EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 4, 2019, 1602-1611
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On Embedding Theorems in Grand Grand
Nikolskii-Morrey Spaces

Alik M. Najafov1,∗, Azizgul M. Gasimova2

1 Azerbaijan University of Architecture and Construction, Baku, Azerbaijan
2 Sumgait State University, Sumgait, Azerbaijan

Abstract. In the paper we introduced a grand grand Nikolskii-Morrey spaces. Some differential
and differential-difference properties of functions from this spaces are proved by means of the
integral representation.

2010 Mathematics Subject Classifications: 46E35, 26B40, 26B40

Key Words and Phrases: Grand grand Nikolskii-Morrey spaces, integral representation, flexible
λ− horn condition, H0̈lder condition

1. Introduction and preliminary notes

It is known that in the middle of the last century, in connection with the study of
the regularity properties of differential equations with partial derivatives of a high (inte-
ger and non-integer) order, it became necessary with the introduction of Sobolev W l

p(G)

(l ∈ Nn) [20] and Nikolskii H l
p(G) (l ∈ (0,∞)n) [15] spaces, etc. These spaces were

further developed and generalized by many mathematicians. Considering that the grand
grand Nikolskii Morrey H l

p)κ),a,α(G,λ) spaces introduced in this paper is wider than all
previously considered spaces of this type, it will be interesting to readers.

In this paper we construct a grand grand Nikolskii-Morrey spaces H l
p)κ),a,α(G,λ) and

we study some differential properties with help of the method of integral representation of
functions in view of embedding theory. Let G ⊂ Rn be a bounded domain, l ∈ (0,∞)n ,
p ∈ (1,∞) ,a ∈ [0, 1],κ ∈ (0,∞)n and α ≥ 0.

Note that the grand Lebesgue spaces Lp)(G) (|G| <∞) introduced in [5] by T.Iwaniec
and C.Sbordone. After a vast amount of research about grand Lebesgue, grand Lebesgue-
Morrey, grand-grand Lebesgue-Morrey, grand-grand Sobolev-Morrey spaces (with different
norms) has been studied by many mathematicians, (see, e.g. [3, 4, 6–11, 13, 16, 18, 19, 21])
e.t.c.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3567

Email addresses: aliknajafov@gmail.com (A. Najafov), ezizgul.qasimova@mail.ru (A. Gasimova)

http://www.ejpam.com 1602 c© 2019 EJPAM All rights reserved.



A. M.Najafov, A. M. Gasimova / Eur. J. Pure Appl. Math, 12 (4) (2019), 1602-1611 1603

Definition 1. By grand grand Nikolskii-Morrey spaces H l
p),κ),a,α(G,λ) we denote the

spaces of all functions f ∈ Lloc1 (G) (mi > li − ki > 0, i = 1, 2, . . . , n) with the finite norm

‖f‖Hl
p),κ),a,α

(G,λ) = ‖f‖p),κ),a,α;G +

+

n∑
i=1

sup
0<t<d0

∥∥∥∆mi
i

(
tλi , Gtλ

)
Dki
i f
∥∥∥
p),κ),a,α

tλi(li−ki)
, (1)

‖f‖p),κ),a,α;G = ‖f‖L p),κ),a,α(G) =

= sup
x ∈ G,
0 < t ≤ d0,
0 < ε < sm

(
1

t|κ|a−αε
ε

|Gtκ (x)|

∫
G
tκ(x)

|f(y)|p−ε dy

) 1
p−ε

, (2)

where, d0−diam G,mi ∈ N, ki ∈ N0, |κ| =
∑n

j=1 κj , sm = min{p− 1, |κ|aα } and x ∈ Rn.

Gtκ (x) = G ∩ Itκ (x) =

= G ∩
{
y : |yj − xj | <

1

2
tκj ; j = 1, 2, . . . , n

}
.

The Nikolskii-Morrey space H l
p,λ (Rn) and Nikolskii-Morrey type space H l

p,ϕ,β (G) stud-
ied in [1, 17]. Also note that in this paper, in theorem 2.2 it was proved that the Holder
”index” is larger than in [1, 12, 14] .

Note that some properties of spaces Lp),κ),a,α(G) and H l
p),κ),a,α (G,λ) .

1) Lp),κ),a,α(G)→ Lp)(G), H l
p),κ),a,α (G,λ)→ H l

p) (G,λ) , i.e.

