Elliptic operators on refined Sobolev scales on vector bundles

Proposition 3.1

Let a function φ ∈ 𝓜 and numbers ε, δ > 0 be given. Put

Then ψ ∈ 𝓑 is an interpolation parameter and

with equality of norms.

Theorem 4.1

Let φ ∈ 𝓜 and ε, δ > 0. Define the interpolation parameter ψ by formula (9). Then

with equivalence of norms.

Theorem 4.2

Let s ∈ ℝ and φ ∈ 𝓜. The Hilbert space H^s,φ(Γ, V) does not depend up to equivalence of norms on the choice of the atlas {α_j} and partition of unity {χ_j} on Γ and on the choice of the local trivializations β_j of V.

Theorem 4.3

Let s ∈ ℝ and φ, φ₁ ∈ 𝓜. The identity mapping u ↦ u, with u ∈ C^∞(Γ, V), extends uniquely (by continuity) to a compact embedding H^{s + ε,φ₁}(Γ, V)\llcorner ↪ H^s,φ(Γ, V) for every ε > 0.

Suppose now that the vector bundle π : V → Γ is Hermitian. Thus, for every x ∈ Γ, a certain inner product 〈⋅,⋅〉_x is defined in the fiber π^{− 1}(x) so that the scalar function Γ ∋ x ↦〈u(x), v(x)〉_X is infinitely smooth on Γ for arbitrary sections u, v ∈ C^∞ (Γ, V). Using the C^∞-density dx on Γ, we define the inner product of these sections by the formula

Theorem 4.4

Let s ∈ ℝ and φ ∈ 𝓜. There exists a number c = c(s, φ) > 0 such that for arbitrary sections u, v ∈ C^∞ (Γ, V) we have the estimate

Thus, the form (11), with u, v ∈ C^∞ (Γ, V), extends by continuity to a sesquilinear form 〈u, v〉_Γ,V defined for arbitrary u ∈ H^s,φ(Γ, V) and v ∈ H^{−s, 1/φ}(Γ, V). Moreover, the spaces H^s,φ(Γ, V) and H^−s,1/φ(Γ, V) are mutually dual (up to equivalence of norms) with respect to the latter form.

In view of this theorem note that φ ∈ 𝓜 ⇔ 1/φ ∈ 𝓜; hence, the space H^−s,1/φ(Γ, V) is well defined.

Theorem 4.5

Let 0 ≤ q ∈ ℤ and φ ∈ 𝓜. Then the condition

is equivalent to that the identity mapping u ↦ u, with u ∈ C^∞ (Γ, V), extends uniquely to a continuous embedding H^{q + n/2,φ}(Γ, V) ↪ C^q(Γ, V). Moreover, this embedding is compact.

Here, of course, C^q(Γ, V) denotes the Banach space of all q times continuously differentiable sections u : Γ → V. The norm in this space is defined by the formula

where each u_j,k ∈ Cbq(Rn) is given by (5). Here, Cbq(Rn) denotes the Banach space of all q times continuously differentiable functions on ℝⁿ whose partial derivatives up to the q-th order are bounded on ℝⁿ. This space is endowed with the norm

of a function w.

We will prove Theorems 4.1–4.5 in Section 7. In the case where π : V → Γ is a trivial vector bundle of rank p = 1, they are established by Mikhailets and Murach [39] (see also their monograph [7, Section 2.1.2]).

Theorem 5.1

Let s ∈ ℝ and φ ∈ 𝓜. The mapping u ↦ Au, u ∈ C^∞(Γ, V₁), extends uniquely (by continuity) to a bounded linear operator

This operator is Fredholm. Its kernel is 𝔑 and its domain

The index of operator (18) is equal to dim 𝔑-dim 𝔑⁺and does not dependent on s and φ.

Recall that a bounded linear operator T : E₁ → E₂ between Banach spaces E₁ and E₂ is called Fredholm if its kernel ker T and co-kernel coker T : = E₂/T(X) are finite-dimensional. If the operator T is Fredholm, then its domain T(X) is closed in E₂ and its index ind T : = dim ker T − dim coker T is finite (see, e.g. [1, Lemma 19.1.1]).

If 𝔑 = {0} and 𝔑⁺ = {0}, then operator (18) is an isomorphism between the spaces H^{s + m,φ}(Γ, V₁) and H^s,φ(Γ, V₂) by virtue of the Banach theorem on inverse operator. In the general situation, this operator induces an isomorphism between their certain subspaces of finite codimension. In this connection consider the following decompositions of these spaces into direct sums of their subspaces:

These decompositions are well defined because the summands in them have the trivial intersection and the finite dimension of the first summand is equal to the codimension of the second one. This equality is due to the following fact: if we consider 𝔑 as a subspace of H^{−s−m,1/φ}(Γ, V₁), then the dual of 𝔑 with respect to the form 〈⋅,⋅〉_Γ,V1 coincides with the second summand in (20) according to Theorem 4.4, analogous reasoning being valid for (21).

