Some new results concerning boundedness and compactness for embeddings between spaces with multiweighted derivatives

DOCTORA L T H E S I S

Department of Mathematics

Some new Results concerning

Boundedness and Compactness

for Embeddings between Spaces with

Multiweighted Derivatives

Zamira Abdikalikova

ISSN: 1402-1544 ISBN 978-91-86233-43-3 Luleå University of Technology 2009

Zamira Abdikalik ov a Some ne w Results concer ning Boundedness and Compactness for Embeddings betw een Spaces with Multiw eighted Der iv ati ves

ISSN: 1402-1544 ISBN 978-91-86233-

XX

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X Se i listan och fyll i siffror där kryssen är

Tryck: Universitetstryckeriet, Luleå

ISSN: 1402-1544 ISBN 978-91-86233-43-3

Luleå www.ltu.se

1 < p ≤ q < ∞

1 < p ≤ q < ∞ 1 ≤ q < p < ∞ Wp,¯α(1, ∞)n q < p p≤ q W_p,¯αn (1, ∞)

 |ν|≤l (−1)|ν|_Dν_(aν(x)Dν_{u(x)) = 0, x ∈ G,}  G  |ν|=l aν(x)[Dν_u(x)]2_{dx, ν ∈ N}n 0, G n Rn N₀n n |ν| ν G aν(x) G m Rm ρ = ρ(x) x∈ G ∂G G p≥ 1 α∈ R u: G → R up:= uLp(G) up,α:= ρα_up. w(r)p,α = wp,α(G)(r) u : G → R fk k ∈ N₀k r G u_w(r) p,α :=  |k|=r f(k)_p,α<∞. α wp,α(r) w(r)p,α≡ w_p,0(r) wp,α(r) α < r−n−m_p ∂G u ∈ w(r)p,α ∂G α u∈ w(r)p,α

G w(r)p,α Ln_p,γ = Ln_p,γ(I) f : I → R I = (0, +∞) I n: fLn_p,γ := xγf(n)p, γ ∈ R 1 ≤ p ≤ ∞ n I = (0, 1) γ <1 −1 p x∈ [0, 1] fj(x) j = 0, 1, . . . , n − 1 γ > n− 1 p f x= 0 (1, +∞) γ < 1 −1 p f f(k) k= 1, 2, . . . , n − 1 x→ +∞ γ > n−1 p f (n−1)_{(∞) =} lim x→+∞f (n−1)_{(x) lim} x→+∞f (i)_{(x) = ∞ i = 0, 1, . . . , n − 2}

Pn−1= a0+ a1x+ . . . + an−1xn−1 lim x→+∞[f(x) − Pn−1(x)] (k)_{= 0, k = 0, 1, . . . , n − 1.} f ∈ Ln_p,γ Pn−1 I (ly)(t) = n  i=0 ai(t)y(i)(t) ai(·) i = 0, 1, . . . , n I l a−1_n (t) = _an1_(t) ai(t) i= 0, 1, . . . , n − 1 I l l I ly = f ly(x) = 0 n− 1 I ly = 0 t→ ∞ Pn−1 n− 1 ly= f x = 1 t df dx = df dt dt dx = −x −2df dt = (−1) 1_t2df dt,

d2f dx2 = d dx(−t 2df dt) = −t 2 d dx( df dt) = −t 2d dt( df dx) = (−1) 2_t2d dtt 2df dt, dnf dxn = (−1) n_t2d dtt 2d dt. . . t 2df dt. f D0_¯αf(t) = tα0_f(t), Di_¯αf(t) = tαi d dtt αi−1 d dt. . . t α1 d dtt α0_f(t), i = 1, 2, . . . , n, ¯α = (α0, α1, . . . , αn) αi∈ R i = 0, 1, . . . , n Di_¯α ¯α f i i= 0, 1, . . . , n W_x(l)_n,2,υ(Rn+) xn ≥ 0 n Rn f2 W_xn,2,υ(l) (Rn +)=  Rn + |f|2_x2υ n dx+ l  k=1  Rn + |Bk xnf| 2_x2υ n dx, Bxn = ∂2 ∂x2_n + 2υ xn ∂ ∂xn, (υ > 0). n= 1 Bxn B1_tf(t) =  d2 dt2 + 2υ t d dt  f(t), B_tkf(t) = Bt[B_tk−1f(t)], k = 1, 2, . . . . αj j = 0, 1, . . . , 2k k D2k_¯α f(t) Bk_tf(t) k= 1 D2_¯αf(t) = tα2 d dtt α1 d dtt α0_f(t) = tα2+α1+α0  d2 dt2 + α1+ 2α0 t d dt + +α0(α1+ α0− 1) t2  f(t).

D2_¯αf(t) = B_t1f(t) α0 = 0 α1 = 2υ α2= −2υ α0 = 2υ − 1 α1= 2 − 2υ α2= −1 k= 2 D4_¯αf(t) = tα4 d dtt α3 d dtt α2 d dtt α1 d dtt α0_f(t) = tα4 d dtt α3 d dtt α1 2+α22 d dtt α1 d dtt α0_{f(t) = t}α4 d dtt α3 d dtt α1 2_B1 tf(t) = tα4+α3+α12  d2 dt2 + α3+ 2α12 t d dt + α12(α3+ α1₂− 1) t2  B_t1f(t) = B2_tf(t), α2= α12+ α22 αi i= 1, 2, 3, 4 α0= 0 α1= 2υ α2= −2υ α3= 2υ α4= −2υ α0= 0 α1= 2υ α2= −1 α3= 2 − 2υ α4 = −1 α0= 2υ − 1 α1= 2 − 2υ α2= −1 α3= 2υ α4 = −2υ α0= 2υ − 1 α1= 2 − 2υ α2= 2υ α3= 2 − 2υ α4= −1 Di_¯α i = 0, 1, . . . , n W_p,¯αn = W_p,¯αn (I) 1 ≤ p < ∞ I = (0, 1) I = (1, +∞) f : I → R fWn p, ¯α:= D n ¯αfp+ n−1  i=0 |Di ¯αf(1)|. W_p,¯αn Ln_p,γ x= 0

(0, 1] [1, +∞) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u0(t) = tα0 u1(t) = tα0 1 t t α1 1 dt1 u2(t) = tα0 1 t t α1 1 1 t1 tα2 2 dt2dt1 . . . un(t) = tα0 1 t tα1 1 1 t1 tα2 2 . . . 1 tn−1 tαn n dtndtn−1. . . dt1 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ v0(t) = tβ0 v1(t) = tβ0 t 1 t β1 1 dt1 v2(t) = tβ0 t 1 tβ1 1 t1 1 tβ2 2 dt2dt1 . . . vn(t) = tβ0 t 1 t β1 1 t1 1 t β2 2 . . . tn−1 1 t βn n dtndtn−1. . . dt1. {ui(·)}n

i=0 {vi(·)}ni=0 {vi(·)}n

i=0

Pn(·) Pn(·) =

n 

i=0ciui(·) {ui(·)} n

Φ[a, b] u(·) ∈ Φ[a, b] P_n0(·) u − P_n0 = inf Pn u − Pn u(·) {ui(·)}n i=0 [a, b] {ui(·)}n

i=0 {vi(·)}ni=0

(n + 1) lim t→0Diu(t) = Diu(0), i = 0, 1, . . . , n, lim t→+∞Diu(t) = Diu(∞), i = 0, 1, . . . , n, D0u(t) = u(t) tα0 , Diu(t) = 1 tαi d dtDi−1u(t), i = 1, 2, . . . , n. {ui(·)}n

i=0 {vi(·)}ni=0

α β Dn+1u(t) = 0. Lp(I) I = (0, 1) I = (1, +∞) uLp := ⎛ ⎝ I |u(t)|p_dt ⎞ ⎠ 1 p , 1 ≤ p < ∞.

Wp,¯α(I) → Wn _{q, ¯}m_β(I), 1 ≤ p, q < ∞ 0 ≤ m < n I = (0, 1) I = (1, ∞) ¯β = (β0, β1, . . . , βm) βi ∈ R i = 0, 1, . . . , m Wp,¯α(I)n 1 < q < p < ∞ 1 < p ≤ q < ∞ Wp,¯α(0, 1)n Wp,¯α(1, +∞)n x= 1_t Wp,¯α(0, 1)n W_p,¯αn (1, +∞) 1 ≤ q < p < ∞ 1 < p ≤ q < ∞ Wp,¯α(I)n ¯α ¯β p q 1 ≤ q < p < ∞

(a, b) 0 ≤ a < b ≤ +∞ 1 ≤ p ≤ q < ∞ v w ⎛ ⎝ b  a w(x)    x  a f(t)dt    q dx ⎞ ⎠ 1 q ≤ Hl ⎛ ⎝ b  a v(t)|f(t)|pdt ⎞ ⎠ 1 p Bl= sup a≤x≤b ⎛ ⎝ b  x w(t)dt ⎞ ⎠ 1 q ⎛ ⎝ x  a v−p(t)dt ⎞ ⎠ 1 p <∞. Hl Bl ≤ Hl ≤ (1 + q p) 1 q_{(1 +}p q) 1 p_B l 1 ≤ q < p < ∞ 1_r = 1_q − 1_p v w Al= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ b  a ⎡ ⎢ ⎢ ⎣ ⎛ ⎝ b  x w(t)dt ⎞ ⎠ 1 q ⎛ ⎝ x  a v1−p(t)dt ⎞ ⎠ 1 q ⎤ ⎥ ⎥ ⎦ r v1−p(x)dx ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ t1 r <∞.

Hl q 1 q pq r 1 q Al≤ Hl≤ q1q_(p₎q1_Al 1 ≤ p ≤ q < ∞ v w ⎛ ⎝ b  a w(x)    b  x f(t)dt    q dx ⎞ ⎠ 1 q ≤ Hr ⎛ ⎝ b  a v(t)|f(t)|pdt ⎞ ⎠ 1 p Br = sup a≤x≤b ⎛ ⎝ x  a w(t)dt ⎞ ⎠ 1 q ⎛ ⎝ b  x v−p(t)dt ⎞ ⎠ 1 p <∞. Hr Br ≤ Hr ≤ (1 + q p) 1 q_{(1 +} p q) 1 p_B r 1 ≤ q < p < ∞ 1 r = 1 q − 1 p v w Ar = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ b  a ⎡ ⎢ ⎢ ⎣ ⎛ ⎝ x  a w(t)dt ⎞ ⎠ 1 q ⎛ ⎝ b  x v1−p(t)dt ⎞ ⎠ 1 q ⎤ ⎥ ⎥ ⎦ r v1−p(x)dx ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ 1 r <∞. Hr q 1 q  pq r 1 q Ar ≤ Hr ≤ q 1 q_(p₎q1_Ar w1q_{(x) = x}μ _vp1_{(x) = x}γ

1 ≤ p ≤ q < ∞ ⎛ ⎝ 1  0 |tμ_{[f(t) − f(1)]|}q_dt ⎞ ⎠ 1 q ≤ H ⎛ ⎝ 1  0  tγdf(t) dt  pdt ⎞ ⎠ 1 p μ >−1 q γ ≤ 1 − 1 p + 1 q + μ 1 ≤ q < p < ∞ ⎛ ⎝ 1  0 |tμ_{[f(t) − f(1)]|}q_dt ⎞ ⎠ 1 q ≤ H ⎛ ⎝ 1  0  tγdf(t) dt  pdt ⎞ ⎠ 1 p μ >−1 q γ <1 − 1 p + 1 q + μ i > j j  k=i c

(0, 1] [1, +∞)

Lp(0, 1) Lp(1, +∞)

