Weighted Sobolev spaces

kuk^p_Hm p,θ(G)

m

X

k=0

|u|^p

H_p,θ^k (G) (2.28)

where

|u|^p

H_p,θ^k (G):= X

α∈N^d0

|α|=k

Z

G

ρ(x)^|α|D^αu(x)

pρ(x)^θ−ddx, (2.29)

fork∈ {0, . . . , m}; see, e.g., [93, Proposition 2.2].

Now we present some useful properties of the spaceH_p,θ^γ (G) taken from [93], see also [80, 81].

Lemma 2.45. LetG⊂R^dbe a domain with non-empty boundary∂G,γ, θ∈R, andp∈(1,∞).

(i) H_p,θ^γ (G) is a separable and reflexive Banach space.

(ii) The space C₀^∞(G) is dense in H_p,θ^γ (G).

(iii) u∈H_p,θ^γ (G) if, and only if, u, ψux∈H_p,θ^γ−1(G) and kuk_H^γ

p,θ(G) ≤Ckψu_xk_Hγ−1

p,θ (G)+Ckuk_Hγ−1

p,θ (G)≤Ckuk_H^γ

p,θ(G). Also, u∈H_p,θ^γ (G) if, and only if, u,(ψu)x∈H_p,θ^γ−1(G) and

kuk_H^γ

p,θ(G)≤Ck(ψu)_xk_H^γ−1

p,θ (G)+Ckuk_H^γ−1

p,θ (G)≤Ckuk_H^γ

p,θ(G). (iv) For any ν, γ ∈R, ψ^νH_p,θ^γ (G) =H_p,θ−νp^γ (G) and

kuk_H^γ

p,θ−νp(G)≤Ckψ^−νuk_H^γ

p,θ(G) ≤Ckuk_H^γ

p,θ−νp(G).

(v) If 0< η <1, γ = (1−η)ν₀+ην₁, 1/p= (1−η)/p₀+η/p₁ and θ= (1−η)θ₀+ηθ₁ with ν₀, ν₁, θ₀, θ₁∈R andp₀, p₁ ∈(1,∞), then

H_p,θ^γ (G) = H_p^ν0

0,θ0(G), H_p^ν1

1,θ1(G)

η (equivalent norms).

Consequently, if γ ∈(ν0, ν1) then, for any ε >0, there exists a constant C, depending on ν₀, ν₁, θ, p, andε, such that

kuk_H^γ

p,θ(G)≤εkuk_H^ν1

p,θ(G)+C(ν0, ν1, θ, p, ε)kuk_H^ν0 p,θ(G).

Also, if θ∈(θ₀, θ₁) then, for anyε >0, there exists a constant C, depending onθ₀, θ₁,γ, p, and ε, such that

kuk_H^γ

p,θ(G) ≤εkuk_H^γ

p,θ0(G)+C(θ₀, θ₁, γ, p, ε)kuk_H^γ

p,θ1(G).

(vi) There exists a constant c₀ >0 depending on p, θ, γ and the function ψ such that, for all c≥c₀, the operator ψ²∆−c is an isomorphism from H_p,θ^γ+1(G) to H_p,θ^γ−1(G).

(vii) If G is bounded, then H_p,θ^γ

1(G),→H_p,θ^γ

2(G) for θ1 < θ2.

(viii) The dual of H_p,θ^γ (G) and the weighted Sobolev space H_p^−γ0,θ⁰(G) with 1/p+ 1/p⁰ = 1 and θ/p+θ⁰/p⁰=dare isomorphic. That is,

H_p,θ^γ (G)∗

'H_p^−γ0,θ⁰(G) where 1 p+ 1

p⁰ = 1 and θ p +θ⁰

p⁰ =d. (2.30) Remark 2.46. Assertions (iv) and (vi) in Lemma 2.45 imply the following: If u ∈ H_p,θ−p^γ (G) and ∆u∈H_p,θ+p^γ (G), thenu∈H_p,θ−p^γ+2 (G) and there exists a constantC, which does not depend on u, such that

kuk_Hγ+2

p,θ−p(G)≤Ck∆uk_H^γ

p,θ+p(G)+Ckuk_H^γ

p,θ−p(G).

