Directory UMM :Journals:Journal_of_mathematics:GMJ:
Georgian Mathematical Journal
1(94), No. 4, 367-376
TWO-WEIGHTED Lp -INEQUALITIES FOR SINGULAR
INTEGRAL OPERATORS ON HEISENBERG GROUPS
V. S. GULIEV
Abstract. Some sufficient conditions are found for a pair of weight
functions, providing the validity of two-weighted inequalities for singular integrals defined on Heisenberg groups.
Estimates for singular integrals of the Calderon–Zygmund type in various spaces (including weighted spaces and the anisotropic case) have attracted a great deal of attention on the part of researchers. In this paper
we will deal with singular integral operators T on the Heisenberg group H n
which have an essentially different character as compared with operators
of the Calderon–Zygmund type. We have obtained the two-weighted Lp inequality with monotone weights for singular integral operators T on H n .
Applications are given.
Let H n be the Heisenberg group (see [1], [2]) realized as a set of points
x = (x0 , x1 , . . . , x2n ) = (x0 , x′ ) ∈ R2n+1 with the multiplication
n
1X
(xi yn+i − xn+i yi ), x′ + y ′ .
xy = x0 + y0 +
2 i=1
The corresponding Lie algebra is generated by the left-invariant vector
fields
∂
∂
, Xi =
+
∂x0
∂xi
∂
1
∂
=
− xi
,
∂xn+i
2 ∂x0
X0 =
Xn+i
∂
1
,
xn+i
2
∂x0
i = 1, . . . , n,
which satisfy the commutation relation
1
X0 ,
4
[X0 , Xi ] = [X0 , Xn+i ] = [Xi , Xj ] = [Xn+i , Xn+j ] = [Xi , Xn+j ] = 0,
[Xi , Xn+i ] =
1991 Mathematics Subject Classification. 42B20, 42B25, 42B30.
367
c 1994 Plenum Publishing Corporation
1072-947X/94/0700-0367$12.50/0
368
V. S. GULIEV
i, j = 1, . . . , n i =
6 j.
The dilation δt : δt x = (t2 x0 , tx′ ), t > 0, is defined on H n . The Haar measure on this group coincides with the Lebesgue measure dx = dx0 dx1 · · · dx2n .
The identity element in H n is e = 0 ∈ R2n+1 , while the element x−1 inverse
to x is (−x).
The function f defined in H n is said to be H-homogeneous of degree m,
on H n , if f (δt x) = tm f (x), t > 0. We also define the norm on H n
2n
X
2 1/4
|x|H = x20 +
xi2
i=1
which is H-homogeneous of degree one. This also yields the distance function, namely, the distance
d(x, y) = d(y −1 x, e) = |y −1 x|H ,
|y −1 x|H =
x0 − y0 −
n
2
1X
(xi yn+i − xn+i yi ) +
2 i=1
2n
2 1/4
X
+
(xi − yi )2
.
i=1
d is left-invariant in the sense that d(x, y) remains unchanged when x and y
are both left-translated by some fixed vector in H n . Furthermore, d satisfies
the triangle inequality d(x, z) ≤ d(x, y) + d(y, z), x, y, z ∈ H n . For r > 0
and x ∈ H n let
B(x, r) = {y ∈ H n ; |y −1 x|H < r} (S(x, r) = {y ∈ H n ; |y −1 x|H = r})
be the H-ball (H-sphere) with center x and radius r.
The number Q = 2n + 2 is called the homogeneous dimension of H n .
Clearly, d(δt x) = tQ dx.
Given functions f (x) and g(x) defined in H n , the Heisenberg convolution
(H-convolution) is obtained by
Z
Z
f (y)g(y −1 x)dy =
(f ∗ g)(x) =
f (xy −1 )g(y)dy,
Hn
Hn
n
where dy is the Haar measure on H .
is summable
The kernel K(x) admitting the estimate |K(x)| ≤ C|x|α−Q
H
in the neighborhood of e for α > 0 and in that case K ∗ g is defined for
the function g with bounded support. If however the kernel K(x) has a
singularity of order Q at zero, i.e., |K(x)| ∼ |x|−Q
H near e, then there arises
a singular integral on H n .