Let P and P⁺ denote the oblique projectors of the spaces H^{s + m,φ}(Γ, V₁) and H^s,φ(Γ, V₂) onto the second summands parallel to the first summands in (20) and (21) respectively. These projectors are independent of s and φ.

Theorem 5.2

Let s ∈ ℝ and φ ∈ 𝓜. The restriction of operator (18) to the subspace P(H^{s + m,φ}(Γ, V₁)) is an isomorphism

The solutions u ∈ H^{s + m,φ}(Γ, V₁) to the elliptic equation Au = f satisfy the following a priori estimate.

Theorem 5.3

Let s ∈ ℝ, φ ∈ 𝓜, and σ < s + m. Besides, let functions χ, η ∈ C^∞(Γ) be chosen so that η = 1 in a neighbourhood of supp χ. Then there exists a number c > 0 that, for any sections u ∈ H^{s + m,φ}(Γ, V₁) and f ∈ H^s,φ(Γ, V₂) satisfying the equation Au = f on Γ, we have the estimate

Remark 5.4

If χ = 1 on Γ, then (23) becomes the global estimate

If s + m − 1 < σ < s + m, then we can take η : = χ in (23); this follows in view of Theorem 4.3 from the estimate

Consider the local regularity of the solutions to the elliptic equation Au = f. Given j ∈{1, 2}, we put

(the latter equality is due to Theorem 4.3). Assume that Γ₀ is an arbitrary open nonempty subset of Γ. We let H1ocs,φ(Γ0,Vj), with s ∈ ℝ and φ ∈ 𝓜, denote the linear space of all sections v ∈ H^{− ∞}(Γ, V_j) such that χ v ∈ H^s,φ(Γ, V_j) for every function χ ∈ C^∞(Γ) with supp χ ⊂ Γ₀. Here, the product χ v ∈ H^{− ∞}(Γ, V_j) is well defined by closer.

Theorem 5.5

Let u ∈ H^{− ∞}(Γ, V₁) be a solution to the elliptic equation Au = f on Γ for a certain section f∈Hlocs,φ(Γ0,V2).Thenu∈Hlocs+m,φ(Γ0,V1).

As we can see, the supplementary regularity φ is inherited by the solution. If Γ₀ = Γ, then the local spaces Hlocs+m,φ(Γ0,V1) and Hlocs,φ(Γ0,V2) becomes H^{s + m,φ}(Γ, V₁) and H^s,φ(Γ, V₂) respectively and then Theorem 5.5 says about the global regularity on Γ.

As an application of the refine Sobolev scale, we give the following result:

Theorem 5.6

Let 0 ≤ q ∈ ℤ. Suppose that a section u ∈ H^{− ∞}(Γ, V₁) is a solution to the elliptic equation Au = f on Γ where f∈Hlocq−m+n/2,φ(Γ0,V2) for a certain function φ ∈ 𝓜 subject to condition (13). Then u∈Clocq(Γ0,V1).

Here, Clocq(Γ0,V1) denotes the linear space of all sections u ∈ H^{− ∞}(Γ, V₁) such that χ u ∈ C^q(Γ, V₁) for arbitrary χ ∈ C^∞(Γ) with supp χ ⊂ Γ₀.

Remark 5.7

Let φ ∈ 𝓜. Condition (13) is sharp in Theorem 5.6. Namely, this condition is equivalent to the implication

We will prove Theorems 5.1–5.6, formula (25) in Remark 5.4, and Remark 5.7 in Section 8. In the case where both π₁ : V₁ → Γ and π₂ : V₂ → Γ are trivial vector bundles of rank p = 1, these theorems are proved by Mikhailets and Murach [39] (see also their monograph [7, Sections 2.2.2 and 2.2.3]).

Proposition 6.1

Let [X0(j),X1(j)], with j = 1, …, r, be a finite collection of admissible couples of Hilbert spaces. Then for every function ψ ∈ 𝓑 we have

with equality of norms.

The second property shows that this interpolation preserves the Fredholm property of the bounded operators that have the same defect (see, e.g., [7, Theorem 1.7]).

