Short communications - Eurasian mathematical journal

EURASIAN MATHEMATICAL JOURNAL ISSN 2077-9879

Volume 6, Number 3 (2015), 93 – 100

EMBEDDINGS AND WIDTHS OF WEIGHTED SOBOLEV CLASSES A.A. Vasil’eva

Communicated by V.D. Stepanov

Key words: weighted Sobolev spaces, John domains, embeddings, widths AMS Mathematics Subject Classification: 26D10, 41A10.

Abstract. In this paper, embedding theorems for reduced weighted Sobolev classes Wˆ_p,g^r (Ω) in Lebesgue spaces L_q,v(Ω) are obtained. Here weight functions have singu- larity at the origin and v /∈Lq(Ω). For some special weight functions order estimates for Kolmogorov, Gelfand and linear widths are obtained.

Let (X, k · kX) be a normed space, let X^∗ be its dual, and let Ln(X), n ∈ Z⁺, be the family of subspaces of X of dimension at most n. Denote by L(X, Y) the space of continuous linear operators from X to a normed space Y. Also, byrkA denote the dimension of the image of an operator A ∈L(X, Y), and by kAkX→Y, its norm.

By the Kolmogorovn-width of a setM ⊂X in the spaceX, we mean the quantity d_n(M, X) = inf

L∈Ln(X)sup

x∈M

y∈Linfkx−yk_X,

by the linear n-width, the quantity

λ_n(M, X) = inf

A∈L(X, X),rkA≤nsup

x∈M

kx−Axk_X,

and by the Gelfand n-width, the quantity dⁿ(M, X) = inf

x^∗₁, ..., x^∗_n∈X^∗sup{kxk: x∈M, x^∗_j(x) = 0, 1≤j ≤n}.

The problem of estimating the widths of Sobolev classes in the Lebesgue space and finite-dimensional balls was studied in the 1960s–1980s (see, e.g., [3, 4, 5, 12, 13]). At the same time, the first results on estimates of n-widths of weighted Sobolev classes in weighted Lebesgue spaces were obtained [2, 14]. The extensive study of this problem began in the 1990s (for details, see [17]).

In [17] estimates forn-widths of the weighted Sobolev class W_p,g^r (Ω) in the weighted Lebesgue space L_q,v(Ω) were obtained, where Ω ⊂ −¹₂, ¹₂d

was a John domain (see definitions below) and 0∈Ω. The weights were defined by

g(x) =|x|^−βg|ln|x||^−αgρ_g(|ln|x||), v(x) =|x|^−βv|ln|x||^−αvρ_v(|ln|x||); (1)

here

β_g +β_v =r+d q − d

p >0, α_g+α_v >

1 q − 1

p

+

, (2)

ρ_g,ρ_v : (0, ∞)→(0, ∞) were absolutely continuous functions such that

y→∞lim

yρ⁰_g(y)

ρ_g(y) = lim

y→∞

yρ⁰_v(y)

ρ_v(y) = 0 (3)

(the case p=q was considered by Triebel [15] and Mieth [7], [8] as well). In addition, β_v satisfied the condition β_v < ^d_q. For d = 1, in [16] similar results were obtained for β_v ∈R\n

1

q, ¹_q + 1, . . . , ¹_q +r−1o

; here estimates for norms of two-weighted Riemann – Liouville operators [11] and their compositions [6] were applied.

In this paper we present estimates of n-widths for d ≥ 2, β_v ≥ ^d_q, β_v ∈/ nd

q, ^d_q + 1, . . . , ^d_q +r−1o .

LetΩ⊂R^dbe a bounded domain (an open connected set), and letg,v : Ω→(0, ∞) be measurable functions. For each measurable vector-valued function ψ : Ω → R^l, ψ = (ψ_k)1≤k≤l, and for each p∈[1, ∞], we put

kψk_Lp(Ω) = max

1≤k≤l|ψ_k| p =



 Z

Ω

1≤k≤lmax|ψ_k(x)|^pdx





1/p

(modifications in the case p=∞are clear). Let β = (β₁, . . . , β_d)∈Z^d+:= (N∪ {0})^d,

|β|=β₁+. . .+β_d. For any distributionf defined onΩwe write∇^rf =

∂^rf /∂x^β

|β|=r

(here the partial derivatives are taken in the sense of distributions), and denote by l_r,d the number of components of the vector-valued distribution ∇^rf. We set

