On solvability of one elliptic system in weighted Sobolev spaces

Selcuk Journal of

Applied Mathematics

Vol. 4, No. 1, pp. 3–24, 2003

On solvability of one elliptic system in

weighted Sobolev spaces

Lina N. Buldygerova

Department of Mathematics, Novosibirsk State University, Pirogova 2, 630090 Novosibirsk, Russia; e-mail: b lina@gorodok.net

Received: March 11, 2003

Summary. In this article we consider an elliptic system in R3 and

prove a theorem on unique solvability in a special scale of weighted Sobolev spaces.

Key words: elliptic systems, weighted Sobolev spaces, uncondi-tional solvability

2000 Mathematics Subject Classification: 35J45, 35C15

1. Introduction

In the present article we consider the following elliptic system of partial differential equations in R3

(1.1)



u++∇u−= f+(x), div u+ = f−(x).

This system arises in various problems of mathematical physics. There are results concerning solvability of the system in various func-tional spaces. In particular, from results by V. N. Maslennikova (see, for example, [1]) it follows that, in the case of f−(x) ≡ 0, the sys-tem (1.1) is uniquely solvable in the Sobolev space

(1.2)

3



1

W_p1(R3)× W_p2(R3)

 _{The research was supported in part by the Russian Foundation for Basic}

for p > 3/2. However, the system (1.1) is not unconditionally solvable in (1.2) for p ≤ 3/2. As is shown in the book [2], in the case of f−(x) ≡ 0 for the system (1.1) to be solvable in (1.2) it is necessary and sufficient that _

R3

f+(x) dx = 0.

To research solvability of the system (1.1) it is more convenient to use the weighted Sobolev spaces W_p,σl introduced by G. V. Demidenko in [3]. In the present article we study solvability of the system (1.1) in the spaces and show that, for suitable values of σ, the system is unconditionally solvable for any p > 1.

By definition [3], the norm in the space

W_p,σl (Rn), l = (l1, . . . , ln), li ∈ N, 1 < p < ∞, σ ∈ R1, is defined by u(x), Wl p,σ(Rn) =  0≤β/l≤1 (1 + x )−σ(1−β/l)_Dβ xu(x), Lp(Rn), where x 2= n  i=1 x2l_i i, β = (β1, . . . , βn), β/l = n  i=1 βi/li.

In the isotropic case l1 = . . . = ln = l the norm is equivalent to the

norm _

0≤|β|≤l

(1 + |x|)−σ(l−|β|)Dβ_xu(x), Lp(Rn).

It was proved in [3] that the set of compactly-supported infinitely differentiable functions is everywhere dense in W_p,σl (Rn) for σ ≤ 1.

Hereinafter, we consider the weighted Sobolev spaces W_p,σl (Rn) for 0 < σ < 1.

We give a result on solvability of the system (1.1) in the weighted Sobolev spaces W_p,σl .

Theorem 1. Let 1 < p < ∞, 1 −_2p3 < σ < _2p3, 1/p + 1/p = 1. Then

for every right-hand side f+(x) ∈ 3  1 W_p,σ1 (R3), (1 +|x|)2(1−σ)∇ f+(x) ∈ 3  1 Lp(R3), (1 +|x|)2(1−σ)f−(x) ∈ Lp(R3),

the system (1.1) has a unique solution u+(x) ∈ 3  1 W_p,σ1 (R3), u−(x) ∈ W_p,σ2 (R3); moreover, the following inequality holds:

(1.3) 3  j=1 u+_j(x), W_p,σ1 (R3) + u−(x), Wp,σ2 (R3) ≤ c(3 j=1 f_j+(x), W_p,σ1 (R3) + (1 + |x|)2(1−σ)∇ f_j+(x), Lp(R3) +(1 + |x|)2(1−σ)f−(x), Lp(R3))

with some constant c > 0 independent of f+(x) and f−(x). Remark 1. It was proved in [3] that the operator

⎛ ⎜ ⎜ ⎝ 1 0 0 Dx1 0 1 0 Dx2 0 0 1 Dx3 Dx1 Dx2 Dx3 0 ⎞ ⎟ ⎟ ⎠: 3  1 W_p,11 (R3)× W_p,12 (R3) → 3 1 W_p,11 (R3)× L_p(R3) is an isomorphism for 1 < p < 3/2.

2. The scheme of the proof of the theorem

We outline the basic points of the proof of solvability of the system (1.1).