TWO-WEIGHTED Lp -INEQUALITIES
369
Let ω(x) be a positive measurable function on H n . Denote by Lp (H n , ω)
a set of measurable functions f (x), x ∈ H n , with the finite norm
1/p
Z
, 1 ≤ p < ∞.
|f (x)|p ω(x)dx
kf kLP (H n ,ω) =
Hn
We say that a locally integrable function ω : H n → (0, ∞) satisfies
Muckenhoupt’s condition Ap = Ap (H n ) (briefly, ω ∈ Ap ), 1 < p < ∞, if
there is a constant C = C(ω, p) such that for any H-ball B ⊂ H n
Z
Z
′
1
1
+ ′ = 1,
ω 1−p (x)dx ≤ C,
ω(x)dx |B|−1
|B|−1
p
p
B
B
where the second factor on the left is replaced by ess sup{ω −1 (x) : x ∈ B}
if p = 1.
Let K(x) be a singular kernel defined on H n \{e} and satisfying the confunction of degree −Q, i.e., K(δt x) =
ditions: K(x) is an H-homogeneous
R
t−Q K(x) for any t > 0 and SH K(x)dσ(x) = 0, where dσ(x) is a measure
element on SH = S(e, 1).
Denote by ωK (δ) the modulus of continuity of the kernel on SH :
ωK (δ) = sup{|K(x) − K(y)| : x, y ∈ SH ,
It is assumed that
Z
1
ωK (t)
0
dt
< ∞.
t
We consider the singular integral operator T :
Z
Z
−1
K(xy )f (y)dy =: lim
T f (x) =
Hn
|y −1 x|H ≤ δ}.
ε→0+
|xy −1 |H >ε
K(xy −1 )f (y)dy.
As is known, T acts boundedly in Lp (H n ), 1 < p < ∞ (see [3], [4]).
For singular integrals with Cauchy–Szeg¨
o kernels the weighted estimates
were established in the norms of Lp (H n , ω) with weights ω satisfying the
condition Ap [5]. These results extend to the more general kernels considered
above [4].
Theorem 1 [4]. Let 1 < p < ∞ and ω ∈ Ap ; then T is bounded in
Lp (H n , ω).
In the sequel we will use
Theorem 2. Let 1 ≤ p ≤ q < ∞ and U (t), V (t) be positive functions
on (0, ∞).
370
V. S. GULIEV
1) The inequality
q 1/q
Z t
Z
Z ∞
≤ K1
ϕ(τ )dτ dt
U (t)
|ϕ(t)|p v(t)dt
0
0
0
∞
1/p
with the constant K1 not depending on ϕ holds iff the condition
p−1
p/q Z t
Z ∞
′
0
is fulfilled;
2) The inequality
Z
Z ∞
U (t)|
t
0
0
t
∞
ϕ(τ )dτ |q dt
1/q
≤ K2
Z
0
∞
1/p
|ϕ(t)|p V (t)dt
with the constant K2 not depending on ϕ holds iff the condition
Z t
p−1
p/q Z ∞
′
V (τ )1−p dτ
0
t
0
is fulfilled.
Note that Theorem 2 was proved by G.Talenti, G.Tomaselli, B.Muckenhoupt [7] for 1 ≤ p = q < ∞, and by J.S.Bradley [8], V.M.Kokilashvili [9],
V.G.Maz’ya [10] for p < q.
epq (γ), γ > 0, if
We say that the weight pair (ω, ω1 ) belongs to the class A
either of the following conditions is fulfilled: a) ω(t) and ω1 (t) are increasing
functions on (0, ∞) and
p/q Z t/2
p−1
Z ∞
′
−1−γq/p′
ω(τ )1−p τ γ−1 dτ
dτ
< ∞;
ω(τ )τ
sup
t>0
0
t
b) ω(t) and ω1 (t) are decreasing functions on (0, ∞) and
p−1
p/q Z ∞
Z t/2
′
′
γ−1
< ∞.
