c Pleiades Publishing, Ltd., 2018. ISSN 1061-9208, Russian Journal of Mathematical Physics, Vol. 25, No. 2, 2018, pp. 248–270. 

Entropy Numbers of Embeddings of Function Spaces on Sets with Tree-Like Structure: Some Generalized Limiting Cases A. A. Vasil’eva Steklov Mathematical Institute of Russian Academy of Sciences; E-mail: vasilyeva [email protected] Received February 15, 2018

DOI 10.1134/S1061920818020103 1. INTRODUCTION ˆ r (Ω) → Lq,v (Ω)) of the embedding In [16, 17], order estimates for the entropy numbers en (I : W p,g ˆ r (Ω) on a John domain Ω ⊂ Rd into a weighted Lebesgue operator I of a weighted Sobolev space W p,g space Lq,v (Ω) were obtained (all necessary notation will be given later in this section). Let the weights be defined by g(x) = ϕg (dist(x, Γ)),

v(x) = ϕv (dist(x, Γ)),

where Γ ⊂ ∂Ω is an h-set with h(t) = tθ | log t|γ τ (| log t|),

0 < θ < d,

ϕg (t) = t−βg | log t|−αg ρg (| log t|,

ϕv (t) = t−βv | log t|−αv ρv (| log t|) (τ , ρg and ρv are “slowly varying” functions), 1 < p < q < ∞,

βg + βv = r +

d d − > 0. q p

In [16], estimates of the entropy numbers were obtained for the following cases: (1) (2)

d−θ , q d−θ , βv = q

βv <

0 < αg + αv = αv >

1−γ , q

1 1 − , p q 1 1 < αg + αv = . q p

In [17], the following critical case was considered: αg + αv = p1 − 1q for (1), and αg + αv = p1 for (2); in addition, ρg ≡ 1, ρv ≡ 1. Here we consider this limiting case for a more general situation: d−θ we do not assume that αv > 1−γ q in the case βv = q , and the functions ρg , ρv are not constants. In order to formulate our result, we recall some notation used in [16, 17]. Let Ω ⊂ Rd be a bounded domain, and let g, v : Ω → (0, ∞) be measurable functions. We denote by lr,d the number of components of the vector-valued distribution ∇rf and set    r (Ω) = f : Ω → R ∃ψ : Ω → Rlr,d : ψ Lp (Ω)  1, ∇rf = g · ψ Wp,g  ∇rf  , the corresponding function ψ is denoted by g f Lq,v (Ω) = f q,v = f v Lq (Ω) ,

Lq,v (Ω) = {f : Ω → R| f q,v < ∞} .

For x ∈ Rd and a > 0, we shall denote by Ba (x) the closed Euclidean ball of radius a in Rd centered at the point x. This work is supported by the Russian Science Foundation under grant 14-50-00005.

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Definition 1.1. Let Ω ⊂ Rd be a bounded domain, and let a > 0. We say that Ω ∈ FC(a) if there is a point x∗ ∈ Ω such that, for any x ∈ Ω, there exist a number T (x) > 0 and a curve γx : [0, T (x)] → Ω with the following properties: (1) γx has the natural parametrization, (2) γx (0) = x, γx (T (x)) = x∗ , (3) Bat (γx (t)) ⊂ Ω for any t ∈ [0, T (x)]. We say that Ω satisfies the John condition (and call Ω a John domain) if Ω ∈ FC(a) for some a > 0. Recall the definition of h-sets. Definition 1.2 (see [4]). Let Γ ⊂ Rd be a nonempty compact set and let h : (0, 1] → (0, ∞) be a nondecreasing function. We say that Γ is an h-set if there are a constant c∗  1 and a finite countably additive measure μ on Rd such that supp μ = Γ and c−1 ∗ h(t)  μ(Bt (x))  c∗ h(t)

for any

x ∈ Γ and

t ∈ (0, 1].

Let | · | be a norm on Rd , and let E ⊂ Rd , x ∈ Rd . We set dist|·| (x, E) = inf{|x − y| : y ∈ E}. Let Ω ∈ FC(a) be a bounded domain, and let Γ ⊂ ∂Ω be an h-set. Below we consider a function h which has the following form near zero: h(t) = tθ | log t|γ τ (| log t|), 0 < θ < d,

(1.1)

where τ : (0, +∞) → (0, +∞) is an absolutely continuous function such that tτ  (t) −→ 0. τ (t) t→+∞ Let 1 < p < q < ∞, r ∈ N, δ := r + v(x) = ϕv (dist|·| (x, Γ)),

d q

ϕg (t) = t−βg | log t|−αg ρg (| log t|),

−

d p

(1.2)

> 0, βg , βv ∈ R, g(x) = ϕg (dist|·| (x, Γ)),

ϕv (t) = t−βv | log t|−αv ρv (| log t|),

(1.3)

where ρg and ρv are absolutely continuous functions such that tρg (t) −→ 0, ρg (t) t→+∞

tρv (t) −→ 0. ρv (t) t→+∞

(1.4)

 d Without loss of generality, we may assume that Ω ⊂ − 12 , 12 . Denote by Pr−1 (Rd ) the space of polynomials on Rd of degree not exceeding r − 1. r (Ω) ⊂ Lq,v (Ω) and there exist M > 0 In Theorem 1, the conditions on weights are such that Wp,g r (Ω), and a linear continuous projection P : Lq,v (Ω) → Pr−1 (Ω) such that, for any function f ∈ Wp,g f − P f Lq,v (Ω)

r ∇ f M ; g Lp (Ω)

(1.5)

the theorem will be proved in Section 4. r r (Ω) = span Wp,g (Ω), Denote Wp,g r r ˆ p,g (Ω) = {f − P f : f ∈ Wp,g (Ω)}. W

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VASIL’EVA

r ˆ p,g Let W (Ω) be equipped with the norm

r ∇ f := . g Lp (Ω)

f W ˆr

p,g (Ω)

r ˆ p,g (Ω) → Lq,v (Ω) the embedding operator. From (1.5), it follows that I is continDenote by I : W uous. We set for j ∈ Z+ d d uj = 2j (βg −r+ p ) (j + 1)−αg ρg (j + 1), w j = 2j (βv − q ) (j + 1)−αv ρv (j + 1). (1.6) Suppose that (1.7) βg + βv = δ and there are C  1 and an absolutely continuous monotone function ω : (0, ∞) → (0, ∞) such  (t) → 0 and, for any j0 ∈ Z+ , that tω ω(t) t→+∞

C −1 (j0 + 1)

1 1 q−p

⎛ ⎞1/q ∞ −j 1 1 h(2 ) ⎠ ω(j0 + 1)  sup uj ⎝ wqi  C(j0 + 1) q − p ω(j0 + 1). −i h(2 ) jj0 i=j

(1.8)

Denote Z = (r, d, p, q, g, v, h, a, c∗ , C, ω, | · |), Z∗ = (Z, R), where a and c∗ are the constants from Definitions 1.1 and 1.2, and R = diam Ω. We use the following notation for order inequalities. Let X, Y be sets, and consider two functions f1 , f2 : X × Y → R+ . We write f1 (x, y)  f2 (x, y) (or f2 (x, y)  f1 (x, y)) y

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)  f2 (x, y)

iff1 (x, y)  f2 (x, y)

y

y

and f2 (x, y)  f1 (x, y). y

Theorem 1. Suppose that (1.1)–(1.4), (1.7), and (1.8) hold. Then ˆ r (Ω) → Lq,v (Ω))  n q − p max{ω(n), ω(log n)}. en (I : W p,g 1

1

Z∗

The upper estimate is based on an abstract result for function classes on sets with tree-like structure; here we refine Theorem 1 from [17]. Then we obtain two-sided estimates for norms of some summation operators on trees in spaces with mixed norms; here we apply methods from Irodova’s papers [7, 6]. The lower estimate follows from constructions in [23]. Example. Let 0 < θ < d, h(t) = tθ , ϕg (t) = t−βg | log t|−αg | log | log t||−λg , near zero, βv =

d−θ , q

βg = r −

d θ + , p q

ϕv (t) = t−βv | log t|−αv | log | log t||−λv αv =

1 , q

αg =

1 1 − , p q

λv >

1 . q

We set λ = λg + λv . Then (1.8) holds with ω(t) = log−λ+ q (t + 2). Hence, 1

r ˆ p,g (Ω) → Lq,v (Ω))  n q − p max{log−λ+ q n, log−λ+ q (log n)}. en (I : W 1

1

1

1

Z∗

The problem of estimating the entropy numbers of two-weighted summation operators on trees was investigated in [9, 10, 11] and then in [16, 17]. We also recall that Triebel [15] and Mieth [12, 13] obtained estimates for the entropy numbers r (B) into Lp (B); here B is a ball and of embedding operators of the weighted Sobolev space Wp,g the weight g has singularity at the origin. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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2. AN ABSTRACT RESULT FOR FUNCTION CLASSES ON SETS WITH TREE-LIKE STRUCTURE In this section, we refine Theorem 1 from [17]. We need notation and definitions from that paper (for more details, see [16, 18]). Given a graph G, we denote by V(G) the vertex set of G. Let (T , ξ0 ) be a tree with a distinguished vertex (or a root) ξ0 . For each j ∈ Z+ , ξ ∈ V(T ), we write Vj (ξ) := VTj (ξ) := {ξ   ξ : ρT (ξ, ξ  ) = j} and denote by Tξ = (Tξ , ξ) the subtree in T with vertex set {ξ  ∈ V(T ) : ξ   ξ}. Suppose that the following objects are defined: (1) a tree (A, ξ0 ) such that card VA 1 (ξ)  c1 ,

