Embedding theorems for spaces with multiweighted derivatives

LICENTIATE T H E S I S

Luleå University of Technology Department of Mathematics

2007:53|: 02-757|: -c -- 07⁄53 -- 

2007:53

Embedding Theorems for Spaces

with Multiweighted Derivatives

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1 < p ≤ q < ∞ 1 < q < p < ∞

 |ν|≤l (−1)ν_(a υ(x)uν(x))ν = 0, x ∈ G,  G  |ν|=l a_ν(x)[uν(x)]2dx, ν∈ N₀n, G n Rn N₀n n |ν| ν G a_ν(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= w(r)p,α(G) u: G → R fk k∈ N₀k r G u_w(r) p,α :=  |k|=r f(k)_p,α<∞. α w(r)p,α w(r)p,α≡ w(r)p,0 w_p,α(r) α < r−n−m_p ∂G u ∈ w(r)p,α ∂G α u∈ w_p,α(r)

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 >1 −1_p f x= 0 (1, +∞) γ < n− 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} P_n−1= a₀+ a₁x+ . . . + a_n−1xn−1 lim x→+∞[f(x) − Pn−1(x)] (k)_{= 0, k = 0, 1, . . . , n − 1.} f ∈ Ln_p,γ

P_n−1 I (ly)(t) = n  i=0 a_i(t)y(i)(t) a_i(·) i = 0, 1, . . . , n I l a−1_n (t) = _a1 n(t) ai(t) i= 0, 1, . . . , n − 1 I l l I ly = f ly(x) = 0 n− 1 I ly= 0 t→ ∞ P_n−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 α1d 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 Di_¯α i = 0, 1, . . . , n W_p,¯αn = W_p,¯αn (0, 1) 1 < p < ∞ f : (0, 1) → R fWn p,¯α := D n ¯αfp+ n−1  i=0 |Di ¯αf(1)|. W_p,¯αn Ln_p,γ x= 0 W_p,¯αn → W_{q, ¯}m_β,

1 ≤ p, q < ∞ 0 ≤ m < n ¯β = (β0, β1, . . . , βm) βi ∈ R i = 0, 1, . . . , m 1 < q < p < ∞ 1 ≤ p ≤ q < ∞ W_p,¯αn (0, 1) W_p,¯αn (1, +∞) x= 1_t W_p,¯αn (0, 1) W_p,¯αn (1, +∞) (0, 1] [1, +∞) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u₀(t) = tα0 u₁(t) = tα01 t tα1 1 dt1 u₂(t) = tα01 t tα1 1 1  t1 tα2 2 dt2dt1 . . . u_n(t) = tα01 t tα1 1 1  t1 tα2 2 . . . 1  tn−1 tαn n dtndtn−1. . . dt1 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ v₀(t) = tβ0 v₁(t) = tβ0t 1 tβ1 1 dt1 v₂(t) = tβ0t 1 tβ₁1 t1  1 tβ₂2dt₂dt₁ . . . v_n(t) = tβ0t 1 tβ₁1 t1  1 tβ₂2. . . tn−1 1 tβn n dtndtn−1. . . dt1.

{ui(·)}n

i=0 {vi(·)}ni=0

(0, 1] [1, +∞) {vi(·)}ni=0 P_n(·) P_n(·) = n i=0ciui(·) {ui(·)}n i=0 Φ[a, b] Φ[a, b] u(·) ∈ Φ[a, b] P_n0(·) u − P_n0 = inf Pn u − Pn u(·) {ui(·)}n i=0 [a, b] {ui(·)}ni=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(·)}ni=0 {vi(·)}ni=0 α β D_n+1u(t) = 0. L_p(I) I = (0, 1) I = (1, +∞) uLp := ⎛ ⎝ I |u(t)|p_dt ⎞ ⎠ 1 p , 1 ≤ 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

B_l= sup a≤x≤b ⎛ ⎝ b  x w(t)dt ⎞ ⎠ 1 q ⎛ ⎝ x  a v−p(t)dt ⎞ ⎠ 1 p <∞. H_l B_l ≤ Hl ≤ (1 + q p) 1 q(1 +p q) 1 p_Bl 1 ≤ q < p < ∞ 1 r = 1q−1p v w A_l= ⎧ ⎪ ⎨ ⎪ ⎩ b  a ⎡ ⎢ ⎣ ⎛ ⎝ b  x w(t)dt ⎞ ⎠ 1 q ⎛ ⎝ x  a v1−p(t)dt ⎞ ⎠ 1 q⎤ ⎥ ⎦ r v1−p(x)dx ⎫ ⎪ ⎬ ⎪ ⎭ 1 r <∞. H_l q1q  p_q r 1 q A_l≤ H_l≤ 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 B_r= sup a≤x≤b ⎛ ⎝ x  a w(t)dt ⎞ ⎠ 1 q ⎛ ⎝ b  x v−p(t)dt ⎞ ⎠ 1 p <∞. H_r B_r ≤ Hr ≤ (1 + q p) 1 q(1 +p q) 1 p_Br 1 ≤ q < p < ∞ 1 r = 1q−1p v w A_r= ⎧ ⎪ ⎨ ⎪ ⎩ b  a ⎡ ⎢ ⎣ ⎛ ⎝ x  a w(t)dt ⎞ ⎠ 1 q ⎛ ⎝ b  x v1−p(t)dt ⎞ ⎠ 1 q⎤ ⎥ ⎦ r v1−p(x)dx ⎫ ⎪ ⎬ ⎪ ⎭ 1 r <∞.

H_r q1q  p_q r 1 q A_r≤ H_r≤ q1q(p)q1_Ar γ <1 −1_p 1 ≤ p < +∞ ⎡ ⎣ b  a   xγ−1 x  a f(t)dt    p dx ⎤ ⎦ 1 p ≤ 1 1 − 1 p− γ ⎛ ⎝ b  a |tγ_f(t)|p_dt ⎞ ⎠ 1 p . p→ +∞

ess sup_x∈(a,b)   xγ−1 x  a f(t)dt  

≤ 1 − γ1 ess supx∈(a,b)|xγf(x)| , γ < 1.

p < ∞ p = q w1q(x) = xγ−1 v1p(x) = xγ _p= +∞ _p→ ∞ γ >1 −1_p 1 ≤ p < +∞ ⎡ ⎣ b  a   xγ−1 b  x f(t)dt    p dx ⎤ ⎦ 1 p ≤ 1 γ− (1 −1_p) ⎛ ⎝ b  a |tγ_f(t)|p_dt ⎞ ⎠ 1 p . p→ +∞

ess sup_x∈(a,b)   xγ−1 b  x f(t)dt  

≤ γ− 11 ess supx∈(a,b)|x

γ_{f(x)| , γ > 1.} w1q(x) = xμ _v1p(x) = xγ 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