ω1 (τ )τ
ω(τ )1−p τ −1−γp /q dτ
dτ
sup
t>0
0
t
ep (Q) ≡
Theorem 3. Let 1 < p < ∞ and the weight pair (ω, ω1 ) ∈ A
n
epp (Q). Then for f ∈ Lp (H , ω(|x|H )) there exists T f (x) for almost all
A
x ∈ H n and
Z
Z
|T f (x)|p ω1 (|x|H )dx ≤ C
|f (x)|p ω(|x|)H )dx,
(1)
Hn
Hn
where the constant C does not depend on f .
TWO-WEIGHTED Lp -INEQUALITIES
371
Corollary. If ω(t), t > 0 is increasing (decreasing) and the function
ω(t)t−β is decreasing (increasing) for some β ∈ (0, Q(p − 1)) (β ∈ (−Q, 0)),
then T is bounded on Lp (H n , ω(|x|H )).
Proof of Theorem 3. Let f ∈ Lp (H n , ω(|x|H )) and ω, ω1 be positive increasing functions on (0, ∞). We will prove that T f (x) exists for almost all
x ∈ H n . We take any fixed τ > 0 and represent the function f in the norm
of the sum f1 + f2 , where
(
f (x), if |x|H > τ /2
f1 (x) =
, f2 (x) = f (x) − f1 (x).
0,
if |x|H ≤ τ /2
Let ω(t) be a positive increasing function on (0, ∞) and f ∈ Lp (H n , ω(|x|H )).
Then f1 ∈ Lp (H n ) and therefore T f1 (x) exists for almost all x ∈ H n . Now
we will show that T f2 converges absolutely for all x : |x|H ≥ τ . Note that
C(K) = supx∈SH |K(x)| < ∞. Hence
Z
|T f2 (x)| ≤ C(K)
|f (y)|
|y|H ≤τ /2
2 Qp
≤
τ
Z
|xy−1 |Q
H
dy ≤
(2)
1
|f (y)|ω(|y|H ) p
1
ω(|y|H ) p
|y|H ≤τ /2
dy,
since |xy −1 |H ≥ |x|H − |y|H ≥ τ /2. Thus, by the H¨older inequality we can
estimate (2) as
|T f2 (x)| ≤ Cτ
−Q/p
kf kLp (H n ,ω(|x|H ))
Z
τ /2
′
ω(t)1−p tQ−1 dt
0
1/p′
.
Therefore T f2 (x) converges absolutely for all x : |x|H ≥ τ and thus T f (x)
exists for almost all x ∈ H n . Assume ω
¯ 1 (t) to be an arbitrary continuous
¯ 1 (t) ≤ ω1 (t), ω
increasing function on (0, ∞) such that ω
¯ 1 (0) = ω1 (0+) and
Rt
ω
¯ 1 (t) = 0 ϕ(τ )dτ + ω
¯ 1 (t) exists;
¯ 1 (0), t ∈ (0, ∞) (it is obvious that such ω
Rt
for example, ω
¯ 1 (t) = 0 ω1′ (τ )dτ + ω1 (t)).
We observe that the condition a) implies
∃C1 > 0, ∀t > 0, ω1 (t) ≤ C1 ω(t/2).
(3)
Indeed, from
Z
∃C2 > 0, ∀t > 0,
p−1
Z t/2
′
ω(τ )1−p τ Q−1 dτ
≤ C2
ϕ(τ )τ −Q(p−1) dτ
∞
t
0
(4)
372
V. S. GULIEV
we obtain (3), since
Z ∞
Z
ω1 (τ )τ −1−Q(p−1) dτ ≥ Cω1 (t)t−Q(p−1) ,
t
t/2
′
ω(τ )1−p τ Q−1 dτ
0
and, besides,
p−1
≤ Cω(t/2)−1 tQ(p−1)
Z ∞
Z ∞
Z ∞
1
−Q(p−1)
dτ =
ϕ(τ )dτ
λ−1−Q(p−1) dλ =
ϕ(τ )τ
Q(p − 1) t
t
τ
Z λ
Z ∞
Z ∞
λ−1−Q(p−1) dλ
ϕ(τ )dτ ≤
ω1 (τ )τ −1−Q(p−1) dτ.