‖f‖p),G ≤ C ‖f‖p),κ),a,α;G ; ‖f‖Hl
p),

(G,λ) ≤ C ‖f‖Hl
p),κ),a,α

(G,λ) (3)

where H l
p(G,λ) is grand Nikolskii space with finite norm

‖f‖Hl
p)

(G,λ) = ‖f‖p),G +

n∑
i=1

sup
0<t<d0

∥∥∆mi
i

(
tλi , Gtλ

)
f
∥∥
p)

tλi(li−ki)
,

‖f‖p),G = ‖f‖Lp(G) = sup
0<ε<p−1

(
ε

|G|

∫
G
|f(x)|p−ε dx

) 1
p−ε

.

2) Lp),κ),a,α(G) and H l
p),κ),a,α (G,λ) are complete.

3) ‖f‖p),κ),0,0;G = ‖f‖p),G and ‖f‖Hl
p),κ),0,0

(G,λ) = ‖f‖Hl
p)

(G,λ) .

Let Mi(·, y) ∈ C∞0 (Rn) be such that

S (Mi) ⊂ I1 =

{
x : |xj | <

1

2
, f = 1, 2, . . . , n

}
,



A. M.Najafov, A. M. Gasimova / Eur. J. Pure Appl. Math, 12 (4) (2019), 1602-1611 1604

Assume 0 < T ≤ 1, λ = (λ1, . . . , λn) , λj > 0 (j = 1, 2, . . . , n) , and put

V =
⋃

0<t≤T

{
y :
( y
tλ

)
∈ S (Mi)

}
.

Clearly, V ⊂ ITλ and let U be an open set contained in the domain G; henceforth
we always assume that U + V ⊂ G. Put GTκ (U) = (U + ITκ (x))

⋂
G. Obviously, if

0 < κj ≤ λj (j = 1, 2, . . . , n), then ITλ ⊂ ITκ and thereby U + V ⊂ GTκ (U) = Q.

Lemma 1. Let 1 < p < q ≤ r ≤ ∞;0 < |κ| ≤ |λ|+αε
1+a ; 0 < t, η ≤ T ≤ d0;0 < γ < γ0;

ν = (ν1, . . . , νn) , νj are integers (j = 1, 2, . . . , n) ; ∆mi
i

(
tλi
)
f ∈ Lp),κ),a,α(G) and let

µ̄i = λili − |ν, λ| − (|λ| − |κ|a− |κ|+ αε)

(
1

p− ε
− 1

q − ε

)
, (4)

Eiη(x) =

∫ η

0
t−1−|λ|−λi−|ν,λ|ϕi(x, t)dt, (5)

Eiη,T (x) =

∫ T

η
t−1−|λ|−λi−|ν,λ|ϕi(x, t)dt, (6)

E(x) =

∫
Rn
f(x+ y + z)Ω

(
y

tλ
,
ρ
(
tλ, x

)
tλ

)
Ω(ν)

(
z

tλ
,
ρ
(
tλ, x

)
tλ

)
dydz, (7)

where

|ν, λ| =
n∑
j=1

νjλj ,

ϕi(x, t) =

∫
Rn

∫ ∞
−∞

Mi

(
y

tλ
,
ρ
(
tλ, x

)
tλ

)
×

×Si

(
u

tλi
,
ρi
(
tλ, x

)
2tλi

,
1

2
ρ′i

(
tλi , x

))
∆mi
i

(
δλiu

)
×

×f (x+ y + uei) dudy. (8)

Then for any x̄ ∈ U the following inequalities

sup
x̄∈U

∥∥Eiη∥∥q−ε,Uγκ (x̄)
≤

≤ C1

∥∥∥t−λili∆mi
i

(
tλi , Gtλ

)
f
∥∥∥
p)κ),a,α;Q

ε
− 1
p−εγ

|κ|(a+1)
q−ε ηµ̄i (µ̄i > 0) , (9)

sup
x̄∈U

∥∥Eiη,T∥∥q−ε,Uγκ (x̄)
≤

≤ C2

∥∥∥t−λili∆mi
i

(
tλi , Gtλ

)
f
∥∥∥
p),κ),a,α;Q

ε
− 1
p−εγ

|κ|(a+1)
q−ε ×



A. M.Najafov, A. M. Gasimova / Eur. J. Pure Appl. Math, 12 (4) (2019), 1602-1611 1605

×


T µ̄i , for µ̄i > 0,

ln T
η , for µ̄i = 0,

ηµ̄i , for µ̄i < 0,

(10)

sup
x̄∈U
‖E‖q−ε,Uγκ (x̄) ≤

≤ C3 ‖f‖p),κ),a,α;Q t
|λ|−(|λ|−|κ|−|κ|a)

(
1
p−ε−

1
q−ε

)
ε
− 1
p−εγ

|κ|(a+1)
q−ε (11)

is hold, where and Uγκ (x̄) =
{
x : |xj − x̄j | < 1

2γ
κj , j = 1, 2, . . . , n

}
, C1 and C2 are con-

stants independent of f, γ, η and T .