∃c1  1 :

ξ ∈ V(A),

(2) a measure space (Ω, Σ, mes), ˆ of Ω into measurable subsets, (3) a countable partition Θ ˆ (4) a bijective mapping Fˆ : V(A) → Θ. Let 1 < p, q < ∞ be arbitrary numbers. We suppose that, for any measurable subset E ⊂ Ω, the following spaces are defined: (1) the space Xp (E) with seminorm · Xp (E) , (2) the Banach space Yq (E) with norm · Yq (E) , which all satisfy the following conditions: (1) Xp (Ω) ⊂ Yq (Ω); (2) Xp (E) = {f |E : f ∈ Xp (Ω)}, Yq (E) = {f |E : f ∈ Yq (Ω)}; (3) if mes E = 0, then dim Yq (E) = dim Xp (E) = 0; (4) if E ⊂ Ω, Ej ⊂ Ω (j ∈ N) are measurable subsets, E = j∈N Ej , then ⎛ ⎞1/p f |Ej pXp (Ej ) ⎠ , f ∈ Xp (E), f Xp (E) = ⎝ j∈N

⎛ f Yq (E) = ⎝



⎞1/q f |Ej qYq (Ej ) ⎠

,

f ∈ Yq (E);

j∈N

(5) if E ∈ Σ, f ∈ Yq (Ω), then f · χE ∈ Yq (Ω). In addition, we suppose that there is a subspace P(Ω) ⊂ Xp (Ω) of finite dimension r0 such that P Xp (Ω) = 0 for any P ∈ P(Ω). Given measurable subset E ⊂ Ω, we write P(E) = {P |E : P ∈ P(Ω)}. Then (2.1) f + P Xp (E) = f Xp (E) , f ∈ Xp (Ω), P ∈ P(Ω), E ∈ Σ. Throughout the paper we suppose that 1 < p < q < ∞. For each subtree A ⊂ A, we set ΩA = ∪ξ∈V(A ) Fˆ (ξ). Let P = {Tj }j∈N be a partition of the tree A. Let ξj be the minimal vertex of Tj . We say that the tree Ts succeeds the tree Tj if ξj < ξs and {ξ ∈ T : ξj  ξ < ξs } ⊂ V(Tj ). We suppose that the assumptions from [17] hold with some modifications. Instead of Assumption 1, we write the weaker condition by modifying formula (2) from [17]. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Assumption 2.1. There exist t0 ∈ Z+ and a partition {At,i }tt0 , i∈Jˆt of the tree A and a number c  1 with the following properties. (1) If the tree At ,i succeeds the tree At,i , then t = t + 1. (2) For each vertex ξ∗ ∈ V(A), there exists a linear continuous projection Pξ∗ : Yq (Ω) → P(Ω) such that, for any function f ∈ Xp (Ω) and for any subtree D ⊂ A rooted at ξ∗ , f − Pξ∗ f qYq (ΩD )  c

∞

⎛ q 1− p

2(

)t uq (2t ) ⎝

t=t0

here



∗

f pXp (ΩD

t,i

i∈Jt,D

Jt,D = {i ∈ Jˆt : V(At,i ) ∩ V(D) = ∅}, u∗ : (0, ∞) → (0, ∞)

⎞q/p )

⎠

;

(2.2)

Dt,i = At,i ∩ D,

is a monotone absolutely continuous function,

(2.3)

xu∗ (x) u∗ (x)

= 0. limx→∞ (3) If f ∈ Yq (Ω), f |Fˆ (ξ∗ ) = 0, then Pξ∗ f = 0. Assumption 2.2. There exist numbers δ∗ > 0 and c2  1 such that, for each vertex ξ ∈ V(A) and for any n ∈ N, m ∈ Z+ , there exists a partition Tm,n (G) of the set G = Fˆ (ξ) with the following properties: (1) card Tm,n (G)  c2 · 2m n. (2) For any E ∈ Tm,n (G), there exists a linear continuous operator PE : Yq (Ω) → P(E) such that for any function f ∈ Xp (Ω), 1 1 f − PE f Yq (E)  c2 (2m n)−δ∗ 2( q − p )t u∗ (2t ) f Xp (E) ,

where t  t0 is such that ξ ∈ ∪j∈Jˆt V(At,j ). (3) For any E ∈ Tm,n (G), card {E  ∈ Tm±1,n (G) : mes(E ∩ E  ) > 0}  c2 . Assumption 2.3. There exist numbers γ∗ > 0, c3  1 and an absolutely continuous function  yψ  (y) ψ∗ : (0, ∞) → (0, ∞) such that limy→∞ ψ∗∗(y) = 0 and, for νt := i∈Jˆt card V(At,i ), the following estimate holds: t t νt  c3 · 2γ∗ 2 ψ∗ (22 ) =: c3 ν t , t  t0 . Notice that, for any function f ∈ Xp (Ω) and for any subtree D ⊂ A rooted at ξ∗ ∈ ∪j∈Jˆt V(At,j ) the following inequality holds: (2.2)

1 1 f − Pξ∗ f Yq (ΩD )  c1/q 2( q − p )t u∗ (2t ) f Xp (ΩD ) .

(2.4)

In particular, if V(D) = {ξ∗ }, ξ∗ ∈ V(Γt ), then 1 1 f − Pξ∗ f Yq (Fˆ (ξ∗ ))  c1/q 2( q − p )t u∗ (2t ) f Xp (Fˆ (ξ∗ )) .

(2.5)

Remark 2.1. We obtain that Assumptions 1–3 from [16] hold with λ∗ = μ∗ = p1 − 1q . ˆ p (Ω) = {f − Pξ f : f ∈ Xp (Ω)} equipped with the norm As in [16, 17], we define the space X 0 ˆ p (Ω) → Yq (Ω), the imbedding operator. Then I is continuous. · X (Ω) , and by I : X p

We set Z0 = (p, q, c1 , c2 , c3 , c, r0 , u∗ , δ∗ , γ∗ , ψ∗ ). The following result is the analog of [17, Theorem 1]. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Theorem 2. Let

253

1 < p < q < ∞,

(2.6)

and let Assumptions 2.1–2.3 hold. Then ˆ p (Ω) → Yq (Ω))  n 1q − p1 max{u∗ (n), u∗ (log n)}. en (I : X Z0

As in [16, 18], we introduce some more notation. Let W ⊂ V(T ). We say that G ⊂ T is a maximal subgraph on the set of vertices W if V(G) = W and any two vertices ξ  , ξ  ∈ W adjacent in T are also adjacent in G. We denote (1) ξˆt,i is the minimal vertex of the tree At,i . (2) Γt is the maximal subgraph on the vertex set ∪i∈Jˆt V(At,i ); hence, card V(Γt ) = νt . (3) Gt = ∪ξ∈V(Γt ) Fˆ (ξ) = ∪i∈Jˆt ΩAt,i . ˜ t is the maximal subgraph on the vertex set ∪jt V(Γj ), t ∈ N. (4) Γ ˜ t. (5) {A˜t,i }i∈J t is the set of connected components of the graph Γ ˆ ˜t,i = ∪ (6) U ˜t,i ) F (ξ). ξ∈V(A ˜t,i = ∪ ˆ ˜t = ∪ U (7) U ˜ F (ξ). i∈J t

ξ∈V(Γt )

Then (see [16, formula (26)]) J t = Jˆt , t  t0 . Let G ⊂ Ω be a measurable subset and let T be a partition of G. We set ST (Ω) = {f : Ω → R : f |E ∈ P(E), f |Ω\G = 0}. If T is finite, then ST (Ω) ⊂ Yq (Ω) (see property 5). For any finite partition T = {Ej }nj=1 of the set E and for each function f ∈ Yq (Ω), we put

f p,q,T

⎛ n =⎝ f |Ej p

⎞ p1

Yq (Ej )

⎠ .

(2.7)

j=1

Denote by Yp,q,T (E) the space Yq (E) with the norm · p,q,T . As in [16], we set t∗ (n) = min{t ∈ N : ν t  n}, t∗∗ (n) = min{t ∈ N : ν t  2n }. Then, by [16, formula (50)], 2t∗ (n)  log n, Z0

2t∗∗ (n)  n.