Ln_p,γ(I) W_p,¯αn R n γ∈ R 1 ≤ p ≤ ∞ 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|tγf(n)(t)|,

p= +∞

Ln_p,γ L_p,γ = L0_p,γ Ln_p = Ln_p,0

f I→ R t₀∈ [0, +∞) f₁ I → R f lim t→t0f1(t) = f1(t0) f t= t₀ f(t₀) f f₁ lim t→+∞f1(t) f t→ ∞ f(+∞) f(+∞) = lim t→+∞f1(t) f₁ f (1, +∞) → R P_n−1(t) =n−1 υ=0aυt υ lim t→+∞[f(t) − Pn−1(t)] (k)_{= 0, k = 0, 1, . . . , n − 1,} f n P_n−1(t) t→ +∞ 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−1p − mixed case, γ > n−1_p − strong degeneration. γ < 1 − 1_p n = 1 L1_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)fp,γ ||f(t0)| − |f(0)|| ≤ c2(γ, p)fp,γ p= 1 (i) (iv) γ= 0

f (0, 1) p = 1 γ < 0 tγ≥ 1 0 < t ≤ 1 1  0 |f_{(t)|dt =} 1  0 |t−γ_tγ_f_{(t)|dt ≤} 1  0 |tγ_f_{(t)|dt = f}_ 1,γ. 1 < p < +∞ 1  0 |f_{(t)|dt =} 1  0 |t−γ_tγ_f_{(t)|dt ≤} ⎛ ⎝ 1  0 |tγ_f_(t)|p_dt ⎞ ⎠ 1 p⎛ ⎝ 1  0 dt tγp ⎞ ⎠ 1 p = _ 1 1 − γ 1−1 p 1−1 pf _p,γ, 1 p+ 1 p = 1. p= +∞ γ <1 1  0 |f_{(t)|dt ≤ t}γ_f_(t) +∞ 1  0 dt tγ = 1 1 − γf+∞,γ. 1  0 |f_{(t)|dt ≤ c} 1(γ, p)fp,γ, 1 ≤ p ≤ +∞, c₁(γ, p) =  1 − γ(1 −1 p) −1 1 p−1 . p= 1 γ= 0 f(t₀) t₀ f(0) f(1) x ∈ (0, 1) y ∈ (0, 1) f(x) = f(y) + x  y f(t)dt

f y → 0 y → 1 f(0) f(1) y= 0 y= 1 1p= 1 y = t₀ p x fp= f(t0) + x  t0 f(t)dt_p≤ f(t₀)_p+  x  t0 f(t)dt_p ≤ [since 0 ≤ x ≤ 1] ≤ |f(t0)|1p+ 1  0 |f_{(t)|dt · 1} p ≤ [by (2.1.6) and (2.1.7)] ≤ |f(t0)| + c1(γ, p)fp,γ. x= t₀ p y f(t₀) = f(y) + t0  y f(t)dt_p≤ fp+ 1  0 |f_{(t)|dt · 1} p. |f(t0)| ≤ fp+ c1(γ, p)fp,γ. |fp− |f(t0)|| ≤ c1(γ, p)fp,γ, ||f(t0)| − |f(0)|| ≤ |f(t0) − f(0)| =    t0  0 f(t)dt   ≤ t0  0 |f_(t)|dt ≤ 1  0 |f_{(t)|dt ≤ c} 1(γ, p)fp,γ.

f ∈ Ln_p,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)_(t 0)| + f(r)p,γ  . P_n−1(t) =n−1 j=0 f(j)₍₀₎ j! tj n− 1 f [f(t) − Pn−1(t)](n−k)_{p,γ−k≤ cf}(n)_{p,γ, k= 1, 2, . . . , n.} f∈ Ln_p,γ f(n−1) ∈ L1_p,γ γ <1 −1_p t₀ ∈ [0, 1] f(n−1)(t₀) f(n−1)_p < +∞ f(n−1) ∈ L_p,0 f(n−2) ∈ L1_p,0 0 < 1 − 1_p 1 < p ≤ +∞ p = 1 α = 0 f(n−2) f(n−2)(t₀) f(n−2)p<+∞ f(n−3)(t₀) f(n−4)(t₀) . . . f(t₀) f(j) j = 0, 1, . . . , n − 1 γ= 0 fp≤ c(|f(t0)| + fp) ≤ c1(|f(t0)| + |f(t0)| + fp) ≤ . . . ≤ cn−2 _n−2  j=0 |f(j)_(t 0)| + f(n−1)p  . γ <1 −1_p p f(n−1) f(n−1)_{p≤ c(|f}(n−1)_(t 0)| + f(n)p,γ), f(j)(t₀) j = 0, 1, . . . , n − 1 t₀ = 0 P_n−1(t) f(n−k)_(t)−n−1 j=0 f(j)(0) j! (t j₎(n−k)_{p,γ−k ≤ cf}(n)_{p,γ, k= 1, 2, . . . , n.}

(tj₎(n−k)_{= j · (j − 1) · · · (j − n + k + 1)t}j−n+k υ= j − n + k f(n−k)_{(t) −}k−1 υ=0 f(n−k+υ)(0) υ! t υ_{p,γ−k ≤ cf}(n)_{p,γ, k= 1, 2, . . . , n.} k = 1 f(n−1)_{(t) − f}(n−1)_(0) p,γ−1= tγ−1 t  0 f(n)(x)dx_p≤ 1 1 −1 p− γ f(n)_p,γ. k < n k+ 1 F(t) = f(n−k−1)(t) − k  υ=1 1 υ!f (n−k−1+υ)_(0)tυ_. F(0) = f(n−k−1)(0) f(n−k−1)(t) − f(n−k−1)(0) − k  υ=1 1 υ!f (n−k−1+υ)_(0)tυ _{= F (t) − F (0),} f(n−k−1)(t) − k  υ=0 1 υ!f (n−k−1+υ)_(0)tυ_{= F (t) − F (0),} F(t) = f(n−k)(t) − k  υ=1 1 (υ − 1)!f(n−k−1+υ)(0)tυ−1= (we replace μ = υ − 1) = f(n−k)_{(t) −}k−1 μ=0 1 μ!f (n−k+μ)_(0)tμ_. f(n−k−1)_{(t) −}k υ=0 1 υ!f (n−k−1+υ)_(0)tυ_ p,γ−k−1 = tγ−k−1_[fn−k−1_{(t) −}k υ=0 f(n−k+υ−1)(0) υ! t υ_] p= tγ−k−1[F (t) − F (0)]p