First, by analogy with [2, Chapter 3], we construct a sequence of approximate solutions to the system (1.1) by using S. V. Uspenskii’s integral representation [6] for summable functions

(2.1) ϕ(x) = lim h→0(2π) −n h−1  h v−|α|−1  Rn  Rn

e(ix−yvα ξ)_{G(ξ)ϕ(y) dξdydv,} where (2.2) G(ξ) = 2mξ 2mexp(−ξ 2m), ξ 2= n  i=1 ξ_i2/αi, m ∈ N.

We use (2.1) and (2.2) for αi = 1/2, n = 3.

To this end we consider the following system with a parameter

ξ ∈ R3 _⎧ ⎪ ⎨ ⎪ ⎩ v++ L(iξ)v−=f+(ξ), M (iξ)v+=f−(ξ), where L(iξ) = ⎛ ⎝iξ_iξ1₂ iξ3 ⎞

⎠_{, M (iξ) = (iξ1, iξ2, iξ3).}

The system is obtained by formal application of the Fourier operator to the system (1.1) under the condition that the vector-functions f+(x) and f−(x) are infinitely differentiable and compactly-support-ed. Since det(M (iξ)L(iξ)) = −|ξ|2 = 0 for ξ ∈ R3\{0}, we obviously

obtain (2.3) v+(ξ) = (I +L(iξ) |ξ|2 M (iξ))f+(ξ) − L(iξ) |ξ|2 f−(ξ), (2.4) v−(ξ) = −M (iξ)_|ξ|2 f+(ξ) + _|ξ|1₂f−(ξ). We construct the vector-functions

(2.5) u+_k(x) = (2π)−3/2 k  1/k v−1  R3 eixξG(ξvα)v+(ξ) dξ dv, (2.6) u−_k(x) = (2π)−3/2 k  1/k v−1  R3 eixξG(ξvα)v−(ξ) dξ dv.

It follows from the definition of the vector-functions v+(ξ) and v−(ξ)

that _⎧ ⎨ ⎩ u+_k(x) + L(Dx)u−_k(x) = f_k+(x), M (Dx)u+_k(x) = f_k−(x), where f_k+(x) = (2π)−3 k  1/k v−|α|−1  R3  R3 e(ix−yvα ξ)_G(ξ)f+_{(y) dξ dy dv,}

f_k−(x) = (2π)−3 k  1/k v−|α|−1  R3  R3 e(ix−yvα ξ)_G(ξ)f−_{(y) dξ dy dv.} By the integral representation (2.1) we can consider the vector-functions (2.5) and (2.6) as an approximate solution to (1.1).

The vector-functions (2.3) and (2.4) can have singularities only at ξ = 0. Therefore, by (2.2) the vector-functions u+_k(x) and u−_k(x) are infinitely differentiable. It is obvious that we can indicate a natural number m1 such that if m ≥ m1 in (2.2) then these vector-functions

are summable with an arbitrary power p ≥ 1. To this end, it suffices to take m1such that the functions k(ξ) ∈ C∞(R3\{0}), homogeneous

in α of degree q ∈ [−1, 0], satisfy the inequality |

R3

eixξξ 2m1_e−ξ2m1_{k(ξ) dξ| ≤ c(1+x )}−(|α|+1)_{, x ∈ R}

3, |α| = 3/2.

Henceforth we suppose that m ≥ m1 in (2.2).

In the following section we prove the estimates

(2.7) 3  j=1 u+_k,j(x), W_p,σ1 (R3) + u−k(x), Wp,σ2 (R3) ≤ c[3 j=1 ( |γ|=1 (1 + |x|)2(1−σ)_Dγ xfj+(x), Lp(R3) + fj+(x), Wp,σ1 (R3)) +f−(x), Lp(R3) + |x|2(1−σ)f−(x), Lp(R3)]

with some constant c > 0 independent of f+(x), f−(x) and k, and also establish the convergence

(2.8) 3  j=1 (u+_k1,j(x) − u+_k2,j(x)), W_p,σ1 (R3) +(u−_k 1(x) − u−k2(x)), Wp,σ2 (R3) → 0, k1, k2 → ∞

under the conditions

1 < p < ∞, 1− 3

2p < σ < 3

2p, 1/p + 1/p

 _{= 1.}

By completeness of the space

3



1

from (2.7) and (2.8) it follows that there exists a vector-function u+(x) ∈ 3  1 W_p,σ1 (R3), u−(x) ∈ Wp,σ2 (R3)

that is a solution to the system (1.1) and satisfies the estimate (1.3). Prove the uniqueness of solutions, i. e. show that if u(x) is a solu-tion to the system

(2.9) ⎧ ⎨ ⎩ u++ L(Dx)u−= 0, x ∈ R3, M (Dx)u+= 0, then u(x) = 0 nearly everywhere in R3.

