

p

dx < ∞ .

For µ + ν = hsi, (1.21) implies K(u

^◦µν

− v

_µν

) ∈ V

_{p,β−s+µ+ν}⁰

(D), i.e. u

^◦µν

β−s+µ+ν,p

∼ v

µν

. Then from (1.21) it follows that

R

D

r

^{p(β−s+µ+ν)}

   

hsi−1−µ−ν

X

i+j=0

(K(u

^◦i+µ,j+ν

− v

_i+µ,j+ν

)) y

ⁱ₁

y

^j₂

i!j!

   

p

dx < ∞ .

For µ + ν = hsi − 1 this yields u

^◦µν

β−s+µ+ν,p

∼ v

µν

. Analogously, by induction on µ + ν we show that u

^◦µν

β−s+µ+ν,p

∼ v

νµ

for µ + ν ≤ hsi − 2.

1.5. Connection between the spaces of traces. We denote by B

_p,β^s−1/p

(Γ

^±

) (s >

1/p, β > −2/p) the space of the traces of functions from W

_p,β^s

(D) on the sides Γ

⁺

and Γ

⁻

, respectively, provided with the norm

kuk

B_p,β^s−1/p(Γ^±)

= inf{kvk

W_p,β^s (D)

: v ∈ W

_p,β^s

(D), v|

_Γ^±

= u} .

By Corollary 1, B

_p,β^s−1/p

(Γ

^±

) ⊂ B

^◦s−1/p_p,β

(Γ

^±

) for β > s − 2/p. Let u ∈ B

_p,β^s−1/p

(Γ

⁺

) and let v ∈ W

_p,β^s

(D) be an extension of u. Then by Theorem 5, u has the repre- sentation

u = v|

_Γ⁺

=

 X

i+j≤hsi

(Kv

ij

) y

ⁱ₁

y

₂^j

i!j! + v

⁰

   

Γ⁺

=

hsi

X

k=0

(Kv

k

) r

^k

k! + u

⁰

where

v

k

=

k

X

i=0

k i



cos ω

0

2



i

 sin ω

0

2



k−i

v

i,k−i

∈ W

_p,β+1/p^s−k

(R

+

× R

ⁿ⁻²

)

and u

⁰

∈ B

^◦s−1/p_p,β

(Γ

⁺

).

Lemma 8. Let v

^k

∈ W

_p,β+1/p^s−k

(R

⁺

× R

ⁿ⁻²

) (k = 0, 1, . . . , hsi). Then

hsi

X

k=0

(Kv

k

) r

^k

k! ∈ V

_p,β−s+1/p⁰

(R

+

× R

ⁿ⁻²

) iff v

k

β−s+k,p

∼ 0 .

P r o o f. 1) If v

k

β−s+k,p

∼ 0 then Kv

k

∈ V

p,β−s+k+1/p⁰

(R

+

×R

ⁿ⁻²

) and (Kv

k

)r

^k

∈ V

_p,β−s+1/p⁰

(R

+

× R

ⁿ⁻²

).

2) If P

hsi

k=0

(Kv

k

)r

^k

/k! ∈ V

_p,β−s+1/p⁰

(R

+

× R

ⁿ⁻²

) then from the properties of K (see Lemma 4) it follows that P

hsi

k=0

(Kv

k

)r

^k

/k! ∈ V

_p,β−s+1/p^s

(R

+

× R

ⁿ⁻²

), i.e.

R

R+×Rⁿ⁻²

r

^{p(β−s+µ)+1}

   

D

_r^µ

hsi

X

k=0

(Kv

k

) r

^k

k!

   

p

dr dz < ∞

for µ = 0, 1, . . . , hsi. Analogously to the proof of Theorem 5, this implies that v

k

β−s+k,p

∼ 0.

As a consequence of Lemma 8 we obtain the following theorem.

Theorem 6. Let u ∈ B

p,β^s−1/p

(Γ

⁺

). Then there exist v

k

∈ W

_p,β+1/p^s−k

(R

+

×R

ⁿ⁻²

) (k = 0, 1, . . . , hsi) such that

(1.22) u −

hsi

X

k=0

(Kv

k

)(r, z) r

^k

k! ∈ B

^◦s−1/p_p,β

(Γ

⁺

) .

The functions v

k

in (1.22) are uniquely determined in the following sense: if w

k

∈ W

_p,β+1/p^s−k

(R

+

× R

ⁿ⁻²

) (k = 0, 1, . . . , hsi) then

u −

hsi

X

k=0

(Kw

k

)(r, z) r

^k

k! ∈ B

^◦s−1/p_p,β

(Γ

⁺

) iff w

k

β−s+k,p

∼ v

k

.

In particular , v

k

β−s+k,p

∼ ∂

_r^k

u for k ≤ hs−1/pi and v

k

β−s+k,p

∼ 0 for k > [s−β −2/p]

(here [κ] denotes the largest integer less than or equal to κ, i.e. [κ] = −h−κi−1).

P r o o f. It remains to show that v

k

β−s+k,p

∼ ∂

_r^k

u for k ≤ hs−1/pi. Differentiating (1.22) we get

∂

_r^ν

 u −

hsi

X

k=0

(Kv

k

) r

^k

k!



∈ B

^{◦s−ν−1/p}_p,β

(Γ

⁺

) . Since ∂

_r^j

Kv

_k

∈ V

_p,β+1/p^s−k−j

(Γ

⁺

) for j ≥ 1 (see Lemma 4) this implies

(1.23) ∂

_r^ν

u −

hsi−ν

X

k=0

(Kv

k+ν

) r

^k

k! ∈ B

^{◦s−ν−1/p}_p,β

(Γ

⁺

) .