Proposition 6.2

Let X = [X₀, X₁] and Y = [Y₀, Y₁] be admissible pairs of Hilbert spaces, and let ψ ∈ 𝓑 be an interpolation parameter. Suppose that a linear mapping T is given on X₀ and satisfies the following property: the restrictions of T to the spaces X_j, where j = 0, 1, are Fredholm bounded operators T : X_j → Y_j that have a common kernel and the same index. Then the restriction of T to the space X_ψ is a Fredholm bounded operator T : X_ψ → Y_ψ with the same kernel and index and, besides, T(X_ψ) = Y_ψ ∩ T(X₀).

The third property reduces the interpolation between the dual or antidual spaces of given Hilbert spaces to the interpolation between these given spaces (see [7, Theorem 1.4]). We need this property in the case of antidual spaces. If H is a Hilbert space, then H′ stands for the antidual of H; namely, H′ consists of all antilinear continuous functionals l : H → ℂ. The linear space H′ is Hilbert with respect to the inner product (l₁, l₂)_H′ : = (v₁, v₂)_H of functionals l₁, l₂ ∈ H′; here v_j, with j ∈ {1, 2}, is a unique vector from H such that l_j(w) = (v_j, w)_H for every w ∈ H. Note that we do not identify H and H′ on the base of the Riesz theorem (according to which v_j exists).

Proposition 6.3

Let a function ψ ∈ 𝓑 be such that the function ψ(t)/t is bounded in a neighbourhood of infinity. Thenfor every admissible pair [X₀, X₁] of Hilbert spaces we have the equality [X1′,X0′]ψ=[X0,X1]χ′ with equality of norms. Here, the function χ ∈ 𝓑 is defined by the formula χ(t) : = t/ψ(t) for t > 0. If ψ is an interpolation parameter, then χ is an interpolation parameter as well.

In view of this theorem we note that if [X₀, X₁] is an admissible pair of Hilbert spaces, then the dual pair [X1′,X0′] is also admissible provided that we identify functions from X0′ with their restrictions on X₁.

Proof of Theorem 4.1

Let s ∈ ℝ. As is known [2, p. 110], the pair of Sobolev spaces on the left of equality (10) is admissible. We will deduce this equality from Proposition 3.1 with the help of some operators of flattening and sewing of the vector bundle π : V → Γ.

We define the flattening operator by the formula

Here, each function u_j,k ∈ C0∞(Rn) is defined by formula (5). According to (4), the norm of u in the space H^s,φ(Γ, V) is equal to the norm of T u in the Hilbert space (H^s,φ(ℝⁿ))^pϰ. Therefore the linear mapping u ↦ Tu, with u ∈ C^∞(Γ, V), extends continuously to the isometric operator

Besides, this mapping extends continuously to the isometric operators

between Sobolev spaces. Since ψ is an interpolation parameter, it follows from the boundedness of the linear operators (29) with σ ∈ {s − ε, s + δ} that the restriction of the operator (29) with σ = s − ε is a bounded operator

Owing to Propositions 3.1 and 6.1, the target space of (30) takes the form

Thus, (30) is a bounded operator between the spaces

Consider now the mapping of sewing

defined on vectors

Here, for each j ∈ {1, …, ϰ}, the section w_j ∈ C^∞(Γ, V) is defined by the formula

in which the function η j ∈ C0∞ (ℝⁿ) is chosen so that η_j = 1 on the set αj−1 (supp χj).

We have the linear mapping

It is left inverse to the flattening mapping (27). Indeed, given u ∈ C^∞(Γ, V), we write

where each section u_j ∈ C^∞(Γ, V) is defined by formula (35) with u instead of w. In this formula, for arbitrary k ∈ {1, …, p} and x ∈ Γ_j, we have the equalities

due to our choice of η_j. Therefore

for every x ∈ Γ_j. Note that if x ∈ Γ \ Γ_j, then u_j(x) = 0 = (χ_ju)(x). Thus, u_j(x) = (χ_ju)(x) for arbitrary x ∈ Γ. Hence,

Let us prove that the mapping (36) extends uniquely to a linear bounded operator between the spaces (H^s,φ(ℝⁿ))^pϰ and H^s,φ(Γ, V). Given a vector (34), we write

Examine the function (χ_lw_j)∘α_l : ℝⁿ↦π⁻¹(Γ_l) with l, j ∈ {1, …, x}. If t ∈ ℝⁿ satisfies α_l(t)∈ Γ_j, then

Here, η_j,l := (χ_l ∘ α_l)η_j ∈ C0∞ (ℝⁿ), whereas α_j,l : ℝⁿ↔ ℝⁿ is an infinitely smooth diffeomorphism such that α_j,l := αj−1 ∘ α_l in a neighbourhood of supp η_j,l and that α_j,l(t) = t whenever |t| ≫ 1. Then, given k ∈ {1, …, p}, we have the equalities

Here, each βl,jk,r is a certain complex-valued function from C^∞(Γ) such that the matrix-valued function (βl,jk,r(x))k,r=1p of x∈supp χl∩ supp(ηj∘αj−1) corresponds to the transition mapping β_l ∘ βj−1 . Thus,

for arbitrary t ∈ ℝ (if α_l(t)∉ Γ_j, then this equality becomes 0 = 0).