= tγ−k−1 t  0 F(x)dx_p≤ [by (1.4.3) and (1.4.4)] ≤ 1 1 −1 p− γ + k tγ−k_F_(t) p=_{1 −} 1 1 p− γ + k tγ−k_[f(n−k)_(t)− − k−1  μ=0 1 μ!f (n−k+μ)_(0)tμ_]

p≤ [by our induction assumption]

≤ c 1 −1 p− γ + k f(n)_p,γ. γ > n− 1_p f ∈ Ln_p,γ γ > n− 1_p f t= 0 f∈ L1_p,γ γ >1 −1_p |fp,γ−1− c|f(1)|| ≤ _{γ− 1 + 1/p}1 fp,γ, c= tγ−1_p γ >1 −1_p c= tγ−1_p= ⎛ ⎝ 1  0 t(γ−1)pdt ⎞ ⎠ 1 p <+∞. fp,γ−1= tγ−1_{[f(1) −} 1  t f(x)dx]_p≤ |f(1)|tγ−1p +tγ−1 1  t f(x)dx_p≤ c|f(1)| + 1 γ− 1 + 1/pf _p,γ.

f(1) = f(t) + 1  t f(x)dx tγ−1 p: |f(1)|tγ−1_{p≤ fp,γ−1+t}γ−1 1  t f(x)dx_p≤ fp,γ−1+ 1 γ− 1 + 1/pf _p,γ. 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. f ∈ Ln_p,γ f(n−1) ∈ L1_p,γ k= 1 f(n−1)_{p,γ−1≤ c |f}(n−1)_{(1)| + f}(n)_p,γ!_. s < n s+ 1 f(n−(s+1))_{p,γ−(s+1)≤ c} 1 |f(n−s−1)(1)| + f(n−s)p,γ−s! ≤ c  |f(n−s−1)_{(1)| +}s j=1 |f(n−j)_{(1)| + f}(n)_ p,γ  . 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+ 1p γ "γ "γ = ⎧ ⎨ ⎩ [γ + 1 p], if γ > 1 −1p, 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 −1_p γ <1 −1_p "γ = 0 −1 p < γ <1 −1p "γ = 0 γ >1 −1p γ+1_p "γ = [γ +1 p] < γ + 1 p. γ_j = ⎧ ⎨ ⎩ γ− n + j, if γ = n − "γ, n − "γ + 1, . . . , n − 1, 0, if γ= 0, 1, . . . , n − "γ − 1. 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 1−1_p < γ < n−1_p f(n−"γ) < +∞ f(n−"γ)∈ L"γ_p,γ γ >"γ −1_p f(n−k)_{p,γ−k ≤ c}  f(n)_p,γ+k j=1 |f(n−j)_(1)|  , k= 1, 2, . . . , "γ. k= 1, 2, . . . , "γ − 1 γ− k > γ − "γ + 1 > 1 −1_p f(n) f(n−1) . . . f(n−"γ+1) t= 0 k= "γ f_Ln−"γ p,γ−"γ = f (n−"γ)_{p,γ−"γ <∞,} γ− "γ < 1 −1_p f(j)(0) j = 0, 1, . . . , n − "γ − 1 f(j)_{p<+∞, j = 0, 1, . . . , n − "γ − 1.} γ_j 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−1p)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 ∈ L1_p,γ lim t→+∞f(t) = limt→+∞ln t = +∞ γ < 1 − 1_p f ∈ Ln_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 |f_p,−1 p−δ− (δp) −1 p|f(1)|| ≤ δf p,γ. f ∈ L1_p,γ γ >1 −1_p f∈ Lp,γ 1 < p < +∞ 1 p+p1 = 1 +∞  1 |f_{(t)|dt ≤} ⎛ ⎝ +∞  1 |tγ_f_(t)|p_dt ⎞ ⎠ 1 p⎛ ⎝ +∞  1 dt tγp ⎞ ⎠ 1 p = cf_{p,γ <+∞,} γp>1 p= 1 γ >0 tγ >1 t >1 +∞  1 |f_{(t)|dt <} +∞  1 |tγ_{f(t)|dt = f}_ 1,γ <+∞. p= +∞ γ >1 +∞  1 |f_{(t)|dt ≤ t}γ_f_(t) +∞· +∞  1 dt tγ = 1 γ− 1f _ +∞,γ. f (1, +∞) f(t) = f(1) + t  1 f(x)dx lim t→+∞f(t) = f(+∞) ||f(+∞)| − |f(1)|| ≤ |f(+∞) − f(1)| ≤ | +∞  1 f(t)dt| ≤ +∞  1 |f_(t)|dt f (1, +∞) ||f(+∞)| − |f(1)|| ≤ cf_ p,γ. f(t) = f(1) + t  1 f(x)dx (∗)

t−1p−δ _{δ >} 0 _p f_p,−1 p−δ = t −1 p−δ_f(t)_p≤ |f(1)|t−p1−δ_p+ t−1p−δ t  1 f(x)dx_p. t−1p−δp= ⎛ ⎝ +∞  1 t(−1p−δ)p_dt ⎞ ⎠ 1 p = (δp)−p1_, lim p→+∞(δp) −1 p = 1. f_p,−1 p−δ ≤ (δp) −1 p|f(1)| + δt1−1p−δ_f(t)_p. γ >1 −1_p >1 −1_p− δ t1−p1−δ _{< t}γ _{t >}1 f_p,−1 p−δ ≤ (δp) −1 p|f(1)| + δf_p,γ. (∗) −f(1) = −f(t)+t 1 f(x)dx (δp)−1p|f(1)| ≤ f p,−1 p−δ+ δf _p,γ. γ > n− 1_p f ∈ Ln_p,γ f(n−1)(+∞) t→ +∞ f f(t) = tn−1 ∈ Ln_p,γ f_Ln 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 . . . a_m−1 s= 1, 2, . . . , m lim t→+∞[f (n−s)_{(t) −}s−1 μ=1 a_m−s+μ μ! t μ_{] = a} m−s, f(n−s)_{(t) −}s−1 μ=0 a_m−s+μ μ! t μ_{p,γ−s≤ cf}(n)_p,γ. γ > 1 − 1_p lim t→+∞f (n−1)_{(t) = a} m−1 s= 1 s= 1 f(n−1)_{(t) − a} m−1p,γ−1= tγ−1[f(n−1)(t) − f(n−1)(+∞)]p = tγ−1 +∞  t f(n)(x)dx_p= [by (1.4.5) and (1.4.6)] ≤ c₁f(n)p,γ. a_m−1 a_m−2 . . . a_m−s s < m s+ 1 s g(t) = f(n−s−1)(t) − s−1  μ=0 a_m−s+μ (μ + 1)!tμ+1. g_{p,γ−s= t}γ−s_[f(n−s)_{(t) −}s−1 μ=0 a_m−s+μ μ! t μ_] p≤ cf(n)p,γ. s≤ m − 1 γ− s ≥ γ − m + 1 > m −1 p− m + 1 = 1 − 1 p, lim t→+∞g(t) = am−s−1