(2.8)

Z0

We also define the operators Qt according to [16, formula (34)]: Qt f (x) = Pξˆt,i f (x) for

˜t,i , x∈U

i ∈ J t,

Qt f (x) = 0 for

˜t . x ∈ Ω\U

(2.9)

ˆ p (Ω). Then Let f ∈ X f=

t∗ (n)−1 j=t0

(Qj+1 f − Qj f )χU˜j+1 +

t∗ (n)−1 j=t0

(f − Qj f )χGj + (f − Qt∗ (n) f )χU˜t

∗ (n)

This can be proved similarly to formula (51) in [16] (we write t∗ (n) instead of t∗∗ (n)). We set t∗ (n)−1 t∗ (n)−1 ˜ (Qj+1 f − Qj f )χU˜j+1 + (f − Qj f )χGj . Pn f = j=t0

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.

(2.10)

(2.11)

254

VASIL’EVA

ˆ p (Ω) → Yq (Ω)) can be estimated as in [16]: applying By Remark 2.1, the entropy numbers en (P˜n : X formula (52), part 2 of Lemma 6 and part 2 of Lemma 7 from [16], we obtain ˆ p (Ω) → Yq (Ω))  n q1 − p1 u∗ (log n). en (P˜n : X

(2.12)

Z0

Notice that, in [16], the condition λ∗ = p1 − 1q holds (see Theorem 5), but Lemmas 6 and 7 are true for λ∗ = p1 − 1q as well. We set ˆ p (Ω); X ˆ pn (Ω) = Rn X ˆ p (Ω). (2.13) Rn f = (f − Qt∗ (n) f )χU˜t (n) , f ∈ X ∗

ˆp (Ω), we set Given f˜ = Rn f , f ∈ X ⎛ f˜ Xˆ n (Ω) := ⎝

⎞1/p



f˜ pX

p

i∈Jˆt∗ (n)

˜

p (Ut∗ (n),i )

⎠

(2.1),(2.9)

=

f Xp (U˜t

∗ (n) )

.

(2.14)

Then Rn Xˆ p (Ω)→Xˆ n (Ω)  1. p

˜:X ˆ n (Ω) → Yq (Ω) by formula We define the operator Q p ˜ | ˆ = Pξ f | ˆ , Qf F (ξ) F (ξ)

ξ ∈ V(A).

(2.15)

ˆ pn (Ω) Then, by (2.13) and assertion (3) of Assumption 2.1, we have for f ∈ X f |Fˆ (ξ) = 0,

Pξ f |Fˆ (ξ) = 0,

˜ | ˆ = 0 for Qf F (ξ)

ξ ∈ V(Γt ), t < t∗ (n).

(2.16)

We have obtained the analog of Lemma 2 from [17]. Lemma 2.1. The following estimate holds: ˜:X ˆ n (Ω) → Yq (Ω))  n 1q − p1 max{u∗ (n), u∗ (log n)}. en (Q p Z0

First we prove some auxiliary assertions. ˜ t (n) ). Then, for any f ∈ X ˆ n (Ω), / V(Γ Proposition 2.1. Let (D, ξ∗ ) be a subtree of A, ξ∗ ∈ p ∗

f − Pξ f qYq (ΩD ) 

Z0

∞

⎛ 2t(

q 1− p

) uq (2t ) ⎝



∗

⎞ pq f pXp (ΩD

t,j

j∈Jt,D

t=t∗ (n)

⎠ .

(2.17)

.

(2.18)

)

This implies that f − Pξ f Yq (ΩD )  (log n) q − p u∗ (log n) f Xˆ p (ΩD ∩U˜t 1

1

Z0

∗ (n) )

Proof. Inequality (2.18) follows from (2.17) by (2.6) and (2.8). Let us prove (2.17). We have f

(2.13)

= (ϕ − Qt∗ (n) ϕ)χU˜t

∗ (n)

,

ˆ p (Ω), ϕ∈X

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Pξ∗ f

255

(2.16)

= 0. Hence, (f − Pξ∗ f )|U˜t

(2.9)

= (ϕ − Pξˆt

∗ (n),i

∗ (n),i

ϕ)|U˜t

∗ (n),i

(f − Pξ∗ f )|Ω\U˜t

,

∗ (n)

= 0.

Therefore,

f − Pξ∗ f qYq (ΩD ) =

ϕ − Pξˆt

i∈Jt∗ (n),D



(2.2)



Z0

∗ (n),i

∞

ϕ qYq (ΩD

t∗ (n),i

⎛



Z0



2t(1− p ) uq∗ (2t ) ⎝ q

i∈Jt∗ (n),D t=t∗ (n)

(2.1),(2.6),(2.19)

)

∞

ϕ pXp (ΩD

t,j

j∈Jt,Dt

⎛ 2t(1− p ) uq∗ (2t ) ⎝ q

⎞ pq



∗ (n),i

⎠

⎞ pq

f pXp (ΩD

t,j

j∈Jt,D

t=t∗ (n)

)

)

⎠ .

This completes the proof. W We need some more notation. Let W ⊂ V(A), ξ0 ∈ W. Given ξ ∈ W, we define the tree D(ξ) with vertex set W ) = {η  ξ : [ξ, η] ∩ W = {ξ}}. (2.20) V(D(ξ)

The partition PW of the tree A is defined by W : ξ ∈ W}. PW = {D(ξ)

(2.21)

Remark 2.2. Let P = {(Tj , ξj )}N j=0 be a partition of A into subtrees. Then P = PW , where N W = {ξj }j=0 . Proposition 2.2. Let W1 , W2 ⊂ V(A), ξ0 ∈ W1 ∩ W2 . Then W1 W2 ∩ D(η) : ξ ∈ W1 , η ∈ W2 }. PW1 ∪W2 = {D(ξ)

(2.22)

Proof. Let ζ ∈ W1 ∪ W2 , ξ = max([ξ0 , ζ] ∩ W1 ),

η = max([ξ0 , ζ] ∩ W2 ).

(2.23)

Then We claim that

ζ = max{ξ, η}.

(2.24)

W1 ∪W2 W1 W2 = D(ξ) ∩ D(η) . D(ζ)

(2.25)

W1 ∪W2 ). Then Indeed, let ξ  ∈ V(D(ζ) (2.23)

[ξ, ζ] ∩ W1 = {ξ},

(2.20)

(2.24)

[ζ, ξ  ] ∩ (W1 ∪ W2 ) = {ζ} = {max(ξ, η)}.

W1 W2 ). Similarly, we obtain ξ  ∈ V(D(η) ). Hence, Therefore, [ξ, ξ  ] ∩ W1 = {ξ} and ξ  ∈ V(D(ξ) W1 ∪W2 W1 W2 ) ⊂ V(D(ξ) ) ∩ V(D(η) ). V(D(ζ)

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VASIL’EVA (2.24)

W1 W2 Let ξ  ∈ V(D(ξ) ) ∩ V(D(η) ). Then ξ   max{ξ, η} = ζ. Since

W1 ∩ [ξ, ξ  ] = {ξ},

W2 ∩ [η, ξ  ] = {η},

we have (W1 ∪ W2 ) ∩ [ζ, ξ  ] = {ζ}. Hence, W1 ∪W2 ) ξ  ∈ V(D(ζ)

W1 ∪W2 W1 W2 and V(D(ζ) ) ⊃ V(D(ξ) ) ∩ V(D(η) ).

This, together with (2.26), implies (2.25). Consequently, W1 W2 ∩ D(η) : ξ ∈ W1 , η ∈ W2 }. PW1 ∪W2 ⊂ {D(ξ)

(2.27)

W1 W2 ) ∩ V(D(η) ) = ∅. Then the vertices ξ and η are comparable. Let Let ξ ∈ W1 , η ∈ W2 , V(D(ξ) ζ = max{ξ, η}. Then (2.23) holds. We apply (2.25), obtain W1 W2 ∩ D(η) : ξ ∈ W1 , η ∈ W2 }. PW1 ∪W2 ⊃ {D(ξ)

This, together with (2.27), implies (2.22). Given t0  t  t∗∗ (n), i ∈ Jˆt , we denote  At,i =

At,i for t < t∗∗ (n), ˜ At∗∗ (n),i for t = t∗∗ (n),

˜ t (n) . Let D be a subtree in A. We set Γt = Γt for t < t∗∗ (n), Γt∗∗ (n) = Γ ∗∗ Jˆt,D = {i ∈ Jˆt : V(At,i ) ∩ V(D) = ∅},

D t,i = At,i ∩ D.

By vmin (D) we denote the minimal vertex of D. Proof of Lemma 2.1. Step 1. As in [17], we denote by Aˆt the subtree in A with vertex set ˜ t+1 ). V(A)\V(Γ ˆ pn (Ω). We define numbers εt  0 and nt ∈ N according to [17, formula (27)]: Let f ∈ B X εt =



f pX

ˆ

p (F (ξ))

nt = n · 2−t εt 

;

if εt > 0;

nt = 1 if

εt = 0.