A proof of the following equivalent characterization of the weighted Sobolev spacesH_p,θ^γ (G) can be found in [93, Proposition 2.2].

Lemma 2.47. Let {ξ_n:n∈Z} ⊆ C₀^∞(G) be such that for all n∈Zand m∈N0,

|D^mξ_n| ≤C(m)c^nm and suppξ_n⊆ {x∈G:c^−n−k0 < ρ(x)< c^−n+k0} (2.31) for some c > 1 and k₀ > 0, where the constant C(m) does not depend on n ∈ Z and x ∈ G.

Then, for any u∈H_p,θ^γ (G), X

n∈Z

c^nθkξ_−n(cⁿ·)u(cⁿ·)k^p

Hp^γ ≤Ckuk^p

H_p,θ^γ (G). If in addition

X

n∈Z

ξ_n(x)≥δ >0 for all x∈G, (2.32) then the converse inequality also holds.

The following sequences{ξ_n:n∈Z}will be useful, when we apply Lemma 2.47 in the proofs of Lemma 3.5(i), Lemma 4.9 and Theorem 6.7, respectively.

Remark 2.48. (i) It can be shown that for anyi, j∈ {1, . . . , d}, both ξ_n⁽¹⁾:=e⁻ⁿ(ζ_n)_xi : n∈Z and

ξ_n⁽²⁾:=e⁻²ⁿ(ζ_n)_xix^j : n∈Z satisfy (2.31) with c:=e. Thus, for any p∈(1,∞) andθ∈R,

X

n∈Z

e^nθ

keⁿ(ζ−n)_xⁱ(eⁿ·)u(eⁿ·)k^p_Hγ

p +ke²ⁿ(ζ−n)_xⁱ_x^j(eⁿ·)u(eⁿ·)k^p_Hγ p

≤Ckuk^p_Hγ p,θ(G). (ii) Let c0>1 andk1 >0. Fix a non-negative function ˜ζ ∈ C₀^∞(R+) with

ζ(t) = 1˜ for all t∈h1

Cc^−k₀ ¹, C(0)c^k₀¹ i

,

where C and C(0) are as in (2.25). Then, the sequence{ξ_n:n∈Z} ⊆ C₀^∞(G) defined by ξn:= ˜ζ(cⁿ₀ψ(·)), n∈Z,

fulfils the conditions (2.31) and (2.32) from Lemma 2.47 withc=c0and a suitablek0 >0.

Furthermore,

ξn(x) = 1 on

x∈G:c^−n−k₀ ¹ ≤ρ(x)≤c^−n+k₀ ¹ .

(iii) Let {ξ_n : n ∈ Z} ⊆ C₀^∞(G) fulfil the conditions (2.31) and (2.32) from Lemma 2.47 for some fixed constantsc >1 andk0 >0. Consider the sequence{ξ_n^∗ :n∈Z} ⊆ C₀^∞(G) given by

ξ_n^∗ := ξ_n P

j∈Zξ_j, n∈Z. Obviously,

X

n∈Z

ξ_n^∗(x) = 1 for all x∈G. (2.33)

By standard calculations one can check that the sequence{ξ^∗_n:n∈Z} ⊆ C₀^∞(G) also fulfils the condition (2.31) from Lemma 2.47. The following fact might be useful: Any x ∈G is contained in at most finitely many stripes

G_n(c, k₀) :=

x∈G:c^−n−k0 < ρ(x)< c^−n+k0 , n∈Z. (2.34) Even more, there exists a finite constant C =C(c, k0) which does not depend on x ∈ G such that

n∈Z:x∈G_n(c, k₀) ≤C. (2.35) Let us also be a little bit more precise on the duality statement from Lemma 2.45(viii).