=
t
t
t
We have
kT f kLp ,¯ω1 (H n ) ≤
¯ 1 (0)
+ ω
Z
Z
|T f (x)|p dx
Hn
|T f (x)|p dx
Hn
Z
1/p
|x|H
ϕ(t)dt
0
1/p
+
= A1 + A2 .
If ω(0+) > 0, then Lp (H n , ω(|x|H )) ⊂ Lp (H n ), and if ω(0+) = 0, then
¯ (t) ≤ ω1 (t) ≤ Cω(t/2) implies ω
ω
¯ 1 (0) = 0. Therefore in the case ω(0+) = 0
we have A2 = 0.
If ω(0) > 0, then f ∈Lp (Rn ) and we have
Z
1/p
Z
1/p
A2 ≤ C ω
≤
¯ 1 (0)
|f (x)|p ω1 (|x|H )dx
≤C
|f (x)|p dx
Hn
Hn
≤ Ckf kLp (H n ,ω(|x|H )) .
Now we can write
Z
A1 ≤
Z
∞
ϕ(t)dt
0
|x|H >t
1/p
≤ A11 + A12 .
|T f (x)p dx
where
p
A11
=
Ap12 =
Z
Z
∞
ϕ(t)dt
0
Z
Z
|x|H >t |y|H >t/2
∞
ϕ(t)dt
0
Z
Z
|x|H >t |y|H t/2
|y|H >t/2
implies f ∈ Lp ({y ∈ H n : |y|H > t}) for any t > 0.
Hence, on account of (3), we have
Z
1/p
Z ∞
|f (x)|p dx
ϕ(t)dt
=
A11 ≤ C
0
=C
≤C
Z
Z
|x|H >t/2
|f (x)|p dx
Hn
Z
2|x|H
ϕ(t)dt
0
|f (x)|p ω1 (2|x|H )dx
Hn
1/p
1/p
≤
≤ Ckf kLp,ω(|x|H ) (H n ).
Obviously, if |x|H > t, |y|H < t/2, then 21 |x|H ≤ |y −1 x|H ≤
Therefore
Z Z
p
K(xy −1 )f (y)dy dx ≤
3
2 |x|H .
|x|H >t |y|H t
≤2
Qp
C(K)
Z
|xy −1 |−Q
H |f (y)|dy
|y|H Q(1 +
Z
0
SH
|x|H >t
1
p′ ),
by virtue of the H¨older inequality, we have
|f (y)|dy = α
Z
dσ(¯
y)
SH
|y|H
1(94), No. 4, 367-376
TWO-WEIGHTED Lp -INEQUALITIES FOR SINGULAR
INTEGRAL OPERATORS ON HEISENBERG GROUPS
V. S. GULIEV
Abstract. Some sufficient conditions are found for a pair of weight
functions, providing the validity of two-weighted inequalities for singular integrals defined on Heisenberg groups.
Estimates for singular integrals of the Calderon–Zygmund type in various spaces (including weighted spaces and the anisotropic case) have attracted a great deal of attention on the part of researchers. In this paper
we will deal with singular integral operators T on the Heisenberg group H n
which have an essentially different character as compared with operators
of the Calderon–Zygmund type. We have obtained the two-weighted Lp inequality with monotone weights for singular integral operators T on H n .
Applications are given.