αi βi i = 0, 1, . . . , n (0, 1] [1, +∞) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u0(t) = tα0 u1(t) = tα01 t t α1 1 dt1 u2(t) = tα0 1 t t α1 1 1 t1 tα2 2 dt2dt1 . . . un(t) = tα0 1 t t α1 1 1 t1 tα2 2 . . . 1 tn−1 tαn n dtndtn−1. . . dt1 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ v0(t) = tβ0 v1(t) = tβ0 t 1 t β1 1 dt1 v2(t) = tβ0 t 1 t β1 1 t1 1 t β2 2 dt2dt1 . . . vn(t) = tβ0 t 1 t β1 1 t1 1 t β2 2 . . . tn−1 1 t βn n dtndtn−1. . . dt1. t = 1 x {vi(x)}n i=0 (0, 1] {ui(t)}n i=0 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ v0(x) = x−β0 v1(x) = x−β01 x x−(β1+2) 1 dx1 v2(x) = x−β0 1 x x −(β1+2) 1 1 x1 x−(β2+2) 2 dx2dx1 . . . vn(x) = x−β0 1 x x −(β1+2) 1 1 x1 x−(β2+2) 2 . . . 1 xn−1 x−(βn+2) n dxndxn−1. . . dx1. k  j=i = 0 i > k X Y X≤ cY c >0

ui i = 0, 1, . . . , n Lp(0, 1) 1 ≤ p < ∞ min 0≤s≤i(α0+ s  j=1 (αj+ 1)) > −1_p. vi i= 0, 1, . . . , n i0= min k∈Mik Mi= {k : 0 ≤ k ≤ i, α0+ k  j=1 (αj+ 1) = min_0≤s≤i(α0+ s  j=1 (αj+ 1))}. ui i= 1, 2, . . . , n 0 < δ < 1 0 < δ1≤ δ t∈ (0, δ1]

ci(δ)t0≤s≤imin(α0+ s

j=1(αj+1)) ≤ ui(t)

ci(δ) → 0 δ → 1 i = 1, 2, . . . , n ui i= 1, 2, . . . , n t∈ (0, 1]

ui(t) t0≤s≤imin(α0+ s j=1(αj+1))| ln t|li li k i0+1 ≤ k ≤ i k  j=i0+1 (αj+1) = 0 i0 < i li = 0 i0 = i 0 ≤ i0 ≤ i i0 = i 0 < i0< i i0= 0 t∈ (0, 1]

i0= i ui(t) = ui0(t) = tα0 1  t tα1 1 1  t1 tα2 2 . . . 1  t_i0−1 tα_i0i0dti0dti0−1. . . dt1. ui(t) = ui0(t) = tα0 1  t tα_i0i0 t_i0  t tα_i0i0−1₋₁ . . . t2  t tα1 1 dt1dt2. . . dti0. tj = tτj j = 1, 2, . . . , i0 ui(t) = ui0(t) = t α0+ i0 j=1(αj+1) 1/t  1 τ_iα₀i0 τ_i0  1 τ_iα₀i0−1₋₁ . . . τ2  1 τα1 1 dτ1dτ2. . . dτi0. 1/t  1 τ_iα₀i0 τ_i0  1 τ_iα₀i0−1₋₁ . . . τ2  1 τα1 1 dτ1dτ2. . . dτi0 t 0 < t < 1 t∈ (0, δ] 0 < δ < 1 1/t  1 τ_iα₀i0 τ_i0  1 τ_iα₀i0−1₋₁ . . . τ2  1 τα1 1 dτ1dτ2. . . dτi0 ≥ ≥ 1/δ  1 τ_iα₀i0 τ_i0  1 τ_iα₀i0−1₋₁ . . . τ2  1 τα1 1 dτ1dτ2. . . dτi0. c(1)_i (δ) t∈ (0, δ] ui(t) = ui0(t) ≥ c (1) i (δ)t α0+i0 j=1(αj+1)= c(1) i (δ)t min 0≤s≤i(α0+ s j=1(αj+1))_, c(1)_i (δ) → 0 δ→ 1

0 < i0< i 0 < t ≤ δ < 1 ui(t) = tα0 1  t tα1 1 1  t1 tα2 2 . . . 1  t_i0−1 tα_i0i0 1  t_i0 tα_i0i0+1₊₁ . . . 1  ti−1 tαi i dtidti−1. . . dt1 ≥ tα0 δ  t tα1 1 δ  t1 tα2 2 . . . δ  t_i0−1 tα_i0i0dti₀dti₀−1. . . dt1 × 1  δ tα_i0i0+1₊₁ 1  t_i0+1 tα_i0i0+2₊₂ . . . 1  ti−1 tαi i dtidti−1. . . dti0+1. c(2)_i (δ) ui(t) ≥ c(2)_i (δ)tα0 δ  t tα1 1 δ  t1 tα2 2 . . . δ  t_i0−1 tα_i0i0dti0dti0−1. . . dt1, c(2)_i (δ) → 0 δ→ 1 ui(t) ≥ c(2)i (δ)tα0 δ  t tα_i0i0 t_i0  t tα_i0i0−1₋₁ . . . t2  t tα1 1 dt1dt2. . . dti0. tj = tτj j = 1, 2, . . . , i0 ui(t) ≥ c(2)_i (δ)tα0+ i0 j=1(αj+1) δ/t  1 τ_iα₀i0 τ_i0  1 τ_iα₀i0−1₋₁ . . . τ2  1 τα1 1 dτ1dτ2. . . dτi0. δ/t  1 τ_iα₀i0 τ_i0  1 τ_iα₀i0−1₋₁ . . . τ2  1 τα1 1 dτ1dτ2. . . dτi0 t 0 < t ≤ δ t∈ (0,1₂δ] δ/t  1 τ_iα₀i0 τ_i0  1 τ_iα₀i0−1₋₁ . . . τ2  1 τα1 1 dτ1dτ2. . . dτi0 ≥

≥ 2  1 τ_iα₀i0 τ_i0  1 τ_iα₀i0−1₋₁ . . . τ2  1 τα1 1 dτ1dτ2. . . dτi0. c(3)_i t ∈ (0,1 2δ] ui(t) ≥ c(2)_i (δ)c(3)_i tα0+ i0 j=1(αj+1)= c(2) i (δ)c(3)i t min 0≤s≤i(α0+ s j=1(αj+1))_, c(2)_i (δ)c(3)_i → 0 δ → 1 i0 = 0 α0 = αi₀ ui ui(t) = tα_i0 1  t tα_i0i0+1₊₁ 1  t_i0+1 tα_i0i0+2₊₂ . . . 1  ti−1 tαi i dtidti−1. . . dti0+1. t 0 < t ≤ δ < 1 1  t tα_i0i0+1₊₁ 1  t_i0+1 tα_i0i0+2₊₂ . . . 1  ti−1 tαi i dtidti−1. . . dti0+1 ≥ 1  δ tα_i0i0+1₊₁ 1  t_i0+1 tα_i0i0+2₊₂ . . . 1  ti−1 tαi i dtidti−1. . . dti0+1. c(4)_i (δ) t∈ (0, δ] ui(t) ≥ c(4)i (δ)tαi0 = c(4)i (δ)t min 0≤s≤i(α0+ s j=1(αj+1))_, c(4)_i (δ) → 0 δ → 1 i0 0 ≤ i0< i α0+ i0  j=1 (αj+ 1) = min_0≤s≤i(α0+ s  j=1 (αj+ 1)).

i0< s≤ i (α0+ i0  j=1 (αj+ 1)) − (α0+ s  j=1 (αj+ 1)) = − s  j=i0+1 (αj+ 1) ≤ 0, i0>0 0 ≤ s ≤ i0− 1 (α0+ i0  j=1 (αj+ 1)) − (α0+ s  j=1 (αj+ 1)) = i0  j=s+1 (αj+ 1) < 0. i0= i 0 < i0< i i0= 0 t∈ (0, 1] i0 = i li = li0 = 0 ui(t) = ui0(t) = tα0 1  t tα1 1 1  t1 tα2 2 . . . 1  t_i0−1 tα_i0i0dti0dti0−1. . . dt1 ≤ tα0 ∞  t tα1 1 ∞  t1 tα2 2 . . . ∞  t_i0−1 tα_i0i0dti0dti0−1. . . dt1 ≤ c(5)i t α0+i0 j=1(αj+1)= c(5) i t min 0≤s≤i(α0+ s j=1(αj+1))|lnt|li_, c(5)_i =  _i0  k=1    i0  j=k (αj+ 1)_ −1 . 0 < i0< i ui ui(t) = tα0 1  t tα1 1 1  t1 tα2 2 . . . 1  t_i0−1 tα_i0i0 1  t_i0 tα_i0i0+1₊₁ . . . 1  ti−1 tαi i dtidti−1. . . dt1 = tα0 1  t tα1 1 1  t1 tα2 2 . . . 1  t_i0−1 tα_i0i0 × ⎛ ⎜ ⎝ 1  t_i0 tαi i ti  t_i0 tαi−1 i−1 . . . t_i0+2 t_i0 tα_i0i0+1₊₁ dti0+1dti0+2. . . dti ⎞ ⎟ ⎠ dti0dti0−1. . . dt1.

I(i,i0)(ti0) = 1  t_i0 tαi i ti  t_i0 tα_i−1i−1. . . t_i0+2 t_i0

tα_i0i0+1₊₁ dti0+1dti0+2. . . dti.

s= i0+ 1 αi₀+1+ 1 ≥ 0 αi0+1+ 1 = 0 αi0+1+ 1 > 0 αi0+1+ 1 = 0 t≤ ti0 ≤ ti0+2 ≤ 1 t_i0+2 t_i0 tα_i0i0+1₊₁ dti₀+1 = t_i0+2 t_i0 t−1_i0+1dti₀+1≤ 1  t t−1_i0+1dti₀+1 = | ln t|. I(i,i0+1)(ti0) ≤ | ln t| 1  t_i0 tαi i ti  t_i0 tαi−1 i−1 . . . t_i0+3 t_i0 tα_i0i0+2₊₂ +αi0+1+1dti0+2dti0+3. . . dti 0 < t ≤ t0<1 αi0+1+ 1 > 0 t_i0+2 t_i0 tα_i0i0+1₊₁ dti0+1≤ t_i0+2 0 tα_i0i0+1₊₁ dti0+1= 1 αi0+1+ 1 tα_i0i0+1₊₂ +1. I(i,i0+1)(ti0) ≤ 1 αi0+1+ 1 1  t_i0 tαi i ti  t_i0 tαi−1 i−1 . . . t_i0+3 t_i0 tα_i0i0+2₊₂ +αi0+1+1dti0+2dti0+3. . . dti. 0 < t ≤ ti0 ≤ 1 I(i,i0)(ti0) ≤ c (6) i | ln t|li, c(6)_i i= 0, 1, . . . , n αi i= 0, 1, . . . , n

ui(t) ≤ c(6)i | ln t|litα0 1  t tα1 1 1  t1 tα2 2 . . . 1  t_i0−1 tα_i0i0dti0dti0−1. . . dt1= c (6) i | ln t|liui0(t). ui(t) ≤ c(5)i c(6)i t min 0≤s≤i(α0+ s j=1(αj+1))|lnt|li_. i0 = 0 ui ui(t) = tαi0 1  t tαi i ti  t tαi−1 i−1 . . . t2  t tα1 1 dt1dt2. . . dti= t min 0≤s≤i(α0+ s j=1(αj+1))_I(i,0)(t). s  j=1(αj+ 1) ≥ 0 1 ≤ s ≤ i I(i,0)(t) ≤ c(7)_i | ln t|li_, c(7)_i i= 0, 1, . . . , n αi i= 0, 1, . . . , n ui(t) ≤ c(7)i t min 0≤s≤i(α0+ s j=1(αj+1))| ln t|li_. u0 i0= 0 l0 = 0 u0= tα0 = t min 0≤s≤0(α0+ s j=1(αj+1))|lnt|l0_. ui ∈ Lp(0, 1) t0≤s≤imin(α0+ s j=1(αj+1))∈ Lp(0, 1).

min 0≤s≤i(α0+ s  j=1 (αj+ 1)) > −1 p. min 0≤s≤i(α0 + s  j=1(αj + 1)) > − 1 p i δi>0 min 0≤s≤i(α0+ s  j=1 (αj+ 1)) − δi >−1 p.

ui(t) t0≤s≤imin(α0+

s j=1(αj+1))| ln t|li _t0≤s≤imin(α0+ s j=1(αj+1))−δi sup 0≤t≤1t δi_{| ln t|}li_. sup 0≤t≤1t δi| ln t|li _<∞ _l i ≥ 0 ui∈ Lp(0, 1) α0+ m  j=1(αj + 1) = α0+ l  j=1(αj+ 1) m = l m, l = 0, 1, . . . , n ui i= 1, 2, . . . , n 0 < δ < 1

ui(t) ≈ t0≤s≤imin(α0+ s j=1(αj+1))_{, t}∈ (0, δ), vi i = 0, 1, . . . , n Lp(1, +∞) 1 ≤ p < ∞ max 0≤s≤i(β0+ s  j=1 (βj+ 1)) < −1_p.

