GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 3, 1995, 259-276

BASIC BOUNDARY VALUE PROBLEMS OF THERMOELASTICITY FOR ANISOTROPIC BODIES

WITH CUTS. II

R. DUDUCHAVA, D. NATROSHVILI, AND E. SHARGORODSKY

Abstract. In the first part [1] of the paper the basic boundary value problems of the mathematical theory of elasticity for threedimensional anisotropic bodies with cuts were formulated. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems were formulated in the Besov (𝔹_{p,q}^{s}) and Bessel-potential (ℍ_{p}^{s}) spaces. In the present part we give the proofs of the main results (Theorems 7 and 8) using the classical potential theory and the nonclassical theory of pseudodifferential equations on manifolds with a boundary.

This paper continues [1]. After recalling some auxiliary results, we prove Theorems 7 and 8 formulated in §3.

§ 4. Auxiliary Results

4.1. Convolution Operators. 𝕊(ℝ^{n}) denotes the space of C^{∞}-smooth fast decaying functions, while 𝕊′(ℝ^{n}) stands for the dual space of tempered distributions. The Fourier transform and its inverse

Fφ(x) = ∫_{ℝ^{n}} e^{ixξ}φ(ξ)dξ, F^{−1}φ(ξ) = (2π)^{−n} ∫_{ℝ^{n}} e^{−ixξ}ψ(x)dx

are continuous operators in both spaces 𝕊(ℝ^{n}) and 𝕊′(ℝ^{n}). Hence the convolution operator

a(D)φ = F^{−1}aFφ, a ∈ 𝕊′(ℝ^{n}), φ ∈ 𝕊(ℝ^{n})  (4.1)

1991 Mathematics Subject Classification. 35C15, 35S15, 73M25. Key words and phrases. Thermoelasticity, anisotropic bodies, cuts, potentials, pseu-

dodifferential equations, boundary integral equations..

259

1072-947X/95/0500-0259$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

is a continuous transformation

a(D) : 𝕊(ℝ^{n}) → 𝕊′(ℝ^{n})

(cf. [2], [3]).

If operator (4.1) has a bounded extension

a(D) : 𝕃_{p}(ℝ^{n}) → 𝕃_{p}(ℝ^{n}), 1 ≤ p ≤ ∞,

we write a ∈ M_{p}(ℝ^{n}) and a(ξ) is called the (Fourier) L_{p}-multiplier. Let

M_{p}^{(r)}(ℝ^{n}) = {(1 + |ξ|^{2})^{r/2}a(ξ) : a ∈ M_{p}(ℝ^{n})}.

Recall that the Bessel potential space ℍ_{p}^{s}(ℝ^{n}) is defined as a subset of 𝕊′(ℝ^{n}) endowed with the norm

\begin{matrix}∥u|ℍ_{p}^{s}(ℝ^{n})∥ = ∥I^{s}(D)u|𝕃_{p}(ℝ^{n})∥, \\ I^{s}(ξ) := (1 + |ξ|^{2})^{s/2}. \end{matrix}  (4.2)

Therefore due to the obvious property

a_{1}(D)a_{2}(D) = (a_{1}a_{2})(D), a_{j} ∈ M_{p}^{(r_{j})}(ℝ^{n})  (4.3)

we easily find that the operator

a(D) : ℍ_{p}^{s}(ℝ^{n}) → ℍ_{p}^{s−r}(ℝ^{n}), s, r ∈ ℝ, 1 ≤ p ≤ ∞,  (4.4)

is bounded if and only if a ∈ M_{p}^{(r)}(ℝ^{n}).

The interpolation property

𝔹_{p,q}^{s}(ℝ^{n}) = [ℍ_{p}^{s_{1}}(ℝ^{n}), ℍ_{p}^{s_{2}}(ℝ^{n})]_{θ,q},
1 < p < ∞, 1 ≤ p ≤ ∞, s_{1}, s_{2} ∈ ℝ,  (4.5)
s = (1 − θ)s_{1} + θs_{2}, 0 ≤ θ ≤ 1

(see [4], [5]) for a ∈ M_{p}^{(r)}(ℝ^{n}) ensures the boundedness of the operator

a(D) : 𝔹_{p,q}^{s}(ℝ^{n}) → 𝔹_{p,q}^{s−r}(ℝ^{n}), 1 ≤ q ≤ ∞.  (4.6)

Equality (4.2) and boundedness (4.4) imply that the operator

I^{r} : ℍ_{p}^{s}(ℝ^{n}) → ℍ_{p}^{s−r}(ℝ^{n})  (4.7)

arranges an isometric isomorphism.

Further, it is well known that the operators

\begin{matrix}I_{+}^{r} : ℍ̃_{p}^{s}(ℝ_{+}^{n}) → ℍ̃_{p}^{s−r}(ℝ_{+}^{n}), \\ I_{−}^{r} : ℍ_{p}^{s}(ℝ_{+}^{n}) → ℍ_{p}^{s−r}(ℝ_{+}^{n}), I_{±}^{r}(ξ) = (ξ_{n} ± i|ξ′| ± i)^{r}, \\ ℝ_{+}^{n} := ℝ^{n−1} × ℝ^{+}, ℝ^{+} := [0, +∞), ξ = (ξ′, ξ^{n}) ∈ ℝ^{n}, ξ′ ∈ ℝ^{n−1}, \end{matrix} (4.8)

BASIC BOUNDARY VALUE PROBLEMS OF THERMOELASTICITY 261

also arrange isomorphisms (though not isometric ones; see, for example, [3], [6]). Isomorphisms similar to (4.8) exist for any smooth manifold with a Lipschitz boundary (for details see [3], [7]).

The equality M_{2}(ℝ^{n}) = 𝕃_{∞}(ℝ^{n}) is well known and trivial. A reasonable description of the class M_{p}^{r}(ℝ^{n}) for p ≠ 2 is less trivial and the problem still remains unsolved.

Theorem 12 (see [8], Theorem 7.9.5; [9]). Let 1 < p < ∞ and

∑_{\begin{matrix}|β|<[n/2]+1 \\ 0≤β≤1 \end{matrix}} sup \{|ξ^{β}D^{β}a(ξ)|, ξ ∈ ℝ^{n}\} ≤ M < ∞,

where for the multi-index β = (β_{1}, …, β_{n}) the inequality 0 ≤ β ≤ 1 reads

as 0 ≤ β_{j} ≤ 1, j = 1, …, n. Then a ∈ ∩_{1<p<∞}M_{p}(ℝ^{n}).

If a ∈ M_{p}^{(r)}(ℝ^{n}), the operators

\begin{aligned}r_{+}a(D) & : ℍ̃_{p}^{s}(ℝ_{+}^{n}) → ℍ_{p}^{s−r}(ℝ_{+}^{n}) \\ & : 𝔹̃_{p,q}^{s}(ℝ_{+}^{n}) → 𝔹_{p,q}^{s−r}(ℝ_{+}^{n}) \end{aligned}  (4.9)

are bounded (1 < p < ∞, s, r ∈ ℝ, 1 ≤ q ≤ ∞); here r_{+}φ = φ|_{ℝ_{+}^{n}} denotes the restriction operator.