(2.28)

ξ∈V(Γt )

Then we construct the partitions Tf,t,l (0  l  log nt ) of the tree Aˆt from [17, proof of Lemma 2, Step 1]. They have the following properties: there exists a C(Z0 ) ∈ N such that card Tf,t,l  C(Z0 )2−l nt ,

(2.29)

card {A ∈ Tf,t,l±1 : V(A ) ∩ V(A ) = ∅}  1,

A ∈ Tf,t,l ,

(2.30)

Z0

ξ∈V(A )∩V(Γ

f pX

ˆ

p (F (ξ))

 C(Z0 )2l n−1 t εt ,

A ∈ Tf,t,l ,

card V(A )  2,

(2.31)

t)

Tf,t, log nt = {Aˆt }. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Now we set t

S t = {ξ ∈ V(Aˆt ) : ∃D ∈ Tf,t,0 : ξ = vmin (D)},

(n)−1

∗∗ S = ∪t=t St. ∗ (n)

(2.33)

As in [17, formulas (42) and (43)], we set Tf = PS

(2.34)

˜ f = {D ∈ Tf : card V(D)  2}. T

(2.35)

(see (2.20), (2.21)). Step 2. We set ˜ f , i.e., V(D) = {ξ∗ }. Then Let (D, ξ∗ ) ∈ Tf \T (2.15)

˜ − Pξ f Y (Ω ) = 0. Qf ∗ q D

(2.36)

If ξ ∈ V(Γt ), t < t∗ (n), then, by (2.16), f |Fˆ (ξ) = 0, ˜ f . We have Let (D, ξ∗ ) ∈ T (2.15) ˜ q f − Pξ f qY f − Qf Yq (ΩD ) =

˜ )| ˆ = 0. (Qf F (ξ)

(2.37)

ˆ

q (F (ξ))

ξ∈V(D) ∞

(2.5),(2.37)



Z0

t=t∗ (n)

(2.3),(2.6),(2.8)





q

2t(1− p ) uq∗ (2t )

f qX

ˆ

p (F (ξ))

ξ∈V(D)∩V(Γt )

⎛



t∗∗ (n)

max{u∗ (log n), u∗ (n)}q

Z0

2t(1− p ) ⎝ q



⎞ pq f pXp (Ω

j∈Jˆt,D

t=t∗ (n)

D t,j

)

⎠ . (2.38)

˜ t (n) ), then If ξ∗ ∈ V(Γ ∗ (n) (2.2) t∗∗

f − Pξ∗ f qYq (ΩD ) 

Z0

2t(1− p ) uq∗ (2t ) ⎝ q

⎞ pq



f pXp (Ω

j∈Jˆt,D

t=t∗ (n)

(2.3),(2.8)



⎛

Z0

)

⎠

⎛



t∗∗ (n)

max{u∗ (log n), u∗ (n)}q

D t,j

2t(1− p ) ⎝ q



⎞ pq f pXp (Ω

j∈Jˆt,D

t=t∗ (n)

˜ t (n) ). By (2.3), (2.6), (2.8), and (2.17), we have Let ξ∗ ∈ V(A)\V(Γ ∗ ⎛ t∗∗ (n) q 2t(1− p ) ⎝ f pXp (Ω f − Pξ∗ f qYq (ΩD )  max{u∗ (log n), u∗ (n)}q Z0

j∈Jˆt,D

t=t∗ (n)

D t,j

)

⎠ . (2.39)

⎞q/p D t,j

)

⎠

.

(2.40)

Consequently, f − Pξ∗ f qYq (ΩD ) ˜f (D, ξ∗ )∈T (2.39),(2.40)



Z0



t∗∗ (n)

max{u∗ (log n), u∗ (n)}q

t=t∗ (n)

⎛



q

2t(1− p )

⎝

˜f (D, ξ∗ )∈T

RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

⎞q/p



f pXp (Ω

j∈Jˆt,D

Vol. 25

No. 2

D t,j

)

2018

⎠

,

(2.41)

258

VASIL’EVA



˜ q f − Qf Yq (ΩD )

˜f (D, ξ∗ )∈T



t∗∗ (n)

(2.38)

 max{u∗ (log n), u∗ (n)}q

Z0

⎛



q

2t(1− p )

⎝

˜f (D, ξ∗ )∈T

t=t∗ (n)



⎞q/p f pXp (Ω

j∈Jˆt,D

D t,j

)

⎠

. (2.42)

By (2.8) and the conditions p < q, f Xp (Ω)  1, we have 2t∗∗ (n)(

q 1− p



) max{u (log n), u (n)}q ∗ ∗

⎛ ⎝

˜f (D, ξ∗ )∈T

⎞q/p



f pXp (Ω

j∈Jˆt∗∗ (n),D

Dt

∗∗ (n),j

)

⎠ (2.43)

q

 n1− p max{u∗ (log n), u∗ (n)}q . Z0

˜ f,t the family of trees T ∈ Tf,t,0 such that Denote by T f pX (Fˆ (ξ))  C(Z0 )n−1 t εt .

(2.44)

p

ξ∈V(T )∩V(Γt )

˜ f (see (2.35)), V(D) ∩ V(Γt ) ⊂ V(T ). Then we write Let t∗ (n)  t < t∗∗ (n), T ∈ Tf,t,0 , D ∈ T D ∈ Dt,T . ˜ f , V(D) ∩ V(Γt ) = ∅, then there exists a tree Assertion. If t∗ (n)  t < t∗∗ (n), (D, ξ∗ ) ∈ T ˜ f,t such that D ∈ Dt,T . T ∈T Proof of Assertion.. By assertion (1) of Assumption 2.1, ξ∗ ∈ Aˆt . Then ξ∗ ∈ V(T ), where (2.35)

ˆ ∈ Tf,t,0 . Since card V(D)  2, we get by (2.33) that card V(T )  2. Then (2.44) follows (T , ξ) ˜ f,t . from (2.31). Hence, T ∈ T If D ∈ / Dt,T , then there exists a vertex ξ ∈ V(D) ∩ V(Γt )\V(T ). We set η = min{ζ ∈ [ξ∗ , ξ] : ζ ∈ V(D)\V(T )}. (2.33)

 T , T  ∈ Tf,t,0 . Hence, η ∈ S. On the Then η ∈ Aˆt is the minimal vertex of some tree T  =  ξ∗ , which leads to a contradiction. This other hand, by (2.33)–(2.35), [ξ∗ , ξ] ∩ S = {ξ∗ } and η = completes the proof of the assertion. Thus,

t∗∗ (n)−1

t=t∗ (n)

2t(

q 1− p



)

⎛ ⎝

˜f (D, ξ∗ )∈T



f pXp (Ω

j∈Jˆt,D

t∗∗ (n)−1



⎞q/p



2t(

q 1− p

)

t∗∗ (n)−1



2t(

q 1− p

)

⎠ ⎛



⎝

⎛



(n)−1 (2.29),(2.44) t∗∗



q



⎞q/p f pXp (Ω

j∈Jˆt,D



⎝

˜ f,t T ∈T

t=t∗ (n)

Z0

)

˜ f,t D∈Dt,T T ∈T

t=t∗ (n)





D t,j

D t,j

)

⎠

⎞q/p f pX

ˆ

p (F (ξ))

⎠

ξ∈V(T )∩V(Γt ) q

q 1− p

2t(1− p ) εtp nt

.

t=t∗ (n)

RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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259

We obtain from (2.28) that nt  n · 2−t εt and

t∗∗ (n)−1

q

q

q (2.28) 1− p

2t(1− p ) εtp nt

q

 n1− p .

Z0

t=t∗ (n)

This, together with (2.36), (2.41), (2.42), and (2.43), implies

q

1− p ˜ − Pξ f q Qf max{u∗ (log n), u∗ (n)}q . ∗ Yq (ΩD )  n

(2.45)

Z0

(D, ξ∗ )∈Tf

ˆ n (Ω), we set Step 3. Given f ∈ B X p

Pf h =

Pξ∗ h · χΩD .

(2.46)

(D, ξ∗ )∈Tf

˜ − Pf f Y (Ω)  n q − p max{u∗ (log n), u∗ (n)}. By (2.45), Qf q 1

1

Z0

Step 4. Arguing as in [17, Steps 3–5], we see that it is sufficient to prove that, for any f ∈ ˆ B Xpn (Ω), ˆ n (Ω) → Yq (Ω))  n 1q − p1 max{u∗ (log n), u∗ (n)}. (2.47) en (Pf : X p Z0

Let t∗ (n)  t < t∗∗ (n), 0  l  log nt . We set 

t∗∗ (n)−1

ˆ f,t = V

{vmin (D) : D ∈ Tf,t ,0 },

Wf,t,l = {vmin (D) : D ∈ Tf,t,l },

(2.48)

t =t+1

ˆ f,t ∪ Wf,t,l . Vf,t,l = V

(2.49)

We claim that, for any tree T ∈ PWf,t,l , card {T  ∈ PWf,t,l±1 : V(T ) ∩ V(T  ) = ∅}  1.