t = 1 x (0, 1] α0 = −β0, αi = −(βi+ 2), i = 1, 2, . . . , n. min 0≤s≤i(α0+ s  j=1 (αj+ 1)) = min_0≤s≤i(−β0+ s  j=1 (−(βj+ 2) + 1)) = − max_0≤s≤i(β0+ s  j=1 (βj+ 1)). i0 = min k∈Lik Li= {k : 0 ≤ k ≤ i, β0+ k  j=1 (βj+ 1) = max 0≤s≤i(β0+ s  j=1 (βj+ 1))}. k  j=i0+1 (αj + 1) = 0 k j=i0+1 (βj + 1) = 0 k  j=i0+1 (αj+ 1) = k  j=i0+1 (−(βj+ 2) + 1) = − k  j=i0+1 (βj+ 1) = 0. vi i= 1, 2, . . . , n λ >1 λ1 ≥ λ > 1 t∈ [λ1,+∞)

ci(λ)t0≤s≤imax(β0+ s

j=1(βj+1))≤ vi(t)

vi i= 1, 2, . . . , n t∈ [1, +∞)

vi(t) t0≤s≤imax(β0+ s j=1(βj+1))| ln t|li li k i0+1 ≤ k ≤ i k  j=i0+1 (βj+1) = 0 i > i0 li= 0 i= i0 0 < δ < 1 0 < δ1 ≤ δ ci(δ)x− max0≤s≤i(β0+

s j=1(βj+1))≤ vi(x), _x∈ (0, δ1], ˜ci(δ) → 0 δ→ 1 λ = 1 δ λ1 = 1 δ1 vi(x) = vi(t) x = 1 t t∈ [λ1,+∞) ci(_λ1)  1 t − max 0≤s≤i(β0+ s j=1(βj+1)) ≤ vi(t), ˜ci(1 λ) → 0 λ→ 1

ci(λ)t0≤s≤imax(β0+ s

j=1(βj+1)) ≤ vi(t),

ci(λ) ≡ ˜ci(1

λ) → 0 λ→ 1

x∈ (0, 1]

vi(x) x− max0≤s≤i(β0+ s j=1(βj+1))| ln x|li_, li k i0 + 1 ≤ k ≤ i k  j=i0+1 (βj+ 1) = 0

vi(x) = vi(t) x = 1 t t ∈ [1, +∞) vi(t)  1 t − max 0≤s≤i(β0+ s j=1(βj+1)) | ln1 t| li = t0≤s≤imax(β0+ s j=1(βj+1))| ln t|li_. v0 i0= 0 l0= 0 v0= tβ0 _{= t}0≤s≤0max(β0+ s j=1(βj+1))|lnt|l0_. max 0≤s≤i(β0+ s  j=1(βj+ 1)) < − 1 p i γi>0 max 0≤s≤i(β0+ s  j=1 (βj+ 1)) + γi<−1 p.

vi(t) t0≤s≤imax(β0+ s j=1(βj+1))| ln t|li _t0≤s≤imax(β0+ s j=1(βj+1))+γi sup 1≤t≤∞t −γi_{| ln t|}li_. sup 1≤t≤∞t −γi| ln t|li _<∞ _l i ≥ 0 vi ∈ Lp(1, +∞) vi ∈ Lp(1, +∞) t0≤s≤imax(β0+ s j=1(βj+1)) ∈ Lp(1, +∞). max 0≤s≤i(β0+ s  j=1(βj+ 1)) < − 1 p

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ w0(t) = tα0 w1(t) = tα0 1 t t α1 1 dt1 w2(t) = tα0 1 t t α1 1 t1 t t α2 2 dt2dt1 . . . wn(t) = tα0 1 t t α1 1 t1 t t α2 2 . . . tn−1 t t αn n dtndtn−1. . . dt1. w0 w1 u0 u1 wi ui i= 2, 3, . . . , n wi wi(t) = tα0 1  t tαi i 1  ti tαi−1 i−1 . . . 1  t2 tα1 1 dt1dt2. . . dti. wi ui wi p (0, 1) wi i= 1, 2, . . . , n 0 < δ < 1 0 < δ1≤ δ t∈ (0, δ1]

ci(δ)t1≤s≤i+1min (α0+ i

j=s(αj+1))≤ wi(t)

ci(δ) → 0 δ→ 1 i = 1, 2, . . . , n wi i= 1, 2, . . . , n t∈ (0, 1]

wi(t) t1≤s≤i+1min (α0+ i j=s(αj+1))| ln t|li li k 1 ≤ k ≤ i0− 1 i0−1 j=k(αj+ 1) = 0 1 < i0 li = 0 i0 = 1

i0= max k∈Nik Ni= {k : 1 ≤ k ≤ i + 1, α0+ i  j=k (αj+ 1) = min 1≤s≤i+1(α0+ i  j=s (αj+ 1))}. w0 l0= 0 w0= tα0 = t min 1≤s≤0+1(α0+ 0 j=s(αj+1))|lnt|l0_. wi i= 0, 1, . . . , n Lp(0, 1) 1 ≤ p < ∞ min 1≤s≤i+1(α0+ i  j=s (αj+ 1)) > −1_p. {yi(·)}n i=0 i yi(t) = tβ0 t  1 tβ1 1 t  t1 tβ2 2 . . . t  ti−1 tβi i dtidti−1. . . dt1.

Ln_p,γ(I) W_p,¯αn R n γ ∈ R 1 ≤ p ≤ ∞ 1 p + 1 p = 1 I = (a, b) 0 ≤ a < b ≤ ∞ Ln_p,γ = Ln_p,γ(I) f : I → R fLn_p,γ = ⎛ ⎝ I |tγ_f(n)_(t)|p_dt ⎞ ⎠ 1 p , 1 ≤ p < ∞ fLn +∞,γ = ess supt∈I|t γ_f(n)_(t)|,

p= +∞ Ln_p,γ Lp,γ = L0p,γ Lnp = Lnp,0 fp,γ = fLp,γ fp = fp,0 I = [0, +∞) f I → R t0 ∈ [0, +∞) f1 I → R f lim t→t0f1(t) = f1(t0) f t = t0 f(t0) I = [0, +∞) f f1 _t→+∞lim f1(t) f t → ∞ f(+∞) f(+∞) = lim t→+∞f1(t) f1 I = (0, +∞) t0 ∈ I f I → R Pn−1(t) =n−1 υ=0aυt υ lim t→t0[f(t) − Pn−1(t)] (k)_{= 0, k = 0, 1, . . . , n − 1,} f n Pn−1(t) t→ t0 Ln_p,γ(0, 1) Ln_p,γ(1, +∞) 1 − 1 p < γ < n− 1p γ+ 1p n= 1 L1_p,γ(0, 1) n >1 Ln_p,γ(0, 1) γ <1 − 1 p − weak degeneration, 1 −1 p < γ < n− 1 p − mixed case, γ > n−1 p − strong degeneration. γ < 1 − 1 p n = 1 L 1 p,γ(0, 1)

f ∈ L1_p,γ(0, 1) γ < 1 −1 p t0∈ [0, 1] f(t0) f(t) = f(t0) + t t0 f(t)dt 0 ≤ t ≤ 1 |fp− |f(t0)|| ≤ c1(γ, p)f_p,γ ||f(t0)| − |f(0)|| ≤ c2(γ, p)f_p,γ p= 1 (i) (iv) γ = 0 f ∈ Lnp,γ(0, 1) γ < 1 − 1 p t0 ∈ [0, 1] f(j)(t0) j = 0, 1, . . . , n − 1 fp≤ c _n−1  j=0 |f(j)_{(t0)| + f}(n)_p,γ  . Pn−1(t) =n−1 j=0 f(j)(0) j! t j _{n− 1} f [f(t) − Pn−1(t)](n−k)_{p,γ−k ≤ cf}(n)_{p,γ, k= 1, 2, . . . , n.} γ > n−1 p f ∈ L n p,γ γ > n−1_p f t= 0 f ∈ L1_p,γ γ >1 −1 p |fp,γ−1− c|f(1)|| ≤ 1 γ− 1 + 1/pf _p,γ, c= tγ−1p f ∈ Ln_p,γ γ > n− 1 p f(n−k)_{p,γ−k ≤ c  k}  j=1 |f(n−j)_{(1)| + f}(n)_p,γ  , k= 1, 2, . . . , n.

1 − 1 p < γ < n− 1 p, n≥ 2, Ln_p,γ(0, 1) γ γ −1 p ≤ γ < γ + 1 − 1 p. 1 ≤ γ ≤ n − 1 γ ≤ γ +1 p <γ + 1 γ = [γ +1_p] γ+ 1 p γ γ γ = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ [γ + 1_p], if γ > 1 −1_p, 0, if γ <1 −1 p. γ ∈ R γ = 1 −1 p γ− γ < 1 −1 p γ >−1 p γ+ 1 p γ− γ + 1 > 1 − 1 p. γ+1 p 1 − 1 p < γ < n− 1 p f ∈ Ln_p,γ f(0), f(0), f(n−γ−1)(0) f(j)_p,γ j <+∞, j = 0, 1, . . . , n − 1.

f ∈ Ln_p,γ f ∈ Ln_p,γ γ+ 1 p 1 − 1 p < γ < n− 1 p f(n−γ) f(n−γ+1) . . . f(n) t= 0 f(t) = tβ β = n − γ + η − ε ε = γ − γ +1 p 0 < η < ε < 1 f ∈ Ln_p,γ(0, 1) fLn_p,γ = ⎛ ⎝ 1  0 |tγ_f(n)_(t)|p_dt ⎞ ⎠ 1 p = c ⎛ ⎝ 1  0 t(n+η−p1)p_dt ⎞ ⎠ 1 p <+∞. η < γ− γ +1 p lim_t→0f(n−γ)(t) = c lim t→0t β−n+γ _{= c lim} t→0t η−γ+γ−1 p = +∞. I = (1, +∞) Ln_p,γ = Ln_p,γ(1, +∞) I = (1, +∞) γ < 1 − 1 p γ >1 − 1 p γ < 1 −1 p f ∈ L1_p,γ γ < 1 − 1 p f t→ +∞ γ < 1 − 1 p f(t) = ln t +∞ 1 t(γ−1)pdt < +∞ f(t) = 1 t ∈ Lp,γ f ∈ L 1 p,γ lim t→+∞f(t) = limt→+∞ln t = +∞ γ < 1 −1 p f ∈ L n p,γ n ≥ 1 f f(k) k= 1, 2, . . . , n − 1 t→ +∞

f(t) = tβ γ = 1 −1_p − ε ε > 0 β 0 ≤ n − 1 < β < n − 1 + ε. +∞  1 |tγ_f(n)_(t)|p_{dt <+∞} (n − γ − β)p > 1 β < n− γ −1 p = n − 1 + ε. β f(t) = tβ ∈ Ln_p,γ f(k)(t) = (tβ₎(k) _{= β(β − 1) · · · (β − k + 1)t}β−k _{k = 0, 1, . . . , n − 1} β− k > 0 f(k)(∞) = c lim t→+∞t β−k_{= +∞} γ > 1 − 1 p f ∈ L1_p,γ γ >1 −1_p lim t→+∞f(t) = f(+∞) < ∞, ||f(+∞)| − |f(1)|| ≤ cf_p,γ, δ >0 |fp,−1 p−δ − (δp) −1 p|f(1)|| ≤ δfp,γ_. γ > n − 1_p f ∈ Ln_p,γ f(n−1)(+∞) t→ +∞ f f(t) = tn−1 ∈ Ln_p,γ fLn_p,γ = 0 f(n−1)(+∞) = c < +∞ f(j)(+∞) = +∞ j = 0, 1, . . . , n − 2 f ∈ Ln_p,γ γ > m− 1_p 1 ≤ m ≤ n a0 a1 . . . am−1 s= 1, 2, . . . , m lim t→+∞[f (n−s)_{(t) −}s−1 μ=0 am−s+μ μ! tμ] = am−s, f(n−s)_{(t) −}s−1 μ=0 am−s+μ μ! t μ_{p,γ−s≤ cf}(n)_p,γ.

f ∈ Ln_p,γ γ > m− 1_p 1 ≤ m ≤ n bν= aν ν!, ν = 0, 1, . . . , m − 1, Pm−1(t) = m−1_ ν=0 bνtν, f ∈ Ln_p,γ lim t→+∞[f (n−m)_{(t) − Pm−1(t)]}(k)_{= 0, k = 0, 1, . . . , m − 1,} c >0 [f(n−m)_{− Pm−1]}(k)_{p,γ−m+k≤ cf}(n)_{p,γ, k = 0, 1, . . . , m − 1,} aν I = (0, 1) I = (1, +∞) ¯α = (α0, α1, . . . , αn) αi ∈ R i = 0, 1, . . . , n n |¯α| =n i=0αi 1 ≤ p < ∞ f I → R D0_¯αf(t) = tα0_f(t), Di_¯αf(t) = tαid dtt αi−1 d dt. . . t α1 d dtt α0_f(t), i = 1, 2, . . . , n, Di_¯αf(t) α f i i= 0, 1, . . . , n W_p,¯αn = Wp,¯α(I)n f I → R α n fWn p, ¯α = D n ¯αfp+ n−1  i=0 |Di ¯αf(1)|,

 · p Lp(I) 1 ≤ p < ∞ αi= 0 i = 0, 1, . . . , n−1 αn = γ W_p,¯αn Ln_p,γ = Lnp,γ(I) fLn_p,γ = tγ_f(n)_p+n−1 i=0|f (i)_(1)| n >1 γ_nα= 1, γ_n−1α = αn, γ_iα= αn+ n−1  k=i+1 (αk− 1), i = 0, 1, . . . , n − 1, γ_imax= max i≤j≤n−1γ α

j, γimin= min_{i≤j≤n−1}γjα, i= 0, 1, . . . , n − 1.