An equality similar to (4.3)

r_{+}a_{1}(D)ℓ_{0}r_{+}a_{2}(D) = r_{+}(a_{1}a_{2})(D), a_{j} ∈ M_{p}^{(r_{j})}(ℝ^{n}),  (4.10)

where ℓ_{0} is extension by 0 from ℝ_{+}^{n} to ℝ^{n}, fails to be fulfilled in general. However, (4.10) holds if there is an analytic extension either a_{1}(ξ′, ξ_{n} − iλ) or a_{2}(ξ′, ξ_{n} + iλ), which can be estimated from above by C(1 + |ξ| + λ)^{N} with N > 0, λ > 0, C = const.

4.2. Pseudodifferential operators. If the symbol a(x, ξ) depends on the variable x, the corresponding convolution (cf. (4.1))

a(x, D)φ(x) := F_{ξ→x}^{−1}a(x, ·)F_{y→ξ}φ(ξ)  (4.11)

is called the pseudodifferential operator (φ ∈ 𝕊(ℝ^{n}), |a(x, ξ)| < C(1 + |ξ|)^{N}, N > 0, C = const).

Let M_{p}^{(s,s−r)}(ℝ^{n}×ℝ^{n}) denote a class of symbols a(x, ξ) for which operator (4.11) can be extended to the bounded mapping

a(x, D) : ℍ_{p}^{s}(ℝ^{n}) → ℍ_{p}^{s−r}(ℝ^{n}).  (4.12)

By S^{r}(Ω × ℝ^{n}) (Ω ⊂ ℝ^{n}, r ∈ ℝ) is denoted the Hörmander class of symbols a(x, ξ) if

|D_{x}^{α}D_{ξ}^{β}a(x, ξ)| ≤ M_{α,β}(1 + |ξ|)^{r−|β|}, ∀x ∈ Ω, ∀ξ ∈ ℝ^{n}, (4.13)

where M_{α,β} is independent of x and ξ.

By S_{r}^{l,m}(Ω × ℝ^{n}) (Ω ⊂ ℝ^{n}, l, m ∈ ℤ^{+}, r ∈ ℝ) we denote the class of symbols a(x, ξ) satisfying the estimates

∫_{Ω} |D_{x}^{α}(ξD_{ξ})^{β}a(x, ξ)|dx ≤ M′_{α,β}(1 + |ξ|)^{r}

∀ξ ∈ ℝ^{n}, |α| ≤ l, |β| ≤ m,

where

(ξD_{ξ})^{β} := (ξ_{1}D_{ξ_{1}})^{β_{1}} … (ξ_{n}D_{ξ_{n}})^{β_{n}}.

If Ω ⊂ ℝ^{n} is compact, then S^{r}(Ω × ℝ^{n}) ⊂ S_{r}^{l,m}(Ω × ℝ^{n}). Such an inclusion does not hold for non-compact Ω.

Theorem 13. Let s, r ∈ ℝ, l, m ∈ ℤ^{+}, m > [n/2] + 1; then

S^{r}(ℝ^{n} × ℝ^{n}) ⊂ M_{p}^{(s,s−r)}(ℝ^{n} × ℝ^{n}).

If, additionally, −l + 1 + 1/p < s − r < l + 1/p, then

S_{r}^{l+n,m}(ℝ^{n} × ℝ^{n}) ⊂ M_{p}^{(s,s−r)}(ℝ^{n} × ℝ^{n}).

Proof. When a symbol a ∈ S^{0}(ℝ^{n}×ℝ^{n}) has a compact support with respect to x, then the continuity of a(x, D) in 𝕃_{p}(ℝ^{n}) follows from Theorem 12, as shown in [10].

For an arbitrary a ∈ S^{0}(ℝ^{n} × ℝ^{n}) the above statement is proved for 𝕃_{p}(ℝ^{n}) using the arguments involved in the proof of Theorem 3.5 from [12]. In the general case the continuity of the mapping ℍ_{p}^{s}(ℝ^{n}) → ℍ_{p}^{s−r}(ℝ^{n}) is established with the aid of the order reduction operator (4.7) (see [4], [10]), while the continuity of the mapping a(x, D) : 𝔹_{p,q}^{s}(ℝ^{n}) → 𝔹_{p,q}^{s−r}(ℝ^{n}) is proved by interpolation (see [4]).

For a different proof of the first claim see [11].

To prove the second claim we shall introduce some notation. For a multiindex μ = (μ_{1}, …, μ_{n}), 0 ≤ μ ≤ 1 we define

dx^{μ} := ∏_{\begin{matrix}μ_{j}=1 \\ j=1,2,...,n \end{matrix}} dx_{j}, (x, h)_{μ} := (z_{1}, …, z_{n}),

z_{j} = \begin{cases}x_{j}, & if μ_{j} = 1, \\ h_{j}, & if μ_{j} = 0, \end{cases} x, h ∈ ℝ^{n}.

Let

a_{(α)}(x, ξ) := D_{x}^{α}a(x, ξ).

By virtue of Theorem 12 the inclusion a ∈ S_{r}^{l,m}(ℝ^{n} × ℝ^{n}) implies

∫_{ℝ^{n}} ∥D_{x}^{α}a(x, ·)|M_{p}^{(r)}(ℝ^{n})∥dx < ∞, |α| ≤ l + n.

From this finiteness and Fubini’s theorem we get

mes_{ℝ^{n}} Δ_{μ,γ} = 0 for any 0 ≤ μ ≤ 1, |γ| ≤ l,

where

Δ_{μ,γ} := {h ∈ ℝ^{n} : ∫_{ℝ^{|μ|}} ∥a_{(μ+γ)}((y, h)_{μ}, ·)|M_{p}^{(r)}(ℝ^{n})∥dy^{μ} = ∞}.

If now

Δ = ⋃_{\begin{matrix}0≤μ≤1 \\ |γ|≤l \end{matrix}} Δ_{μ,γ}

then, obviously, mes_{ℝ^{n}} Δ = 0. There exists a vector h_{0} ∈ ℝ^{n}\Δ. Then we have

∫_{ℝ^{n}} ∥a_{(μ+γ)}((y, h_{0})_{μ}, ·)|M_{p}^{(r)}(ℝ^{n})∥dy^{μ} < ∞.

With these conditions we can use Theorem 5.1 and Remark 5.5 from [20] where the claimed inclusion a ∈ M_{p}^{(s,s−r)}(ℝ^{n} × ℝ^{n}) is proved.