Let H n be the Heisenberg group (see [1], [2]) realized as a set of points
x = (x0 , x1 , . . . , x2n ) = (x0 , x′ ) ∈ R2n+1 with the multiplication
n
1X
(xi yn+i − xn+i yi ), x′ + y ′ .
xy = x0 + y0 +
2 i=1
The corresponding Lie algebra is generated by the left-invariant vector
fields
∂
∂
, Xi =
+
∂x0
∂xi
∂
1
∂
=
− xi
,
∂xn+i
2 ∂x0
X0 =
Xn+i
∂
1
,
xn+i
2
∂x0
i = 1, . . . , n,
which satisfy the commutation relation
1
X0 ,
4
[X0 , Xi ] = [X0 , Xn+i ] = [Xi , Xj ] = [Xn+i , Xn+j ] = [Xi , Xn+j ] = 0,
[Xi , Xn+i ] =
1991 Mathematics Subject Classification. 42B20, 42B25, 42B30.
367
c 1994 Plenum Publishing Corporation
1072-947X/94/0700-0367$12.50/0
368
V. S. GULIEV
i, j = 1, . . . , n i =
6 j.
The dilation δt : δt x = (t2 x0 , tx′ ), t > 0, is defined on H n . The Haar measure on this group coincides with the Lebesgue measure dx = dx0 dx1 · · · dx2n .
The identity element in H n is e = 0 ∈ R2n+1 , while the element x−1 inverse
to x is (−x).
The function f defined in H n is said to be H-homogeneous of degree m,
on H n , if f (δt x) = tm f (x), t > 0. We also define the norm on H n
2n
X
2 1/4
|x|H = x20 +
xi2
i=1
which is H-homogeneous of degree one. This also yields the distance function, namely, the distance
d(x, y) = d(y −1 x, e) = |y −1 x|H ,
|y −1 x|H =
x0 − y0 −
n
2
1X
(xi yn+i − xn+i yi ) +
2 i=1
2n
2 1/4
X
+
(xi − yi )2
.
i=1
d is left-invariant in the sense that d(x, y) remains unchanged when x and y
are both left-translated by some fixed vector in H n . Furthermore, d satisfies
the triangle inequality d(x, z) ≤ d(x, y) + d(y, z), x, y, z ∈ H n . For r > 0
and x ∈ H n let
B(x, r) = {y ∈ H n ; |y −1 x|H < r} (S(x, r) = {y ∈ H n ; |y −1 x|H = r})
be the H-ball (H-sphere) with center x and radius r.
The number Q = 2n + 2 is called the homogeneous dimension of H n .
Clearly, d(δt x) = tQ dx.
Given functions f (x) and g(x) defined in H n , the Heisenberg convolution
(H-convolution) is obtained by
Z
Z
f (y)g(y −1 x)dy =
(f ∗ g)(x) =
f (xy −1 )g(y)dy,
Hn
Hn
n
where dy is the Haar measure on H .
is summable
The kernel K(x) admitting the estimate |K(x)| ≤ C|x|α−Q
H
in the neighborhood of e for α > 0 and in that case K ∗ g is defined for
the function g with bounded support. If however the kernel K(x) has a
singularity of order Q at zero, i.e., |K(x)| ∼ |x|−Q
H near e, then there arises
a singular integral on H n .
TWO-WEIGHTED Lp -INEQUALITIES
369
Let ω(x) be a positive measurable function on H n . Denote by Lp (H n , ω)
a set of measurable functions f (x), x ∈ H n , with the finite norm
1/p
Z
, 1 ≤ p < ∞.
|f (x)|p ω(x)dx
kf kLP (H n ,ω) =
Hn
We say that a locally integrable function ω : H n → (0, ∞) satisfies
Muckenhoupt’s condition Ap = Ap (H n ) (briefly, ω ∈ Ap ), 1 < p < ∞, if
there is a constant C = C(ω, p) such that for any H-ball B ⊂ H n
Z
Z
′
1
1
+ ′ = 1,
ω 1−p (x)dx ≤ C,
ω(x)dx |B|−1
|B|−1
p
p
B
B
where the second factor on the left is replaced by ess sup{ω −1 (x) : x ∈ B}
if p = 1.