αi= γ_i−1α − γ_iα+ 1, i = 1, 2, . . . , n − 1. W_p,¯αn γi i= 0, 1, . . . , n − 1 γ_maxα = max 0≤j≤n−1γ α j <1 − 1p γ_minα = min 0≤j≤n−1γ α j >1 −1p γ_minα <1 − 1_p < γ_maxα i, j = 0, 1, . . . , n − 1

Ki+1,j(t, x) ≡ Ki+1,j(t, x, ¯α) K¯i+1,j(x, t) ≡ ¯Ki+1,j(x, t, ¯α) Ki+1,j(t, x) = x t t −αi+1 i+1 x ti+1 t−αi+2 i+2 . . . x tj−1 t−αj j dtjdtj−1. . . dti+1 i < j, Ki+1,j(t, x) ≡ 1 i= j, Ki+1,j(t, x) ≡ 0 i > j 0 < t ≤ x < ∞; ¯ Ki+1,j(x, t) =t x t−αi+1 i+1 ti+1 x t−αi+2 i+2 . . . tj−1 x t−αj j dtjdtj−1. . . dti+1 i < j, ¯ Ki+1,j(x, t) ≡ 1 i= j, ¯Ki+1,j(x, t) ≡ 0 i > j 0 < t ≤ x < ∞. wj(t, x) = t−α0_{K1,j(t, x), ¯w} j(x, t) = t−α0K¯1,j(t, x), j = 0, 1, . . . , n − 1.

{wj(t, x)}n−1 j=0 { ¯wj(x, t)}n−1j=0 x ∈ R Dn_¯αw(t) = 0 (0, x) (x, +∞) x > 0 t= 0 I = (0, 1) γ_maxα <1 −1 p. ∀f ∈ Wn p,¯α {wj(t, 1) = wj(t)}n−1 j=0 Pn(t; f; α) = n−1  j=0 aj(f)wj(t), an−1(f) = (−1)n−1_lim t→0D n−1 ¯α f(t), ai(f) = (−1)ilim t→0D i ¯α " f(t) − n−1  j=i+1 aj(f)wj(t) # , i= 0, 1, . . . , n − 2, x∈ (0, 1] Di_{¯α(f − Pn)p,γα} i−1,(0,x)≤ cD n ¯αfp,(0,x), i= 0, 1, . . . , n − 1, sup 0≤t≤x|t −(1−1/p−γα i)_Di ¯α[f(t) − Pn(t; f; α)]| ≤ c1Dn¯αfp,(0,x), i= 0, 1, . . . , n − 1. i= 0, 1, . . . , n − 1 Di_¯αf(t) = n−1  j=i aj(f)Di¯αwj(t) + t  0 s−αn_Dn ¯αf(s) · Di¯αw¯n−1(s, t)ds. f ∈ W_p,¯αn Pn(t; f; α) t→ 0 lim t→0t −(1−1/p−γα i)_Di ¯α " f(t) − n−1  j=0 aj(f)wj(t) # = 0, i = 0, 1, . . . , n − 1.

1 < p < ∞ t0>0 f ∈ W_p,¯αn ai= ai(t0, f) i = 0, 1, . . . , n − 1 lim t→0D i ¯α(f(t) − ˜Pn(t, f, α)) = 0, i = 0, 1, . . . , n − 1, ˜ Pn(t, f, α) =n−1 i=0(−1) i_{aiwi(t, t0)} lim t→0B t0 i f(t) ≡ Bit0f(0), Bt0 i f(t) = n−1 j=iKi+1,j(t, t0)D j ¯αf(t) i = 0, 1, . . . , n − 1 Bt0 i f(0) = ai i= 0, 1, . . . , n − 1 f ∈ W_p,¯αn t→ 0 W_p,¯αn I = (0, 1) γ_maxα <1 − 1_p δ δ >1 − 1_p− γ_minα f(1)Wn p, ¯α = D n ¯αfp+ n−1  i=0 Di ¯αfp,γiα−1+δ W_p,¯αn I = (0, 1) W_p,¯αn γ_minα >1 −1_p f(2)Wn p, ¯α = n−1  i=0 Di ¯αfp,γiα−1+ D n ¯αfp I = (0, 1) W_p,¯αn γ_minα <1 −1_p f(3)W_{p, ¯}n_α = n−1  i=n1 Di ¯αfp,γα i−1+ n1−1 i=0 Di ¯αfp,γα i−1+δ+ D n ¯αfp, n1= max{k = i + 1 : 0 ≤ i ≤ n − 1, γiα<1 −1p} δ > 1 − 1p− γαmin

¯β = (β0, β1, . . . , βm) βi ∈ R i = 0, 1, . . . , m W_p,¯αn (I) W_{q, ¯}m_β = W_{q, ¯}m_β(I) W_p,¯αn (I) → W_{q, ¯}m_β(I) 1 ≤ p, q < ∞ 0 ≤ m < n I = (0, 1) I = (1, +∞) W_p,¯αn (I) → W_{q, ¯}m_β(I) I = (0, 1) p≤ q q < p Wp,¯α(1, +∞)n X Y X → Y X ⊂ Y c > 0 x ∈ X xY ≤ cxX. c >0

1 < p ≤ q < ∞

f (0, 1) → R n Dk_β¯f(t) = k  i=0 ck,itγ β 0−γ0α+β0−α0+γαi−γkβ_Di_¯αf(t), k = 0, 1, . . . , m, ck,k = 1 k = 0, 1, . . . , m ck,i i= 0, 1, . . . , k − 1 k= 0, 1, . . . , m ck,0= ck−1,0(γ₀β− γ_k−1β + β0− α0),

ck,i= ck−1,i−1+ ck−1,i(γ₀β− γα₀ + β0− α0+ γ_iα− γk−1), i = 1, 2, . . . , k − 1.β γ_minα >1 −1 p I = (0, 1) 1 < p ≤ q < ∞ 0 ≤ m < n γ_minα >1 − 1 p γ₀β− γ₀α+ β0− α0≥ 1 p− 1 q. W_{q, ¯}m_β fWm q, ¯β = D m ¯ βfq + m−1_ i=0 |Di ¯ βf(1)|, c > 0 f ∈ W_p,¯αn fWm q, ¯β ≤ cfWp, ¯nα. Dm ¯ βfq ≤ c1fWp, ¯nα, ∀f ∈ W n p,¯α,

m−1_ i=0 |Di ¯ βf(1)| ≤ c2fWp, ¯nα, ∀f ∈ W n p,¯α, c1, c2>0 f k= m γ_mβ = 1 Dm_β¯f(t) = m  i=0 cm,i(α, β)tγ0β−γ0α+β0−α0+γiα−1_Di ¯αf(t). q Dm ¯ βfq ≤ c3 m  i=0 ⎛ ⎝ 1  0 |tγβ₀−γα 0+β0−α0+γiα−1_Di ¯αf(t)|qdt ⎞ ⎠ 1 q ≤ c3 ⎡ ⎢ ⎢ ⎣ m  i=0 ⎛ ⎝ 1  0 |tγ₀β−γα 0+β0−α0+γαi−1[Di ¯αf(t) − Di¯αf(1)]|qdt ⎞ ⎠ 1 q + + m  i=0 ⎛ ⎝ 1  0 |tγβ₀−γα 0+β0−α0+γiα−1_Di ¯αf(1)|qdt ⎞ ⎠ 1 q ⎤ ⎥ ⎥ ⎦ , c3 = max_0≤i≤m|cm,i|

γ_iα− 1 + γβ₀ − γ₀α+ β0− α0 ≥ 1 p − 1 q + γ α i − 1, γ_minα >1 − 1 p γβ₀ − γ₀α+ β0− α0+ γ_iα− 1 > −1 q, i= 0, 1, . . . , n − 1. 1  0 |tγβ 0−γα0+β0−α0+γiα−1|q_dt≤ c∗_{3, i}= 0, 1, . . . , n − 1,

c∗₃= 1 (γ0β− γ0α+ β0− α0+ γiα− 1)q + 1 Dm ¯ βfq ≤ c3 ⎡ ⎢ ⎢ ⎣ m  i=0 ⎛ ⎝ 1  0 |tγβ 0−γ0α+β0−α0+γαi−1(Di_¯αf(t) − Di_¯αf(1))|q_dt ⎞ ⎠ 1 q + + m  i=0 |Di ¯αf(1)| # , c₃= max{c3, c∗3} γ_iα = αi+1 + γ_i+1α − 1 γ_i+1α − 1 + αi+1 ≤ 1 −1 p+ 1 q+ γ β 0 − γ0α+ β0− α0+ γiα− 1, i = 0, 1, . . . , n − 1. Dm ¯ βfq ≤ c3 ⎡ ⎢ ⎢ ⎣ m  i=0 Hi ⎛ ⎝ 1  0 |tγα i+1+αi+1−1 d dtD i ¯αf(t)|pdt ⎞ ⎠ 1 p + m  i=0 |Di ¯αf(1)| ⎤ ⎥ ⎥ ⎦ ≤ c4 ⎡ ⎢ ⎢ ⎣ m  i=0 ⎛ ⎝ 1  0 |tγα i+1−1_Di+1 ¯α f(t)|pdt ⎞ ⎠ 1 p + m  i=0 |Di ¯αf(1)| ⎤ ⎥ ⎥ ⎦ , c4= c₃max{1, Hi, i= 0, 1, . . . , m} n > m≥ 0 Dm ¯ βfq ≤ c4 " _n  i=1 Di ¯αfp,γαi−1+ m  i=0 |Di ¯αf(1)| # . Dm ¯ βfq ≤ c4 " fWn p, ¯α+ m  i=0 |Di ¯αf(1)| + Dn¯αfp # = 2c4fWn p, ¯α = c1fWp, ¯nα,

c1 = 2c4 Dk_β¯f(1) = k  i=0 Di_¯αf(1), k = 0, 1, . . . , m. m  k=0 |Dk ¯ βf(1)| ≤ m  k=0 k  i=0 |ck,i||Di ¯αf(1)| = m  i=0 m  k=i |ck,i||Di ¯αf(1)| = m  i=0 |Di ¯αf(1)| m  k=i |ck,i|. n > m ≥ 0 m  k=0 |Dk ¯ βf(1)| ≤ c5 n−1  i=0 |Di ¯αf(1)| ≤ c2fWp, ¯nα, c2= max 0≤i≤m m  k=i|ck,i| f0(t) = t−γ0α−α0+ 1_p+ε _{ε > 0} 1 p + p1 = 1 Di_¯αf0(t) = tαi d dtt αi−1 d dt. . . t α2 d dtt α1 d dt(t α0−γα0−α0+ 1_p+ε₎ = (−γα 0 +_p1_ + ε)tαi_dtdtαi−1_dtd . . . tα2_dtdtα1−γ α 0+ 1p+ε−1 = (−γα 0 +_p1 + ε)(−γ1α+_p1 + ε)tαi d dtt αi−1d dt. . . d dtt α2−γα1+ 1_p+ε−1_{= . . .} = i−2  j=0 (−γα j +_p1 + ε)tαi d dtt αi−1−γαi−2+ 1p+ε−1₌ i−1  j=0 (−γα j +_p1+ ε)t αi−γi−1α + 1p+ε−1 = i−1  j=0 (−γα j +_p1 + ε)t −γα i+ 1_p+ε_, αi− γi−1α − 1 = −γiα i= 1, 2, . . . , n − 1

i= 1, 2, . . . , n − 1 (−γ_jα+1 p+ ε) f0∈ W_p,¯αn i Dn_¯αf0(t) = n−1  j=0 (−γα j +p1 + ε)t−1+ 1 p+ε_. (−1 + 1 p + ε)p + 1 = εp > 0 1  0 t(−1+ 1p+ε)p_{dt <∞,} f0 ∈ W_p,¯αn βi = γβ_i−1− γ_iβ + 1 γ_mβ = 1 β f0(t) Dm_β¯f0(t) = tβm d dtt βm−1_{. . .} d dtt β2 d dtt β1 d dtt β0−γ0α−α0+ 1_p+ε = (β0− γα 0 − α0+_p1_ + ε)tβm_dtdtβm−1. . ._dtdtβ2_dtdtβ1+β0−γ α 0−α0+ 1_p+ε−1 = (β0− γα 0 − α0+_p1_ + ε)(γ0β− γ0α+ β0− α0− γ1β+_p1_ + ε)tβm_dtdtβm−1. . . . . . d dtt β2+γβ0−γα0+β0−α0−γ1β+ 1p+ε−1 _{= . . .} = m−1 i=0 (γ0β− γ0α+ β0− α0− γiβ+_p1 + ε)t βm+γ0β−γα0+β0−α0−γm−1β + 1p+ε−1 = m−1 i=0 (γ0β− γα0 + β0− α0− γiβ+_p1_ + ε)tγ β 0−γα0+β0−α0−1+ 1_p+ε_. ε0 >0 ε∈ (0, ε0) m−1 i=0 (γ0β− γ0α+ β0− α0− γiβ+_p1_ + ε) = 0. f0 ∈ W_{q, ¯}m_β 1  0 t(γ0β−γ0α+β0−α0−1+ 1_p+ε)q_{dt <∞,}