Let

A, B : ℍ_{p}^{s}(ℝ^{n}) → ℍ_{p}^{s−r}(ℝ^{n})

be the bounded operators; they are called locally equivalent at x_{0} ∈ ℝ^{n} (see [3], [13]) if

inf {∥χ(A − B)∥ : χ ∈ C_{x_{0}}(ℝ^{n})} = inf {∥(A − B)χI∥ : χ ∈ C_{x_{0}}(ℝ^{n})} = 0,

where I is the identity operator and C_{x_{0}}(ℝ^{n}) = {χ ∈ C_{0}^{∞}(ℝ^{n}) : χ(x) = 1 in some neighborhood of x_{0}}. In such a case we write A ∼^{x_{0}} B. In a similar manner we define the equivalence A_{0} ∼^{x_{0}} B_{0} for operators

A_{0}, B_{0} : ℍ̃_{p}^{s}(ℝ_{+}^{n}) → ℍ_{p}^{s−r}(ℝ_{+}^{n}).

Assume now that \overline{S} = S ∪ ∂S is a compact n-dimensional C^{∞}-smooth manifold with a C^{∞}-smooth boundary ∂S and

S = ∪^{N}_{j=1}V_{j}, κ_{j} : X_{j} → V_{j}, X_{j} ⊂ ℝ_{+}^{n}  (4.14)

are coordinate diffeomorphisms. Let {χ_{j}}_{1}^{N} ⊂ C_{0}^{∞}(S) be a partition of the unity subordinated to the covering of S in (4.14); also let

κ_{j_{∗}}φ(t) = χ_{j}^{0}φ(χ_{j}(t)), κ_{j^{∗}}^{−1}ψ(x) = χ_{j}ψ(κ_{j}^{−1}(x)),

where χ_{j}^{0}(t) := χ_{j}(κ_{j}(t)), t ∈ ℝ_{+}^{n}, x ∈ S. The following mapping properties

κ_{j_{∗}} : ℍ_{p}^{r}(S) → ℍ_{p}^{r}(ℝ_{+}^{n}), supp κ_{j}^{−1} ∩ ∂S ≠ ∅,
κ_{j_{∗}} : ℍ̃_{p}^{r}(S) → ℍ̃_{p}^{r}(ℝ_{+}^{n}), supp κ_{j}^{−1} ∩ ∂S ≠ ∅,  (4.15)
κ_{j_{∗}} : ℍ_{p}^{r}(S) → ℍ_{p}^{r}(ℝ^{n}), supp κ_{j}^{−1} ∩ ∂S = ∅.

are almost evident.

A bounded operator

A : ℍ̃_{p}^{ν}(S) → ℍ_{p}^{ν−r}(S)  (4.16)

is called pseudodifferential (of order r) if:

(i) χ_{1}Aχ_{2}I is a compact operator in ℍ̃_{p}^{r}(S) → ℍ_{p}^{ν−r}(S) for any χ_{1}, χ_{2} ∈ C_{0}^{∞}(S) with disjoint supports supp χ_{1} ∩ supp χ_{2} = ∅;

(ii)

\begin{matrix}κ_{j^{∗}}Aκ_{j^{∗}}^{−1} ∼^{x_{0}} a(x_{0}, D), x_{0} ∈ S, \\ κ_{j^{∗}}Aκ_{j^{∗}}^{−1} ∼^{x_{0}} r_{+}a(x_{0}, D), x_{0} ∈ ∂S, \end{matrix}  (4.17)

where a(x_{0}, ·) ∈ M_{p}^{(r)}(ℝ^{n}) for any x_{0} ∈ \overline{S}.

Example 14 (see [3], Example 3.19]). . Let \overline{Ω} ⊂ ℝ^{n} be a compact domain with a smooth boundary ∂Ω ≠ ∅.

The operator r_{Ω}a(x, D), where a(x, ξ) ∈ S^{r}(Ω × ℝ^{n}) and r_{Ω}φ = φ|_{Ω} denotes the restriction, is a pseudodifferential one of order r and

r_{Ω}a(x, D) ∼^{x_{0}} a(x_{0}, D), x_{0} ∉ ∂Ω,

r_{Ω}a(x, D) ∼^{x_{0}} r_{+}a(x_{0}, D), x_{0} ∈ ∂Ω.

(4.18)

If a(x_{0}, ξ) has the radial limits

a^{∞}(x_{0}, ξ) = lim_{λ→∞} λ^{−r}a(x_{0}, λξ)  (4.19)

which are nontrivial bounded functions of ξ, then a^{∞}(x_{0}, ξ) is a homogeneous function of order r with respect to ξ:

a^{∞}(x_{0}, λξ) = λ^{r}a^{∞}(x_{0}, ξ), λ > 0.

Let

a^{0}(x_{0}, ξ) = a^{∞}(x_{0}, (1 + |ξ′|)|ξ′|^{−1}ξ′, ξ_{n})  (4.20)

represent the modified symbol (see [6], Section 3). Assume that a^{0} ∈ M_{p}^{(r)}(ℝ^{n}); then using (4.17) and the relation

lim_{R→∞} sup_{|ξ|≥R} |ξ|^{−r}|a(x_{0}, ξ) − a^{0}(x_{0}, ξ)| = 0

we obtain

\begin{matrix}κ_{j_{∗}}Aκ_{j^{∗}}^{−1} ∼^{x_{0}} a^{0}(x_{0}, D), x_{0} ∉ Ω, \\ κ_{j_{∗}}Aκ_{j^{∗}}^{−1} ∼^{x_{0}} r_{+}a^{0}(x_{0}, D), x_{0} ∈ Ω. \end{matrix}  (4.21)

Thus the operators χ[a(x_{0}, D) − a^{0}(x_{0}, D)], [a(x_{0}, D) − a^{0}(x_{0}, D)]χI with χ ∈ C_{0}^{∞}(ℝ^{n}) are compact in ℍ_{p}^{ν}(ℝ^{n}) → ℍ_{p}^{ν−r}(ℝ^{n}) (see [3]). As for the compact operator T : ℍ_{p}^{ν}(ℝ^{n}) → ℍ_{p}^{ν−r}(ℝ^{n}), the equivalence T ∼^{x_{0}} 0 holds automatically.

The functions a^{∞}(x_{0}, ξ) (see (4.19)) and a^{0}(x_{0}, ξ) (see (4.20)) are respectively called the homogeneous principal symbol and the modified principal symbol of the operator A.

Theorem 15 (see [3])). Let (4.16) be a pseudodifferential operator (r, ν ∈ ℝ, 1 < p < ∞). A is a Fredholm operator if and only if the following conditions are fulfilled:

(i) inf\{| det a^{∞}(x_{0}, ξ)| : x_{0} ∈ \overline{S}, ξ ∈ ℝ^{n}\} > 0;

(ii) r_{+}a_{ν,r}(x_{0}, D) is a Fredholm operator in the space 𝕃_{p}(ℝ_{+}^{n}) for any x_{0} ∈ ∂S, where

a_{ν,r}(x_{0}, ξ) = (ξ_{n} − i|ξ′| − i)^{ν−r}a^{0}(x_{0}, ξ)(ξ_{n} + i|ξ′| + i)^{−ν},
ξ = (ξ′, ξ_{n}), ξ′ ∈ ℝ^{n−1}.