Proof. Applying sequentially the generalized the Minkowskii inequality for any x̄ ∈ U∥∥Eiη∥∥q−ε,Uγκ (x̄)
≤
∫ η

0
t−1−|λ|−|ν,λ|−λi ‖ϕi (·, t)‖q−ε,Uγκ (x̄) dt, (12)

and from the Hölder inequality (q ≤ r) we obtain

‖ϕi (·, t)‖q−ε,Uγκ (x̄) ≤ ‖ϕi (·, t)‖r−ε,Uγκ (x̄) γ
|κ|

(
1
q−ε−

1
r−ε

)
. (13)

Now estimate the norm ‖ϕi (·, t)‖r−ε,Uγκ (x̄) . Let X be a characteristic function of the

set S (Mi). Noting that 1 < p < r ≤ ∞, s ≤ r
(

1
s = 1− 1

p−ε + 1
r−ε

)
and

∣∣∣∣Mi

∫ +∞

−∞
Si∆

mi
i fdu

∣∣∣∣ =

(∣∣∣∣∫ +∞

−∞
Si∆

mi
i fdu

∣∣∣∣p−ε |Mi|s
) 1

r−ε

×

×

(∣∣∣∣∫ +∞

−∞
Si∆

mi
i fdu

∣∣∣∣p−ε x
) 1

p−ε−
1
r−ε

(|Mi|s)
1
s
− 1
r−ε

and apply to |ϕi| the Holder inequality
(

1
r−ε +

(
1
p−ε −

1
r−ε

)
+
(

1
s −

1
r−ε

)
= 1
)
,

‖ϕi (·, t)‖r−ε,Uγκ (x̄) ≤

≤ C1 sup
x∈Uγκ(x̄)

(∫
Rn

∣∣∣∣∣
∫ +∞

−∞
Si

(
u

tλi
,
ρi
(
tλi , x

)
tλi

,
1

2
ρ′i

(
tλi , x

))
∆mi
i

(
tλi
)
f (x+ y + uei) du

∣∣∣∣∣
p−ε

×

×X
( y

tλi

)
dy
) 1
p−ε−

1
r−ε

sup
y∈v
×

(∫
Uγκ(x̄)

∣∣∣∣∣
∫ +∞

−∞
Si

(
u

tλi
,
ρi
(
tλi , x

)
tλi

,
1

2
ρ′i

(
tλi , x

))
∆mi
i

(
tλi
)
f (x+ y + uei) du

∣∣∣∣∣
1
p−ε

×

×
(∫

Rn

∣∣∣M1
i

( y
tλ

)∣∣∣s dy)s , (14)



A. M.Najafov, A. M. Gasimova / Eur. J. Pure Appl. Math, 12 (4) (2019), 1602-1611 1606

suppose that |Mi (x, y, z)| ≤ C1
∣∣M1

i (x)
∣∣ .

Obviously, if |κ| ≤ |λ|
1+a , 0 < t ≤ 1, then Qtλ(x) ⊂ Qtκ (x). For every x ∈ U we have

∫
Rn

∣∣∣∣∣
∫ +∞

−∞
Si

(
u

tλi
,
ρi
(
tλi , x

)
tλi

,
1

2
ρ′i

(
tλi , x

))
∆mi
i

(
tλi
)
f(x+ y + ueidu)

∣∣∣∣∣
p−ε

X
( y
tλ

)
dy ≤

≤
∫
Qtκ (x)

∣∣∣∣∣
∫ +∞

−∞
Si

(
u

tλi
,
ρi
(
tλi , x

)
tλi

,
1

2
ρ′i

(
tλi , x

))
∆mi
i

(
tλi
)
f(y + uei)

∣∣∣∣∣
p−ε

dy ≤

≤
∥∥∥t−λili∆mi

i

(
tλi
)
f
∥∥∥p−ε
p−ε,Qtκ (x)

tλili(p−ε) ≤

≤
∥∥∥t−λi∆mi

i

(
tλi
)
f
∥∥∥p−ε
p),κ),a,α;Q

ε−1t|κ|+|κ|a+λili(p−ε)−αε, (15)

for y ∈ V∫
Uγκ (x̄)