(γ0β− γ0α+ β0− α0− 1 + _p1_ + ε)q + 1 > 0. ε∈ (0, ε0) γ₀β−γ₀α+β0−α0≥ 1 p− 1 q I = (0, 1) 1 ≤ p < ∞ n ≥ 1 γ_minα >1 − 1 p Wp,¯α(I) → Wn _{p, ¯}n_β(I) γ₀β− γ₀α+ β0− α0≥ 0 p = q m < n k= n D_βn¯f(t) = n  i=0 cn,i(α, β)tγ₀β−γα 0+β0−α0+γαi−1_Di ¯αf(t). p Dn ¯ βfp≤ c3 ⎡ ⎢ ⎢ ⎣ ⎛ ⎝ 1  0 |tγ₀β−γα 0+β0−α0_Dn ¯αf(t)|pdt ⎞ ⎠ 1 p + n−1  i=0 ⎛ ⎝ 1  0 |tγ₀β−γα 0+β0−α0_Di ¯αf(t)|pdt ⎞ ⎠ 1 p ⎤ ⎥ ⎥ ⎦ , c3 = max 0≤i≤n|cn,i| γ₀β− γ₀α+ β0− α0≥ 0 γ_minα >1 − 1 p γ₀β− γ₀α+ β0− α0+ γ_iα>1 −1 p, i= 0, 1, . . . , n. 1  0 |tγβ 0−γ0α+β0−α0+γiα−1|p_dt≤ c∗_{3, i}= 0, 1, . . . , n − 1,

c∗₃= 1 (γ0β− γ0α+ β0− α0+ γiα− 1)p + 1 Dn ¯ βfp≤ c3 ⎡ ⎢ ⎢ ⎣ ⎛ ⎝ 1  0 |tγ₀β−γα 0+β0−α0_Dn ¯αf(t)|pdt ⎞ ⎠ 1 p + + n−1  i=0 ⎛ ⎝ 1  0 |tγβ₀−γα 0+β0−α0+γiα−1(Di ¯αf(t) − Di¯αf(1))|pdt ⎞ ⎠ 1 p + n−1  i=0 |Di ¯αf(1)| ⎤ ⎥ ⎥ ⎦ , c₃= max{c3, c∗3} p= q |tγ₀β−γα 0+β0−α0| ≤ 1 γ₀β− γ₀α+ β0− α0 ≥ 0 0 < t < ∞ Dn ¯ βfp≤ c4 " Dn ¯αfp+ n−1  i=0 Di+1 ¯α fp,γi+1α −1+ n−1  i=0 |Di ¯αf(1)| # ≤ c4 " fWn p, ¯α+ n  i=0 Di ¯αfp,γiα−1 # , c4 = c3max{1, Hi, i = 0, 1, . . . , n − 1} Dn ¯ βfp≤ c1fWp, ¯nα, m = n W_p,¯αn (I) → W_{p, ¯}n_β(I) W_p,¯αn W_{p, ¯}n_β(I) |¯α| = | ¯β| W_p,¯αn = Ln_p,γ γ = |¯α| > n−1 p ¯β βn <1 −1 p βi< n− i + 1 − 1 p− n  k=i+1βk i= 1, 2, . . . , n − 1 β0= γ − n  k=1βk

γ_maxβ <1 −1 p | ¯β| = γ W_p,¯αn = Ln_p,γ → W_{p, ¯}n_β f ∈ W_p,¯αn = Ln_p,γ Pn(t; f; ¯β) lim t→0t −(1−1/p−γ_iβ)_Di ¯ β $ f(t) − Pn(t; f; ¯β)%= 0, i = 0, 1, . . . , n − 1, f ∈ W_p,¯αn = Ln_p,γ t= 0 W_{p, ¯}n_β I = (0, 1) 1 ≤ p < ∞ γ_minα > 1 − 1 p γ β min > 1 −1 p W n p,¯α(I) Wp, ¯nβ(I) |¯α| = | ¯β|. Wp,¯α(I) → Wn _{p, ¯}n_β(I) W_{p, ¯}n_β(I) → Wp,¯α(I)n γ₀β − γα₀ + β0− α0 ≥ 0 γ0α− γ0β+ α0− β0 ≥ 0 γ₀β− γ₀α+ β0− α0= 0.

γ_i−1β = βi+ γ_iβ− 1 γ_i−1α = αi+ γ_iα− 1 i = 0, 1, . . . , n − 1 γβ_n = 1 γ_nα= 1 β0+ γ0β= α0+ γ0α⇒ β0+ β1+ γ1β= α0+ α1+ γ1α⇒ ⇒ β0+ β1+ β2+ . . . + βn+ γβ n = α0+ α1+ α2+ . . . + αn+ γαn. n  i=0βi= n  i=0αi βn = γ βi = 0 i = 0, 1, . . . , n − 1 γ_minβ >1 −1 p γ_minβ = γ₀β = β1+γ₁β−1 = γβ₁−1 = β2+γ₂β−2 = . . . = βn−1+γ_n−1β −(n−1) = = γβ n−1− (n − 1) = βn− (n − 1) = γ − (n − 1) > 1 −1_p.

γ_minβ >1−1 p γ > n− 1 p W_{p, ¯}n_β(I) Ln_p,γ(I) I = (0, 1) 1 ≤ p < ∞ γ_minα > 1 − 1 p γ > n− 1 p W n p,¯α(I) Lnp,γ(I) n  i=0αi = γ W_p,¯αn Ln_p,γ(I) γ_minα <1 − 1 p I = (0, 1) 1 < p ≤ q < ∞ 0 ≤ m < n γ_minα < 1 − 1 p γ₀β− γ₀α+ β0− α0>1 −1 q − γ α min, m+1 ≤ n1 − 1 n ≥ m + 1 > n1 − 1 n1 = max{k = i + 1 : 0 ≤ i ≤ n− 1, γ_iα<1 −1 p} m+ 1 ≤ n1− 1 Dm ¯ βfq ≤ c4 _m+1  i=0 Di ¯αfp,γα i−1+δ+ m  i=0 |Di ¯αf(1)|  , δ >1 − 1 p− γ α min γ₀β− γ₀α+ β0− α0+1 q − 1 p >1 − 1 p− γ α min.

δ γ₀β− γ₀α+ β0− α0+1 q − 1 p ≥ δ > 1 − 1 p − γ α min. γ_i+1α −1+αi+1+δ ≤ 1−1 p+ 1 q+(γ β 0−γ0α+β0−α0+γiα−1), i = 0, 1, . . . , n−1. γ₀β− γ₀α+ β0− α0+ γ_minα − 1 > −1 q, Dm ¯ βfq ≤ c3 ⎡ ⎢ ⎢ ⎣ m  i=0 ⎛ ⎝ 1  0 |tγβ₀−γα 0+β0−α0+γiα−1(Di ¯αf(t) − Di¯αf(1))|qdt ⎞ ⎠ 1 q + + m  i=0 |Di ¯αf(1)| # . Dm ¯ βfq ≤ c3 ⎡ ⎢ ⎢ ⎣ m  i=0 Hi ⎛ ⎝ 1  0 |tγα i+1−1+αi+1+δ d dtD i ¯αf(t)|pdt ⎞ ⎠ 1 p + m  i=0 |Di ¯αf(1)| ⎤ ⎥ ⎥ ⎦ ≤ c4 ⎡ ⎢ ⎢ ⎣ m  i=0 ⎛ ⎝ 1  0 |tγα i+1−1+δ_Di+1 ¯α f(t)|pdt ⎞ ⎠ 1 p + m  i=0 |Di ¯αf(1)| ⎤ ⎥ ⎥ ⎦ = c4 "_m+1  i=1 Di ¯αfp,γiα−1+δ+ m  i=0 |Di ¯αf(1)| # , c4 = c3max{1, Hi, i= 0, 1, . . . , m}

n≥ m + 1 > n1− 1 Dm ¯ βfq ≤ c3 ⎡ ⎢ ⎢ ⎣ m  i=n1−1 ⎛ ⎝ 1  0 |tγβ₀−γα 0+β0−α0+γ_iα−1_(Di ¯αf(t) − Di¯αf(1))|qdt ⎞ ⎠ 1 q + n1−2 i=0 ⎛ ⎝ 1  0 |tγ₀β−γα 0+β0−α0+γα_i−1_(Di ¯αf(t) − Di¯αf(1))|qdt ⎞ ⎠ 1 q + m  i=0 |Di ¯αf(1)| ⎤ ⎥ ⎥ ⎦ , n1−2 i=0 n1<2 Dm ¯ βfq ≤ c3 ⎡ ⎢ ⎢ ⎣ m  i=n1−1 Hi ⎛ ⎝ 1  0 |tγα i+1+αi+1−1 d dtD i ¯αf(t)|pdt ⎞ ⎠ 1 p + n1−2 i=0 Hi ⎛ ⎝ 1  0 |tγα i+1+αi+1−1+δ d dtD i ¯αf(t)|pdt ⎞ ⎠ 1 p + m  i=0 |Di ¯αf(1)| ⎤ ⎥ ⎥ ⎦ ≤ c4 ⎡ ⎢ ⎢ ⎣ m  i=n1−1 ⎛ ⎝ 1  0 |tγα i+1−1_Di+1 ¯α f(t)|pdt ⎞ ⎠ 1 p + n1−2 i=0 ⎛ ⎝ 1  0 |tγα i+1−1+δ_Di+1 ¯α f(t)|pdt ⎞ ⎠ 1 p + m  i=0 |Di ¯αf(1)| # = c4 "_m+1  i=n1 Di ¯αfp,γiα−1+ n1−1 i=1 Di ¯αfp,γiα−1+δ+ m  i=0 |Di ¯αf(1)| # ≤ c4 "_m+1  i=n1 Di ¯αfp,γαi−1+ n1−1 i=0 Di ¯αfp,γαi−1+δ+ m  i=0 |Di ¯αf(1)| # .

I = (0, 1) 1 < p ≤ q < ∞ 0 ≤ m < n γ_minα <1−1 p γ_minα −γ₀α+β0−α0= 0 γmin−γα α₀+β0−α0+γ₀β−γ_iβ= 0 i = 1, 2, . . . , m−1 γ₀β − γ₀α + β0 − α0 > 1 −1 q − γ α min f0(t) = tγminα −γ0α−α0 Di_¯αf0(t) = tαi d dtt αi−1 d dt. . . t α2 d dtt α1 d dt(t α0+γαmin−γα0−α0) = (γα min− γ0α)tαi d dtt αi−1 d dt. . . t α2 d dtt α1+γminα −γ0α−1 = (γα min− γ0α)tαi_dtdtαi−1_dtd . . ._dtdtα2_dtdtγ α min−γα1 = (γα min− γ0α)(γαmin− γ1α)tαi d dtt αi−1 d dt. . . d dtt α2+γαmin−γα1−1= . . . = i−1  j=0 (γα min− γjα)tγ α min−γiα_{, i}= 1, 2, . . . , n. γ_iα₀ = γ_minα 0 ≤ i0≤ n − 1 Di_¯αf0(t) = 0 ∀i ≤ i0 Di0+1 ¯α f0(t) = 0 Dn¯αf0(t) = 0 f0 ∈ Wp,¯αn f0∈ W_{q, ¯}m_β β f0(t) D_βm¯f0(t) = tβm d dtt βm−1_{. . .} d dtt β2 d dtt β1 d dtt β0+γminα −γ0α−α0 = (β0− α0+ γα min− γ0α)tβm_dtdtβm−1. . ._dtdtβ2_dtdtβ1+β0−α0+γ α min−γα0−1 = (β0− α0+ γα min− γ0α)tβm d dtt βm−1 d dt. . . t β2 d dtt γ₀β−γ₁β+β0−α0+γαmin−γ0α = (β0− α0+ γα min− γ0α)(γminα − γα0 + γ0β+ β0− α0+ γ1β)tβm d dtt βm−1_{. . .} . . . d dtt β2+γ_minα −γ0α+γ0β+β0−α0−γ1β−1 = . . . = (β0−α0+ γα min−γ0α) m−1 i=1 (γα min−γ0α+ γ0β+ β0−α0+ γβi)tγ α min−1+β0−α0+γ0β−γ0α_.