Theorem 16 (see [3]). Let a(x, D) be a pseudodifferential operator of the order r ∈ ℝ with the N × N matrix symbol a(x, ·) ∈ S^{r}(ℝ^{n}) for any x ∈ \overline{S}. If a(x, ξ) is positive definite, i.e.,

(a(x, ξ)η, η) ≥ δ_{0}|ξ|^{r}|η|^{2} for some δ_{0} > 0

and any ξ ∈ ℝ^{n}, x ∈ \overline{S}, η ∈ ℂ^{N}, (4.22)

then

a(x, D) : ℍ̃_{2}^{\frac{r}{2} +ν}(S) → ℍ_{2}^{− \frac{r}{2} +ν}(S)  (4.23)

is a Fredholm operator for any |ν| < \frac{1}{2} and

Ind a(x, D) = 0.  (4.24)

4.3. Further Auxiliary Results. Let H^{r}(ℝ^{n}) denote the class of functions with the properties

(i) a(λξ) = λ^{r}a(ξ), λ > 0, ξ ∈ ℝ^{n};

(ii) a ∈ C^{∞}(S^{n−1}), S^{n−1} := {ω ∈ ℝ^{n} : |ω| = 1};

(iii) if a(ξ) = a_{0}(ω′, t, ξ_{n}), where ω′ = |ξ′|^{−1}ξ′, t = |ξ′|, ξ = (ξ′, ξ_{n}) ∈ ℝ^{n}, then

\begin{matrix}lim_{t→0} D_{t}^{k}a_{0}(ω′, t, −1) = (−1)^{k} lim_{t→0} D_{t}^{k}a_{0}(ω′, t, 1), \\ ω′ ∈ S^{n−2}, k = 0, 1, 2, …. \end{matrix}  (4.25)

For r = 0 condition (4.25) coincides with the well-known transmission property (see [6,14]).

Lemma 17. Let a ∈ H^{r}(ℝ^{n}) be a positive definite N×N matrix-function (cf. (4.22))

(a(ξ)η, η) ≥ δ_{0}|ξ|^{r}|η|^{2} for some δ_{0} > 0

and any ξ ∈ ℝ^{n}, η ∈ ℂ^{N}. (4.26)

Then a(ξ) admits the factorization

a(ξ) = a_{−}(ξ)a_{+}(ξ), a_{±}(ξ) = (ξ_{n} ± i|ξ′|)^{− \frac{r}{2}} b_{±}(ξ),  (4.27)

where b_{+}^{±1}(ξ′, ξ_{n} + iλ), b_{−}^{±1}(ξ′, ξ_{n} − iλ) have uniformly bounded analytic extensions for λ > 0, ξ′ ∈ ℝ^{n−1}, ξ_{n} ∈ ℝ and

∑_{|α|≤m} sup {|ξ^{α}D^{α}b_{±}^{±1}(ξ)| : ξ ∈ ℝ^{n}} ≤ M_{m} < ∞, m = 0, 1, 2, …. (4.28)

Proof. For the proof of this lemma see [2,9,15].

Remark 18. A lemma similar to the above one but for a general elliptic symbol was proved in [2,9] (see [6] for the scalar case N = 1). In [15, §2] a similar but more general assertion is proved when a(x, ξ) depends smoothly on a parameter x ∈ S.

A pair of Banach spaces {𝕏_{0}, 𝕏_{1}} embedded in some topological space 𝔼 is called an interpolation pair. For such a pair we can introduce the following two spaces: 𝕏_{min} = 𝕏_{0} ∩ 𝕏_{1} and 𝕏_{max} = 𝕏_{0} + 𝕏_{1} := {x ∈ 𝔼 : x = x_{0} + x_{1}, x_{j} ∈ 𝕏_{j}, j = 0, 1}; 𝕏_{min} and 𝕏_{max} become Banach spaces if they are endowed with the norms

∥x|𝕏_{min}∥ = max {∥x|𝕏_{0}∥, ∥x|𝕏_{1}∥},

∥x|𝕏_{max}∥ = inf {∥x_{0}|𝕏_{0}∥ + ∥x_{1}|𝕏_{1}∥ : x = x_{0} + x_{1}, x_{j} ∈ 𝕏_{j}, j = 0, 1},

respectively.

Moreover, we have the continuous embeddings

𝕏_{min} ⊂ 𝕏_{0}, 𝕏_{1} ⊂ 𝕏_{max}.  (4.29)

For any interpolation pairs {𝕏_{0}, 𝕏_{1}} and {𝕐_{0}, 𝕐_{1}} the space L({𝕏_{0}𝕏_{1}}, {𝕐_{0}𝕐_{1}}) consists of all linear operators from 𝕏_{max} into 𝕐_{max} whose restrictions to 𝕏_{j} belong to L(𝕏_{j}, 𝕐_{j}) (j = 0, 1). The notation L(𝕏, 𝕐) is used for the space of all linear bounded operators A : 𝕏 → 𝕐.

Lemma 19. Assume {𝕏_{0}, 𝕏_{1}} and {𝕐_{0}, 𝕐_{1}} to be interpolation pairs and the embeddings 𝕏_{min} ⊂ 𝕏_{max}, 𝕐_{min} ⊂ 𝕐_{max} to be dense. Let an operator A ∈ L(𝕏_{0}, 𝕐_{0}) ∩ L(𝕏_{1}, 𝕐_{1}) have a common regularizer: let R ∈ L(𝕐_{0}, 𝕏_{0}) ∩ L(𝕐_{1}, 𝕏_{1}) and RA − I ∈ L(𝕏_{0}𝕏_{0}) ∩ L(𝕏_{1}, 𝕏_{1}) be compact. Then

A : 𝕏_{min} → 𝕐_{min}, A : 𝕏_{max} → 𝕐_{max}

are Fredholm operators and

Ind_{𝕏_{min}→𝕐_{min}} A = Ind_{𝕏_{max}→𝕐_{max}} A = Ind_{𝕏_{j}→𝕐_{j}} A, j = 0, 1. (4.30)

If y ∈ 𝕐_{j}, then any solution x ∈ 𝕏_{max} of the equation Ax = y belongs to 𝕏_{j}. In particular,

ker_{𝕏_{min}} A = ker_{𝕏_{j}} A = ker_{𝕏_{max}} A, j = 0, 1.  (4.31)

Proof. We begin by noting that the definition of a norm in 𝕏_{min}, …, 𝕐_{max} implies

∥A|L(𝕏_{min}, 𝕐_{min})∥ ≤ max {∥A|L(𝕏_{j}, 𝕐_{j})∥ : j = 0, 1},
∥A|L(𝕏_{max}, 𝕐_{max})∥ ≤ max {∥A|L(𝕏_{j}, 𝕐_{j})∥ : j = 0, 1}.