s s+ 1 f(n−s−1)_{(t) −}s μ=0 a_m−s−1+μ μ! t μ_ p,γ−s−1 = [f(n−s−1)_{(t) −}s μ=1 a_m−s−1+μ μ! t μ_{] − am−s−1p,γ−s−1} = [f(n−s−1)_{(t) −}s−1 μ=0 a_m−s+μ (μ + 1)!tμ+1] − am−s−1p,γ−s−1. lim t→+∞g(t) = am−s−1 f(n−s−1)_{(t) −}s μ=0 a_m−s−1+μ μ! t μ_ p,γ−s−1= g(t) − g(+∞)p,γ−s−1 = tγ−s−1 +∞  t g(x)dx_p. γ−s > 1−1_p f(n−s−1)_{(t) −}s μ=0 a_m−s−1+μ μ! t μ_{p,γ−s−1= c} 1gp,γ−s≤ c2f(n)p,γ. f ∈ Ln_p,γ γ > m−1_p 1 ≤ m ≤ n b_ν = aν ν!, ν = 0, 1, . . . , m − 1, P_m−1(t) = m−1  ν=0 b_νtν, f∈ Ln_p,γ lim t→+∞[f (n−m)_{(t) − P} m−1(t)](k)= 0, k = 0, 1, . . . , m − 1, c >0 [f(n−m)_{(t) − P} m−1](k)p,γ−m+k≤ cf(n)p,γ, k= 0, 1, . . . , m − 1, a_ν

P_m−1 μ= ν − k P_m−1(k) (t) = m−k−1_ μ=0 (k + μ)(k + μ − 1) · · · (μ + 1)bk+μtμ, k= 0, 1, . . . , m − 1. s= m − k [f(n−m)_{(t) − P} m−1(t)](k)= [f(n−m)(t) − Pm−1(t)](m−s) = f(n−s)_(t)−s−1 μ=0 (m−s+μ)(m−s+μ−1) · · · (μ+1)bm−s+μtμ= [see (2.1.35)] = f(n−s)_{(t) −}s−1 μ=0 (m − s + μ)(m − s + μ − 1) · · · (μ + 1) (m − s + μ)! am−s+μtμ = f(m−s)_{(t) −}s−1 μ=0 a_m−s+μ μ! t μ_{, s= 1, 2, . . . , m.} [f(n−m)_{(t) − P} m−1(t)](k)= f(m−s)(t) − s−1  μ=0 a_m−s+μ μ! t μ_{, s= 1, 2, . . . , m,} I = (0, 1) ¯α = (α₀, α₁, . . . , α_n) α_i ∈ R i = 0, 1, . . . , n n |¯α| = n i=0αi 1 < p < ∞ f I→ R D0_¯αf(t) = tα0_f(t), Di_¯αf(t) = tαi d dtt αi−1d dt. . . t α1 d dtt α0_{f(t), i = 1, 2, . . . , n,}

Di_¯αf(t) α f i i= 0, 1, . . . , n α_j j = 0, 1, . . . , 2k k D2k_¯αf(t) B_tkf(t) B_t1f(t) =  d2 dt2+ 2ν + 1 t d dt  f(t), B_tkf(t) = B_t[B_tk−1f(t)], k = 1, 2, . . . . k= 1 D2_¯αf(t) = tα2 d dtt α1d dtt α0_f(t) = tα2+α1+α0  d2 dt2+ α₁+ 2α₀ t d dt +α0(α1+ α0− 1) t2  f(t). D2_¯αf(t) = B1_tf(t) α₀= 0 α₁= 2ν +1 α₂= −2ν −1 α₀= 2ν α₁= 1 − 2ν α₂= −1 k= 2 D4_¯αf(t) = tα4d dtt α3d dtt α2 d dtt α1d dtt α0_f(t) = tα4d dtt α3 d dtt α1 2+α22d dtt α1d dtt α0_{f(t) = t}α4d dtt α3 d dtt α1 2_B1 tf(t) = tα4+α3+α12  d2 dt2+ α₃+ 2α1₂ t d dt+ α1₂(α₃+ α₂1− 1) t2  B_t1f(t) = B_t2f(t), α₂= α1₂+ α2₂ α_i i= 1, 2, 3, 4 α₀= 0 α₁= 2ν + 1 α₂= −2ν − 1 α₃= 2ν + 1 α₄= −2ν − 1 α₀= 0 α₁= 2ν + 1 α₂= −1 α₃= 1 − 2ν α₄= −1 α₀= 2ν α₁= 1 − 2ν α₂= −1 α₃= 2ν + 1 α₄= −2ν − 1 α₀= 2ν α₁= 1 − 2ν α₂= 2ν − 1 α₃= 1 − 2ν α₄= −1 W_p,¯αn = W_p,¯αn (I) f I → R α n fWn p,¯α = Dn¯αfp+ n−1  i=0 |Di ¯αf(1)|,

 · p Lp(I) 1 ≤ p < ∞ n >1 γ_nα= 1, γ_n−1α = α_n, γ_iα= α_n+ n−1  k=i+1 (αk− 1), i = 0, 1, . . . , n − 1, γ_imax= max i≤j≤n−1γj, γ min i = min_{i≤j≤n−1}γj, i= 0, 1, . . . , n − 1. α_i= γ_i−1α − γα_i + 1, i = 1, 2, . . . , n − 1. W_p,¯αn γ_i 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< γα_max j = 1, 2, . . . , n − 1 w₀(t, x) ≡ 1, w_j(t, x) = t−α0 x  t t−α1 1 x  t1 t−α2 2 . . . x  tj−1 t−α_j jdt_jdt_j−1. . . dt₁ ¯ w₀(x, t) ≡ 1, ¯w_j(x, t) = t−α0 t  x t−α1 1 t1  x t−α2 2 . . . tj−1  x t−α_j jdt_jdt_j−1. . . dt₁. {wj(t, x)}n−1 j=0 { ¯wj(x, t)}n−1j=0 x ∈ R Dn_¯αw(t) = 0 (x, +∞) (0, x) x > 0 t= 0