First, we assume that u+(x) and u−(x) are compactly-supported. Applying the Fourier operator to the system (2.9), we obtain the following system of linear equations:

⎧ ⎪ ⎨ ⎪ ⎩  u++ L(iξ)u−= 0, x ∈ R3, M (iξ)u+= 0.

We can rewrite the system in the matrix form L(iξ)u = 0, where L(iξ) =  I L(iξ) M (iξ) 0  .

Since det(M (iξ)L(iξ)) = 0 for ξ ∈ R3\{0} thenu(ξ) = 0 nearly every-

where in R3; moreover, by continuityu(ξ) ≡ 0, ξ ∈ R 3. Hence, u(x) is the zero solution, and the system (1.1) has a unique compactly-supported solution. Then, taking into account the estimates from Lemma 2 of [4], for β, |β| = 2, the solution to (1.1) satisfies the inequality

(2.10) Dβ_xu−(x), Lp(R3) ≤ c(3

j=1

f_j+(x), W_p,σ1 (R3) + f−(x), Lp(R3)) with some constant c > 0 independent of f+(x) and f−(x).

Now we consider a more general case. Let u(x) ∈ 3

1 W

1

p,σ(R3)× W_p,σ2 (R3) be a solution to the system (2.9). Prove that u(x) = 0 nearly

everywhere in R3. Since u−(x) ∈ Wp,σ2 (R3) it follows that for every

ε > 0 there is a compactly-supported function u−_ε(x) ∈ W_p,σ2 (R3) such that

u−(x) − u−_ε(x), W_p,σ2 (R3) ≤ ε.

Consider the function u+_ε(x) = −L(Dx)u−_ε(x). Obviously, u+_ε(x) is compactly-supported too. We have the following system

⎧ ⎨ ⎩

u+_ε(x) + L(Dx)u−ε(x) ≡ 0, x ∈ R3,

M (Dx)u+ε(x) ≡ − u−ε(x).

According to (2.10), for β, |β| = 2, the estimate holds: Dβ

xu−ε(x), Lp(R3) ≤ c − u−ε(x) Lp(R3)

with some constant c > 0 independent of ε. Since u+(x) and u−(x) satisfy the system (2.9) then u−(x) ≡ 0. Consequently,

 − u− ε(x), Lp(R3) = (u−(x) − u−_ε(x)), Lp(R3) ≤ c  |γ|=2 Dγ_x(u−(x) − u−_ε(x)), Lp(R3) ≤ cε. Hence, for β, |β| = 2, we have

Dβ

xu−ε(x), Lp(R3) ≤ cε,

Dβ

xu−(x), Lp(R3) ≤ Dβx(u−(x) − u−ε(x)), Lp(R3)

+D_xβu−_ε(x), Lp(R3) ≤ ε + cε.

Since the inequality holds for every ε > 0 then Dβ_xu−(x), Lp(R3) =

0 for |β| = 2. From the estimate [7]

|x|−2σu−(x), Lp(R3) ≤ c|x|2(1−σ) u−(x), Lp(R3)

it follows that u−(x) = 0 nearly everywhere in R3. Then u+(x) = 0

and u(x) = 0 nearly everywhere in R3. Thus, we proved the

3. Estimates for approximate solutions

To obtain estimates for approximate solutions to the system (1.1), we rewrite the vector-functions (2.5) and (2.6) as follows:

u+_k(x) = u+,+_k (x) + u+,−_k (x) = (2π)−3/2 k  1/k v−1  R3 eixξG(ξvα)(v+,+(ξ) + v+,−(ξ)) dξ dv, u−_k(x) = u−,+_k (x) + u−,−_k (x) = (2π)−3/2 k  1/k v−1  R3 eixξG(ξvα)(v−,+(ξ) + v−,−(ξ)) dξ dv, where

v+,+(ξ) = (I +L(iξ)_|ξ|2 M (iξ))f+(ξ), v+,−(ξ) = −L(iξ)_|ξ|2 f−(ξ),

v−,+(ξ) = −M (iξ)_|ξ|2 f+(ξ), v−,−(ξ) = _|ξ|1₂f−(ξ).

Estimates for the higher-order derivatives of the vector-functions u+_k(x) and u−_k(x) follow from the lemmas [4].