Dm_β¯f0(t) = 0 0 < t ≤ 1 f0 ∈ W_{q, ¯}m_β 1  0 t(γminα −1+β0−α0+γβ0−γα0)q_{dt <}∞, γ₀β− γ₀α+ β0− α0>1 −1_q − γ_minα γα_min< 1−1 p γ β 0−γ0α+β0−α0>1−1_q−γminα γβ0−γ0α+β0−α0> 1_p−1_q γ_minα < 1 − 1 p 1 − 1 q − γ α min > 1_p − 1_q γ_iα i = 0, 1, . . . , n − 1 γ₀β β0 α0 γminα < 1 − 1_p 1−1 q−γ α min> γ0β−γ0α+β0−α0> 1_p−1_q f0(·) W_p,¯αn W_{q, ¯}m_β

1 ≤ q < p < ∞

γ_minα >1 − 1 p I = (0, 1) 1 ≤ q < p < ∞ 0 ≤ m < n γ_minα > 1 − 1 p γ₀β− γ₀α+ β0− α0> 1 p − 1 q, γ₀β− γ₀α+ β0− α0≥ 1 p− 1 q

1 < p ≤ q < ∞ 1 ≤ q < p < ∞ γ_minα <1 −1 p I = (0, 1) 1 ≤ q < p < ∞ 0 ≤ m < n γ_minα < 1 −1_p γ₀β− γ₀α+ β0− α0 >1 −1 q − γ α min, Dm ¯ βfq ≤ c1fWp, ¯nα, ∀f ∈ W n p,¯α. m+ 1 ≤ n1− 1 n− 1 < m + 1 ≤ n n1= max{k = i + 1 : 0 ≤ i ≤ n − 1, γiα<1 − 1p} m+1 ≤ n1−1 Dm ¯ βfq ≤ c4 _m+1  i=0 Di ¯αfp,γiα−1+δ+ m  i=0 |Di ¯αf(1)|  δ >1 −1 p − γ α min δ γ₀β− γα₀ + β0− α0+1 q − 1 p > δ >1 − 1 p − γ α min. γ_iα+ δ < 1 −1 p + 1 q + (γ β 0 − γα0 + β0− α0+ γiα− 1), i = 0, 1, . . . , n − 1, γ_i+1α −1+αi+1+δ < 1−1 p+ 1 q+(γ β 0−γ0α+β0−α0+γiα−1), i = 0, 1, . . . , n−1.

γ₀β− γ₀α+ β0− α0+ γ_minα − 1 > −1 q, Dm ¯ βfq ≤ c3 ⎡ ⎢ ⎢ ⎣ m  i=0 ⎛ ⎝ 1  0 |tγβ 0−γ0α+β0−α0+γαi−1(Di_¯αf(t) − Di_¯αf(1))|q_dt ⎞ ⎠ 1 q + + m  i=0 |Di ¯αf(1)| # . Dm ¯ βfq ≤ c3 ⎡ ⎢ ⎢ ⎣ m  i=0 Hi ⎛ ⎝ 1  0 |tγα i+1−1+αi+1+δ d dtD i ¯αf(t)|pdt ⎞ ⎠ 1 p + m  i=0 |Di ¯αf(1)| ⎤ ⎥ ⎥ ⎦ ≤ c4 ⎡ ⎢ ⎢ ⎣ m  i=0 ⎛ ⎝ 1  0 |tγα i+1−1+δ_Di+1 ¯α f(t)|pdt ⎞ ⎠ 1 p + m  i=0 |Di ¯αf(1)| ⎤ ⎥ ⎥ ⎦ = c4 "_m+1  i=1 Di ¯αfp,γα i−1+δ+ m  i=0 |Di ¯αf(1)| # , c4= c₃max{1, Hi, i= 0, 1, . . . , m} Dm ¯ βfq ≤ c4 "_m+1  i=0 Di ¯αfp,γiα−1+δ+ m  i=0 |Di ¯αf(1)| # . n1− 1 < m + 1 ≤ n Dm ¯ βfq ≤ c3 ⎡ ⎢ ⎢ ⎣ n1−2 i=0 ⎛ ⎝ 1  0 |tγ₀β−γα 0+β0−α0+γαi−1(Di ¯αf(t) − Di¯αf(1))|qdt ⎞ ⎠ 1 q

+ m  i=n1−1 ⎛ ⎝ 1  0 |tγβ 0−γ0α+β0−α0+γαi−1(Di_¯αf(t) − Di_¯αf(1))|q_dt ⎞ ⎠ 1 q + m  i=0 |Di ¯αf(1)| ⎤ ⎥ ⎥ ⎦ , n1−2 i=0 n1<2 Dm ¯ βfq ≤ c3 ⎡ ⎢ ⎢ ⎣ n1−2 i=0 Hi ⎛ ⎝ 1  0 |tγα i+1+αi+1−1+δ d dtD i ¯αf(t)|pdt ⎞ ⎠ 1 p + m  i=n1−1 Hi ⎛ ⎝ 1  0 |tγα i+1+αi+1−1d dtD i ¯αf(t)|pdt ⎞ ⎠ 1 p + m  i=0 |Di ¯αf(1)| ⎤ ⎥ ⎥ ⎦ ≤ c4 ⎡ ⎢ ⎢ ⎣ n1−2 i=0 ⎛ ⎝ 1  0 |tγα i+1−1+δ_Di+1 ¯α f(t)|pdt ⎞ ⎠ 1 p + m  i=n1−1 ⎛ ⎝ 1  0 |tγα i+1−1_Di+1 ¯α f(t)|pdt ⎞ ⎠ 1 p + m  i=0 |Di ¯αf(1)| # = c4 "_n1−1  i=1 Di ¯αfp,γαi−1+δ+ m+1_ i=n1 Di ¯αfp,γαi−1+ m  i=0 |Di ¯αf(1)| # ≤ c4 "_n1−1  i=0 Di ¯αfp,γiα−1+δ+ m+1_ i=n1 Di ¯αfp,γiα−1+ m  i=0 |Di ¯αf(1)| # . αk = 1 k = 0, 1, . . . , n − 1 αn = n βk = 1 k= 0, 1, . . . , m − 1 βm = m 0 ≤ m < n γ_kα= αn+ n−1  i=k+1 (αi− 1) = n, k = 0, 1, . . . , n − 1, γ0β = βm+ m−1_ i=1 (βi− 1) = m.

γ_minα = min 0≤k≤n−1γ α k = n > 1 −1_p. γ₀β− γα₀ + β0− α0= m − n + 1 − 1 = m − n < 0, 1 ≤ q < p < ∞ α0 = 1 − n2 αk = 2k + 1 k = 1, 2, . . . , n − 1 αn = n − 1_p β0 = −m2 βk = 2k k = 1, 2, . . . , m − 1 βm = 2m − 1_q 0 ≤ m < n γ_kα= n − 1 p+ n−1  i=k+1 (2i) = n2₋ 1 p− k(k + 1), k = 0, 1, . . . , n − 1, γ_minα = γ_n−1α = n −1 p >1 − 1 p. γ₀β = 2m −1 q + m−1_ i=1 (2i − 1) = m2_{+ 1 −} 1 q, γ₀β− γ₀α+ β0− α0= m2+ 1 − 1 q − n 2₊1 p − m 2_{− 1 + n}2 ₌ 1 p− 1 q. 1 < p ≤ q < ∞

W

_p,n_¯α

(1, ∞)

Wp,¯α(0, 1)n Wp,¯α(1, ∞)n x = 1 t f ∈ W_p,¯αn (1, ∞) f˜(x) = f(1 x) W_p,¯˜αn (0, 1) ¯˜α = (˜α0,˜α1, . . . ,˜αn) ˜αn = −αn + 2 − 2 p ˜αi = −αi+ 2

i= 1, 2, . . . , n − 1 ˜α0= −α0 Dn ¯αfp,(1,+∞)= ⎛ ⎝ +∞  1 |Dn ¯αf(t)|pdt ⎞ ⎠ 1 p = = ⎛ ⎝ +∞  1 |tαn d dtt αn−1 d dt. . . t α1 d dtt α0_f(t)|p_dt ⎞ ⎠ 1 p = = ⎛ ⎝ 1  0 |x−αn d x−2dxx −αn−1 d x−2dx. . . x −α1 d x−2dxx −α0_f(1 x)| pdx x2 ⎞ ⎠ 1 p = = ⎛ ⎝ 1  0 |x−αn+2−2_p d dxx −αn−1+2 d dx. . . x −α1+2 d dxx −α0f(1 x)| p_dx ⎞ ⎠ 1 p = = ⎛ ⎝ 1  0 |x˜αn d dxx ˜αn−1 d dx. . . x ˜α1 d dxx ˜α0_f˜_(x)|p_dx ⎞ ⎠ 1 p = Dn ¯˜αf˜p,(0,1), Di_¯˜αf(1) = Di_¯αf(1) i = 0, 1, . . . , n − 1 W_{q, ¯}m_β(1, +∞) W_{q, ¯˜}m_β(0, 1) W_p,¯˜αn (0, 1) → W_{q, ¯˜}m_β(0, 1), W_p,¯˜αn (0, 1) W_p,¯αn (1, +∞) γ_i˜α = −αn+ 2 − 2 p+ n−1 k=i+1(−αi+ 2 − 1) = −(αn+ n−1 k=i+1(αi− 1)) + 2 − 2 p = −γα i + 2 − 2_p i= 0, 1, . . . , n − 1. γ_min˜α > 1 − 1 p γ_maxα <1 − 1 p

γ₀β˜− γ₀˜α+ ˜β0− ˜α0≥ 1 p− 1 q γ₀β− γ₀α+ β0− α0≤ 1 q − 1 p. γ₀β˜−γ₀˜α+ ˜β0−˜α0= −γ₀β+2−2 p+γ α 0−2+2_p−β0+α0= −γ0β+γ0α−β0+α0 ≥ 1_p−1_q. I = (1, +∞) 1 < p ≤ q < ∞ 0 ≤ m < n γ_maxα <1 −1 p γ₀β− γ₀α+ β0− α0≤ 1 q − 1 p. m= n p= q I = (1, ∞) 1 ≤ p < ∞ γ_maxα <1 − 1 p W_p,¯αn (I) → W_{p, ¯}n_β(I) γ₀β− γ₀α+ β0− α0≤ 0 γ₀β− γ₀α+ β0− α0= 0 n  i=0αi= n  i=0βi I = (1, ∞) 1 ≤ p < ∞ γ_maxα < 1 − 1 p γ β max < 1 − 1_p Wp,¯α(I)n W_{p, ¯}n_β(I) |¯α| = | ¯β|.

βn = μ βi = 0 i = 0, 1, . . . , n − 1 γ_maxβ < 1 −1 p μ < n− 1 p W n p, ¯β(I) Lnp,μ(I) I = (1, ∞) 1 ≤ p < ∞ γ_maxα < 1 − 1 p μ > n−1 p W n p,¯α(I) Lnp,μ(I) n  i=0αi = μ γ_min˜α < 1 − 1 p γ α max > 1 − 1_p γ₀β˜− γ₀˜α+ ˜β0− ˜α0 >1 −1 q − γ ˜α min γ0β− γ0α+ β0− α0 < γ_maxα + 1 q − 1 I = (1, ∞) 1 < p ≤ q < ∞ 0 ≤ m < n γ_maxα > 1 −1 p γ β 0 − γ0α+ β0− α0< γmaxα +1_q − 1 I = (1, +∞) 1 < p ≤ q < ∞ 0 ≤ m < n γ_maxα > 1 − 1 p γ α max− γ0α+ β0− α0 = 0 γmaxα − γ0α+ β0− α0+ γ0β− γiβ = 0 i= 1, 2, . . . , m−1 W_p,¯αn (I) → W_{q, ¯}m_β(I) γ₀β− γ₀α+ β0− α0< γ_maxα +1 q − 1

W_p,¯αn (I) → W_{q, ¯}m_β(I) 1 ≤ p, q < ∞ 0 ≤ m < n I = (0, 1) I = (1, +∞) Wp,¯α(0, 1)n Wp,¯α(1, +∞)n i, j = 0, 1, . . . , n − 1 Ki+1,j(t, x) ≡ Ki+1,j(t, x, ¯α) = x t t −αi+1 i+1 x ti+1 t−α_i+2i+2. . . x tj−1 t−α_j jdtjdtj−1. . . dti+1 i < j, Ki+1,j(t, x) ≡ Ki+1,j(t, x, ¯α) ≡ 1 i= j, Ki+1,j(t, x) ≡ Ki+1,j(t, x, ¯α) ≡ 0 i > j 0 < t ≤ x.