γ_maxα <1 − 1 p. ∀f ∈ Wn p,¯α {wj(t, 1) = wj(t)}n−1j=0 P_n(t; f) = n−1  j=0 a_j(f)w_j(t), a_n−1(f) = (−1)n−1lim t→0D n−1 ¯α f(t), a_i(f) = (−1)ilim t→0D i ¯α # f(t) − n−1  j=i+1 a_j(f)w_j(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)]| ≤ c1Dn¯αfp,(0,x), i= 0, 1, . . . , n − 1. i= 0, 1, . . . , n − 1 Di_¯αf(t) = n−1  j=i a_j(f)Di_¯αw_j(t) + t  0 s−αn_Dn ¯αf(s) · Di¯αw¯n−1(s, t)ds. Dn−1_¯α f∈ W_p,α1 _n, α_n= γα_n−1<1 −1_p lim t→0D n−1 ¯α f(t) = ˜an−1= (−1)n−1an−1 Dn−1_¯α f(t) = (−1)n−1a_n−1+ t  0 Dn_¯αf(s)s−αn_ds,

Dn−1 ¯α f− (−1)n−1an−1p,γα n−1−1,(0,x)≤ cD n ¯αfp,(0,x), sup 0≤t≤x t−(1−1/p−γn−1α )[Dn−1 ¯α f(t) − (−1)n−1an−1] ≤c1Dn¯αfp,(0,x). Dn−1_¯α w_n−1(t) = (−1)n−1, Dn−1_¯α w_j(t) = 0, j = 0, 1, . . . , n − 2, Dn−1 ¯α (f − Pn)p,γα n−1−1,(0,x)≤ cD n ¯αfp,(0,x), sup 0≤t≤x t−(1−1/p−γn−1α )_Dn−1 ¯α [f(t) − Pn(t)] ≤c1Dn¯αfp,(0,x). i= n − 1 i= n − 1, n − 2, . . . , m > 0 i = m − 1 i= m ∀x ∈ (0, 1] d dtD m−1 ¯α # f− n−1  j=m a_jw_j $ p,γα m−1,(0,x) = D m ¯α # f− n−1  j=m a_jw_j $ p,γ_mα_−1,(0,x) = Dm ¯α [f − Pn] p,γ_mα_−1,(0,x)≤ c₁Dn_¯αfp,(0,x). Dm−1_¯α # f− n−1  j=m a_jw_j $ ∈ W1 p,γα m−1. γ_m−1α <1 −1_p lim t→0D m−1 ¯α # f(t) − n−1  j=m a_jw_j(t) $ = ˜am−1= (−1)m−1am−1 Dm−1 ¯α # f− n−1  j=m a_jw_j $ − (−1)m−1_{am−1p,γα} m−1−1,(0,x)

≤ cd dtD m−1 ¯α # f− n−1  j=m a_jw_j $ p,γα m−1,(0,x) sup 0≤t≤x   t−(1−1/p−γ α m−1) % Dm−1_¯α # f(t) − n−1  j=m a_jw_j(t) $ − (−1)m−1_a m−1 &   ≤ c1d dtD m−1 ¯α # f− n−1  j=m a_jw_j $ p,γα m−1,(0,x). Dm−1_¯α w_m−1(t) = (−1)m−1, Dm−1_¯α w_j(t) = 0, j = 0, 1, . . . , n − 2, i = m − 1 Dm−1_¯α # f(t) − n−1  j=m a_jw_j(t) $ = (−1)m−1_a m−1+ + t  0 d dsD m−1 ¯α # f(s) − n−1  j=m a_jw_j(s) $ ds Dm−1_¯α f(t) = n−1  j=m−1 a_jDm−1_¯α w_j(t) + t  0 s−αm # Dm_¯αf(s) − n−1  j=m a_jDm_¯αw_j(s) $ ds. i= m Dm_¯αf(s) − n−1  j=m a_jDm_¯αw_j(t) = s  0 Dn_¯αf(τ)τ−αn_Dm ¯αw¯n−1(τ, s)dτ. Dm−1_¯α f(t) = n−1  j=m−1 a_jDm−1_¯α w_j(t) + t  0 Dn_¯αf(τ)τ−αn t  τ s−αm_Dm ¯αw¯n−1(τ, s)dsdτ

= n−1 j=m−1 a_jDm−1_¯α w_j(t) + t  0 Dn_¯αf(τ)τ−αn_Dm−1 ¯α w¯n−1(τ, t)dτ. i= m − 1 W_p,¯αn γ_maxα < 1 − 1_p δ δ >1 −1_p− γα_min f(1)_Wn p,¯α = D n ¯αfp+ n−1  i=0 Di ¯αfp,γiα−1+δ W_p,¯αn γ_maxα <1 − 1_p δ >1 − 1_p− γ_minα  Di−1 ¯α fp,γα i−1−1+δ− ci|D i−1 ¯α f(1)| ≤ ¯ciDi¯αfp,γα i−1+δ, i= 1, 2, . . . , n, c_i ¯c_i f i= n Dn−1_¯α f(t) = Dn−1_¯α f(1) − 1  t s−αn_Dn ¯αf(s)ds tαn−1+δ = tγαn−1−1+δ _p Dn−1 ¯α fp,γn−1α −1+δ≤ tγ α n−1−1+δp|Dn−1 ¯α f(1)|+tγ α n−1−1+δ 1  t s−αn_Dn ¯αf(s)dsp.