Remark 2.49. Fix γ ∈ R,p ∈ (1,∞), θ ∈ R and let p⁰ and θ⁰ be as in (2.30). Fix {ξ_n : n ∈ Z} ⊆ C₀^∞(G) with P

n∈Zξn = 1 on G satisfying (2.31) from Lemma 2.47 for some fixed c >1 and k₀ >0. Simultaneously, assume that we have a sequence {ξ˜_n :n∈Z} ⊆ C₀^∞(G) such that for every n∈Z, ˜ξn equals one on the support of ξn, i.e.,

ξ˜_n suppξn

= 1, (2.36)

and satisfying (2.31)—with the same c > 1 but possibly different k0 > 0—and (2.32) from Lemma 2.47. By Remark 2.48(ii) and (iii), it is clear that we can construct such sequences.

The assertion of Lemma 2.45(viii) has been proven in [93, Proposition 2.4] by showing that the mapping

Ψ :H_p^−γ0,θ⁰(G)→ H_p,θ^γ (G)∗

u7→Ψ(u) := [u,·]

with

[·,·] :H_p^−γ0,θ⁰(G)×H_p,θ^γ (G)→R, [u, v] :=X

n∈Z

c^nd ξ˜−n(cⁿ·)u(cⁿ·), ξ_−n(cⁿ·)v(cⁿ·)

(2.37) is an isomorphism; see (2.24) for the meaning of (·,·) on H_p^−γ0 (R^d)×Hp^γ(R^d). From now on we will use this notation also onH_p^−γ0,θ⁰(G)×H_p,θ^γ (G) and define

(·,·) := [·,·] on H_p^−γ0,θ⁰(G)×H_p,θ^γ (G), (2.38) with [·,·] as in (2.37). This is justified by the following calculation: Let u ∈H_p^−γ0,θ⁰(G) ⊆ D⁰(G) and ϕ∈ C₀^∞(G). Then, sinceP

n∈Zξn= 1 and ˜ξnis constructed in such a way that (2.36) holds, we obtain

(u, ϕ) = u,X

n∈Z

ξ−nϕ

=X

n∈Z

u, ξ−nϕ

=X

n∈Z

ξ˜−nu, ξ−nϕ

= [u, ϕ].

After presenting and discussing these fundamental properties of weighted Sobolev spaces, we prove now that they satisfy the following geometric Banach space properties.

Lemma 2.50. Let G be an arbitrary domain in R^d with non-empty boundary. Letγ, θ ∈R and p∈(1,∞). Then H_p,θ^γ (G) is a umd space with typer := min{2, p}.

Proof. First we prove the umd property. Obviously, the linear operator S :H_p,θ^γ (G)→Lp

Z,P(Z),X

n∈Z

e^nθδn;H_p^γ(R^d)

u7→

n7→ζ−n(eⁿ·)u(eⁿ·) .

is isometric. Therefore, and since H_p,θ^γ (G) is complete, the range of S is a closed subspace of Lp Z,P(Z),P

n∈Ze^nθδn;Hp^γ(R^d)

, which satisfies theumd property by Lemma 2.6(iii). Thus, H_p,θ^γ (G) is isomorphic to a closed subspace of aumdBanach spaces. Hence, due to Lemma 2.6(i) and Remark 2.39, the umd property ofH_p,θ^γ (G) follows.

In order to prove thatH_p,θ^γ (G) has typer = min{2, p} we argue as follows: Fix an arbitrary Rademacher sequence (rk)^∞_k=1 and a finite set {u₁, . . . , uK} ⊆ H_p,θ^γ (G). Then, by the Kahane-Khintchine inequality, see Lemma 2.10, we have

E

"

K

X

k=1

r_ku_k

r H_p,θ^γ (G)

#!¹_r

≤C E

"

K

X

k=1

r_ku_k

p H_p,θ^γ (G)

#!_p¹ .

By the definition of the weighted Sobolev norm, this yields

E

"

K

X

k=1

r_ku_k

r H_p,θ^γ (G)

#!¹_r

≤C E

"

X

n∈Z

e^nθ

ζ−n(eⁿ·)X^K

k=1

r_ku_k (eⁿ·)

p Hp^γ(R^d)

#!¹_p .