W_p,g^r (Ω) =

f : Ω→R

∃ψ : Ω →R^lr,d: kψk_Lp(Ω)≤1, ∇^rf =g·ψ

we denote the corresponding function ψ by ∇^rf g

,

kfk_Lq,v(Ω)=kfk_q,v=kf vk_Lq(Ω), L_q,v(Ω) ={f : Ω→R| kfk_q,v <∞}. We call the set W_p,g^r (Ω) a weighted Sobolev class.

Denote by AC[t₀, t₁] the space of absolutely continuous functions on an interval [t0, t1]. For x ∈ R^d and a > 0 we shall denote by Ba(x) the closed Euclidean ball of radius a in R^d centered at the point x.

Definition 1. Let Ω ⊂ R^d be a bounded domain, and let a > 0. We say that Ω ∈ FC(a) if there exists a point x∗ ∈ Ω such that, for any x ∈ Ω, there exist a number T(x)>0 and a curveγ_x : [0, T(x)]→Ωwith the following properties:

1) γ_x ∈AC[0, T(x)],

dγx(t) dt

= 1 a.e., 2) γ_x(0) =x, γ_x(T(x)) =x∗,

3) B_at(γ_x(t))⊂Ωfor any t ∈[0, T(x)].

Definition 2. We say that Ω satisfies the John condition (and callΩ a John domain) if Ω∈FC(a) for somea >0.

In [9, 10], Reshetnyak found an integral representation for smooth functions on a John domain Ω in terms of their rth order derivatives. This integral representation implies that forp > 1,1≤q <∞and ^r_d+¹_q−¹_p ≥0(correspondingly ^r_d+¹_q−¹_p >0) the class W_p^r(Ω) is continuously (correspondingly, compactly) embedded inL_q(Ω); i.e., the conditions of a continuous and compact embedding are the same as for Ω = [0, 1]^d).

Given nonnegative sequences u = (u_j)j≥0, w = (w_j)j≥0, 1 ≤ p, q ≤ ∞, we denote by S^p,q_u,w and S˜^p,q_u,w the minimal constants C in the inequalities

∞

X

j=0

w^q_j

j

X

i=0

u_if_i

!^q!¹_q

≤C

∞

X

j=0

|f_j|^p

!¹_p

, (f_j)^∞_j=0 ∈l_p, f_j ≥0,

and

∞

X

j=0

w^q_j

∞

X

i=j

u_if_i

!q!¹_q

≤C

∞

X

j=0

|f_j|^p

!¹_p

, (f_j)^∞_j=0 ∈l_p, f_j ≥0,

correspondingly. Two-sided sharp estimates for S^p,q_u,w and S˜^p,q_u,w were obtained by Ben- nett [1].

We use the following notation for order inequalities. Let X, Y be sets, and let f1, f2 : X ×Y → R⁺. We write f1(x, y) .

y

f2(x, y) (or f2(x, y) &

y

f1(x, y)) if for any y ∈ Y there exists c(y) > 0 such that f1(x, y) ≤ c(y)f2(x, y) for any x ∈ X;

f₁(x, y)

y f₂(x, y) if f₁(x, y).

y

f₂(x, y) and f₂(x, y).

y

f₁(x, y).

Theorem A. [1]. Let 1< p ≤ ∞, 1 ≤q <∞, and let u={u_n}n∈Z+, w= {w_n}n∈Z+

be nonnegative sequences. We set

Mu,w := sup

m∈Z+

X^∞

n=m

w^q_n

¹_qX^m

n=0

u^p_n⁰ _p¹0

<∞ if 1< p ≤q <∞,

M_u,w :=





∞

X

m=0

X^∞

n=m

w_n^q_p¹X^m

n=0

u^p_n⁰_p¹0

!_p−q^pq w_m^q





1 q−¹

p

<∞

if 1≤q < p≤ ∞, M˜_u,w := sup

m∈Z+

X^m

n=0

w^q_n¹_qX^∞

n=m

u^p_n⁰_p¹0

<∞ if 1< p ≤q <∞,

M˜_u,w :=





∞

X

m=0

X^∞

n=m

u^p_n⁰_q¹0X^m

n=0

w_n^q1_q

!_p−q^pq u^p_m⁰





1 q−¹_p

<∞

if 1≤q < p≤ ∞.