∣∣∣∣∣
∫ +∞

−∞
Si

(
u

tλi
,
ρi
(
tλi , x

)
tλi

,
1

2
ρ′i

(
tλi , x

))
∆mi
i

(
tλi
)
f(x+ y + ueidu)

∣∣∣∣∣
p−ε

dx ≤

≤
∫
Qγκ (x̄+y)

∣∣∣∣∣
∫ +∞

−∞
Si

(
u

tλi
,
ρi
(
tλi , x

)
tλi

,
1

2
ρ′i

(
tλi , x

))
∆mi
i

(
tλi
)
f(x+ uei)du

∣∣∣∣∣
p−ε

dx ≤

≤ tλili(p−ε)
∥∥∥t−λili∆mi

i

(
tλi
)
f
∥∥∥p−ε
p−ε,Qγκ (x)

≤

≤
∥∥∥t−λili∆mi

i

(
tλi
)
f
∥∥∥p−ε
p),κ),a,α;Q

tλili(p−ε)γ|κ|+|κ|a−αεε−1. (16)∫
Rn

∣∣∣M1
i

( y
tλ

)∣∣∣s dy = t|λ| ‖M1‖ss (17)

From inequalities (13)-(17) for r = q that∥∥Eiη∥∥q−ε,Uγκ (x̄)
≤ C1

∥∥∥t−λili∆mi
i (tλi)f

∥∥∥
p),κ),a,α,Q

ε
− 1
p−εγ

|κ|a+|κ|−αε
q−ε ×

×t|λ|−(|λ|−|κ|−|κ|a+αε)
(

1
p−ε−

1
q−ε

)
(18)

Unseating this inequality in (12), for all x̄ ∈ U, we see that∥∥Eiη∥∥q−ε,Uγκ (x̄)
≤ C2

∥∥∥t−λili∆mi
i (tλi)f

∥∥∥
p),κ),a,α;Q

ε
− 1
p−εγ

|κ|(1+a)
q−ε ηµ̄i(µi > 0)

Similarly, we can prove (10) and (11).



A. M.Najafov, A. M. Gasimova / Eur. J. Pure Appl. Math, 12 (4) (2019), 1602-1611 1607

2. Main results

.
We proved two theorems on the properties of the functions from spaces H l

p),κ),a,α (G,λ).

Theorem 1. Let G ⊂ Rnbe an open bounded set satisfy the flexible λ− horn condition
(see [2]); 1 < p < q ≤ ∞; |κ| ≤ λ+αε

1+a ; ν = (ν1, . . . , νn), νj ≥ 0 are integers (j = 1, . . . , n);

µ̄i > 0(i = 1, 2, . . . , n) and let f ∈ H l
p),κ),a,α (G,λ).

Then Dν : H l
p),κ),a,α (G,λ) → Lq−ε(G) hold for any ε ∈ (0, sm), and moreover, the

following inequality is valid

‖Dνf‖q−ε,G ≤ C(ε)
(
T µ̄0 ‖f‖p),κ),a,α;G +

+

n∑
i=1

T µ̄i sup
0<t<d0

∥∥∥∥∥∆mi
i

(
tλi , Gtλ

)
f

tλili

∥∥∥∥∥
p),κ),a,α

 (19)

In particular, if µ̄i,0 = λili − |ν, λ| − (|λ| − |κ| − |κ| a+ αε) 1
p−ε > 0 (i = 1, 2, . . . , n) if

Dνf is continuous on G and

sup
x∈G
|Dνf(x)| ≤ C(ε)

(
T µ̄0,0 ‖f‖p),κ),a,α;G +

+
∑

T µ̄i,0 sup
0<t<d0

∥∥∥∥∥∆mi
i

(
tλi , Gtλ

)
f

tλili

∥∥∥∥∥
p),κ),a,α

 (20)

moreover 0 < T ≤ d0, C(ε) = Cε
− 1
p−ε and C is a constant independent of f, T and ε.

Proof. At first note that in the conditions of our theorem there exists a generalized
derivatives Dνf on G Indeed, from the condition µ̄i > 0 (i = 1, 2, . . . , n) it follows that for
f ∈ H l

p),κ),a,α (G,λ) → H l
p)(G,λ) → H l

p−ε(G,λ) (p − ε > 1). Then Dνf exists on G and

belongs to Lp−ε(G) and for almost each point x ∈ G the integral representation in [2].