Using Fubini’s theorem and the fact that ζ−n(eⁿ·) PK k=1rkuk

(eⁿ·) =PK

k=1rkζ−n(eⁿ·)u_k(eⁿ·), we obtain the estimate

E

"

K

X

k=1

r_ku_k

r H_p,θ^γ (G)

#!¹_r

≤C X

n∈Z

e^nθE

K

X

k=1

r_kζ−n(eⁿ·)u_k(eⁿ·)

p Hp^γ(R^d)

!¹_p .

Thus, using again the Kahane-Khintchine inequality, we have

E

"

K

X

k=1

rkuk

r H_p,θ^γ (G)

#!¹

r

≤C X

n∈Z

e^nθ

E

K

X

k=1

rkζ−n(eⁿ·)u_k(eⁿ·)

r Hp^γ(R^d)

^p_r!¹

p

.

Since Hp^γ(R^d) has type r, see Remark 2.39, this leads to

E

"

K

X

k=1

r_ku_k

r H_p,θ^γ (G)

#!¹_r

≤C X

n∈Z

e^nθ K

X

k=1

ζ−n(eⁿ·)u_k(eⁿ·)

r Hp^γ(R^d)

^p_r!_p¹ .

Therefore, applying the triangle inequality in L_p/r(Z,P(Z),P

ne^nθδ_n;R) yields E

"

K

X

k=1

rkuk

r H_p,θ^γ (G)

#!¹

r

≤C

K

X

k=1

n7→

ζ−n(eⁿ·)u_k(eⁿ·)

r Hp^γ(R^d)

1 r

L_p/r Z,P(Z),P

ne^nθδn;R

≤C K

X

k=1

n7→

ζ−n(eⁿ·)u_k(eⁿ·)

r Hp^γ(R^d)

L_p/r Z,P(Z),P

ne^nθδn;R

¹_r

=C K

X

k=1

X

n∈Z

e^nθkζ−n(eⁿ·)u_k(eⁿ·)k^p_Hγ p(R^d)

^r_p¹_r

=C K

X

k=1

ku_kk^r_Hγ p,θ(G)

¹_r . The assertion follows.

Remember that in this thesis we are mainly concerned with SPDEs on bounded Lipschitz domainsO ⊂R^d. As already mentioned, the weighted Sobolev spaces introduced above will serve as state spaces for the solutions processes u = (u(t))_t∈[0,T] of the SPDEs under consideration.

Therefore, since we are interested in solutions fulfilling a zero Dirichlet boundary condition, we need to check whether the elements of the the weighted Sobolev spaces introduced above

‘vanish at the boundary’ in an adequate way. In order to answer this question for the relevant range of parameters (this will be done in Remark 4.3) we will need the following lemma. It is an immediate consequence of [87, Theorem 9.7]. Let us mention that this result holds for a wider class of domains. E.g. it stays true for bounded domains with H¨older continuous boundary, see [87, Remark 9.8(ii)] for details. However, in the course of this thesis, we will not need these generalizations.

Lemma 2.51. For a bounded Lipschitz domainO ⊂R^d andk∈N0, W˚_p^k(O) =H_p,d−kp^k (O)

with equivalent norms.

In order to formulate the stochastic equations under consideration, we will use the following spaces H_p,θ^γ (G;`2). They are counterparts of the spaces Hp<sup>γ</sup>(R<sup>d</sup>;`2) introduced in the previous subsection. We define and discuss them for the general case of arbitrary domains with non-empty boundary, although later on we are mainly interested in the case of bounded Lipschitz domains.

Definition 2.52. Let G be an arbitrary domain in R^d with non-empty boundary. For γ ∈ R, p∈(1,∞) andθ∈R, we define

H_p,θ^γ (G;`2) :=

g= (g^k)k∈N∈ H_p,θ^γ (G)_N : kgk^p

Hp^γ(G;`2):=X

n∈Z

e^nθ

ζ−n(eⁿ·)g^k(eⁿ·)

k∈N

p

Hp^γ(`2) <∞

, with ζ_n,n∈Z, from above, cf. (2.27).