Then S^p,q_u,w

p,qM_u,w, S˜^p,q_u,w

p,q

M˜_u,w.

Leta >0,Ω∈FC(a),0∈Ω. Consider the weight functions

g(x) =ϕ_g(|x|), v(x) =ϕ_v(|x|), (4)

where ϕ_g,ϕ_v : (0, ∞)→(0, ∞). Suppose that there existsc₀ ≥1 such that ϕ_g(t)

ϕ_g(s) ≤c₀, ϕ_v(t)

ϕ_v(s) ≤c₀, t, s ∈[2^−j−1, 2^−j+1], j ∈Z. (5)

Let r∈N, 1< p≤ ∞, 1≤q <∞. We set u= (u_j)^∞_j=0, w= (w_j)^∞_j=0, uj =ϕg(2^−j)·2⁻(^r−dp)^j, wj =ϕv(2^−j)·2^−djq.

Denote byPr−1(R^d)the space of polynomials on R^d of degree not exceeding r−1.

For a measurable set E ⊂R^d we put Pr−1(E) = {f|_E : f ∈ Pr−1(R^d)}.

If S^p,q_u,w < ∞, r + ^d_q − ^d_p > 0, then W_p,g^r (Ω) ⊂ L_q,v(Ω) and there exists a linear continuous projection P :L_q,v(Ω)→ Pr−1(Ω) such that for each function f ∈W_p,g^r (Ω)

kf −P fk_Lq,v(Ω) .

p,q,r,d,a,c0

S^p,q_u,w

∇^rf g

Lp(Ω)

(see [17, proofs of Lemmas 5, 6]). Observe that by Theorem A it follows that in this case

∞

P

j=0

w^q_j < ∞; i.e., v ∈ L_q(Ω) (the last property follows from [17, formula (37), Corollary 1]).

In this paper we consider the case v /∈ L_q(Ω). Since W_p,g^r (Ω) ⊃ Pr−1(Ω), the inclusionW_p,g^r (Ω) ⊂L_q,v(Ω)fails. Therefore, instead ofW_p,g^r (Ω)we consider the reduced Sobolev class Wˆ_p,g^r (Ω).

Definition 3. We denote by Wˆ_p,g^r (Ω) the completion of the set W_p,g^r (Ω)∩ {f ∈C^∞(Ω) : ∃ε >0 : f|_Bε(0) = 0}

under the norm kfk_Wp,g^r _(Ω) :=

∇^rf g

_L

p(Ω).

Proposition 1. Let the functions ϕ_g, ϕ_v satisfy (5), and let

∞

P

j=0

ϕ^q_v(2^−j)·2^−jd = ∞.

Then there exists a John domain Ω such that for any n ∈Z+

d_n( ˆW_p,g^r (Ω), L_q,v(Ω)) =∞.

Therefore, there exists a John domain Ω such that for any finite-dimensional sub- space L⊂L_q,v(Ω) and for any bounded subsetM ⊂L_q,v(Ω) we getWˆ_p,g^r (Ω) 6⊂L+M;

i.e., the class Wˆ_p,g^r (Ω) cannot be imbedded continuously inLq,v(Ω). For this reason we need to impose some more assumptions on domains Ω.

Definition 4. Let b > 0, c ≥ 1. We say that Ω ∈ FC⁰(b, c) if for any x ∈ Ω there exists a curve γ˜_x : [0, T˜(x)]→Ω∪ {0} with the following properties:

1) γ˜_x is absolutely continuous, ^d˜_dt^γx

= 1 a.e., 2) γ˜_x(0) = 0,γ˜_x( ˜T(x)) =x,

3) c⁻¹|x| ≤T˜(x)≤c|x|,

4) B_b·min{t,T˜(x)−t}(˜γ_x(t))⊂Ω for any t∈(0, T˜(x)).