Whence we find

L(𝕏_{0}, 𝕐_{0}) ∩ L(𝕏_{1}, 𝕐_{1}) ⊂ L(𝕏_{min}, 𝕐_{min}) ∩ L(𝕏_{max}, 𝕐_{max}).

Next we shall prove that A is a Fredholm operator in the spaces 𝕏_{min} → 𝕐_{min} and 𝕏_{max} → 𝕐_{max}. For this it suffices to show that AR−I, RA−I are compact in the spaces 𝕏_{min} and 𝕏_{max}, since by the conditions of the lemma they are compact in 𝕏_{0} and 𝕏_{1}. Let us prove a more general inclusion

Com(𝕏_{0}, 𝕐_{0}) ∩ Com(𝕏_{1}, 𝕐_{1}) ⊂ Com(𝕏_{min}, 𝕐_{min}) ∩ Com(𝕏_{max}, 𝕐_{max}),

that implies the claimed assertion.

Assume T : 𝕏_{j} → 𝕐_{j} (j = 0, 1) to be compact and {x_{k}}_{k∈ℕ} to be an arbitrary bounded sequence in 𝕏_{min}. Then {x_{k}}_{k∈ℕ} is bounded in both spaces 𝕏_{0} and 𝕏_{1}. It can be assumed without loss of generality that the sequences {Tx_{k}}_{k∈ℕ} are convergent in both 𝕐_{0} and 𝕐_{1} (otherwise we can select subsequences). Then {Tx_{k}}_{k∈ℕ} is convergent in 𝕐_{min} and therefore T ∈ Com(𝕏_{min}, 𝕐_{min}).

If S_{0}, S_{1}, and S_{max} denote the unit balls in 𝕏_{0}, 𝕏_{1}, and 𝕏_{max}, respectively, then S_{max} ⊂ S_{0} + S_{1}. Due to the compactness of T : 𝕏_{j} → 𝕐_{j} (j = 0, 1), there exist ε/2-grids {y_{k}^{(j)}}_{k=1}^{m_{j}} ⊂ T(S_{j}) (j = 0, 1), ε > 0. Then {y_{k}^{(0)} + y_{n}^{(1)}}_{k,n} ⊂ T(S_{0}) + T(S_{1}) defines an ε-grid in T(S_{max})(⊂ T(S_{0}) + T(S_{1})). Since ε > 0 is arbitrary, T : 𝕏_{max} → 𝕐_{max} is compact.

Now we shall show that the density of the embedding 𝕐_{min} ⊂ 𝕐_{max} implies the density of 𝕐_{min} ⊂ 𝕐_{j} (j = 0, 1). For the sake of definiteness assume that j = 0. By the condition of the lemma for any ε > 0 and a ∈ 𝕐_{0} there exists b ∈ 𝕐_{min} with the property

∥(a − b)|𝕐_{max}∥ < ε;

i.e., there exist a_{0} ∈ 𝕐_{0}, a_{1} ∈ 𝕐_{1} such that a − b = a_{0} + a_{1},

∥a_{0}|𝕐_{0}∥ + ∥a_{1}|𝕐_{1}∥ < ε.

Since a ∈ 𝕐_{0} and b ∈ 𝕐_{min} ⊂ 𝕐_{0}, we obtain a − b ∈ 𝕐_{0} and a_{1} = (a−b)−a_{0} ∈ 𝕐_{0}, so that a_{1} ∈ 𝕐_{0} ∩𝕐_{1} = 𝕐_{min} and a_{1} +b ∈ 𝕐_{min}. Therefore

∥[a − (a_{1} + b)]|𝕐_{0}∥ = ∥a_{0}|𝕐_{0}∥ < ε,

which proves that the embedding 𝕐_{min} ⊂ 𝕐_{0} is dense.

The density of the embeddings 𝕐_{min} ⊂ 𝕐_{j} ⊂ 𝕐_{max}, j = 0, 1, yields

𝕐_{max}^{∗} ⊂ 𝕐_{j}^{∗} ⊂ 𝕐_{min}^{∗}, j = 0, 1.

Since 𝕏_{min} ⊂ 𝕏_{j} ⊂ 𝕏_{max} and A^{∗} : 𝕐_{j}^{∗} → 𝕏_{j}^{∗} (j = 0, 1), A^{∗} : 𝕐_{min}^{∗} → 𝕏_{min}^{∗}, A^{∗} : 𝕐_{max}^{∗} → 𝕏_{max}^{∗} are Fredholm, we have

ker_{𝕏_{min}} A ⊂ ker_{𝕏_{j}} A ⊂ ker_{𝕏_{max}} A,  (4.32)
ker_{𝕐_{max}^{∗}} A^{∗} ⊂ ker_{𝕐_{j}^{∗}} A^{∗} ⊂ ker_{𝕐_{min}^{∗}} A^{∗}.  (4.33)

The dimensions of the kernels (dim ker A) in appropriate spaces will be denoted by α_{min}, α_{j}, α_{max}, while the notation β_{min}, β_{j}, β_{max} will be used for the dimensions of cokernels (dim Coker A). Note that for a Fredholm operator we have

dim Coker A = dim ker A^{∗}.

Embeddings (4.32) and (4.33) imply

α_{min} ≤ α_{j} ≤ α_{max}, j = 0, 1,  (4.34)
β_{max} ≤ β_{j} ≤ β_{min}, j = 0, 1.  (4.35)

By the definition of Ind A we obtain

Ind_{𝕏_{min}→𝕐_{min}} A ≤ Ind_{𝕏_{j}→𝕐_{j}} A ≤ Ind_{𝕏_{max}→𝕐_{max}} A.  (4.36)

A similar inequality for indices of the regularizer R is proved just in the same manner. Since Ind R = − Ind A, the inequalities inverse to (4.36) are valid and therefore (4.30) holds. Now from (4.34) and (4.35) we obtain α_{min} = α_{j} = α_{max}. The latter equality and (4.32) give (4.31).

Remark 20. Similar statements under different conditions on spaces and operators are well known (see, for example, [16], [17], [18]).

§ 5. Proofs of Theorems

5.1. Proof of Theorem 7. In the first place we shall prove that P_{S}^{1} (see (3.2), (3.6), (3.7)) is a pseudodifferential operator according to the definition given in Subsection 4.2.