Remark 2.53. Forp∈(1,∞) andγ, θ ∈R,H_p,θ^γ (G;`₂) is a Banach space. This can be proven by following the lines of the proof of the completeness ofH_p,θ^γ (G) presented in [93, Proposition 2.4.1].

The details are left to the reader.

In Chapter 6 we will need the fact that H_p,θ^γ (G) is isomorphic to the corresponding class of γ-radonifying operators from`2(N) toH<sub>p,θ</sub><sup>γ</sup> (G). We prove this now. As in Subsection 2.2.2, from now on{γ<sub>k</sub>:k∈N}denotes a Gaussian sequence, see Definition 2.11. Remember that forh∈`2

and u∈E, whereE is a Banach space, we use to writeh⊗ufor the rank one operator hh,·i_`2u, see also (2.7).

Theorem 2.54. Let G be an arbitrary domain in R^d with non-empty boundary. Furthermore, let γ, θ ∈R and p∈[2,∞). Then, the operator

Φ :H_p,θ^γ (G;`2)→Γ(`2, H_p,θ^γ (G)) (g^k)k∈N7→

∞

X

k=1

ek⊗g^k (convergence in Γ(`2, H_p,θ^γ (G))) is an isomorphism, and therefore,

H_p,θ^γ (G;`<sub>2</sub>)'Γ(`₂, H_p,θ^γ (G)).

Proof. First of all we show that Φ is well-defined and bounded. Fix g ∈ H_p,θ^γ (G;`2). Then, using the equality (2.9) from Theorem 2.18 together with the norm equivalence (2.8), for any m₁, m₂ ∈N, we can estimate the norm of the finite rank operatorPm2

k=m1e_k⊗g^k as follows

m2

X

k=m1

e_k⊗g^k

p

Γ(`2,H_p,θ^γ (G))

≤CE

"

m2

X

k=m1

γ_kg^k

p

H_p,θ^γ (G)

# . Since for everyω ∈Ω,

m2

X

k=m1

γ_k(ω)g^k

p

H_p,θ^γ (G)

=X

n∈Z

e^nθ

m2

X

k=m1

γ_k(ω)ζ−n(eⁿ·)g^k(eⁿ·)

p

Hp^γ(R^d)

, with {ζ_n:n∈Z}as defined in (2.27), an application of Beppo-Levi’s theorem yields

m2

X

k=m1

ek⊗g^k

p

Γ(`2,H_p,θ^γ (G))

≤C X

n∈Z

e^nθE

"

m2

X

k=m1

γkζ−n(eⁿ·)g^k(eⁿ·)

p

Hp^γ(R^d)

#

. (2.39)

For every n∈Z, we can apply Equality (2.9) from Theorem 2.18 to the finite rank operator

m2

X

k=m1

e_k⊗ ζ−n(eⁿ·)g^k(eⁿ·)

∈ L_f(`2, H<sub>p</sub><sup>γ</sup>(R<sup>d</sup>))⊆Γ(`2, H_p^γ(R^d)),

followed by the norm equivalence (2.8), and obtain E

"

m2

X

k=m1

γ_kζ−n(eⁿ·)g^k(eⁿ·)

p

Hp^γ(R^d)

#

=C

m2

X

k=m1

e_k⊗ ζ−n(eⁿ·)g^k(eⁿ·)

p

Γp(`2,Hp^γ(R^d))

≤C

m2

X

k=m1

e_k⊗ ζ−n(eⁿ·)g^k(eⁿ·)

p

Γ(`2,Hp^γ(R^d))

. Thus, if we set

˜

g_n^k(m₁, m₂) :=

(ζ−n(eⁿ·)g^k(eⁿ·), ifk∈ {m₁, . . . , m₂}

0 , else

) ,

obviously, ˜g_n^k(m₁, m₂)

k∈N∈H_p^γ(R^d;`₂), and an application of Theorem 2.42 leads to E

"

m2

X

k=m1

γ_kζ−n(eⁿ·)g^k(eⁿ·)

p

Hp^γ(R^d)