(2.50)

Z0

Indeed, let V(T ) ∩ V(T  ) = ∅, T  ∈ PWf,t,l±1 . Notice that T ∩ Aˆt ∈ Tf,t,l , T  ∩ Aˆt ∈ Tf,t,l±1 (see Remark 2.2) and V(T ) ∩ V(T  ) ∩ V(Aˆt ) = ∅. This, together with (2.30), implies (2.50). Let ˆ f,t,l = PV . (2.51) T f,t,l

Notice that ξ0 ∈ Wf,t,l for all t, l. By (2.32), ˆ f,t+1,0 , ˆ f,t, log n = T T t

t∗ (n)  t < t∗∗ (n) − 1.

(2.52)

ˆ f,t,l , Assertion. If 0  l  log nt , 0  l ± 1  log nt , then, for any D ∈ T ˆ f,t,l±1 : V(D) ∩ V(D  ) = ∅}  1. card {D  ∈ T

(2.53)

Z0

Proof of Assertion.. If t = t∗∗ (n) − 1, then Vf,t,l = Wf,t,l , and the assertion follows from (2.50). RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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260

VASIL’EVA

ˆ f,t,l , D  ∈ T ˆ f,t,l±1 , there are Let t < t∗∗ (n) − 1. By Proposition 2.2 and (2.49), for any D ∈ T   trees B ∈ PV ˆ f,t , B ∈ PV ˆ f,t , T ∈ PWf,t,l , and T ∈ PWf,t,l±1 such that V(D  ) = V(B  ) ∩ V(T  ).

V(D) = V(B) ∩ V(T ),

If V(D) ∩ V(D  ) = ∅, then B = B  and V(T ) ∩ V(T  ) = ∅. Hence, ˆ f,t,l±1 : V(D) ∩ V(D  ) = ∅} card {D  ∈ T (2.50)

 card {T  ∈ PWf,t,l±1 : V(T ) ∩ V(T  ) = ∅}  1. Z0

This completes the proof of (2.53). ˆ pn (Ω) we set For h ∈ X



ˆ f,t,l h = P

Pξ h · χΩD .

(2.54)

ˆ f,t,l (D, ξ)∈T (2.32),(2.48),(2.49) ˆ f,t (n)−1, log n ˆ f,t (n),0 (2.33),(2.34),(2.48),(2.49),(2.51) = Tf , T = {A}, we Since T ∗ ∗∗ t∗∗ (n)−1

obtain (2.16) ˆ f,t (n)−1, log n ˆ f,t (n),0 h (2.46) ˆ pn (Ω). = Pf h, P h = 0, h ∈ X P ∗ ∗∗ t∗∗ (n)−1

Therefore, (2.52)

Pf h =

t∗∗ (n)−2 nt −1





t=t∗ (n)

l=0

ˆ f,t,l h − P ˆ f,t,l+1 h). (P

ˆ f,t,l , D  ∈ T ˆ f,t,l+1 }. Then, for h ∈ X ˆ pn (Ω), Let T∗f,t,l = {D ∩ D  : D ∈ T ˆ f,t,l+1 h ∈ ST∗ ˆ f,t,l h − P P f,t,l

(2.55)

and ˆ f,t,l+1 h p,q,T∗ ˆ f,t,l h − P P f,t,l ⎛ (2.7),(2.54)



Z0

⎛ +⎝

(2.7),(2.53)



⎝



Z0

ˆ f,t,l h ˆ ˆ h − P ˆ f,t,l+1 p,q,Tf,t,l + h − Pf,t,l+1 h p,q,T ⎞1/p ⎛ +⎝

h − Pξ h pYq (ΩD ) ⎠

ˆ f,t,l , ξ∈V(Γ ˜ t (n) ) (D, ξ)∈T ∗



h − Pξ h pYq (ΩD ) ⎠

ˆ f,t,l , ξ ∈V( ˜ t (n) ) (D, ξ)∈T / Γ ∗

⎞1/p ⎛

h − Pξ h pYq (ΩD ) ⎠

⎞1/p





+⎝

⎞1/p

h − Pξ h pYq (ΩD ) ⎠

ˆ f,t,l+1 , ξ ∈V( ˜ t (n) ) (D, ξ)∈T / Γ ∗

ˆ f,t,l+1 , ξ∈V(Γ ˜ t (n) ) (D, ξ)∈T ∗

(2.4),(2.8),(2.14),(2.18)



(log n) q − p u∗ (log n) h Xˆ n (Ω) . 1

1

p

Z0

Therefore, ˆ f,t,l+1 h p,q,T∗  (log n) q − p u∗ (log n), ˆ f,t,l h − P P f,t,l 1

1

ˆ n (Ω). h ∈ BX p

(2.56)

Z0

We claim that

ˆ f,t,l+1 h : h ∈ X ˆ pn (Ω)}  2−l nt . ˆ f,t,l h − P dim{P

(2.57)

Z0

RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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261

ˆ pn (Ω), (D, ξ) ∈ T ˆ f,t,l , (D  , ξ  ) ∈ T ˆ f,t,l+1 , (P ˆ f,t,l h − P ˆ f,t,l+1 h)|Ω ∩Ω  = 0, Indeed, let h ∈ X D D  V(D) ∩ V(D ) = ∅. We shall prove that ξ ∈ Wf,t,l

ξ  ∈ Wf,t,l+1 .

or

(2.58)

It is sufficient to consider t < t∗∗ (n) − 1. If ξ = ξ  , then ˆ f,t,l h|Ω ∩Ω  P D D

(2.54)

= Pξ h|ΩD ∩ΩD = Pξ h|ΩD ∩ΩD

(2.54)

ˆ f,t,l+1 h|Ω ∩Ω  ; = P D D

ˆ f,t,l+1 h)|Ω ∩Ω  = 0. Therefore, ξ = ξ  . By (2.49) and (2.51), we have D ∈ ˆ f,t,l h − P i.e., (P D D    PV ˆ f,t ∪Wf,t,l , D ∈ PV ˆ f,t ∪Wf,t,l+1 . By Proposition 2.2, there exist trees (B, η), (B , η ) ∈ PV ˆ f,t such that V(D) ⊂ V(B), V(D  ) ⊂ V(B  ). Since V(D) ∩ V(D  ) = ∅, η = η  . Then ξ = η or ξ  = η. This implies (2.58). Hence, (2.55)

ˆ f,t,l h − P ˆ f,t,l+1 h : h ∈ X ˆ f,t,l , (D  , ξ  ) ∈ T ˆ f,t,l+1 , ˆ pn (Ω)}  card {(D, D  ) : (D, ξ) ∈ T dim{P Z0

ˆ f,t,l h − P ˆ f,t,l+1 h)|Ω ∩Ω  = 0} ˆ pn (Ω) : (P ∃h ∈ X D D (2.58)

ˆ f,t,l , (D  , ξ  ) ∈ T ˆ f,t,l+1 , ξ ∈ Wf,t,l , V(D) ∩ V(D  ) = ∅}  {(D, D  ) : (D, ξ) ∈ T ˆ f,t,l , (D  , ξ  ) ∈ T ˆ f,t,l+1 , ξ  ∈ Wf,t,l+1 , V(D) ∩ V(D  ) = ∅} + {(D, D  ) : (D, ξ) ∈ T (2.48),(2.53)



(2.29)

card Tf,t,l + card Tf,t,l+1  2−l nt .