Let U_{1}, …, U_{N} be a covering of S ⊂ ℝ^{3} (see (4.14), where n = 2), κ_{1}, …, κ_{N} be coordinate diffeomorphisms, and

\begin{matrix}κ̃_{j} : X̃_{j} → Ũ_{j}, X̃_{j}, Ũ_{j} ⊂ ℝ^{3}, Ũ_{j} ∩ S = V_{j}, \\ X̃_{j} = (−ε, ε) × X_{j}, κ̃_{j}|_{X_{j}} = κ_{j}, j = 1, …, N, \end{matrix}  (5.1)

be extensions of diffeomorphisms (4.14). By dκ_{j}(t) = κ′_{j}(t) and dκ̃_{j}(t̃) = κ′_{j}(t̃) (t = (t_{1}, t_{2}) ∈ ℝ_{+}^{2}, t̃ = (t_{0}, t_{1}, t_{2}) ∈ ℝ_{+}^{3}) we denote the corresponding Jacobian matrices of orders 3 × 2 and 3 × 3. κ′_{j}(t) will coincide with κ′_{j}(0, t)(t ∈ X_{j} ⊂ ℝ_{+}^{2}) if the first column in these matrices is deleted.

Let further

Γ_{χ_{j}}(t) = ( det ∥(∂_{k}κ_{j}, ∂_{l}κ_{j})∥_{2×2})^{1/2}, ∂_{k}κ_{j} = (∂_{k}κ_{j1}, ∂_{k}κ_{j2}, ∂_{k}κ_{j3})

denote the square root of the Gramm determinant of the vector-function κ_{j} = (κ_{j1}, κ_{j2}, κ_{j3}).

If the operator P_{S}^{1} is lifted locally from the manifold S onto the halfspace ℝ_{+}^{2} by means of operators (4.15), then we obtain the operator (cf. (4.17))

P_{s,κ_{j}}^{1}v(t) = κ_{j∗}P_{s}^{1}κ_{j∗}^{−1}v(t) = χ_{j}^{0}(t) ∫_{ℝ_{+}^{2}} Φ((κ_{j}(t) −

−κ_{j}(θ), τ)χ_{j}^{0}(θ)Γ_{κ_{j}}(θ)v(θ)dθ, t ∈ ℝ_{+}^{2}, χ_{j}^{0} ∈ C_{0}^{∞}(ℝ_{+}^{2}).

From the last equality it follows that operator (3.7) is bounded. Moreover,

K_{j}v(t) := χ_{j}^{0}(t) ∫_{ℝ_{+}^{2}} [Φ(κ_{j}(t) − κ_{j}(θ), τ)Γ_{κ_{j}}(θ) −

−Φ(κ′_{j}(t)(t − θ), τ)Γ_{κ_{j}}(t)]χ_{j}^{0}(θ)v(θ)dθ

has the order −2, i.e., the operator

K_{j} : ℍ̃_{p}^{ν}(ℝ_{+}^{2}) → ℍ_{p}^{ν+2}(ℝ_{+}^{2})  (5.2)

is bounded for any ν ∈ ℝ (see [19, Section 33.2 and Theorem 13]). Due to (5.2) the operator

K_{j} : ℍ̃_{p}^{ν}(ℝ_{+}^{2}) → ℍ_{p}^{ν+1}(ℝ_{+}^{2})  (5.3)

is compact, since χ_{j}^{0} ∈ C_{0}^{∞}(ℝ_{+}^{2}) [see (4.19)]. From (5.3), Example 14, and (2.1), it follows that the symbol of the pseudodifferential operator P_{S}^{1} reads (x ∈ \overline{S}, ξ ∈ ℝ^{2})

P_{S}^{1}(x, ξ) = Γ_{κ_{j}}(t) ∫_{ℝ^{2}} e^{iξη}Φ(κ′_{j}(t)η, τ)dη =

= Γ_{κ_{j}}(t) ∫_{ℝ^{2}} e^{iξη}Φ(κ′_{j}(0, t)(0, η), τ)dη =

= \frac{Γ_{κ_{j}}(t)}{(2π)^{3}} ∫_{ℝ^{2}} e^{iξη} ∫_{ℝ^{3}} e^{−i(κ′_{j}(0,t)(0,η),_{y})}A^{−1}(ỹ, τ)dỹdη =

= \frac{Γ_{κ_{j}}(t)}{(2π)^{3} det κ′_{j}(0, t)} ∫_{ℝ^{2}} e^{iξη} ∫_{ℝ^{2}} e^{−iηy} ∫_{−∞}^{∞} A^{−1}([(κ′_{j}(0, t))^{T} ]^{−1}ỹ, τ)dy_{0}dydη =

= \frac{Γ_{κ_{j}}(t)}{2π det κ′_{j}(0, t)} ∫_{−∞}^{∞} A^{−1}([(κ′_{j}(0, t))^{T} ]^{−1}ζ, τ)dy_{0}.  (5.4)

for t = κ_{j}^{−1}(x), x ∈ S, t ∈ ℝ_{+}^{2}, ξ ∈ ℝ^{2}, ỹ = (y_{0}, y) ∈ ℝ^{3}, ζ = (y_{0}, ξ). By (2.3) the principal homogeneous symbol of P_{S}^{1} (see (2.18)) is written in the form

(P_{S}^{1})^{∞}(x, ξ) = \frac{Γ_{κ_{j}}(t)}{2π det κ′_{j}(0, t)} ∫_{−∞}^{∞} A_{0}^{−1}([(κ′_{j}(0, t))^{T} ]^{−1}ζ)dy_{0}, (5.5)
x ∈ \overline{S}, ξ ∈ ℝ^{2}, t = κ_{j}^{−1}(x) ∈ ℝ_{+}^{2}, ζ = (y_{0}, ξ),
A_{0}^{−1}(ξ̃) = ∥\begin{matrix}C^{−1}(ξ̃) & 0 \\ 0 & Λ^{−1}(−iξ̃) \end{matrix}∥ , ξ̃ ∈ ℝ^{3},  (5.6)

where C(ξ̃) and Λ(ξ̃) are defined by (2.4). Since −C(ξ̃) and −Λ(−iξ̃) are positive-definite (see (1.12) and (1.14)), the same is true for −A_{0}^{−1}(ξ̃):

( − A_{0}^{−1}(ξ̃)η, η) ≥ δ_{2}|η|^{2}|ξ̃|^{−2}, δ_{2} > 0, η ∈ ℂ^{4}, ξ̃ ∈ ℝ^{3}.

Applying this fact, we proceed as follows:

((−P_{S}^{1})^{∞}(x, ξ)η, η) =

= \frac{Γ_{κ_{j}}(t)}{2π det κ′_{j}(t)} ∫_{−∞}^{+∞}( − A_{0}^{−1}([(κ′_{j}(0, t))^{T} ]^{−1}ζ)η, η)dy_{0} ≥

≥ δ_{2}|η|^{2} ∫_{−∞}^{+∞}|κ′_{j}(0, t)ζ|^{−2}dy_{0} ≥

≥ δ_{3}|η|^{2} ∫_{−∞}^{+∞}\frac{dy_{0}}{y_{0}^{2} + |ξ|^{2}} = δ_{4}|η|^{2}|ξ|^{−1},  (5.7)
η ∈ ℂ^{4}, ξ ∈ ℝ^{2}, ζ = (y_{0}, ξ), δ_{k} = const > 0, k = 2, 3, 4.