γ_minα ≤ γ_n−1α ≤ γ_maxα <1 − 1_p 1 −1_p− γ_n−1α ≤ 1 −1_p− γ_minα < δ γ_n−1α + δ > 1 −1_p Dn−1 ¯α fp,γn−1α −1+δ≤ cn|D n−1 ¯α f(1)| + 1 γ_n−1α + δ − 1 + 1/p ⎧ ⎨ ⎩ 1  0 |sγα n−1+δ_s−αn_Dn ¯αf(s)|pds ⎫ ⎬ ⎭ 1 p , c_n= tγαn−1−1+δp_<+∞  Dn−1 ¯α fp,γα n−1−1+δ− cn|Dn−1¯α f(1)|   ≤ ¯cnDn¯αfp,γα n−1+δ, γ_n−1α = α_n γ_nα ≡ 1 Dn−1 ¯α fp,γα n−1−1+δ<+∞. i= k +1, k +2, . . . , n Dk ¯αfp,γα k−1+δ<+∞. i= k Dk−1_¯α f(1) − Dk−1_¯α f(t) = 1  t sγkα−γk−1α −1_Dk ¯αf(s)ds tγαk−1−1+δ _p |tγα k−1−1+δ_p|Dk−1 ¯α f(1)| − Dk−1¯α fp,γα k−1−1+δ| ≤ ≤ tγα k−1−1+δ 1  t sγkα−γαk−1−1_Dk ¯αf(s)dsp. |Dk−1 ¯α fp,γk−1α −1+δ− ck|D k−1 ¯α f(1)|| ≤_γα 1 k−1+ δ − 1 + 1/pD k ¯αfp,γαk−1+δ,

i= k Di ¯αfp,γα i−1+δ≤ ¯ci ' |Di ¯αf(1)| + Di+1¯α fp,γα i+1−1+δ ( , i= 0, 1, . . . , n − 1, ¯ci= ci+1· ¯ci+1 i= 0, 1, . . . , n − 1 γ_nα≡ 1 δ > 0 Di ¯αfp,γiα−1+δ≤ ¯cn #_n−1  j=i |Dj ¯αf(1)| + Dn¯αfp $ , i= 0, 1, . . . , n − 1. n−1  i=0 Di ¯αfp,γα i−1+δ+ D n ¯αfp≤ ¯c #_n−1  i=0 n−1  j=i |Dj¯αf(1)| + Dn¯αfp $ , f(1)_Wn p,¯α ≤ ¯cfWp,¯nα. i= 0, 1, . . . , n − 1 |Di ¯αf(1)| ≤ ci ' Di+1 ¯α fp,γα i+1−1+δ+ D i ¯αfp,γα i−1+δ ( , c_i= ¯ci+1 ci+1 i= 0, 1, . . . , n − 1 n−1  i=0 |Di ¯αf(1)| ≤ cn n−1  i=0 Di ¯αfp,γiα−1+δ, fWn p,¯α ≤ cf (1) Wn p,¯α. W_p,¯αn γ_minα > 1 −1_p f(2)_Wn p,¯α = n−1  i=0 Di ¯αfp,γαi−1+ D n ¯αfp

i= 0, 1, . . . , n − 1 Di ¯αfp,γiα−1≤ ci #_n−1  j=i |Dj ¯αf(1)| + Dn¯αfp $ n−1  j=i |Dj ¯αf(1)| ≤ ¯ci #_n−1  j=i |Dj ¯αfp,γα j−1+ D n ¯αfp $ . i = n − 1 Dn−1_¯α f ∈ L1_p,αn α_n = γ_n−1α ≥ γ_minα >1 −_p1 Dn−1 ¯α fp,γαn−1−1≤ cn−1 ' |Dn−1 ¯α f(1)| + (Dn−1¯α f)p,γn−1α ( = cn−1  |Dn−1 ¯α f(1)| + tαn d dtD n−1 ¯α fp  = cn−1)|Dn−1¯α f(1)| + Dn¯αfp * , |Dn−1 ¯α f(1)| ≤ ¯cn−1 ' Dn−1 ¯α fp,γα n−1−1+ (D n−1 ¯α f)p,γα n−1 ( = ¯cn−1 ' Dn−1 ¯α fp,γn−1α −1+ D n ¯αfp ( . i= k, k + 1, . . . , n − 1 i = k − 1 i = k Dk−1_¯α f∈ Lp,γα k−1 γ α k + αk− 1 = γk−1α Dk−1 ¯α f1Lp,γα_k−1= (D k−1 ¯α f)p,γα k+αk−1 = tαk d dtD k−1 ¯α fp,γα k−1−1 = Dk ¯αfp,γkα−1<+∞, γ_k−1α ≥ γα_min>1 −1_p Dk−1 ¯α fp,γα k−1−1≤ ck−1 ' |Dk−1 ¯α f(1)| + (Dk−1¯α f)p,γα k−1 ( = ck−1)|Dk−1¯α f(1)| + Dk¯αfp,γαk−1 * , |Dk−1 ¯α f(1)| ≤ ¯ck−1 ' Dk−1 ¯α fp,γk−1α −1+ D k ¯αfp,γαk−1 ( .

i= k Dk−1 ¯α fp,γα k−1−1≤ ck # |Dk−1 ¯α f(1)| + n−1  j=k |Dj ¯αf(1)| + Dn¯αfp $ = ck # _n−1  j=k−1 |Dj¯αf(1)| + Dn¯αfp $ , i= k − 1 i = k i= k − 1 n−1  i=0 Di ¯αfp,γα i−1+ D n ¯αfp≤ c1 #_n−1  i=0 n−1  j=i |Dj¯αf(1)| + Dn¯αfp $ ≤ c2 #_n−1  j=0 |Dj ¯αf(1)| + Dn¯αfp $ . i= 0 n−1  j=0 |Dj¯αf(1)| + Dn¯αfp≤ ¯c1 #_n−1  i=0 Di ¯αfp,γi−1α + D n ¯αfp $ . W_p,¯αn γ_minα <1 −1_p f(3)_Wn p,¯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{i = i + 1 : 0 ≤ i ≤ n − 1, γα_i <1 −1_p} δ > 1 −1_p− γ_minα n₁ γ_iα>1−1_p i= n₁, n₁+1, . . . , n−1 m = n − n₁ ¯α(1) = (α₀(1), α(1)₁ , . . . , α(1)_m) α(1)_i = α_n1+i i = 0, 1, . . . , m γ_jα(1) = α(1)_m+ m−1_ k=j+1 (α_k(1)−1) = αn+ n−1  k=n1+j+1 (αk−1) = γαn1+j, j= 0, 1, . . . , m−1,