Z0

Z0

This completes the proof of (2.57). Now, repeating arguments from [17, Lemma 2, Substep 6.4] and taking into account (2.56) and (2.57), we obtain (2.47). This completes the proof of the lemma. Lemma 2.2. The following inequality holds: ˜:X ˆ n (Ω) → Yq (Ω))  n q − p max{u∗ (n), u∗ (log n)}. en (I − Q p 1

1

Z0

In order to prove this lemma, we obtain analog of formulas (23) and (92) from [17]. The arguments are almost the same, but at the last step, we estimate from above the quantity 

En := en (id : l∞ (2δ∗ m lp (wϕ )) → l1 (lq ) with

ϕ(x) = (log(2 + x)) p − q u−1 ∗ (log(2 + x)). 1

1



Here wϕ (s) = 1 for 0  s  1 and wϕ (s) = ϕ(s) for s > 1; lr (2δ∗ m lp (wϕ )) is the space of sequences  x such that x|lr (2δ∗ m lp (wϕ )) < ∞, ⎛ ⎛ ⎞ pr ⎞ r1 ∞   ⎜ m δ∗ r ⎝ ⎟ 2 |xm ,k wϕ (2−m k)|p ⎠ ⎠ , x = (xm ,k )m ,k∈Z+ x|lr (2δ∗ m lp (wϕ )) := ⎝ m =0

k∈Z+

(appropriately modified if p = ∞ or r = ∞). Denote by Φ0 the class of non decreasing functions ϕ : [1, ∞) → (0, ∞) that satisfy the following condition: there exist c > 0 and α > 0 such that, for any 1  τ  t < ∞,  α 1 + log t ϕ(t) c . ϕ(τ ) 1 + log τ RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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2018

262

VASIL’EVA

Theorem A [8, p. 11]. Let 0 < p < q  ∞, 0 < r, s  ∞, δ > 0, ϕ ∈ Φ0 . If ϕ(t) c ϕ(τ )



1 + log t 1 + log τ

 p1 − 1q

1  τ  t < ∞,

,

then en (id : lr (2δm lp (wϕ )) → ls (lq )) If ϕ(t) c ϕ(τ )



1 + log t 1 + log τ

 p1 − 1q



p,q,r,s,ϕ

1 . ϕ(2n )

1  τ  t < ∞,

,

then

(2.60)

(log n) p − q 1

en (id : lr (2δm lp (wϕ )) → ls (lq ))

(2.59)



p,q,r,s,ϕ

1

ϕ(n)n p − q 1

1

.

If u∗ is increasing, then (2.59) holds and 1 1 1  n q − p u∗ (n). n ϕ(2 ) Z0

En  Z0

If u∗ is decreasing, then (2.60) holds and (log n) p − q 1

En  Z0

ϕ(n)n

(log n) p − q

1

1 1 p−q

1



Z0

(log n)

1 1 p−q

1

= n q − p u∗ (log n). 1

u−1 ∗ (log n)n

1 1 p−q

1

Proof of Theorem 2. Combining (2.10), (2.11), (2.12), Lemmas 2.1 and 2.2, we obtain Theorem 2. 3. SOME DISCRETE INEQUALITIES ON TREES Given f : V(A) → R, 1  p, q, s < ∞, we set ⎛ f ls (A) = ⎝



⎛

⎞1/s |f (ξ)|s ⎠

,

⎛

∞ ⎜ ⎜ f lq (lp (A)) = ⎜ ⎝ ⎝ j=0

ξ∈V(A)



ξ∈VA (ξ0 ) j

⎞ pq ⎞ q1 ⎟ ⎟ |f (ξ)|p ⎠ ⎟ ⎠ .

Let u, w : V(A) → R+ . Given f : V(A) → R, we set Su,w,Af (ξ) = w(ξ)



u(η)f (η),

ξ ∈ V(A).

ηξ

ˆ p,q By Sp,q A,u,w we denote the operator norm of Su,w,A : lp (A) → lq (A), and by SA,u,w we denote the operator norm of Su,w,A : lq (lp (A)) → lq (A). If p  q, then f lp (A)  f lq (lp (A)) and p,q ˆ p,q S A,u,w  SA,u,w .

(3.1)

The two-sided sharp estimates of Sp,q A,u,w were obtained by Arcozzi, Rochberg, and Sawyer [1, 2]. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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ENTROPY NUMBERS OF EMBEDDINGS OF FUNCTION SPACES

Let u(ξ) = uj ,

ξ ∈ VA j (ξ0 ).

w(ξ) = wj ,

263

(3.2)

In addition, we suppose that there exist a number C∗  1 and a function S : Z+ → (0, ∞) satisfying the following conditions: C∗−1

S(j  ) S(j  )  card VA , (ξ)  C  ∗ j −j S(j) S(j) S(0) = 1;

ξ ∈ VA j (ξ0 ),

j   j;

(3.3) (3.4)

there exist R0  R > 1 such that R0 

S(j + 1)  R. S(j)

(3.5)

Theorem 3.1. Suppose that 1 < p < q < ∞ and conditions (3.2), (3.3), (3.4), (3.5) hold. Then ⎛ ˆ p,q S A,u,w

sup uj ⎝



p,q,C∗ ,R,R0 j∈Z+

ij

⎞ 1q S(i) ⎠ wiq S(j)



sup u(ξ) w lq (Aξ ) .

p,q,C∗ ,R,R0 ξ∈V(A)

The following result was proved by Bennett [3]. Theorem B (see [3]). Let 1 < p  q < ∞. ˆ = {w ˆn }n∈Z+ be nonnegative sequences such that (1) Let u ˆ = {ˆ un }n∈Z+ , w Muˆ,wˆ := sup

∞ 

m∈Z+ n=m

w ˆnq

m  q1 



u ˆpn

 1 p

< ∞.

n=0

Let Sp,q u ˆ ,w ˆ be the minimal constant C in the inequality 

⎛ ⎞1/p  q 1/q ∞  n    u ˆk fk  C⎝ |fn |p ⎠ , ˆ w  n 

n=0

k=0

{fn }n∈Z+ ∈ lp .

n∈Z+

Then Sp,q ˆ ,w ˆ. u ˆ ,w ˆ  Mu p,q

ˆ = {w ˆn }n∈Z+ be nonnegative sequences such that (2) Let u ˆ = {ˆ un }n∈Z+ , w Muˆ,wˆ := sup

m 

m∈Z+ n=0

w ˆnq

∞  1q 



u ˆpn

 1 p

< ∞.

n=m

Let Sp,q u ˆ ,w ˆ be the minimal constant C in the inequality 

⎛ ⎞1/p  q 1/q ∞  ∞    u ˆk fk  C⎝ |fn |p ⎠ , ˆn w  

n=0

k=n

{fn }n∈Z+ ∈ lp .

n∈Z+

Then Sp,q ˆ ,w ˆ. u ˆ ,w ˆ  Mu p,q

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Proof of Theorem 3.1. By (3.1), (3.2), (3.3) and [1, Th. 4], it is sufficient to prove that

ˆ p,q S A,u,w

 p,q,C∗ ,R,R0

⎞ 1q ⎛ q S(i) ⎠ . sup uj ⎝ wi S(j) j∈Z+ ij

Denote by {ηj,i }i∈Ij the set of vertices VA j (ξ0 ). Let h : V(A) → R+ , h lp (A) = 1, f (ξ) =



u(η)h(η).

(3.6)

ηξ

We estimate from above the quantity ⎛ ⎞q wq (ξ) ⎝ u(η)h(η)⎠ = wq (ξ)f q (ξ). ηξ

ξ∈V(A)

ξ∈V(A)

Let n ∈ Z+ . Denote by Xn the disjoint union of intervals Δn,i (i ∈ In ) of unit length. Let μn be a measure on Xn such that μn (Δn,i ) = 1 and the restriction of μn on Δn,i is the Lebesgue measure. Then (3.3),(3.4)  S(n). (3.7) μn (Xn ) = card VA n (ξ0 ) C∗

We define the function ϕ : Xn → R by ϕ|Δn,i ≡ f (ηn,i ).

(3.8)

Given 0  k  n, s ∈ Ik , we set Qk,s = ∪ηn,i ηk,s Δn,i . Then (3.3)

μn (Qk,s ) = card {i ∈ In : ηn,i  ηk,s } 

C∗



Let Pk ϕ|Qk,s =

u(η)h(η),

S(n) . S(k)

(3.9)

Ek ϕ = ϕ − Pk ϕ.

(3.10)

η<ηk,s

In addition, we set Pn+1 ϕ := ϕ,

En+1 ϕ := 0.

(3.11)

Let us estimate from above the value

(3.8)

|f (ηn,i )|q = ϕ qLq (Xn ) .

(3.12)

i∈In

To this end, we argue similarly as in [6, Th. 4.1], [7, Subsec. 1.4]. Denote by ϕ∗ : [0, μn (Xn )] → R+ the non increasing rearrangement of the function |ϕ|. Then, for any t ∈ [0, μn (Xn )], there exists a set Yt ⊂ Xn such that μn (Yt ) = t,

∀x ∈ Yt

Then ϕ∗ (t) 

|ϕ(x)|  ϕ∗ (t).

(3.13)

ϕ Lp (Yt ) . [μn (Yt )]1/p

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Let k ∈ Z+ , μn (Xn ) μn (Xn )
(3.15)

We set kn = min{k, n + 1}. Then ϕ Lp (Yt )

(3.10),(3.11)



Ekn ϕ Lp (Yt ) + sup |Pkn ϕ(x)|[μn (Yt )]1/p . x∈Xn

This, together with (3.14), yields ϕ∗ (t)  Ekn ϕ Lp (Yt ) [μn (Yt )]−1/p + sup |Pkn ϕ(x)|. x∈Xn

From (3.5), (3.13), and (3.15), we obtain ϕ∗ (t)

Ekn ϕ Lp (Yt ) [μn (Xn )]−1/p [S(k)]1/p + sup |Pkn ϕ(x)|.