Lemma 1. Let β = (β1, β2, β3),|β| = 1. Then the following estimates

hold: (3.1) 3  j=1 Dβ xu+,+k,j (x), Lp(R3) ≤ c 3  j=1  |γ|=1 Dγ xfj+(x), Lp(R3), (3.2) 3  j=1 Dβ xu+,−k,j (x), Lp(R3) ≤ cf−(x), Lp(R3)

with some constant c > 0 independent of f+(x), f−(x) and k; more-over, (3.3) 3  j=1 Dβ_xu+_k1,j(x) − Dβ_xu_k+_2,j(x), Lp(R3) → 0, k₁, k2→ ∞.

Lemma 2. Let β = (β1, β2, β3),|β| = 2. Then the following estimates hold: (3.4) Dβ_xu−,+_k (x), Lp(R3) ≤ c 3  j=1  |γ|=1 Dγ xfj+(x), Lp(R3), (3.5) Dβ_xu−,−_k (x), Lp(R3) ≤ cf−(x), Lp(R3)

with some constant c > 0 independent of f+(x), f−(x) and k; more-over,

(3.6) Dβ_xu−_k1(x) − D_xβu−_k2(x), Lp(R3) → 0, k₁, k2 → ∞.

To obtain estimates for u+_k(x) and D_xγu−_k(x), |γ| < 2, we consider the functions (3.7) Kh(x) = h_−1 h v−1  R3 eixξG(ξvα)k(ξ) dξ dv, 0 < h < 1, where the function k(ξ) ∈ C∞(R3\{0}) is homogeneous in α of degree

q < 0. The derivatives of k(ξ) satisfy the estimates |D_ξβk(ξ)| ≤ cβξ q−βα, ξ = 0,

where cβ > 0 are constants.

We will use the following lemma [4].

Lemma 3. Let |α| + q > 0. There is m₀ such that if m ≥ m0 in

the definition (2.2) of the function G(ξ) then the following uniform estimate holds:

x |α|+q_|K

h(x)| ≤ c, x ∈ Rn,

with some constant c > 0 independent of h.

Henceforth we deal with integrals of the form (3.7) for q ∈ [−1, 0]. Now we establish estimates for the vector-functions u+_k(x) and Dγ_xu−_k(x), |γ| < 2. Lemma 4. Let 1 < p < ∞, 1− 3 2p < σ < 3 2p, 1/p + 1/p _{= 1.}

Then the following inequalities hold: (3.8) 3  j=1 |x|−σ_u+,+ k,j (x), Lp(R3)

≤ c3 j=1  |γ|=1 |x|(1−σ)D_xγf_j+(x), Lp(R3), (3.9) 3  j=1 |x|−σu+,−_k,j (x), Lp(R3) ≤ c|x|(1−σ)f−(x), Lp(R3)

with some constant c > 0 independent of f+(x), f−(x) and k; more-over, (3.10) 3  j=1 (1 + |x|)−σ_(u+ k1,j(x) − u+k2,j(x)), Lp(R3) → 0, k1, k2 → ∞.

Proof. From the definition of the vector-function u+,+_k (x) we have (3.11) u+,+_k (x) = 3  l=1 (2π)−3/2 k  1/k v−1  R3 eixξG(ξvα) ×(iξl)−1(iξl)v+,+(ξ) dξ dv = 3  l=1 (2π)−3/2 k  1/k v−1  R3 eixξG(ξvα)(iξl)−1

×(I + L(iξ)_|ξ|2 M (iξ))Dylf+(ξ) dξ dv = 3  l=1 (2π)−3 k  1/k v−1  R3  R3 ei(x−y)ξG(ξvα)(iξl)−1 ×(I +L(iξ) |ξ|2 M (iξ))dξ  Dylf+(y) dy dv. Introduce the notation

Kk,l(x) = k  1/k v−1  R3 eixξG(ξvα)Ql(ξ) dξ dv,

Then u+,+_k (x) = 3  l=1 (2π)−3  R3

Kk,l(x − y)Dylf+(y) dy.

The entries of the matrix L(iξ)_|ξ|2 M (iξ) are homogeneous in α = (1/2, 1/2, 1/2) of degree 0. Therefore, all entries Ql(ξ) are homogeneous

in α of degree q = −1/2 < 0. Since |α| + q = 3/2 − 1/2 = 1 > 0 then the conditions of Lemma 3 are satisfied. Hence, applying this lemma, we obtain the inequality

3  j=1 |x|−σ_u+,+ k,j (x), Lp(R3) ≤ c3 j=1 3  l=1 x qσ  R3 x − y −|α|−q|Dylfj+(y)| dy, Lp(R3) = c 3  j=1 3  l=1 |x|qσ  R3 x − y −|α|−qy q(1−σ) ×y q(σ−1)|Dylfj+(y)| dy, Lp(R3). Taking into account the inequality