min 0≤j≤m−1γ α(1) j >1 −1p f ∈ Wp,¯αn f1= Dn¯α1f f1Wm p,¯α(1) = D m ¯α(1)f1p+ m−1_ i=0 |Di ¯α(1)f1(1)| = Dm ¯α(1)Dn_¯α1fp+ m−1  i=0 |Di ¯α(1)Dn_¯α1f(1)| = Dn_¯αfp+ n−1  i=n1 |Di ¯αf(1)| < +∞, f₁∈ W_p,¯αm(1) γα (1) min >1−1p Dm ¯α(1)f1p+ m−1_ i=0 |Di ¯α(1)f1(1)| Dm ¯α(1)f1p+ m−1_ i=0 Di ¯α(1)f1_p,γα(1) i −1 W_p,¯αn Dn ¯αfp+ n−1 i=n1 |Di ¯αf(1)|, Dn ¯αfp+ n−1 i=n1 Di ¯αfp,γαi−1. ¯α(2)_{= (α}(2) 0 , α(2)1 , . . . , α(2)n1) α(2)_n1 = γ_nα₁− 1 + αn₁ = γ_nα₁₋₁, α(2)_i = α_i, i= 0, 1, . . . , n₁− 1. Dn1 ¯α(2)fp= tγ α n1−1+αn1 d dtD n1−1 ¯α fp= Dn¯α1fp,γα n1−1<∞, f ∈ W_p,¯αn f ∈ Wn1 p,¯α(2) γα (2) i < 1 − 1p i = 0, 1, . . . , n1−1 W_p,¯αn1(2) max 0≤i≤n1−1γ α(2) i < 1 −1 p Dn1 ¯α(2)fp+ n1−1 i=0 |Di ¯α(2)f(1)|

Dn1 ¯α(2)fp+ n1−1 i=0 Di ¯α(2)f_p,γα(2) i −1+δ γ_iα(2)− 1 + δ = γ_nα₁₋₁+ n1−1 k=i+1 (αk− 1) − 1 + δ = γiα− 1 + δ, i = 0, 1, . . . , n − 1, W_p,¯αn Dn1 ¯α fp,γα n1−1+ n 1−1 i=0 |D i ¯αf(1)|, Dn1 ¯α fp,γα n1−1+ n 1−1 i=0 D i ¯αfp,γα i−1+δ.

W_p,¯αn → W_{q, ¯}m_β 1 ≤ p, q < ∞ 0 ≤ m < n X Y X → Y X ⊂ Y c > 0 x∈ X xY ≤ cxX. c >0 f (0, 1) → R n D_βk_¯f(t) = k  i=0 c_k,itγ0β−γα0+β0−α0+γαi−γβk_Di ¯αf(t), k = 0, 1, . . . , m,

c_k,k = 1 k = 0, 1, . . . , m 0 ≤ m ≤ n c_k,i i =

0, 1, . . . , k − 1 k = 0, 1, . . . , m

c_k,0= c_k−1,0(γ₀β− γ_k−1β + β₀− α₀),

c_k,i= c_k−1,i−1+c_k−1,i(γ₀β−γ₀α+β₀−α₀+γ_iα−γ_k−1β ), i = 1, 2, . . . , k−1. k= 0 D0_β¯f(t) = tβi_{f(t) = t}β0−α0_(tα0_{f(t)) = t}β0−α0_D0 ¯αf(t) = c0,0tβ0−α0D0¯αf(t), c_0,0= 1 k= 1 D1_β¯f(t) = tβ1d dt(t β0_{f(t)) = t}β1 d dt(t β0−α0_{· t}α0_f(t)) = tβ1+β0−α0−α1_(tα1 d dtD 0 ¯αf(t)) + (β0− α0)tβ1+β0−α0−1D0¯αf(t) = tγβ 0−γ1β+β0−α0−γ0α+γα1_D1 ¯αf(t) + c1,0tγ β 0−γβ1+β0−α0_D0 ¯αf(t) = c1,1tγ β 0−γα0+β0−α0+γα1−γβ1_D1 ¯αf(t) + c1,0tγ β 0−γ1β+β0−α0_D0 ¯αf(t), c_1,1= 1 c_1,0= β₀− α₀= γ₀β− γ₁β+ β₀− α₀ k= 0, 1, . . . , i − 1 ≤ m − 1 i≤ m Di_β¯f(t) = tβi d dtD i−1 ¯ β f(t) = tβi d dt _i−1  j=0 c_i−1,jtγ0β−γ0α+β0−α0+γαj−γi−1β _Dj ¯αf(t)  = i−1  j=0 c_i−1,jtβi+γ0β−γ0α+β0−α0+γαj−γi−1β d dtD j ¯αf(t)+ + i−1  j=0

c_i−1,j(γ₀β− γ₀α+ β₀− α₀+ γ_jα− γ_i−1β )tγ0β−γα0+β0−α0+γjα−γβi−1−1_Dj

¯αf(t) = i−1  j=0 c_i−1,jtγ0β−γα0+β0−α0+γj+1α −γiβ_Dj+1 ¯α f(t)+

+ i−1  j=0 c_i−1,j(γ₀β− γ₀α+ β₀− α₀+ γ_jα− γ_i−1β )tγ0β−γα0+β0−α0+γjα−γβi_Dj ¯αf(t) = i  j=1 c_i−1,j−1tγ0β−γ0α+β0−α0+γjα−γiβ_Dj ¯αf(t)+ + i−1  j=0 c_i−1,j(γ₀β− γ₀α+ β₀− α₀+ γ_jα− γ_i−1β )tγ0β−γα0+β0−α0+γjα−γβi_Dj ¯αf(t) = tγ0β−γα0+β0−α0+γiα−γβi_Di ¯αf(t)+ i−1  j=1

[ci−1,j−1+ci−1,j(γ0β−γ0α+β0−α0+γjα−γi−1β )] ×tγ0β−γα0+β0−α0+γαj−γβi_Dj ¯αf(t) + ci−1,0(γβ0− γi−1β + β0− α0)tγ β 0−γiβ+β0−α0_D0 ¯αf(t) =i j=0 c_i,jtγβ0−γ0α+β0−α0+γjα−γiβ_Dj ¯αf(t),

c_i,i= 1, c_i,j= c_i−1,j−1+c_i−1,j(γβ₀−γ₀α+β₀−α₀+γ_jα−γ_i−1β ), j = 1, 2, . . . , i−1, c_i,0= c_i−1,0(γ₀β− γ_i−1β + β₀− α₀).