It can be easily verified that K(r

^k

v)∈V

_p,β−k+1/p^s

(R

+

× R

ⁿ⁻²

) for v∈W

_p,β+1/p^s

(R

+

× R

ⁿ⁻²

) and positive integers k. Then (1.23) yields that K(∂

_r^ν

u) − KKv

ν

∈ V

_p,β+ν+1/p^s

(R

+

× R

ⁿ⁻²

), i.e. ∂

^ν_r

u

^β−s+ν,p

∼ Kv

_ν^β−s+ν,p

∼ v

ν

.

The following lemma gives a connection between the spaces B

_p,β^s−1/p

(R

+

× R

ⁿ⁻²

) and W

_p,β^s−1/p

(R

+

× R

ⁿ⁻²

).

Lemma 9. Let s − 1/p be non-integer , s > 1/p and β > −1/p. Then B

_p,β^s−1/p

(R

+

× R

ⁿ⁻²

) = W

_p,β^s−1/p

(R

+

× R

ⁿ⁻²

) .

P r o o f. 1) If u ∈ B

_p,β^s−1/p

(R

+

× R

ⁿ⁻²

) then there exist v

k

∈ W

_p,β+1/p^s−k

(R

+

× R

ⁿ⁻²

) such that

u −

hsi

X

k=0

(Kv

k

) r

^k

k! ∈ B

^◦s−1/p_p,β

(R

+

× R

ⁿ⁻²

) = V

_p,β^s−1/p

(R

+

× R

ⁿ⁻²

) . Since (Kv

k

)r

^k

∈ W

_p,β^s−1/p

(R

+

× R

ⁿ⁻²

) we get u ∈ W

_p,β^s−1/p

(R

+

× R

ⁿ⁻²

).

2) Let u ∈ W

_p,β^s−1/p

(R

+

× R

ⁿ⁻²

). Then analogously to Theorem 5 it can be shown that

v = u −

hs−1/pi

X

k=0

(Ku

k

) r

^k

k! ∈ V

_p,β^s−1/p

(R

+

× R

ⁿ⁻²

) = B

^◦s−1/p_p,β

(R

+

× R

ⁿ⁻²

)

where u

k

= ∂

_r^k

u ∈ W

_p,β^s−k−1/p

(R

+

× R

ⁿ⁻²

). Without loss of generality we can assume that Γ

⁺

is the half-plane Γ

⁺

= {x = (y, z) : y

1

> 0, y

2

= 0, z ∈ R

ⁿ⁻²

} which can be identified with R

+

× R

ⁿ⁻²

. If v

⁰

∈ V

_p,β^s

(D) is an extension of v then

u

⁰

= v

⁰

+

hs−1/pi

X

k=0

(Ku

k

) y

₁^k

k!

is an extension of u which lies in W

_p,β^s

(D). Hence, u ∈ B

_p,β^s−1/p

(R

+

× R

ⁿ⁻²

).

2. Applications to boundary value problems

2.1. Compatibility conditions for boundary data on Γ

^±

. For the spaces V

_p,β^s

(D) the following lemma has been proved (see [4], [9]).

Lemma 10. Let B

k^±

(k = 1, . . . , p

^±

) be homogeneous differential operators with constant coefficients of order m

^±_k

< s − 1/p. Assume that {B

_k⁺

} and {B

_k⁻

} are normal on Γ

⁺

and Γ

⁻

, respectively. Then for all g

_k^±

∈ B

^◦s−m

± k−1/p

p,β

(Γ

^±

) there exists u ∈ V

_p,β^s

(D) such that B

_k^±

u = g

_k^±

on Γ

^±

for k = 1, . . . , p

^±

.

We will investigate conditions on g

^±_k

(g

^±_k

∈ B

^s−m

± k−1/p

p,β

(Γ

^±

)) under which there exists u ∈ W

_p,β^s

(D) satisfying the boundary conditions

(2.1) B

_k^±

u = X

µ+ν+|γ|=m^±_k

b

^±_k;µνγ

D

^µ_y1

D

_y^ν₂

D

_z^γ

u = g

_k^±

on Γ

^±

(k = 1, . . . , p

^±

). Here we restrict ourselves to the case n = 3.

By Theorem 6, B

_k^±

u|

_Γ^±

− g

_k^±

(u ∈ W

_p,β^s

(D), g

_k^±

∈ B

^s−m

± k−1/p

p,β

(Γ

^±

)) belongs to B

^◦s−m

± k−1/p

p,β

(Γ

^±

) iff

(2.2) ∂

_r^j

(B

_k^±

u|

Γ^±

− g

_k^±

)

^β−s+m

± k+j,p

∼ 0

for j = 0, 1, . . . , [s − β − 2/p] − m

^±_k

. Using the representation u =

[s−β−2/p]

X

i+j=0

(Ku

^◦ij

) y

ⁱ

y

^j

i!j! + u

⁰

where u

^◦ij

∈ W

_p,β+1/p^s−i−j

(R

+

×R

ⁿ⁻²

), u

⁰

∈ V

_p,β^s

(D), D

^µ_r

D

^ν_z

Ku

^◦_ij

∈ V

_p,β+1/p^{s−i−j−µ−ν}

(R

+

× R

ⁿ⁻²

) if µ ≥ 1 or ν ≥ s − β − 2/p − i − j (see Theorem 5) we get the following equivalences:

(2.3) X

µ+ν+γ=m^±_k

(−i)

^µ+ν

b

^±_k;µνγ

j

X

σ=0

 j σ



cos ω

0

2



σ



± sin ω

0

2



j−σ

×D

_z^γ

u

^◦µ+σ,ν+j−σ

β−s+m^±_k+j,p

∼ ∂

_r^j

g

_k^±