(3.16)

x∈Xn

p,R,R0

Let kn  1 (otherwise, Pkn ϕ = 0), and let x ∈ Qkn −1,s . For any 0  j  kn − 1, there exists the unique lj ∈ Ij such that ηj,lj  ηkn −1,s . Then (Pj+1 ϕ − Pj ϕ)|Qj,lj

(3.8),(3.10),(3.11)

=

const,

0  j  kn − 1.

(3.17)

Observe that P0 ϕ = 0. Hence, |Pkn ϕ(x)| 

k n −1

|Pj+1 ϕ(x) − Pj ϕ(x)| 

j=0 (3.17)

=

k n −1

Pj+1 ϕ − Pj ϕ C(Qj,lj )

j=0

k n −1

Pj+1 ϕ − Pj ϕ Lp (Qj,lj ) [μn (Qj,lj )]−1/p

j=0

1/p  n −1 (3.9),(3.10),(3.11) k S(j) 

p,C∗

S(n)

j=0

1/p kn  (3.5) S(j) 

S(n)

p,R,R0 j=0

( Ej ϕ Lp (Qj,lj ) + Ej+1 ϕ Lp (Qj,lj ) ) (3.11)

min{k,n} 

Ej ϕ Lp (Xn ) =

j=0

S(j) S(n)

1/p Ej ϕ Lp (Xn ) .

This, together with (3.7), (3.11), and (3.16), implies that ∗

ϕ (t)

min{k, n} 



 p,C∗ ,R,R0

j=0

S(j) S(n)

1/p Ej ϕ Lp (Xn )

if

μn (Xn ) μn (Xn )
(3.18)

Therefore, 

 |ϕ(x)|q dx = Xn

μn (Xn )

|ϕ∗ (t)|q dt

0 (3.4),(3.5),(3.7),(3.18)



p,q,C∗ ,R,R0

⎛ ⎞q 1/p k  S(j) ⎝ Ej ϕ Lp (Xn ) ⎠ S(k) j=0 S(n)

n S(n) k=0

=: A. (3.19) RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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We claim that

 −1 n  S(j) p q



A

S(n)

p,q,C∗ ,R,R0 j=0

 Indeed, let ψj = ⎛ sup ⎝ 0kn

S(j) S(n)

 p1 − 1q 

S(n)

We have

⎞ 1 ⎛ ⎞ q1 q n S(n) ⎠ ⎠ ⎝ S(j)



 −1 n  S(j) p q



p,q,R,R0 ,C∗ j=0 n 



p,q,R,R0 j=0

S(n)

S(j) S(n)

sup

p,q,R,R0 0kn

j=k

From (3.19) and (3.20), we derive ϕ qLq (Xn )



(3.5)

n (3.10),(3.11) = (Pi+1 ϕ − Pi ϕ) i=j

Ej ϕ Lp (Xn )

(3.20)

Ej ϕ Lp (Xn ) . Then (3.20) follows from Theorem B and the estimate

q /q k  S(j) j=0

Ej ϕ qLp (Xn ) .



S(k) S(n)

n

 1q  1 S(n) q · = 1. S(k)

Pi+1 ϕ − Pi ϕ Lp (Xn ) .

i=j

Lp (Xn )

⎛ ⎞q n ⎝ Pi+1 ϕ − Pi ϕ Lp (Xn ) ⎠ i=j

 pq −1

(3.21)

Pj+1 ϕ − Pj ϕ qLp (Xn ) .

The last inequality follows from Theorem B, since ⎞1/q  j  1/q ⎛ n  q  1 1 S(i)  p −1 S(n) q ( p − q ) ⎠ ⎝ sup S(n) S(i) 0jn i=0 i=j Further, Pj+1 ϕ − Pj ϕ pLp (Xn )

(3.6),(3.8),(3.10),(3.11)

=

 s∈Ij

(3.2),(3.9)





p,q,R,R0 s∈I j

upj |h(ηj,s )|p

(3.5)



1.

p,q,R,R0

p       u(η)h(η) − u(η)h(η) dμn  Qj,s ηη  η<ηj,s j,s

S(n) p S(n) = u |h(ηj,s )|p . S(j) S(j) j s∈Ij

This, together with (3.21), implies ϕ qLq (Xn )



n S(n)

C∗ ,p,q,R,R0 k=0

S(k)

 uqk



 pq |h(ηk,s )|p

.

s∈Ik

Hence, ∞

wnq

n=0

=



|f (ηn,i )|

k=0





∞

p,q,R,R0 ,C∗ n=0

i∈In ∞

(3.12)

q



 pq

|h(ηk,s )|p

s∈Ik

uqk

∞ n=k

wnq

n S(n) k=0

S(k) 

S(n)  wnq S(k)

 uqk



 pq |h(ηk,s )|

p

s∈Ik

sup uqk

0k<∞

∞ n=k

 ∞   pq S(n) wnq |h(ηk,s )|p . S(k) k=0

s∈Ik

This completes the proof. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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4. ENTROPY NUMBERS FOR EMBEDDINGS OF WEIGHTED SOBOLEV SPACES ON JOHN DOMAINS Proof of Theorem 1. Let us prove the upper estimate. Consider the tree A with vertex set {ηj,i }jjmin ,i∈I˜j and the partition of the domain Ω into subdomains Ω[ηj,i ], as defined in [20, 21]. Let the number s = s(a, d)  4 be as defined in [20]. We have {ηj,i }i∈I˜j = VA j−jmin (ηjmin ,1 ),

card VA j  −j (ηj,i ) 

a,d,c0

h(2−sj ) , h(2−sj  )

j  j

(4.1)

(see [21, formula (5.4)]). For t ∈ Z+ , we denote by Γt the maximal subgraph in A on the vertex set {ηj,l : 2t−1 < sj  2t , l ∈ I˜j } for t ∈ N, {ηj,l : 0  sj  1, l ∈ I˜j } for t = 0,

(4.2)

and by {At,i }i∈Jˆt , the set of connected components of Γt . Let t0 = min{t ∈ Z+ : V(Γt ) = ∅}. By (1.1) and (4.1), t (4.3) card V(Γt )  2θ·2 2−γt τ −1 (2t ). Z0 r (Ω), Yq (Ω) = Lq,v (Ω). We set Fˆ (ηj,i ) = Ω[ηj,i ], Xp (Ω) = Wp,g Let (1.3)

u(ηj,i ) = ϕg (2−sj ) · 2−(r− p )sj = 2sj (βg −r+ p ) (sj + 1)−αg ρg (sj + 1), d

w(ηj,i ) = ϕv (2−sj ) · 2−

dsj q

(1.3)

d

(4.4)

= 2sj (βv − q ) (sj + 1)−αv ρv (sj + 1). d

(4.5)

Repeating the arguments from the proof of Theorem 2 in [21] (without summation over vertices ξ ∈ V(Aξ∗ )), we see that, for any vertex ξ∗ ∈ V(A), there exists a linear continuous projection Pξ∗ : Yq (Ω) → P(Ω) such that, for any vertex ξ  ξ∗ and for any function f ∈ Xp (Ω), f − Pξ∗ f Yq (Fˆ (ξ))  w(ξ) Z



u(ξ  ) f Xp (Fˆ (ξ )) .

(4.6)

ξ∗ ξ  ξ

r (Ω), f |Fˆ (ξ∗ ) = 0, then by (4.6), Notice that if f ∈ Wp,g

Pξ∗ f Yq (Fˆ (ξ∗ ))  w(ξ∗ )u(ξ∗ ) f Xp (Fˆ (ξ∗ )) = 0. Z

Hence, Pξ∗ f = 0 and we conclude that assertion (3) of Assumption 2.1 holds. Assumption 2.3 follows from (4.3), and the assertion (1) of Assumption 2.1 follows from (4.1) and (4.2). Assumption 2.2 holds with δ∗ = dδ (see [18, Lemma 8]). It remains to prove assertion (2) of Assumption 2.1 with u∗ = ω (see (1.8)). Notice that it implies (1.5). By (4.6), it is sufficient to prove that, for any subtree (D, ξ∗ ) of A and for any f ∈ Xp (Ω), ξ∈V(D)

⎛



wq (ξ) ⎝ ξ∗

ξ  ξ

⎞q u(ξ  ) f Xp (Fˆ (ξ )) ⎠ 

∞

⎛ q 2(1− p )t ω q (2t ) ⎝

f pXp (ΩD

t,j

Z t=t 0

RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS



⎞ pq

i∈Jt,D

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)

⎠ .

268

VASIL’EVA

To this end, we prove that, for any function ϕ : V(A) → R+ , ⎛



⎞q



wq (ξ) ⎝

u ˜(ξ  )ϕ(ξ  )⎠ 

⎛



⎝

Z k=0

ξ∗ ξ  ξ

ξ∈V(D)

∞

⎞ pq ϕp (ξ)⎠ ,

ξ∈VD (ξ∗ ) k

where 1 1 ˜j = uj · 2( p − q )t ω −1 (2t ), u ˜(ηj,i ) := u

ηj,i ∈ V(Γt ).