Dνf(x) = f
(ν)

Tλ
(x) + (−1)|ν|

∫ T

0

n∑
i=1

∫
Rn

∫ ∞
−∞

t−1−|λ|−λi−|ν,λ|×

×Ψ
(ν)
i

(
y

tλ
,
ρ(tλ, x)

tλ

)
Si

(
u

tλi
,
ρ(tλ, x)

2tλi
,
1

2
ρ′i(t

λi , x)

)
×

×∆mi
i (δλiu)f(x+ y + uei)dudydt, (21)

f
(ν)

Tλ
(x) = (−1)|ν|T−2|λ|−|ν,λ|

∫
Rn

∫
Rn
f(x+ y + z)×

×Ω

(
y

tλ
,
ρ
(
tλ, x

)
tλ

)
Ω(ν)

(
z

tλ
,
ρ
(
tλ, x

)
tλ

)
dydz, (22)



A. M.Najafov, A. M. Gasimova / Eur. J. Pure Appl. Math, 12 (4) (2019), 1602-1611 1608

0 < T ≤ d0 and Ω (·, y) ,Ψi(·, y) ∈ C∞0 (Rn) , Si (·, y, z) ∈ C∞0 (R).
Recall that the flexible λ− horn and x+ V is the support of the representation (21) and
(22). Applying the Minkowski inequality, from identities (21) and (22) we get

‖Dνf‖q−ε,G ≤
∥∥∥f (ν)

Tλ

∥∥∥
q−ε,G

+
n∑
i=1

∥∥EiT∥∥q−ε,G . (23)

By (11) for U = G, Mi = Ω, t = T we get∥∥∥f (ν)

Tλ

∥∥∥
q−ε,G

≤ C1(ε) ‖f‖p),κ),a,α;G · T
µ̄0 , (24)

by (11) for U = G, Mi = Ψi , η = T we get∥∥EiT∥∥q−ε,G ≤ C2(ε)
∥∥∥t−λili∆mi

i

(
tλi , Gtλ

)
f
∥∥∥
p),κ),a,α

T µ̄i . (25)

Substituting (25) and (24) in (23), we get inequality (19).
Now let conditions µ̄i,0 > 0 (i = 1, 2, . . . , n). Show that Dνf is continuous on G. By

(21) and (22), using (23) for q =∞ and µ̄i(q =∞) = µ̄i,0 > 0 (i = 1, 2, . . . , n) we obtain

∥∥∥Dνf − f (ν)

Tλ

∥∥∥
∞,G
≤ C(ε)

n∑
i=1

T µ̄i,0 sup
0<t<d0

∥∥∥∥∥∆mi
i

(
tλi , Gtλ

)
f

tλili

∥∥∥∥∥
p),κ),a,α

.

As T → 0, the left side of this inequality tends to zero, since f
(ν)

Tλ
(x) is continuous on

G and the convergence in L∞(G) coincides with the uniform convergence. Then the limit
function Dνf is continuous on G. Theorem 2.1 is proved.

Let ξ be an n− dimensional vector.

Theorem 2. Suppose that the domain G the parameters p, q and vector ν satisfy the
condition of theorem 2.1. If µ̄i > 0 (i = 1, . . . , n) then Dνf satisfies the Holder condition
with exponent σ on G in the metric of Lq−ε; more exactly

‖∆ (ξ,G)Dνf‖q−ε,G ≤ C(ε) ‖f‖Hl
p),κ),a,α

(G,λ) |ξ|
σ , (26)

σ is an arbitrary number satisfying the inequalities:

0 ≤ σ ≤ 1, if
µ̄0

λ0
> 1;

0 ≤ σ < 1, if
µ̄0

λ0
= 1; (27)

0 ≤ σ ≤ µ̄0

λ0
, if

µ̄0

λ0
< 1,

where µ̄0 = min (µ̄1, µ̄2, . . . , µ̄n) , λ0 = maxλj
j=1,...,n

.



A. M.Najafov, A. M. Gasimova / Eur. J. Pure Appl. Math, 12 (4) (2019), 1602-1611 1609

If µ̄i,0 > 0 (i = 1, . . . , n), then

sup
x∈G
|∆(ξ,G)Dνf(x)| ≤ C(ε) ‖f‖Hl

p),κ),a,α
(G,λ) |ξ|

σ0 , (28)

where σ0 satisfy the some conditions as σ with µ̄i,0 instead of µ̄i and C(ε) = Cε
− 1
p−ε and

C is a constant independent of f and ε.