#

≤C

˜g_n^k(m₁, m₂)

k∈N

p

H^γp(R^d;`2),

the constant C being independent of n ∈ Z and g. Inserting this into the estimate (2.39), we obtain

m2

X

k=m1

e_k⊗g^k

p

Γ(`2,H_p,θ^γ (G))

≤C X

n∈Z

e^nθ

g˜_n^k(m₁, m₂)

k∈N

p

Hp^γ(R^d;`2).

Since g ∈ H_p,θ^γ (G;`2), the right hand side converges to zero for m1, m2 → ∞ by Lebesgue’s dominated convergence theorem. Thus, the sequence

R_m

m∈N:=

m

X

k=1

e_k⊗g^k

m∈N

⊆ L_f(`<sub>2</sub>, H<sub>p,θ</sub><sup>γ</sup> (G)) converges in Γ(`2, H_p,θ^γ (G)) and its limitP∞

k=1e_k⊗g^kis well-defined. The boundedness of Φ can now be proven by repeating the calculations above with m₁ = 1 and m₂ =∞.

By the open mapping theorem, showing that

Φ : Γ(`e <sub>2</sub>, H<sub>p,θ</sub><sup>γ</sup> (G))→H<sub>p,θ</sub><sup>γ</sup> (G;`₂) R7→(Re_k)k∈N

is the inverse of Φ, would finish the proof. Let us check whether this operator is well-defined. If so, then the fact that it is the inverse of Φ follows by simple calculations. FixR∈Γ(`₂, H_p,θ^γ (G)).

Since for everyn∈Z, the operator

S_n:H_p,θ^γ (G)→H_p^γ(R^d) u7→ζ−n(eⁿ·)u(eⁿ·)

is obviously bounded, the composition S_nR is γ-radonifying, i.e., S_nR ∈ Γ(`₂, Hp^γ(R^d)), see Theorem 2.17. Furthermore, by Theorem 2.42,

S_nRe_k

k∈N

Hp^γ(R^d;`2)=

ζ−n(eⁿ·)Re_k(eⁿ·)

k∈N

Hp^γ(R^d;`2)≤C S_nR

Γ(`2,Hp^γ(R^d))

with a constant C independent of n∈ Z and R. Using this together with Equality (2.9) from Theorem 2.18 together with the norm equivalence (2.8), yields

X

n∈Z

e^nθ

ζ−n(eⁿ·)Re_k(eⁿ·)

k∈N

p

Hp^γ(R^d;`2)

≤C X

n∈Z

e^nθ S_nR

p

Γ(`2,Hp^γ(R^d))

≤C X

n∈Z

e^nθE

"

∞

X

k=1

γ_kS_nRe_k

p

Hp^γ(R^d)

# . Applying Beppo-Levi’s theorem and using the definitions of the norms we obtain

X

n∈Z

e^nθ

ζ−n(eⁿ·)Re_k(eⁿ·)

k∈N

p

Hp^γ(R^d;`2)≤CE

"

X

n∈Z

e^nθ

∞

X

k=1

γ_kS_nRe_k

p

Hp^γ(R^d)

#

=CE

"

X

n∈Z

e^nθ

∞

X

k=1

γ_kζ−n(eⁿ·)Re_k(eⁿ·)

p

Hp^γ(R^d)

#

=CE

"

∞

X

k=1

γ_kRe_k

p

H^γ_p,θ(G)

# .

Therefore, another application of Equality (2.9) from Theorem 2.18 followed by the norm equiv-alence (2.8), leads to

X

n∈Z

e^nθ

ζ−n(eⁿ·)Re_k(eⁿ·)

k∈N

p

Hp^γ(R^d;`2)≤CkRk^p

Γ(`2,H<sub>p,θ</sub><sup>γ</sup> (G)). Thus, (Rek)k∈N∈H<sub>p,θ</sub><sup>γ</sup> (G;`2).