(4.7)

Hence, it remains to obtain the inequality ˆ p,q S D,˜ u,w  1.

(4.8)

Z

ˆ p,q ˆ p,q Since S D,˜ u,w  SAξ∗ ,˜ u,w , by B. Levi’s theorem, it is sufficient to prove (4.8) for D = Aξ∗ ,N , where N is an arbitrary natural number and Aξ∗ ,N is the subtree of Aξ∗ with the vertex set V(Aξ∗ ,N ) = ∪0kN VA k (ξ∗ ). ˆ ζ0,1 ) with the following Repeating arguments from [22, Lemma 2], we construct the tree (A, properties: (1) for any j   j   0, ζ ∈ VA j  (ζ0,1 ), ˆ



ˆ card VA j  −j  (ζ)



a,d,c0

h(2−sj ) , h(2−sj  ) ˆ

A (2) for any N ∈ N, j0  jmin , ξ∗ ∈ VA j0 −jmin (ηjmin ,1 ), there exists a vertex ζ∗ ∈ Vj0 (ζ0,1 ) such that ˆ p,q ˆ p,q  S , S u,w Aξ∗ ,N ,˜ ˆ A ,ˆ u, w ˆ ζ∗ ,N

a,d,c0 ,p,q

ˆ = wj for ζ ∈ VA where u ˆ(ζ) = u ˜j , w(ζ) j (ζ0,1 ). ˆ

ˆ p,q Now we estimate from above the quantity S ˆ A

u,w ˆ ζ∗ ,N ,ˆ

. Since θ > 0, there exist a function S :

Z+ → (0, ∞) and numbers R0 (Z)  R(Z) > 1 such that 1  S(j), h(2−sj ) Z

S(0) = 1,

R0 

S(j + 1)  R. S(j)

By Theorem 3.1, ⎛ ˆ p,q S ˆ A

u, w ˆ ζ∗ ,ˆ

 sup u ˆj ⎝ Z jj0

ij

⎞1/q w ˆiq

S(i) ⎠ S(j)

⎛  sup u ˜j ⎝ Z jj0



wiq

ij

−sj

⎞1/q

h(2 ) ⎠ h(2−si )

=: M.

Let 2t  sj0 < 2t+1 . Then by (1.6), (1.8), (4.4), (4.5), (4.7), we obtain 1 1 1 1 M 2( q − p )t ω(2t ) · 2( p − q )t ω −1 (2t ) = 1.

Z

This completes the proof of the upper estimate. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Let us prove the lower estimate. To this end, we argue as in [16, 23]. Let β = βg +βv , α = αg +αv . We claim that there are numbers k∗∗ = k∗∗ (Z∗ ) and functions {ψt,j }j∈Jt ∈ C ∞ (Rd ) such that card Jt  2θk∗∗ t (k∗∗ t)−γ τ −1 (k∗∗ t), Z∗ r ∇ ψt,j = 1, g Lp (Ω)



ψt,j Lq,v (Ω)  sup uk∗∗ l Z∗ lt

∞

h(2−k∗∗ l ) w qk∗∗ i −k∗∗ i h(2

i=l

)

1/q .

(4.9)

The last inequality can be obtained as follows. If βv <

d−θ , q

(4.10)

then inequality (144) from [16] holds: ψt,j Lq,v (Ω)  2k∗∗ t(β−δ) (k∗∗ t)−α ρ(k∗∗ t) = (k∗∗ t)−α ρ(k∗∗ t) Z∗

(β = δ by the conditions of theorem). Notice that  t

1 1 q−p

(1.8)

ω(t)  sup ul Z

Hence, α =

lt

∞ i=l

wqi

h(2−l ) h(2−i )

1/q (1.1),(1.6),(4.10)

 Z

sup ul w l = sup l−α ρ(l). lt

lt

1 1 p − q , ρ = ω. d−θ q . We construct

the functions ψt,j as in [23, Proposition 1], but instead of the lower Let βv = estimate (31) from [23], we obtain (4.9). The arguments are almost the same, but we notice that the quantity

ˆ

ˆ

ˆ −αv q ρq (m + kl) ˆ · 2−md−(d−θ)kl 2βv q(m+kl) (m + kl) v

l∈Z+

mγ τ (m) ˆ γ τ (m + kl) ˆ (m + kl)

from [23, p. 137, line 4 from above] equals to l∈Z+

w qm+kl ˆ

h(2−m ) ˆ h(2−m−kl )

by (1.1), (1.6). Then we apply [16, Lemma 12], argue as in [16, proof after formula (150)] and obtain the desired estimate. This completes the proof. REFERENCES 1. N. Arcozzi, R. Rochberg, and E. Sawyer, “Carleson Measures for Analytic Besov Spaces,” Rev. Mat. Iberoamericana 18, 443–510 (2002). 2. N. Arcozzi,“Carleson Measures for Analytic Besov Spaces: the Upper Triangle Case,” J. Inequal. Pure Appl. Math. 6 (1), article 13 (2005). 3. G. Bennett, “Some Elementary Inequalities. III,” Quart. J. Math. Oxford Ser. 42 (2), 149–174 (1991). 4. M. Bricchi, “Existence and Properties of h-Sets,” Georgian Math. J. 9 (1), 13–32 (2002). RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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5. D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators (Cambridge University Press, Cambridge Tracts in Mathematics 120, 1996). 6. I. P. Irodova, “Dyadic Besov Spaces,” St. Petersburg Math. J. 12 (3), 379–405 (2001). 7. I. P. Irodova, “Piecewise Polynomial Approximation Methods in the Theory Nikol’skii–Besov Spaces,” J. Math. Sci. 209 (3), 319–480 (2015). 8. Th. K¨ uhn, H.-G. Leopold, W. Sickel, and L. Skrzypczak, “Entropy Numbers of Embeddings of Weighted Besov Spaces III. Weights of Logarithmic Type,” Math. Z. 255 (1), 1–15 (2007). 9. M. A. Lifshits, “Bounds for Entropy Numbers for Some Critical Operators,” Trans. Amer. Math. Soc. 364 (4), 1797-1813 (2012). 10. M.A. Lifshits and W. Linde, “Compactness Properties of Weighted Summation Operators on Trees,” Studia Math. 202 (1), 17–47 (2011). 11. M.A. Lifshits and W. Linde, “Compactness Properties of Weighted Summation Operators on Trees — the Critical Case,” Studia Math. 206 (1), 75–96 (2011). 12. T. Mieth, “Entropy and Approximation Numbers of Embeddings of Weighted Sobolev Spaces,” J. Approx. Theory 192, 250–272 (2015). 13. T. Mieth, “Entropy and Approximation Numbers of Weighted Sobolev Spaces via Bracketing,” J. Funct. Anal. 270 (11), 4322–4339 (2016). 14. C. Sch¨ utt, “Entropy Numbers of Diagonal Operators between Symmetric Banach Spaces,” J. Approx. Theory 40, 121–128 (1984). 15. H. Triebel, “Entropy and Approximation Numbers of Limiting Embeddings, an Approach via Hardy Inequalities and Quadratic Forms,” J. Approx. Theory 164 (1), 31–46 (2012). 16. A. A. Vasil’eva, “Entropy Numbers of Embedding Operators of Weighted Sobolev Spaces with Weights That Are Functions of Distance from Some h-Set,” Izv. Math. 81 (6), 1095–1142 (2017). 17. A. A. Vasil’eva, “Estimates for the Entropy Numbers of Embedding Operators of Function Spaces on Sets with Tree-Like Structure: Some Limiting Cases,” J. Complexity, 36, 74–105 (2016). 18. A. A. Vasil’eva, “Widths of Function Classes on Sets with Tree-Like Structure,” J. Approx. Theory 192, 19–59 (2015). 19. A. A. Vasil’eva, “Widths of Weighted Sobolev Classes on a John Domain,” Proc. Steklov Inst. Math. 280, 91–119 (2013). 20. A. A. Vasil’eva, “Embedding Theorem for Weighted Sobolev Classes on a John Domain with Weights That Are Functions of the Distance to Some h-Set,” Russ. J. Math. Phys. 20 (3), 360–373 (2013). 21. A. A. Vasil’eva, “Embedding Theorem for Weighted Sobolev Classes on a John Domain with Weights That Are Functions of the Distance to Some h-Set,” Russ. J. Math. Phys. 21 (1), 112–122 (2014). 22. A. A. Vasil’eva, “Some Sufficient Conditions for Embedding a Weighted Sobolev Class on a John Domain,” Sib. Math. J. 56 (1), 54–67 (2015). 23. A. A. Vasil’eva, “Widths of Weighted Sobolev Classes with Weights That Are Functions of the Distance to Some h-Set: Some Limit Cases,” Russ. J. Math. Phys. 22 (1), 127–140 (2015).

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