Suppose thatΩ∈FC(a)∩FC⁰({0}; b, c)and the weights g,v : Ω→(0, ∞)satisfy (4), (5). Let 1< p≤ ∞, 1≤q <∞,r ∈N,δ :=r+^d_q − ^d_p >0.

Denote Z= (p, q, r, d, a, b, c, c₀).

For each 0≤k ≤r−1 we set u_(k) ={u_k,j}j∈Z+,w_(k) ={w_k,j}j∈Z+, where u_k,j = 2^−j(^r−k−dp)ϕ_g(2^−j), w_k,j = 2^−j(^k+dq)ϕ_v(2^−j).

Theorem 1. Let S˜^p,q_u

(r−1),w(r−1) < ∞. Then Wˆ_p,g^r (Ω) ⊂ L_q,v(Ω) and for any function f ∈Wˆ_p,g^r (Ω)

kfk_Lq,v(Ω).

Z

S˜^p,q_u,w

∇^rf g

Lp(Ω)

. Theorem 2. Let 0 ≤k ≤ r−2, S˜^p,q_u

(k),w_(k) <∞, S^p,q_u

(k+1),w_(k+1) < ∞. Then Wˆ_p,g^r (Ω) ⊂ L_q,v(Ω) and there exists a linear continuous projection P : L_q,v(Ω) → Pr−1(Ω) such that for any function f ∈Wˆ_p,g^r (Ω)

kf −P fk_Lq,v(Ω) .

Z

max{S˜^p,q_u

(k),w(k),S^p,q_u

(k+1),w(k+1)}

∇^rf g

Lp(Ω)

. LetΩ∈FC(a)∩FC⁰({0}; b, c), Ω⊂ −¹₂, ¹₂d

, R= diam Ω, and let the weightsg and v be defined by (1). We setZ∗ = (p, q, r, d, a, b, c, g, v, R). Denoteα=α_g+α_v, ρ(y) =ρ_g(y)ρ_v(y).

In estimating the Kolmogorov, linear, and Gelfand widths we set, respectively, ϑ_l(M, X) = d_l(M, X) and qˆ = q, ϑ_l(M, X) = λ_l(M, X) and qˆ = min{q, p⁰}, ϑ_l(M, X) =d^l(M, X) and qˆ=p⁰.

Theorem 3. Let r ∈ N, 1< p ≤ ∞, 1 ≤ q < ∞, δ :=r+ ^d_q − ^d_p > 0, Ω ∈ FC(a)∩ FC⁰({0}; b, c). Suppose that (1), (2) and (3) hold,β_v ∈R\n

d

q, ^d_q + 1, . . . , ^d_q +r−1o . 1. Let either p ≥ q or p < q, qˆ≤ 2. Suppose that α 6= ^δ_d. We set θ₁ = ^δ_d, θ₂ = α,

σ₁ = 0, σ₂ = 1. Let j_∗ ∈ {1, 2} be such that θ_j∗ = min{θ₁, θ₂}. Then ϑ_n( ˆW_p,g^r (Ω), L_q,v(Ω))

Z∗

n(¹_q⁻¹_p)₊^−θj∗

ρ(n^σj∗).

2. Let p < q, q >ˆ 2. We set θ1 = ^δ_d + min n1

2 − ¹_qˆ, ¹_p − ¹_qo

, θ2 = ^qδ_2d^ˆ , θ3 = α + minn

1

2 − ¹_qˆ, ¹_p − ¹_qo

, θ₄ = ^qαˆ₂ , σ₁ = σ₂ = 0, σ₃ = 1, σ₄ = ^q₂^ˆ. Suppose that there exists j∗ ∈ {1, 2, 3,4} such that θ_j∗ <minj6=j∗θ_j. Then

ϑn( ˆW_p,g^r (Ω), Lq,v(Ω))

Z∗

n^−θj∗ρ(n^σj∗).

Acknowledgments

This work was supported by the Russian Science Foundation (project 14-11-00443).

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Anastasia Andreevna Vasil’eva Steklov Mathematical Institute Russian Academy of Sciences 8 Gubkina St

Moscow 119991, Russia

E-mail: vasilyeva_nastya@inbox.ru

Received: 17.09.2015