i≤ m

1 < p ≤ q < ∞

W_p,¯αn → W_{q, ¯}m_β c >0 f ∈ W_p,¯αn fWm q, ¯β ≤ cfWp,¯nα. Dm ¯ βfq≤ c1fWn p,¯α, ∀f ∈ Wp,¯αn m−1  i=0 |Di ¯ βf(1)| ≤ c2fWn p,¯α, ∀f ∈ Wp,¯αn , c₁, c₂>0 f k= m γ_mβ = 1 D_βm¯f(t) = m  i=0 c_m,i(α, β)tγ0β−γα0+β0−α0+γiα−1_Di ¯αf(t). q Dm ¯ βfq≤ c3 m  i=0 ⎛ ⎝ 1  0 |tγβ 0−γ0α+β0−α0+γiα−1_Di ¯αf(t)|qdt ⎞ ⎠ 1 q ≤ c3 ⎡ ⎢ ⎣ m  i=0 ⎛ ⎝ 1  0 |tγ0β−γα0+β0−α0+γαi−1[Di ¯αf(t) − Di¯αf(1)]|qdt ⎞ ⎠ 1 q + + m  i=0 ⎛ ⎝ 1  0 |tγβ0−γ0α+β0−α0+γiα−1_Di ¯αf(1)|qdt ⎞ ⎠ 1 q⎤ ⎥ ⎦ , c₃= max 0≤i≤m|cm,i| γ_iα− 1 + γ₀β− γ₀α+ β₀− α₀≥ 1 p− 1 q+ γ α i − 1,

γ_minα >1 −1_p γ₀β− γ₀α+ β₀− α₀+ γα_i − 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 |tγ0β−γα0+β0−α0+γαi−1(Di ¯αf(t) − Di¯αf(1))|qdt ⎞ ⎠ 1 q + +m i=0 |Di ¯αf(1)| $ , c₃= max{c₃, c∗₃} γ_iα = α_i+1+ γ_i+1α − 1 γ_i+1α − 1 + αi+1≤ 1 −1 p+ 1 q+ γ β 0− γα0+ β0− α0+ γiα− 1, i = 0, 1, . . . , n − 1. Dm ¯ βfq≤ c3 ⎡ ⎢ ⎣ m  i=0 H_i ⎛ ⎝ 1  0 |tγα i+1+αi+1−1d 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, H_i, i= 0, 1, . . . , m} n > m≥ 0 Dm ¯ βfq ≤ c4 # _n  i=1 Di ¯αfp,γα i−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 D_βk_¯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)| ≤ c2fWn p,¯α, c₂= max 0≤i≤m m k=i|ck,i| W_p,¯αn → W_{q, ¯}m_β f₀(t) = t−γ0α−α0+_p1+ε _{ε > 0} 1 p+p1 = 1 Di_¯αf₀(t) = tαi d dtt αi−1d dt. . . t α2 d dtt α1d dt(t α0−γα0−α0+_p1+ε₎ = (−γα 0 +_p1 + ε)tαi d dtt αi−1d dt. . . t α2 d dtt α1−γ0α+p1+ε−1 = (−γα 0 +_p1+ ε)(−γ1α+_p1 + ε)tαi d dtt αi−1d dt. . . d dtt α2−γα1+p1+ε−1= . . . = i−2 + j=0 (−γα j +_p1_+ ε)tαi d dtt αi−1−γi−2α +p1+ε−1= i−1 + j=0 (−γα j +_p1_+ ε)tαi−γ α i−1+p1+ε−1

= i−1 + j=0 (−γα j +_p1_ + ε)t−γ α i+p1+ε_, α_i− γα_i−1− 1 = −γ_iα i= 1, 2, . . . , n − 1 i= 1, 2, . . . , n − 1 (−γ_jα+_p1+ 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+p1+ε)p_{dt <}∞, f₀ ∈ W_p,¯αn β_i = γ_i−1β − γ_iβ + 1 γ_mβ = 1 β f₀(t) Dm_β¯f₀(t) = tβmd dtt βm−1_{. . .} d dtt β2d dtt β1 d dtt β0−γ0α−α0+_p1+ε = (β0− γα0 − α0+_p1_ + ε)tβm_dtdtβm−1. . ._dtdtβ2_dtdtβ1+β0−γ α 0−α0+_p1+ε−1 = (β0− γα0 − α0+_p1_ + ε)(γ0β− γ0α+ β0− α0− γβ1+_p1_+ ε)tβm d dtt βm−1_{. . .} . . . d dtt β2+γ0β−γα0+β0−α0−γβ1+p1+ε−1= . . . = m−1₊ i=0 (γβ 0 − γ0α+ β0− α0− γβi +_p1+ ε)tβm+γ β 0−γ0α+β0−α0−γβm−1+p1+ε−1 = m−1₊ i=0 (γ₀β− γα 0 + β0− α0− γiβ+_p1+ ε)tγ β 0−γ0α+β0−α0−1+_p1+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(γβ0−γ0α+β0−α0−1+_p1+ε)q_{dt <∞,} (γβ 0− γ0α+ β0− α0− 1 +_p1_ + ε)q + 1 > 0. ε∈ (0, ε₀) γ₀β−γ₀α+β₀−α₀≥ 1_p−1_q 1 < p < ∞ n ≥ 1 γ_minα >1 −_p1 W_p,¯αn → W_{p, ¯}n_β γ₀β− γ₀α+ β₀− α₀≥ 0 p= q m < n k= n D_βn_¯f(t) = n  i=0 c_n,i(α, β)tγ0β−γα0+β0−α0+γiα−1_Di ¯αf(t). p Dn ¯ βfp≤ c3 ⎡ ⎢ ⎣ ⎛ ⎝ 1  0 |tγβ0−γ0α+β0−α0_Dn ¯αf(t)|pdt ⎞ ⎠ 1 p + n−1  i=0 ⎛ ⎝ 1  0 |tγβ 0−γ0α+β0−α0_Di ¯αf(t)|pdt ⎞ ⎠ 1 p⎤ ⎥ ⎦ , c₃= max 0≤i≤n|cn,i| γβ₀− γ₀α+ β₀− α₀≥ 0 γ_minα >1 −1_p γβ₀− γ₀α+ β₀− α₀+ γ_iα>1 −1 p, i= 0, 1, . . . , n.

1  0 |tγβ 0−γα0+β0−α0+γαi−1|p_dt≤ c∗ 3, i= 0, 1, . . . , n − 1, c∗₃= 1 (γβ 0−γ0α+β0−α0+γiα−1)p+1 Dn ¯ βfp≤ c3 ⎡ ⎢ ⎣ ⎛ ⎝ 1  0 |tγ0β−γ0α+β0−α0_Dn ¯αf(t)|pdt ⎞ ⎠ 1 p + + n−1  i=0 ⎛ ⎝ 1  0 |tγβ 0−γ0α+β0−α0+γαi−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−γα0+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, H_i, i = 0, 1, . . . , n − 1} Dn ¯ βfp≤ c1fWp,¯nα, m = n W_p,¯αn → W_{p, ¯}n_β 1 < p < ∞ γα min>1 − 1p γminβ >1 − 1p W_p,¯αn W_{p, ¯}n_β |¯α| = | ¯β|.