i < j Ki+1,j Ki+1,j(zt, zx) = zx  zt t−αi+1 i+1 zx  ti+1 t−αi+2 i+2 . . . zx  tj−1 t−αj j dtjdtj−1. . . dti+1 = = [tk= zτk, dtk = zdτk] = x  t (zτi+1)−αi+1 x  τi+1 (zτi+2)−αi+2_{. . .} . . . x  τj−1 (zτj)−αj_zj−i_dτ jdτj−1. . . dτi+1 = z j k=i+1(1−αk)_Ki+1,j(t, x). x= 1 t= 1 Ki+1,j(zt, z) = z j k=i+1(1−αk)Ki+1,j(t, 1), Ki+1,j(z, zx) = z j k=i+1(1−αk)_Ki+1,j(1, x), 0 ≤ i ≤ j ≤ n − 1 ki,j = min{k : i ≤ k ≤ j, k  s=i+1 αs− k = max i≤ξ≤j( ξ  s=i+1 αs− ξ)}, Mi,j = max i≤s≤j(j − s + 1 − j+1  k=s+1 αk). Mi γ_imin Mi γimin Mi = max_{i≤s≤n−1}(n−s− n  k=s+1 αk) = − min i≤s≤n−1(αn+ n−1  k=s+1 (αk−1))+1 = 1−γmin i ki ≡ ki,n−1 Mi = Mi,n−1 Mi ≥

Mi+1 M0 = max_{0≤i≤n−1}Mi

n− 1 ui(t) = tα0K1,i(t, 1, −¯α) i = 0, 1, . . . , n − 1 Ki+1,j(t, 1) Ki+1,j(1, t) min i≤s≤j  α0+ s  k=i+1 (1 − αk)  = min_i≤s≤j " α0+ j − i + 1 − j+1  k=i+1 αk− −(j − s + 1 − j+1  k=s+1 αk) # = α0+ j − i + 1 − j+1  k=i+1 αk− Mi,j. 0 ≤ i ≤ j ≤ n − 1 Ki+1,j(t, 1) << tj−i+1− j+1

k=i+1αk−Mi,j|lnt|li,j_{, t∈ (0, 1],}

li,j k ki,j + 1 ≤ k ≤ j k

s=ki,j+1

(αs− 1) = 0 ki,j < j li,j = 0 ki,j = j

0 ≤ i ≤ n − 1 δ 0 < δ < 1 t∈ (0, δ] Ki+1,n−1(t, 1) >> tn−i− n k=i+1αk−Mi 0 ≤ i ≤ n − 1 t−αn_{Ki+1,n−1(1, t) << t}Mi−1|lnt|li_{, t}≥ 1, li k i+ 1 ≤ k ≤ ki− 1 ki−1 s=k(αs− 1) = 0 ki > i+ 1 li = 0 ki= i + 1 fs(t) = t−α0K1,s(t, 1, ¯α) 0 ≤ m ≤ s ≤ n D_βm¯f(t) = 0, ∀t ∈ (0, 1].

fs(t) = t−α0K1,s(t, 1, ¯α) 0 ≤ m ≤ s≤ n fi(t) = t−β0_{K1,i(t, 1, ¯β), i = 0, 1, . . . , m − 1,} fs(t) = m−1_ i=0 cit−β0K1,i(t, 1, ¯β), ∀t ∈ (0, 1], m−1 i=0 c 2 i = 0 ci∈ R i = 0, 1, . . . , m − 1 ¯α k k = 0, 1, . . . , m − 1 Dk_¯αfs(t) = m−1_ i=0 ciDk_¯α(t−β0K1,i(t, 1, ¯β)), ∀t ∈ (0, 1]. ¯α k 0 ≤ k < m ¯β Dk_¯αf(t) = k  j=0 dk,jtγk,jDj_β¯f(t), γk,j =k i=0αi− j  i=0βi+ j − k dk,k ≡ 1 0 ≤ j ≤ k < m k 0 ≤ k < m Dk_¯αfs(t) = m−1_ i=0 ci k  j=0 dk,jtγk,jD_βj¯(t−β0K1,i(t, 1, ¯β)) = = k  j=0 (−1)j_d k,jtγk,j m−1_ i=j ciKj+1,i(t, 1, ¯β). Dk_¯αfs(t) = Dk_¯α(t−α0_K1,s(t, 1, ¯α)) = (−1)k_Kk+1,s(t, 1, ¯α), k= 0, 1, . . . , m − 1; s = m, m + 1, . . . , n.

(−1)k_{Kk+1,s(t, 1, ¯α) =}k j=0 (−1)j_d k,jtγk,j m−1_ i=j ciKj+1,i(t, 1, ¯β) = = k  j=0 (−1)j_dk,jcjtγk,jKj+1,j(t, 1, ¯β) + k  j=0 (−1)j_dk,jtγk,j m−1  i=j+1 ciKj+1,i(t, 1, ¯β), k= 0, 1, . . . , m − 1; s = m, m + 1, . . . , n. Kj+1,j(t, 1, ¯β) = 1 Kj+1,i(t, 1, ¯β) = 0 i = j + 1, j + 2, . . . , m − 1 (−1)k_{Kk+1,s(t, 1, ¯α) =}k j=0 (−1)j_d k,jcjtγk,j, k= 0, 1, . . . , m − 1 s = m, m + 1, . . . , n. t = 1 m k  j=0 (−1)j_d k,jcj = 0, k = 0, 1, . . . , m − 1. k= 0 d0,0c0 = 0 d0,0 = 1 c0 = 0 k = 1, 2, . . . , m − 1 dk,k = 0 ck = 0 k= 0, 1, . . . , m − 1 ck k= 0, 1, . . . , m − 1 Lp Lq 1 ≤ q < p < ∞ f (0, 1) → R n Dβk¯f(t) = k  i=0 ck,itμk,iDi¯αf(t), k = 0, 1, . . . , m,

μk,i = k  j=0βj − i  j=0αj + i − k i = 0, 1, . . . , k k = 0, 1, . . . , m ck,i i = 0, 1, . . . , k − 1 k = 0, 1, . . . , m ck,k= 1, ck,0= ck−1,0 _k−1  j=0 βj− α0− k + 1  ,

ck,i= ck−1,i−1+ ck−1,i _k−1  j=0 βj− i  j=0 αj+ i − k + 1  , i= 1, 2, . . . , k − 1, f ∈ W_p,¯αn Di_¯αf(t) = n−1  j=i (−1)j−i_K i+1,j(t, 1)Dj¯αf(1) + 1  t x−αn_{Ki+1,n−1(t, x)D}n ¯αf(x)dx, i= 0, 1, . . . , n − 1

q < p

i0 = min{i : 0 ≤ i ≤ m, cm,i = 0} cm,i i= 0, 1, . . . , m

I = (0, 1) 1 ≤ q < p < ∞ 0 ≤ m < n | ¯β| − |¯α| + n − m +1_q >max{1 p, Mi0}. i) ⇒ ii) i) f ∈ W_p,¯αn fWm q, ¯β ≤ cfWp, ¯nα

W_{q, ¯}m_β Dm ¯ βfq ≤ cfWp, ¯nα c >0 f ∈ W_p,¯αn k= m Dm_β¯f(t) = m  i=i0 cm,itμm,i n−1  j=i (−1)j−i_{Ki+1,j(t, 1)D}j ¯αf(1)+ + m  i=i0 cm,itμm,i 1  t x−αnKi+1,n−1(t, x)Dn ¯αf(x)dx. L W_p,¯αn f ∈ L Dj¯αf(1) = 0, j = 0, 1, . . . , n − 1. L W_p,¯αn F ∈ Lp(0, 1) f ∈ L Dn_¯αf(t) = F (t) fWn p, ¯α = F p Dn_¯α L⊂ W_p,¯αn Lp(0, 1) m  i=i0

cm,ix−αntμm,iKi+1,n−1(t, x) = ¯K(t, x).

f ∈ L Dm_β¯f(t) = 1  t ¯ K(t, x)Dn_¯αf(x)dx = ¯KDn_¯αf(t). f ∈ L  ¯KDn_¯αfq ≤ cDn_¯αfp,  ¯KFq ≤ cF p, ¯ K Lp Lq 1 ≤ q < p < ∞ K¯ Lp Lq

W_p,¯αn Lq ii) iii) ⇒ i) iii) f ∈ W_p,¯αn t= 1 m−1_ k=0 |Dk ¯ βf(1)| << n−1  k=i0 |Dk ¯αf(1)|. 1  0 |tμm,i_K i+1,j(t, 1)|qdt <∞, i = i0, i0+ 1, . . . , m; j = i, i + 1, . . . , n − 1, KiDn_¯αf(t) = tμm,i 1  t x−αnKi+1,n−1(t, x)Dn ¯αf(x)dx, i = i0, i0+ 1, . . . , m, Lp(0, 1) Lq(0, 1) 0 ≤ i ≤ j ≤ n − 1 1  0 |tμm,i_K i+1,j(t, 1)|qdt << 1  0

tq[μm,i− maxi≤s≤j( s

k=i+1αk+i−s)]|lnt|qli,j_dt.

i0 ≤ i ≤ j ≤ m ≤ n − 1 μm,i− max i≤s≤j( s  k=i+1 αk+ i − s) + 1 q >0, | ¯β| − |¯α| + n − m +1_q > max i≤s≤j( s  k=i+1 αk− s) − n  k=i+1 αk+ n = = max i≤s≤j(n − s − n  k=s+1 αk). Mi0 ≥ max_i≤s≤j(n−s− n  k=s+1αk) i0 ≤ i ≤ j ≤ n−1 i= 0, 1, . . . , m j = i, i + 1, . . . , n − 1

1 ≤ q < p < ∞ Lp(0, 1) Lq(0, 1)

Bn = max_i

0≤i≤mi≤j≤n−1max B

n i,j <∞, B_i,jn = ⎧ ⎨ ⎩ 1  0  1 t |x −αn_Kj+1,n−1(t, x)|p_dx q(p−1) p−q × × ⎛ ⎝ t  0 |sμm,i_{Ki+1,j(s, t)|}q_ds ⎞ ⎠ q p−q d ⎛ ⎝ t  0 |sμm,i_{Ki+1,j(s, t)|}q_ds ⎞ ⎠ ⎫ ⎪ ⎬ ⎪ ⎭ p−q pq . Bi,j Ki+1,j t  0 |sμm,i_K i+1,j(s, t)|qds= [s = tz, ds = tdz] = = tμm,iq+1 ⎛ ⎝ 1  0 |zμm,i_{Ki+1,j(tz, t)|}q_dz ⎞ ⎠ = = tμm,iq+1+q j k=i+1(1−αk) ⎛ ⎝ 1  0 |zμm,i_K i+1,j(z, 1)|qdz ⎞ ⎠ . | ¯β| − |¯α| + n − m +1 q = m  k=0βk− i  k=0αk+ i − m + n − i − n  k=i+1αk+ 1 q > > Mi₀ ≥ n − j − n  k=j+1 αk. μm,i+ j − i − j  k=i+1 αk+ 1 q >0

1 + qμm,i+ q j  k=i+1 (1 − αk) > 0, d ⎛ ⎝ t  0 |sμm,iKi+1,j(s, t)|q_ds ⎞ ⎠ = = c · d ⎛ ⎝t1+qμm,i+q j k=i+1(1−αk) ⎞ ⎠ = c1· tq(μm,i+ j k=i+1(1−αk))_dt, c= 1  0 |sμm,i_{Ki+1,j(s, 1)|}q_{ds, c} 1 = c ·  1 + qμm,i+ q j  k=i+1 (1 − αk)  , i= i0, i0+ 1, . . . , m, j = i, i + 1, . . . , n − 1. B_i,jn << ⎧ ⎨ ⎩ 1  0 t(q(μm,i+ j k=i+1(1−αk))+1) q p−q +q(μm,i+ j k=i+1(1−αk))× × ⎛ ⎝ 1  t |x−αn_{Kj+1,n−1(t, x)|}p_dx ⎞ ⎠ q(p−1) p−q dt ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ p−q pq = = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1  0 t(μm,i+ j k=i+1(1−αk)+ 1p) pq p−q ⎛ ⎝ 1  t |x−αn_{Kj+1,n−1(t, x)|}p_dx ⎞ ⎠ q(p−1) p−q dt ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ p−q pq . p− 1 p = 1 p B_i,jn << ⎧ ⎨ ⎩ 1  0 ⎛ ⎝tμm,i+ j k=i+1(1−αk)+ 1p×

× ⎛ ⎝ 1  t |x−αnKj+1,n−1(t, x)|p_dx ⎞ ⎠ 1 p ⎞ ⎟ ⎟ ⎠ pq p−q dt ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ p−q pq . Ki+1,j ₁ t |x −αn_Kj+1,n−1(t, x)|p_dx 1 p = t−αn+ 1_p 1 t 1 |x −αn_Kj+1,n−1(t, tx)|p_dx 1 p = = t−αn+ 1p+ n−1 k=j+1(1−αk) ⎛ ⎜ ⎝ 1 t  1 |x−αn_Kj+1,n−1(1, x)|p_dx ⎞ ⎟ ⎠ 1 p << << t− 1p+ n k=j+1(1−αk) ⎛ ⎜ ⎝ 1 t  1 |xp_(Mj−1) |lnx|p_lj dx ⎞ ⎟ ⎠ 1 p , j= i0, i0+ 1, . . . , n − 1. ∞  1 xp(Mj−1)|lnx|plj_{dx <}∞ _M j < 1 p, j = i0, i0+ 1, . . . , n − 1, t >0 ⎛ ⎝ 1  t |x−αn_{Kj+1,n−1(t, x)|}p_dx ⎞ ⎠ 1 p << ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t n k=j+1(1−αk)−Mj|lnt|lj _M j> 1 p, t n k=j+1(1−αk)− 1p _Mj< 1 p, t n k=j+1(1−αk)− 1p|lnt|lj+ 1p _M j= 1 p.