Owing to (38) and (39) we write

here, each

Thus

As is known [1, Theorem B.1.7, B.1.8], the operator of change of variables v ↦ v ∘ α_j,l and the operator of the multiplication by a function from C0∞ (ℝⁿ) are bounded on each Sobolev space H^σ(ℝⁿ) with σ ∈ ℝ. Therefore the linear operator v↦ηj,lk,r⋅(v∘αj,l) is bounded on H^σ(ℝⁿ). Hence, owing to Proposition 3.1, this operator is also bounded on the Hörmander space H^s,φ(ℝⁿ). Thus, formula (40) implies that

for a certain number c > 0 that does not depend on w∈(C0∞(Rn))pϰ. Therefore the mapping (36) extends uniquely (by continuity) to a linear bounded operator

Besides, this mapping extends uniquely to a bounded linear operator

Taking here σ ∈ {s − ε, s + δ} and using the interpolation with the function parameter ψ, we conclude that the restriction of the operator (43) with σ = s − ε to the space (31) is a bounded operator

It follows from equality (37) and from the boundedness of operators (44) and (28) that

where c₁ is the norm of the product of these operators, and

Besides, the boundedness of operators (42) and (32) implies that

with c₂ being the norm of the product of the last two operators. Thus, the norms in the spaces H^s,φ(Γ, V) and X_ψ are equivalent on the linear manifold C^∞(Γ, V). Since this manifold is dense in these spaces, they coincide up to equivalence of norms (the set C^∞(Γ, V) is dense in X_ψ due to (8)). □

The proofs of Theorems 4.2–4.4 are quite similar to the proofs of assertions (i), (iii) and (v) of Theorem 2.3 from monograph [7, Section 2.1.2], where the case of trivial vector bundle of rank p = 1 is considered. We will give these proofs for the sake of the readers convenience and completeness of the presentation.

Proof of Theorem 4.2

Consider two triplets 𝓐₁ and 𝓐₂ each of which is formed by an atlas of the manifolod Γ, appropriate partition of unity on Γ, and collection of local trivializations of the total space V. Let H^s,φ (Γ, V ; 𝓐_j) and H^σ(Γ, V ; 𝓐_j) respectively denote the Hörmander space H^s,φ(Γ, V) and the Sobolev space H^σ(Γ, V) corresponding to the triplet 𝓐_j with j ∈ {1, 2}. The conclusion of Theorem 4.2 holds true in the Sobolev case of φ ≡ 1 (see, e.g., [2, p. 110]). Hence, the identity mapping is an isomorphism

for each σ ∈ ℝ. Considering this isomorphism for σ := {s − ε, s + δ} and using the interpolation with the function parameter ψ defined by formula (9), we conclude that the identity mapping is an isomorphism

According to Theorem 4.1,

for each j ∈ {1, 2} with equivalence of norms in the spaces. Thus, the spaces H^s,φ (Γ, V ; 𝓐₁) and H^s,φ(Γ, V ; 𝓐₂) are equal up to equivalence of norms. □

Proof of Theorem 4.3

Let ε > 0. According to Theorem 4.1 there exist interpolation parameters χ, η∈ 𝓑 such that

with equivalence of norms. Hence, owing to (8), we have the continuous embeddings

They are extensions by continuity of the identity mapping u ↦ u, u ∈ C^∞(Γ, V). Here, the middle embedding

is compact (see, e.g., [2, Proposition 1.2]). Therefore, the embedding

is also compact. □

Proof of Theorem 4.4

This theorem is known in the Sobolev case of φ = 1 (see, e.g., [2, p. 110]). Hence, for every σ∈ ℝ, the lineal mapping Q : v ↦ 〈v, ⋅〉_{Γ, V}, with v ∈ H^σ(Γ, V), is an isomorphism Q : H^σ(Γ, V) ↔ (H^−σ(Γ, V))…. Considering the latter for σ = s ∓ 1 and using the interpolation with the function parameter ψ defined by formula (9) with ε = δ = 1, we obtain an isomorphism