Formulas (1.6), (5.5) and (5.6) also imply

\begin{matrix}D_{x}^{α}D_{ξ_{1}}^{m}(P_{S}^{1})^{∞}(x, λξ) = |λ|^{−1}λ^{−m}D_{x}^{α}D_{ξ_{1}}^{m}(P_{S}^{1})^{∞}(x, ξ), \\ |α| < ∞, m = 0, 1, …, ξ ∈ ℝ^{2}, λ ∈ ℝ. \end{matrix}  (5.8)

Hence we have the equivalences (see (4.18), (4.21), (5.1), (5.2))

κ_{j∗}P_{S}^{1}κ_{j∗}^{−1} ∼^{x_{0}} (P_{S}^{1})^{0}(x_{0}, D), x_{0} ∈ U_{j} ⊂ S, x_{0} ∉ ∂S,

κ_{j∗}P_{S}^{1}κ_{j∗}^{−1} ∼^{x_{0}} r_{+}(P_{S}^{1})^{0}(x_{0}, D), x_{0} ∈ U_{j} ∩ ∂S,

where (see (4.20))

(P_{S}^{1})^{0}(x, ξ) := (P_{S}^{1})^{∞}(x, (1 + |ξ_{1}|)|ξ_{1}|^{−1}ξ_{1}, ξ_{2}).

Due to (5.7) the symbol (P_{S}^{1})^{0}(x, ξ) is an elliptic one,

inf\{| det(P_{S}^{1})^{∞}(x, ξ)| : x ∈ \overline{S}, |ξ| = 1\} > 0.

Since condition (5.8) implies the continuity property (4.25) for the symbol (P_{S}^{1})^{∞}(x, ξ), by virtue of Lemma 17 it admits the factorization

(P_{S}^{1})^{0}(x, ξ) = [(ξ_{2} − i|ξ_{1}| − i)^{−1/2}P_{−}(x, ξ)][(ξ_{2} + i|ξ′| + i)^{−1/2}P_{+}(x, ξ)],

P_{−}^{±1}(x, ·), P_{+}^{±1}(x, ·) ∈ M_{p}(ℝ^{2}), x ∈ ∂S,

where P_{−}^{±1}(x, ξ_{1} −iλ), P_{+}^{±1}(x, ξ_{1} +iλ) have bounded analytic extensions for λ > 0. According to Theorem 15 operator (3.7) is a Fredholm one if and only if the operators r_{+}(P_{S}^{1})_{ν,−1}(x_{0}, D) are Fredholm ones in 𝕃_{p}(ℝ_{+}^{2}) for all x_{0} ∈ ∂S, where

(P_{S}^{1})_{ν,−1}(x_{0}, ξ) = \frac{(ξ_{2} − i|ξ_{1}| − i)^{ν+1}}{(ξ_{2} + i|ξ_{1}| + i)^{ν}} (P_{S}^{1})^{0}(x_{0}, ξ) =

= (\frac{ξ_{2} − i|ξ_{1}| − i}{ξ_{2} + i|ξ_{1}| + i})^{ν+1/2}P_{−}(x_{0}, ξ)P_{+}(x_{0}, ξ), x_{0} ∈ ∂S.  (5.9)

Therefore (see (4.10), (5.9))

r_{+}(P_{S}^{1})_{ν,−1}(x_{0}, D) = r_{+}P_{−}(x_{0}, D)ℓ_{0}r_{+}G_{ν}(D)P_{+}(x_{0}, D), (5.10)

with

G_{ν}(ξ) = (\frac{ξ_{2} − i|ξ_{1}| − i}{ξ_{2} + i|ξ_{1}| + i})^{ν+1/2}  (5.11)

and since r_{+}P_{±}(x_{0}, D) are invertible (according to (4.10) the inverses read r_{+}P_{±}^{−1}(x, D)). The proof will be completed if we find invertibility conditions for r_{+}G_{ν}(D) in 𝕃_{p}(ℝ_{+}^{2}); the latter is invertible if and only if

1/p − 1 < ν + 1/2 < 1/p  (5.12)

and the inverse reads (r_{+}G_{ν}(D))^{−1} = I_{+}^{ν+1/2}(D)ℓ_{0}r_{+}I_{−}^{−ν−1/2}(D) (see [2], §2). Conditions (5.12) coincide with (3.8).

The local inverses to P_{S}^{1} : ℍ̃_{p}^{ν}(S) → ℍ_{p}^{ν+1}(S) are, therefore, independent of the parameters p and ν if conditions (3.8) are fulfilled.

In fact, the operator

(r_{+}P_{S}^{1})_{ν,−1}^{−1}(x_{0}, D) := P_{+}^{−1}(x_{0}, D)I_{+}^{ν+1/2}(D)ℓ_{0}r_{+}I_{−}^{−ν−1/2}(D)P_{+}^{−1}(x_{0}, D)

is inverse to (r_{+}P_{S}^{1})_{ν,−1}(x_{0}, D) in L_{p}(ℝ_{+}^{2}); if we “lift” these operators from the space L_{p}(ℝ_{+}^{2}) to the Bessel potential spaces by means of the Bessel potentials I_{±}^{μ}(D) defined by (4.8), we shall come to the following conclusion: if (3.8) holds, the operator

I_{+}^{−ν}(D)ℓ_{0}(r_{+}P_{S}^{1})_{ν,−1}^{−1}(x_{0}, D)I_{−}^{ν+1}(D) =
= (D)P_{+}^{−1}(x_{0}, D)I_{+}^{1/2}(D)ℓ_{0}r_{+}I_{−}^{1/2}P_{−}^{−1}(x_{0}, D)

inverts the operator

I_{+}^{−ν}(D)ℓ_{0}(r_{+}P_{S}^{1})_{ν,−1}(x_{0}, D)I_{−}^{ν+1}(D) =
= P_{S}^{1}(x_{0}, D) : ℍ̃_{p}^{ν}(ℝ_{+}^{2}) → ℍ_{p}^{ν+1}(ℝ_{+}^{2}), x_{0} ∈ ∂S

which is a local representation of P_{S}^{1} = P_{S}^{1}(x, D) (x ∈ S, x_{0} ∈ ∂S).

Thus the regularizer constructed by means of the local inverses (see, for example, [2], [3], [13]) can be chosen independent of p and ν if (3.8) holds. Now we can take p = 2 and by Theorem 16 and Lemma 19 we get Ind P_{S}^{1} = 0.

To complete the proof for the space H_{p}^{ν}(S) it remains to check that ker P_{S}^{1} = 0. We need to do this only for ν = −1/2 and p = 2, since ker P_{S}^{1} is also independent of the parameters p and ν (see Lemma 19).