B_i,jn << ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎝1 0 t (μm,i+ j k=i+1(1−αk)+ 1 p−Mj)p−qpq _|lnt|lj_p−qpq dt ⎞ ⎠ p−q pq Mj > 1 p, ⎛ ⎝1 0 t (μm,i+ j k=i+1(1−αk)) pq p−q dt ⎞ ⎠ p−q pq Mj < 1 p, ⎛ ⎝1 0 t(μm,i+ j k=i+1(1−αk)) pq p−q |lnt|(lj+ 1_p)_p−qpq _dt ⎞ ⎠ p−q pq Mj = 1 p. Bn_i,j i0≤ i ≤ m i ≤ j ≤ n − 1 μm,i+ n  k=i+1 (1 − αk) +1_p − Mj > q− p pq , | ¯β| − |¯α| + n − m +1_q > Mj Mj > 1 p, μm,i+ n  k=i+1 (1 − αk) > q_pq− p, | ¯β| − |¯α| + n − m +1 q > 1 p Mj ≤ 1 p. i j Mi i = i0, i0+ 1, . . . , n−1 Bn = max_i

0≤i≤mi≤j≤n−1max B

n i,j ii) ⇒ iii) ii) f ∈ W_p,¯αn f0(t) = t−α0_K1,n−1(t, 1) _Dn ¯αf0(t) = 0 t∈ (0, 1) Di_¯αf0(1) = 0 i = 0, 1, . . . , n − 2 |Dn−1¯α f0(1)| = 1 f0 ∈ Wp,¯αn f0Wn p, ¯α = 1 Dm ¯ βf0q ≤ c.

Dm ¯ βf0q > 0 1  0 | m  i=i0 (−1)i_c

m,itμm,iKi+1,n−1(t, 1)|qdt≤ cq.

Ki+1,n−1(t, 1) >> tn−i− n k=i+1αk−Mi 0 ≤ i ≤ n − 1 t >0 tμm,i_Ki+1,n−1(t, 1) >> t| ¯β|−|¯α|+n−m−Mi_{, i}= i0_{, i} 0+ 1, . . . , m, t >0 cm,i0 = 0 Mi0 ≥ Mi i0≤ i ≤ m Mi0 > 1_p t| ¯β|−|¯α|+n−m−Mi0 _t(| ¯β|−|¯α|+n−m−Mi0)q t= 0 | ¯β| − |¯α| + n − m +1 q > Mi0. f1(t) = tn−|¯α|− εp 0 < ε < 1 Dn_¯αf1(t) = n−1 j=0 (n − j − n  k=j+1 αk− ε p)t− εp. f1∈ Wp,¯αn Dm_β¯f1(t) = m−1 i=0  _i  k=0 βk− |¯α| + n − i −ε p  t| ¯β|−|α|+n−m− εp. ε0 > 0 ε∈ (ε0,1) m−1 i=0  _i  k=0 βk− |¯α| + n − i − ε p  = 0. D_βm¯f1∈ Lq(0, 1) | ¯β| − |¯α| + n − m − ε p+ 1 q >0 ε∈ (ε0,1). ε→ 1 | ¯β| − |¯α| + n − m + 1 q ≥ 1 p.

Mi0 < 1 p | ¯β| − |¯α| + n − m +1 q − 1 p = 0. fε(t) = cεt−α0 1  t K1,n−1(t, x)x−αn_χ0,ε(x)x− εp_{dx, ε} 0< ε <1, cε χ0,ε(·) (0, ε) Dn_¯αfε(t) = cε(−1)nχ(0,ε)(t)t− εp fε∈ W_p,¯αn ε∈ (0, 1) cε fεW_p1 = Dn_¯αfεp= 1 cε = (1 − ε) 1 p_εε−1p _. fε 0 < ε < 1 ε→ 0 W_p,¯αn

Lp(I) × Rn (W_p,¯αn )∗= (Lp(I) × Rn)∗= Lp(I) × Rn

Di_¯αfε(1) = 0 i = 0, 1, . . . , n − 1 G= (g, a) ∈ Lp(I) × Rn | < fε, G >| = | 1  0 Dn_¯αfε(t)g(t)dt| = cε| ε  0 t− εpg(t)dt| ≤ ≤ cε ⎛ ⎝ ε  0 t−εdt ⎞ ⎠ 1 p⎛ ⎝ ε  0 |g(t)|p dt ⎞ ⎠ 1 p = ⎛ ⎝ ε  0 |g(t)|p dt ⎞ ⎠ 1 p . < fε, G >→ 0 ε→ 0 G∈&Wp,¯αn '∗ fε 0 < ε < 1 ε→ 0 W_{q, ¯}m_β D_βm¯fε(t) = m  i=i0 cm,itμm,iDi¯αfε(t) = = m  i=i0 (−1)i_c m,itμm,i 1  t Ki+1,n−1(t, x)x−αn_χ0,ε(x)x− εp_dx.

i= i0, i0+ 1, . . . , m ε∈ (0, 1) 1  0 |tμm,i 1  t Ki+1,n−1(t, x)x−αn_χ0,ε(x)x− εp_dx|q_{dt <∞} 1  0 |tμm,i 1  t Ki+1,n−1(t, x)x−αn− εpdx|qdt << << 1  0 |tμm,i−αn− εp+1+ n−1 k=i+1(1−αk) 1 t  1 zMi−1− εp|lnz|li_dz|q_dt. Mi0 < 1 p Mi ≤ M0 i= i0, i0+ 1, . . . , m ε∈ (0, 1) Mi− 1 − ε p <0 i = 0, 1, . . . , m 1 t  1 zMi−1− εp|lnz|li_dz ≤ 1 t  1 |lnz|li_dz ≤ 1 t|lnt| li_, 1  0 |tμm,i 1  t Ki+1,n−1(t, x)x−αn− εpdx|qdt << 1  0 t(μm,i−αn− εp+ n−1 k=i+1(1−αk))q|lnt|qli_dt. μm,i− αn−_pε+ n−1  k=i+1 (1 − αk) > −1_q, ∀ε ∈ (0, 1). Dm ¯ βfεq = cε ₁ 0 | m  i=i0 (−1)i_cm,itμm,i 1 t Ki+1,n−1(t, x)x −αn− εp_{χ0,ε(x)dx|q} dt 1 q = = cε ⎛ ⎝ ε  0 | m  i=i0 (−1)i_c m,itμm,i ε  t Ki+1,n−1(t, x)x−αn− εpdx|qdt ⎞ ⎠ 1 q .

t → εt x → εx Dm ¯ βfεq = ε | ¯_{β|−|¯α|+n−m+ 1q−p}1 Tε = Tε, Tε = (1 − ε)1p ⎛ ⎝ 1  0 | m  i=i0 (−1)i_cm,itμm,i 1  t Ki+1,n−1(t, x)x−αn− εp dx|qdt ⎞ ⎠ 1 q . Tε <∞ ε∈ (0, 1) T0= lim_ε→0Tε = lim ε→0(1 − ε) 1 p ⎛ ⎝ 1  0 | m  i=i0 (−1)i_c m,itμm,i 1  t Ki+1,n−1(t, x)x−αn− εpdx|qdt ⎞ ⎠ 1 q = = ⎛ ⎝ 1  0 | m  i=i0 (−1)i_c m,itμm,i 1  t Ki+1,n−1(t, x)x−αn_dx|q_dt ⎞ ⎠ 1 q = = ⎛ ⎝ 1  0 |Dm ¯ β(t−α0K1,n(t, 1))|qdt ⎞ ⎠ 1 q = 0, D_βm¯(t−α0K1,n(t, 1)) = 0 t ∈ (0, 1] Dm_β¯fεq → 0 ε → 0 fε W_{q, ¯}m_β ε→ 0 Mi0 < 1 p | ¯β| − |¯α| + n − m +1_q > 1 p, I = (0, 1) αk = 0 k = 0, 1, . . . , n − 1 αn = γ βi= 0 i = 0, 1, . . . , m − 1 βm = υ Ln_p,γ Lm_q,υ Mi0 = max_i 0≤s≤n−1(n−s−γ) = n−γ−i0

I = (0, 1) 0 ≤ m < n 1 ≤ q < p < ∞ Ln_p,γ(I) → W_{q, ¯}m_β(I) Ln_p,γ(I) → W_{q, ¯}m_β(I) | ¯β| − γ + n − m + 1 q >max{n − γ − i0, 1 p} I = (0, 1) 0 ≤ m < n 1 ≤ q < p < ∞ Wp,¯α(I) → Ln m_q,υ(I) W_p,¯αn (I) → Lm_q,υ(I) υ− |¯α| + n − m +1 q >max{Mi0, 1 p}

p

≤ q

I = (0, 1) 1 < p ≤ q < ∞ 0 ≤ m < n Mi0 ≥ 1 p | ¯β| − |¯α| + n − m +1 q > Mi0. i) ⇒ iii) Dm ¯ βfq ≤ cfWp, ¯nα, ∀f ∈ W n p,¯α. f0(t) = t−α0_K1,n−1(t, 1) _Dn ¯αf0(t) = 0 ∀t ∈ (0, 1) Di_¯αf0(1) = 0 i = 0, 1, . . . , n − 2 |Dn−1_¯α f0(1)| = 1 f0∈ Wp,¯αn f0Wn p, ¯α = 1 Dm ¯ βf0q ≤ c. Dm ¯ βf0q >0 1  0 | m  i=i0 (−1)i_c

t >0 Ki+1,n−1(t, 1) >> tn−i− n k=i+1αk−Mi_{, i}= 0, 1, . . . , n − 1. tμm,i_Ki+1,n−1(t, 1) >> t| ¯β|−|¯α|+n−m−Mi_{, i}= i0_{, i} 0+ 1, . . . , m, t = 0 cm,i0 = 0 Mi0 ≥ Mi i0 ≤ i ≤ m Mi0 > 1 p t= 0 t| ¯β|−|¯α|+n−m−Mi0 t(| ¯β|−|¯α|+n−m−Mi0)q _t= 0 | ¯β| − |¯α| + n − m +1 q > Mi0. i) ⇒ iii)

iii) ⇒ ii) iii)

f ∈ W_p,¯αn D_βm¯f(t) = m  i=i0 cm,itμm,i n−1  j=i (−1)j−i_{Ki+1,j(t, 1)D}j ¯αf(1)+ + m  i=i0 cm,itμm,i 1  t x−αn_Ki+1,n−1(t, x)Dn ¯αf(x)dx. m−1_ k=0 |Dk ¯ βf(1)| << n−1  k=i0 |Dk ¯αf(1)|, 1  0

KiF(t) = tμm,i 1  t x−αn_{Ki+1,n−1(t, x)F (x)dx, i = i0, i} 0+ 1, . . . , m, Lp(0, 1) Lq(0, 1) Lp(0, 1) Lq(0, 1) i0 ≤ i ≤ m 1  0 |tμm,i_Ki+1,j(t, 1)|q_{dt <<} 1  0 tq(μm,i+j−i+1− j+1

k=i+1αk−Mi,j)|lnt|qli,j_dt.

μm,i+j−i+1− j+1  k=i+1 αk−Mi,j+1 q >0, i = i0, i0+1, . . . , m, j = i, i+1, . . . , n−1, | ¯β| − |¯α| + n − m + 1 q >i≤s≤jmax  j− s + 1 − j+1  k=s+1 αk  − j − n  k=j+2 αk+ n = = max_i≤s≤j  n− s + 1 − n  k=s+1 αk  . Mi0 ≥ max_i≤s≤j  n− s + 1 − n k=s+1αk  i0 ≤ i ≤ m i≤ j ≤ n − 1 Lp(0, 1) Lq(0, 1) 1 ≤ p ≤ q < ∞ max

i≤j≤n−1_0<z<1sup Ai,j(z) < ∞, i = i0, i0+ 1, . . . , m,

lim