Proof. By Lemma 8.6 of [2] there is a domain Gω ⊂ G(ω = k rλ(x), k > 0, rλ(x) =
ρλ(x, ∂G), x ∈ G). Suppose that |ξ|λ < ω, then segment joining the points of the segment
with the some kernels. Making simple transformations, we obtain

|∆(ξ,G)Dνf(x)| ≤ C1T
−2|λ|−|ν,λ|×

×
∫
Rn

∫
Rn

|f(x+ y + z)|
∣∣∣∣Ω(ν)

(
y − ξ
tλ

,
ρ(tλ, x)

tλ

)
− Ω(ν)

(
y

tλ
,
ρ(tλ, x)

tλ

)∣∣∣∣ dydz+

C2

n∑
i=1


|ξ|

1
λ0∫

0

t−1−|λ|−λi−|ν,λ|
∫
Rn

+∞∫
−∞

∣∣∣∣Si( u

tλi
,
ρ(tλi , x)

tλi
,
1

2
ρ′i(t

λi , x)

)∣∣∣∣ ×
×
∥∥∥∥Ψ

(ν)
i

(
y

tλ
,
ρ(tλ, x)

tλ

)∥∥∥∥ ∣∣∣∆mi
i

(
δλiu

)
f(x+ y + uei)

∣∣∣ dudydt+
+

T∫
|ξ|

1
λ0

t−|λ|−λi−|ν,λ|
∫
Rn

+∞∫
−∞

∣∣∣∣Si( u

tλi
,
ρ(tλi , x)

tλi
,
1

2
ρ′i(t

λi , x)

)∣∣∣∣

×
∥∥∥∥Ψ

(ν)
i

(
y

tλ
,
ρ(tλ, x)

tλ

)∥∥∥∥
1∫

0

∣∣∣∆mi
i

(
δλiu

)
f(x+ y + uei + ωξ)

∣∣∣ dudydtdω
 =

= C1A(x, ξ) + C2

n∑
i=1

(B(x, ξ) + F (x, ξ)) , (29)

where 0 < T < t0. We also assume that |ξ| < T λ, and consequently |ξ| ≤ min(ωλ0 , T λ0).
If x ∈ G\Gω then by definition ∆ (ξ,G)Dνf(x) = 0.
By (29)

‖∆ (ξ,G)Dνf‖q−ε,G = ‖∆ (ξ,G)Dνf‖q−ε,Gω ≤ C1 ‖A (·, ξ)‖q−ε,Gω +

+C2

∑(
‖B (·, ξ)‖q−ε,Gω + ‖F (·, ξ)‖q−ε,Gω

)
(30)

A(x, ξ) ≤
n∑
j=1

T−λj−2|λ|−|ν,λ|
|ξ|∫

0

dγ×



REFERENCES 1610

×
∫
Rn

∫
Rn

|f(x+ y + z + ξeγ)|

∣∣∣∣∣DjΩ
(ν)

(
y

T λ
,
p
(
T λ, x

)
T λ

)
Ω

(
z

T λ
,
p
(
T λ, x

)
T λ

)∣∣∣∣∣ dydz.
taking into account ξeγ +Gω ⊂ G and applying the generalized Minkowski inequality and
by (11) for U = G we have

‖A(·, ξ)‖q−ε,Gω ≤ C1 (ε) |ξ| ‖f‖p),κ),a;G . (31)

By means of inequality (10) for U = G, Mi = Ψi, η = |ξ|
1
λ0 we obtain

‖B(·, ξ)‖q−ε,Gω ≤ C2 (ε)
∥∥∥t−λili∆mi

i (tλi , Gtλ)f
∥∥∥
p),κ),a,α

|ξ|
µ̄i
λ0 , (32)

and by means inequality (10) for U = G, Mi = Ψi, η = |ξ|
1
λ0 we obtain

‖F (·, ξ)‖q−ε,Gω ≤ C3 (ε) |ξ|σ
∥∥∥t−λili∆mi

i (tλi , Gtλ)f
∥∥∥
p),κ),a,α

. (33)

From inequalities (30)-(33) we get the required inequality.
Now suppose that |ξ| > min(ωλ0 , T λ0), then

‖∆ (ξ,G)Dνf‖q−ε,G ≤ 2 ‖Dνf‖q−ε,G ≤ C (ω, T ) ‖Dνf‖q−ε,G |ξ|
σ .

Estimating ‖Dνf‖q−ε,G by means of (19) we obtain the sought inequality in this case
as well. The theorem is proved.

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