We occasionally use the following properties of the spaces H_p,θ^γ (G;`2) in this thesis. In several publications like [73, 75], these properties are stated and used without proof. Since we did not find any proof in the literature, we sketch a proof based on the isomorphy from Theorem 2.54 above. The details are left to the reader.

Lemma 2.55. Let G be an arbitrary domain in R^d with non-empty boundary, p ∈ (1,∞) and γ, θ ∈R.

(i) g= (g^k)k∈N∈H_p,θ^γ (G;`2) if, and only if, g,(ψg<sub>x</sub><sup>k</sup>)k∈N∈H<sub>p,θ</sub><sup>γ−1</sup>(G;`2) and kgk_H^γ

p,θ(G;`2)≤C

(ψg^k_x)k∈N

H_p,θ^γ−1(G;`2)+Ckgk_H^γ−1

p,θ (G;`2) ≤Ckgk_H^γ

p,θ(G;`2). Also, g= (g<sup>k</sup>)k∈N∈H<sub>p,θ</sub><sup>γ</sup> (G;`2) if, and only if, g,((ψg^k)x)k∈N∈H_p,θ^γ−1(G;`2) and

kgk_H^γ

p,θ(G;`2)≤C

((ψg^k)_x)_k∈N

H_p,θ^γ−1(G;`2)+Ckgk_H^γ−1

p,θ (G;`2)≤Ckgk_H^γ

p,θ(G;`2). (ii) For any ν, γ ∈R, ψ<sup>ν</sup>H<sub>p,θ</sub><sup>γ</sup> (G;`₂) =H_p,θ−νp^γ (G;`₂) and

kgk_H^γ

p,θ−νp(G;`2)≤C

(ψ^−νg^k)k∈N

H_p,θ^γ (G;`2)≤Ckgk_H^γ

p,θ−νp(G;`2).

(iii) There exists a constant c0 >0 depending on p, θ, γ and the function ψ such that, for all c≥c₀, the operator

ψ²∆−c:H_p,θ^γ+1(G;`2)→H<sub>p,θ</sub><sup>γ−1</sup>(G;`2)

g= (g^k)k∈N7→(ψ²∆−c)g:= (ψ²∆−c)g^k

k∈N

is an isomorphism.

(iv) If G is bounded, then H_p,θ^γ

1(G;`2),→H_p,θ^γ

2(G;`2) for θ1< θ2.

(v) If 0< η <1, γ = (1−η)ν₀+ην₁, 1/p= (1−η)/p₀+η/p₁ and θ= (1−η)θ₀+ηθ₁ with ν0, ν1, θ0, θ1∈R andp0, p1 ∈(1,∞), then

H_p,θ^γ (G;`2) = H_p^ν0

0,θ0(G;`2), H_p^ν1

1,θ1(G;`2)

η (equivalent norms).

Sketch of proof. The assertions can be proven by using the isomorphism from Theorem 2.54 together with the ideal property ofγ-radonifying operators (Theorem 2.17) and the correspond-ing properties of the spaces H_p,θ^γ (G),γ, θ ∈R,p∈(1,∞), from Lemma 2.45. In order to prove the interpolation statement (v) one additionally needs the fact that

Γ(`₂, H_p^ν0

0,θ0(G)),Γ(`₂, H_p^ν1

1,θ1(G))

η = Γ(`₂, H_p,θ^γ (G)).

This follows from [114, Theorem 2.1] with H0=H1 =`2,X0 =H_p^ν0

0,θ0(G) and X1=H_p^ν1

1,θ1(G).

(Note that, by a result of G. Pisier, the B-convexity ofX0andX1assumed in [114, Theorem 2.1]

is equivalent to the fact that the Banach spaces have non-trivial type, see, e.g., [49, Theo-rem 13.10] for a proof.)

Im Dokument Besov regularity of stochastic partial differential equations on bounded Lipschitz domains (Seite 47-56)