3  i=1 |yi|(1−σ)/|α|≤ cy 1−σ, 0≤ σ ≤ 1, we have 3  j=1 |x|−σu+,+_k,j (x), Lp(R3) ≤ c3 j=1 3  l=1  R3 3  i=1 |xi|σq/|α||xi− yi|−1−q/|α||yi|q/|α|(1−σ) ×y q(σ−1)_|D ylfj+(y)| dy, Lp(R3) = c 3  j=1 3  l=1 (  R1  R1  R1 3  i=1 1/|xi|−σq/|α|1/|xi− yi|1+(q/|α|)

×1/|yi|q/|α|(σ−1)Fl,j(y) dy, Lp(R1)pdx2dx3)1/p = c 3  j=1 3  l=1 (  R1  R1  R1  R1 [  R1 1/|x1|−σq/|α|1/|x1− y1|1+(q/|α|)

×1/|y1|q/|α|(σ−1)Fl,j(y) dy1] 2  i=1 1/|xi|−σq/|α|1/|xi− yi|1+(q/|α|) ×1/|yi|q/|α|(σ−1)dy2dy3, Lp(R1)pdx2dx3)1/p = c 3  j=1 3  l=1 (  R1  R1   R1  R1 KFl,j(x1, y2, y3) ×2 i=1 1/|xi|−σq/|α|1/|xi− yi|1+(q/|α|) ×1/|yi|q/|α|(σ−1)dy2dy3, Lp(R1)pdx2dx3)1/p.

Applying Minkowski’s inequality, we obtain

3  j=1 |x|−σu+,+_k,j (x), Lp(R3) ≤ c3 j=1 3  l=1 (  R1  R1 [  R1  R1 KFl,j(x1, y2, y3), Lp(R1) ×2 i=1 1/|xi|−σq/|α|1/|xi− yi|1+(q/|α|) ×1/|yi|q/|α|(σ−1)dy2dy3]pdx2dx3)1/p. As σ < _2p3 then −qσ/|α| < −q/p < 1/p. From the conditions of the lemma we find that

σ − 1 > − 3

2p, q(σ − 1)/|α| < −q/p

_{< 1/p}_.

Since

−qσ/|α| < 1/p, q(σ − 1)/|α| < 1/p, −q/|α| > 0, using the Hardy–Littlewood inequality [5], we have

3  j=1 |x|−σ_u+,+ k,j (x), Lp(R3) ≤ c3 j=1 3  l=1 (  R1  R1 [  R1  R1 Fl,j(x1, y2, y3), Lp(R1)

×2 i=1 1/|xi|−σq/|α|1/|xi− yi|1+(q/|α|) ×1/|yi|q/|α|(σ−1)dy2dy3]pdx2dx3)1/p = c 3  j=1 3  l=1 (  R1   R1  R1 Fl,j(x1, y2, y3), Lp(R1) ×2 i=1 1/|xi|−σq/|α|1/|xi− yi|1+(q/|α|) ×1/|yi|q/|α|(σ−1)dy2dy3, Lp(R1)pdx3)1/p.

Applying Minkowski’s inequality, we obtain

3  j=1 |x|−σ_u+,+ k,j (x), Lp(R3) ≤ c3 j=1 3  l=1 (  R1 [  R1  R1 Fl,j(x1, y2, y3), Lp(R1) ×1/|x2|−σq/|α|1/|x2− y2|1+(q/|α|)1/|y2|q/|α|(σ−1)dy2, Lp(R1) ×1/|x3|−σq/|α|1/|x3− y3|1+(q/|α|)1/|y3|q/|α|(σ−1)dy3]pdx3)1/p.

Using the Hardy–Littlewood inequality two times, we have

3  j=1 |x|−σ_u+,+ k,j (x), Lp(R3) ≤ c3 j=1 3  l=1 (  R1 [  R1 Fl,j(x1, x2, y3), Lp(R1), Lp(R1) ×1/|x3|−σq/|α|1/|x3− y3|1+(q/|α|)1/|y3|q/|α|(σ−1)dy3]pdx3)1/p = c 3  j=1 3  l=1   R1 Fl,j(x1, x2, y3), Lp(R1), L_p(R1) ×1/|x3|−σq/|α|1/|x3− y3|1+(q/|α|)1/|y3|q/|α|(σ−1)dy3, Lp(R1) ≤ c3 j=1 3  l=1 Fl,j(x1, x2, x3), Lp(R1), Lp(R1), Lp(R1)

= c 3  j=1 3  l=1 (  R1 Fl,j(x), Lp(R1), L_p(R1)pdx3)1/p = c 3  j=1 3  l=1 (  R1  R1 Fl,j(x), Lp(R1)pdx2dx3)1/p = c 3  j=1 3  l=1 Fl,j(x), Lp(R3) = c 3  j=1 3  l=1 x q(σ−1)Dxlfj+(x) Lp(R3).