The equality ker P_{S}^{1} = 0, in turn, follows from the triviality of a solution of the homogeneous Problem D. Actually, formula (1.17) implies that for any solution U = (u_{1}, …, u_{4}) of the homogeneous Problem D we have

∫_{ℝ_{S}^{3}} \{c_{ijkl}D_{l}u_{k}D_{j}\overline{u}_{i} + ρτ^{2}u_{k}\overline{u}_{k} + \frac{1}{\overline{τ}T_{0}} λ_{ij}D_{j}u_{4}D_{i}\overline{u}_{4} + \frac{c_{0}}{T_{0}} u_{4}\overline{u}_{4}\}dx = 0;

recalling that τ = σ + iω and separating the real and the imaginary part, we obtain

∫_{ℝ_{S}^{3}} \{c_{ijkl}D_{l}u_{k}D_{j}\overline{u}_{i} + ρ(σ^{2} − ω^{2})u_{k}\overline{u}_{k} +

+\frac{σ}{|τ|^{2}T_{0}} λ_{ij}D_{j}u_{4}D_{i}\overline{u}_{4} + \frac{c_{0}}{T_{0}} u_{4}\overline{u}_{4}\}dx = 0,  (5.13)

\frac{ω}{T_{0}} ∫_{ℝ_{S}^{3}} \{2σT_{0}u_{k}\overline{u}_{k} + λ_{ij}D_{j}u_{4}D_{i}\overline{u}_{4}\}dx = 0.

Whence by (1.12) and (1.14) we find U = 0 for an arbitrary τ with Re τ > 0. For τ = 0 we obtain

D_{j}u_{k}(x) + D_{k}u_{j}(x) = 0, u_{4} = 0, k, j = 1, 3, x ∈ ℝ_{S}^{3}. (5.14)

The general solution of this system is (see [1])

U = [a × x] + b,

where a and b are the constant three-dimensional vectors with complex entries and [·×·] denotes the vector product of two vectors. From conditions (1.10) and (5.14) it follows that U = 0.

Thus the homogeneous Problem D has only a trivial solution and ker P_{S}^{1} = {0}.

To prove the theorem for the Besov space 𝔹_{p,p}^{ν}(S) recall the following interpolation property from (4.5):

If A : ℍ̃_{p}^{ν}(S) → ℍ_{p}^{ν+r}(S) is bounded for any ν_{0} < ν < ν_{1} and some 1 < p < ∞, then the operator A : 𝔹̃_{p,q}^{ν}(S) → 𝔹_{p,q}^{ν+r}(S) is also bounded for any ν_{0} < ν < ν_{1}, 1 ≤ q ≤ ∞.

Let conditions (3.8) be fulfilled. Then the operator P_{S}^{1} : ℍ̃_{p}^{ν}(S) → ℍ_{p}^{ν+1}(S) has the bounded inverse (P_{S}^{1})^{−1} : ℍ_{p}^{ν+1}(S) → ℍ̃_{p}^{ν}(S); due to the above-mentioned interpolation property the operator (P_{S}^{1})^{−1} : 𝔹_{p,q}^{ν+1}(S) → 𝔹̃_{p,q}^{ν}(S) will also be bounded and therefore the operator P_{S}^{1} in (3.6) has the bounded inverse.

5.2. Proof of Theorem 8. After the localization and local transformation of variables (see (5.1)–(5.9)) we obtain the equivalences

κ_{j∗}P_{S}^{4}κ_{j∗}^{−1} ∼^{x_{0}} (P_{S}^{4})^{0}(x_{0}, D), x_{0} ∈ U_{j} ⊂ S, x_{0} ∉ ∂S,

κ_{j∗}P_{S}^{4}κ_{j∗}^{−1} ∼^{x_{0}} r_{+}(P_{S}^{4})^{0}(x_{0}, D), x_{0} ∈ U_{j} ∩ ∂S,

where

(P_{S}^{4})^{0}(x_{0}, ξ) = B^{0}(x_{0}, ξ)(P_{S}^{1})^{0}(x_{0}, ξ)(B^{0})^{T} (x_{0}, ξ)  (5.15)

and B^{0}(x_{0}, ξ) represents the modified principal symbol of the operators B(D_{x}, n(x)) and Q(D_{x}, n(x)) (whose principal symbols coincide). The order of B^{0}(x_{0}, ξ) is 1 and therefore (5.15), (5.7) yield

((P_{S}^{4})^{∞}(x_{0}, ξ)η, η) ≥ δ_{5}|ξ||η|^{2}, ξ ∈ ℝ^{2}, η ∈ ℂ^{4}, δ_{5} > 0.

The homogeneity property

D_{ξ_{1}}^{m}D_{x}^{α}(P_{S}^{4})^{∞}(x, λξ) = |λ|λ^{−m}D_{ξ_{1}}^{m}D_{x}^{α}(P_{S}^{4})^{∞}(x, ξ),
|α| < ∞, m = 0, 1, …, ξ ∈ ℝ^{2}, λ ∈ ℝ

holds as well (see (5.8)).

Thus the symbol (P_{S}^{4})^{∞}(x, ξ) is elliptic

inf\{| det(P_{S}^{4})^{∞}(x, ξ)| : x ∈ \overline{S}, |ξ| = 1\} > 0

and operator (3.10) is a Fredholm one if and only if the operators r_{+}(P_{S}^{4})_{ν+1,1}^{0}(x_{0}, D) are Fredholm in 𝕃_{p}(ℝ_{+}^{2}) for all x_{0} ∈ ∂S; here

(P_{S}^{4})_{ν+1,1}^{0}(x_{0}, ξ) = \frac{(ξ_{2} − i|ξ_{1}| − i)^{ν}}{(ξ_{2} + i|ξ_{1}| + i)^{ν+1}} (P_{S}^{4})^{0}(x_{0}, ξ) =

= (\frac{ξ_{2} − i|ξ_{1}| − i}{ξ_{2} + i|ξ_{1}| + i})^{ν+1/2}P_{−}^{4}(x_{0}, ξ)P_{+}^{4}(x_{0}, ξ),

(P_{+}^{4})^{±1}(x, ·), (P_{−}^{4})^{±1}(x, ·) ∈ M_{p}(ℝ^{2}), x_{0} ∈ ∂S,

and (P_{+}^{4})^{±1}(x_{0}, ξ_{1}, ξ_{2} + iλ), (P_{−}^{4})^{±1}(x_{0}, ξ_{1}, ξ_{2} − iλ) have bounded analytic extensions for λ > 0. The proof is completed similarly to that of Theorem 7.

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20. E. Shargorodsky, Some remarks on the boundedness of pseudodifferential operators. Math. Nachr. (to appear).

(Received 01.10.1993)

Authors’ addresses:

R. Duduchava A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, Z. Rukhadze Str., 380093 Tbilisi Republic of Georgia

D. Natroshvili Georgian Technical University Department of Mathematics (99) 77, M. Kostava Str., 380075 Tbilisi Republic of Georgia

E. Shargorodsky Tbilisi State University Department of Mechanics and Mathematics 2, University Str., 380043 Tbilisi Republic of Georgia