Since q = −1/2 and x 1/2 ∼ |x| it follows that

3  j=1 |x|−σu+,+_k,j (x), Lp(R3) ≤ c 3  j=1 3  l=1 |x|(1−σ)Dxlfj+(x) Lp(R3). Prove (3.9). From the definition of the vector-function u+,−_k (x) we obtain u+,−_k (x) = (2π)−3/2 k  1/k v−1  R3 eixξG(ξvα)(−L(iξ) |ξ|2 )f−(ξ) dξ dv = (2π)−3 k  1/k v−1  R3  R3 ei(x−y)ξG(ξvα)(−L(iξ) |ξ|2 ) dξ  f−(y) dy dv. All entries of the matrix L(iξ)_|ξ|2 are homogeneous in α = (1/2, 1/2, 1/2) of degree q = −1/2 < 0. Therefore, as above we have the estimate

3  j=1 |x|−σ_u+,− k,j (x), Lp(R3) ≤ c_ R3 3  i=1 |xi|qσ/|α||xi− yi|−1−q/|α||yi|q(1−σ)/|α|

×y q(σ−1)|f−(y)| dy, Lp(R3).

Applying the Hardy–Littlewood inequality three times, we obtain the estimate (3.9): 3  j=1 |x|−σ_u+,− k,j (x), Lp(R3) ≤ cx q(σ−1)f−(x), Lp(R3) ≤ c|x|(1−σ)f−(x), Lp(R3).

To prove the convergence (3.10), we have to prove that (3.13) (1 + |x|)−σ(u+,+_k1,j(x) − u+,+_k2,j(x)), Lp(R3) → 0, (3.14) (1 + |x|)−σ(u+,−_k1,j(x) − u+,−_k2,j(x)), Lp(R3) → 0

for every j = 1, 2, 3 as k1, k2 → ∞. Obviously, it suffices to establish

(3.13) and (3.14) for compactly-supported f+(x).

Prove (3.13). By the representation (3.11) and the notation (3.12) for the matrix Ql(ξ) = (Qj,il (ξ)), it suffices to consider the function

wk(x) = (2π)−3 k  1/k v−1  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  Dylfi+(y) dy dv and prove the convergence

(3.15) (1 + |x|)−σ(wk1(x) − wk2(x)), Lp(R3) → 0, k1, k2 → ∞. We have wk1(x) − wk2(x) = (2π)−3 k1  1/k1 v−1 ×  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  Dylfi+(y) dy dv −(2π)−3 k2  1/k2 v−1  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  Dylfi+(y) dy dv = (2π)−3 1/k_ 2 1/k1 v−1  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  Dylfi+(y) dy dv +(2π)−3 k1  1/k2 v−1  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  Dylfi+(y) dy dv −(2π)−3 k1  1/k2 v−1  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  Dylfi+(y) dy dv −(2π)−3 k2  k1 v−1  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  Dylfi+(y) dy dv

= (2π)−3 1/k_ 2 1/k1 v−1  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  Dylfi+(y) dy dv −(2π)−3 k2  k1 v−1  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  Dylfi+(y) dy dv. Using Minkowski’s inequality, we obtain

(1 + |x|)−σ(wk1(x) − wk2(x)), Lp(R3) ≤ | 1/k_ 2 1/k1 v−1(1 + |x|)−σ  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  ×Dylfi+(y) dy, Lp(R3) dv| +| k2  k1 v−1(1 + |x|)−σ  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  ×Dylfi+(y) dy, Lp(R3) dv| = Ik11,k2 + Ik21,k2.

Consider the first summand. As the functions Qj,i_l (ξ) are homoge-neous in α of degree q = −1/2 < 0 and (1 + |x|)−σ≤ 1, from Young’s inequality we obtain I_k1_1,k2 ≤ | 1/k 2 1/k1 v−1  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  ×Dylfi+(y) dy, Lp(R3) dv| ≤ | 1/k_ 2 1/k1 v−1  R3

eixξG(ξvα)Qj,i_l (ξ) dξ, L1(R3) dv|Dylfi+(y), Lp(R3)

(change of variables: s = ξvα, z = x/vα) =| 1/k 2 1/k1 v−1−qdv|  R3

eiszG(s)Qj,i_l (s) dξ, L1(R3)D_ylf_i+(y), Lp(R3). Recalling that q < 0, we find that

Consider the second summand I_k2_1,k2. Using the inequalities x − y (1 + x )−1≤ a(1 + y ), |x|−σ ≤ ax qσ, q = −1/2 < 0, we have I_k2_1,k2 ≤ | k2  k1 v−1(1 + x )qσ  R3  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ  ×Dylfi+(y) dy, Lp(R3) dv| ≤ c| k2  k1 v−1  R3 x − y qσ|  R3 ei(x−y)ξG(ξvα)Qj,i_l (ξ) dξ| ×(1 + y )−qσ|Dylfi+(y)| dy, Lp(R3) dv|.

As the functions Qj,i_l (ξ) are homogeneous in α of degree q = −1/2 < 0, from Young’s inequality we obtain

I_k2_1,k2 ≤ c| k2  k1 v−1x qσ  R3 eixξG(ξvα)Qj,i_l (ξ) dξ, Lp(R3) dv| ×(1 + y )−qσ_|D ylfi+(y)|, L1(R3) (change of variables: s = ξvα, z = x/vα) = c| k2  k1 v−1−|α|/p−q+qσdv|z qσ  R3 eiszG(s)Qj,i_l (s) dξ, Lp(R3) ×(1 + y )−qσ_|D ylfi+(y)|, L1(R3). Since 1 +|α|/p+ q(1 − σ) > 1 + q(σ − 1) + q(1 − σ) = 1 and the functions f_i+(y) are compactly-supported, we find that

I_k2_1,k2 → 0, k1, k2→ ∞.

The above arguments yield (3.15) and consequently (3.13). Simi-larly, we can prove (3.14).

Lemma 5. Let β = (β1, β2, β3), α = (1/2, 1/2, 1/2), βα < 1, 1 < p < ∞, 1− 3 2p < σ < 3 2p, 1/p + 1/p _{= 1.}

Then the following inequalities hold:

(3.16) x −σ(1−βα)Dβ_xu−,+_k (x), Lp(R3) ≤ c1 3  j=1 |x|1−σDxjfj+(x), Lp(R3) +c2 3  j=1 |x|2(1−σ)_D xjfj+(x), Lp(R3), (3.17) x −σ(1−βα)Dβ_xu−,−_k (x), Lp(R3) ≤ c(|x|1−σf−(x), Lp(R3) + |x|2(1−σ)f−(x), Lp(R3))

with constants c1, c2 > 0 independent of f+(x), f−(x) and k; more-over,

(3.18) (1 + x )−σ(1−βα)(Dβ_xu−_k1(x) − D_xβu−_k2(x)), Lp(R3) → 0, k1, k2 → ∞.

Proof. From the definition of u−,+_k (x) we have Dβ_xu−,+_k (x) = (2π)−3/2 k  1/k v−1  R3

eixξG(ξvα)(iξ)β(−M (iξ)

|ξ|2 )f+(ξ) dξ dv = (2π)−3/2 3  j=1 k  1/k v−1  R3 eixξG(ξvα)−(iξ)β |ξ|2 Dyjfj+(ξ) dξ dv = (2π)−3 3  j=1 k  1/k v−1  R3  R3 ei(x−y)ξG(ξvα)−(iξ) β |ξ|2 dξ)Dyjfj+(y) dy dv = (2π)−3 3  j=1 (  R3 _k 1/k v−1  R3 ei(x−y)ξG(ξvα)−(iξ) β |ξ|2 dξ dv  Dyjfj+(y) dy).

Introduce the notation KQ,k(x) = k  1/k v−1  R3 eixξG(ξvα)Q(ξ) dξ dv,

where Q(ξ) = −(iξ)_|ξ|2β. Then

Dβ_xu−,+_k (x) = (2π)−3 3  j=1  R3

KQ,k(x − y)Dyjfj+(y) dy.

Since the function Q(ξ) is homogeneous in α of degree q = βα−1 < 0, we obtain |α| + q = 3/2 + βα − 1 ≥ 1/2 > 0. Hence, using Lemma 3, we have x −σ(1−βα)D_xβu−,+_k (x), Lp(R3) ≤ c3 j=1 x qσ R3

x−y −|α|−q_y q(1−σ)_y q(σ−1)_|D

yjfj+(y)| dy, Lp(R3) ≤ c3 j=1   R3 3  i=1 |xi|qσ/|α||xi− yi|−1−q/|α|

×|yi|q(1−σ)/|α|y q(σ−1)|Dyjfj+(y)| dy, Lp(R3).

As σ < _2p3 then

−qσ/|α| < −q/p < 1/p. From the conditions of the lemma we find that

σ − 1 > − 3

2p, q(σ − 1)/|α| < −q/p

_{< 1/p}_.

Since

−qσ/|α| < 1/p, q(σ − 1)/|α| < 1/p_{, −q/|α| > 0,}

by the Hardy–Littlewood inequality, we have x −σ(1−βα)_Dβ xu−,+k (x), Lp(R3) ≤ c3 j=1 x (1−σ)(1−βα)_D xjfj+(x), Lp(R3) and the estimate (3.16)

Prove (3.17). From the definition of the vector-functions u−,−_k (x) we obtain D_xβu−,−_k (x) = (2π)−3/2 k  1/k v−1  R3 eixξG(ξvα)(iξ)β |ξ|2 f−(ξ) dξ dv = (2π)−3 k  1/k v−1  R3  R3 ei(x−y)ξG(ξvα)(iξ)β |ξ|2 dξ  f−(y) dy dv = (2π)−3  R3

KQ,k(x − y)f−(y) dy,

where KQ,k(x) = k  1/k v−1  R3 eixξG(ξvα)Q(ξ) dξ dv, Q(ξ) = (iξ)β |ξ|2 .

Since the function Q(ξ) is homogeneous in α of degree q = βα−1 < 0, |α| + q = 3/2 + βα − 1 ≥ 1/2 > 0, then, as above, by Lemma 3 we have the inequality

x −σ(1−βα)D_xβu−,−_k (x), Lp(R3) = (2π)−3x qσ



R3

KQ,k(x − y)f−(y) dy, Lp(R3)

≤ cx qσ R3

x − y −|α|−q_y q(1−σ)_y q(σ−1)_|f−_{(y)| dy, L} p(R3) ≤ c_ R3 3  i=1 |xi|qσ/|α||xi− yi|−1−q/|α|

|yi|q(1−σ)/|α|y q(σ−1)|f−(y)| dy, Lp(R3).

Therefore, by the Hardy–Littlewood inequality, we arrive at (3.17): x −σ(1−βα)Dβ_xu−,−_k (x), Lp(R3) ≤ x q(σ−1)|f−(x)|, Lp(R3)

≤ c(|x|1−σf−(x), Lp(R3) + |x|2(1−σ)f−(x), Lp(R3)). The convergence (3.18) can be established in the same way as (3.10).

Taking into account Lemmas 1, 2, 4 and 5, for the approximate solutions we obtain the following estimates

u−_k(x), W_p,σ2 (R3) ≤ c[ 3  j=1 ( |γ|=1 D_xγf_j+(x), Lp(R3) +(|x|1−σDxjfj+(x), Lp(R3) + |x|2(1−σ)Dxjfj+(x), Lp(R3)) +f−(x), Lp(R3) + |x|1−σf−(x), Lp(R3) +|x|2(1−σ)f−(x), Lp(R3)], 3  j=1 u+ k,j(x), Wp,σ1 (R3) ≤ c[3 j=1 ( |γ|=1 |x|(1−σ)_Dγ xfj+(x), Lp(R3) + Dγxfj+(x), Lp(R3)) +f−(x), Lp(R3) + |x|1−σf−(x), Lp(R3)] for p ∈ (1, ∞) and σ ∈ (1 −_2p3,_2p3 ). Since |x|1−σ ≤ c(1 + |x|)(1−σ)≤ c(1 + |x|)2(1−σ)

and for the Laplace operator the following inequality holds [7]: |x|−2σ_{u, L} p(R3) + |x|−σ∇u, Lp(R3) ≤ c|x|2(1−σ) u, Lp(R3), we have 3  j=1 u+_k,j(x), W_p,σ1 (R3) ≤ c[3 j=1 ( |γ|=1 (1 + |x|)2(1−σ)D_xγf_j+(x), Lp(R3) + f_j+(x), W_p,σ1 (R3)) +f−(x), Lp(R3) + |x|2(1−σ)f−(x), Lp(R3)], u−_k(x), W_p,σ2 (R3) ≤ c[3 j=1 ( |γ|=1 (1 + |x|)2(1−σ)D_xγf_j+(x), Lp(R3) + fj+(x), Wp,σ1 (R3)) +f−(x), Lp(R3) + |x|2(1−σ)f−(x), Lp(R3)].

Hence, we obtain the estimate (2.7). From Lemmas 1, 2, 4, 5 we have also the convergence (2.8) for p ∈ (1, ∞) and σ ∈ (1 −_2p3,_2p3).

The theorem is proved.

Acknowledgment. The author is very grateful to Prof. G. V. Demi-denko for statement of the problem and his assistance.

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