Let K(x) be a singular kernel defined on H n \{e} and satisfying the confunction of degree −Q, i.e., K(δt x) =
ditions: K(x) is an H-homogeneous
R
t−Q K(x) for any t > 0 and SH K(x)dσ(x) = 0, where dσ(x) is a measure
element on SH = S(e, 1).
Denote by ωK (δ) the modulus of continuity of the kernel on SH :
ωK (δ) = sup{|K(x) − K(y)| : x, y ∈ SH ,
It is assumed that
Z
1
ωK (t)
0
dt
< ∞.
t
We consider the singular integral operator T :
Z
Z
−1
K(xy )f (y)dy =: lim
T f (x) =
Hn
|y −1 x|H ≤ δ}.
ε→0+
|xy −1 |H >ε
K(xy −1 )f (y)dy.
As is known, T acts boundedly in Lp (H n ), 1 < p < ∞ (see [3], [4]).
For singular integrals with Cauchy–Szeg¨
o kernels the weighted estimates
were established in the norms of Lp (H n , ω) with weights ω satisfying the
condition Ap [5]. These results extend to the more general kernels considered
above [4].
Theorem 1 [4]. Let 1 < p < ∞ and ω ∈ Ap ; then T is bounded in
Lp (H n , ω).
In the sequel we will use
Theorem 2. Let 1 ≤ p ≤ q < ∞ and U (t), V (t) be positive functions
on (0, ∞).
370
V. S. GULIEV
1) The inequality
q 1/q
Z t
Z
Z ∞
≤ K1
ϕ(τ )dτ dt
U (t)
|ϕ(t)|p v(t)dt
0
0
0
∞
1/p
with the constant K1 not depending on ϕ holds iff the condition
p−1
p/q Z t
Z ∞
′
0
is fulfilled;
2) The inequality
Z
Z ∞
U (t)|
t
0
0
t
∞
ϕ(τ )dτ |q dt
1/q
≤ K2
Z
0
∞
1/p
|ϕ(t)|p V (t)dt
with the constant K2 not depending on ϕ holds iff the condition
Z t
p−1
p/q Z ∞
′
V (τ )1−p dτ
0
t
0
is fulfilled.
Note that Theorem 2 was proved by G.Talenti, G.Tomaselli, B.Muckenhoupt [7] for 1 ≤ p = q < ∞, and by J.S.Bradley [8], V.M.Kokilashvili [9],
V.G.Maz’ya [10] for p < q.
epq (γ), γ > 0, if
We say that the weight pair (ω, ω1 ) belongs to the class A
either of the following conditions is fulfilled: a) ω(t) and ω1 (t) are increasing
functions on (0, ∞) and
p/q Z t/2
p−1
Z ∞
′
−1−γq/p′
ω(τ )1−p τ γ−1 dτ
dτ
< ∞;
ω(τ )τ
sup
t>0
0
t
b) ω(t) and ω1 (t) are decreasing functions on (0, ∞) and
p−1
p/q Z ∞
Z t/2
′
′
γ−1
< ∞.
ω1 (τ )τ
ω(τ )1−p τ −1−γp /q dτ
dτ
sup
t>0
0
t
ep (Q) ≡
Theorem 3. Let 1 < p < ∞ and the weight pair (ω, ω1 ) ∈ A
n
epp (Q). Then for f ∈ Lp (H , ω(|x|H )) there exists T f (x) for almost all
A
x ∈ H n and
Z
Z
|T f (x)|p ω1 (|x|H )dx ≤ C
|f (x)|p ω(|x|)H )dx,
(1)
Hn
Hn
where the constant C does not depend on f .
TWO-WEIGHTED Lp -INEQUALITIES
371
Corollary. If ω(t), t > 0 is increasing (decreasing) and the function
ω(t)t−β is decreasing (increasing) for some β ∈ (0, Q(p − 1)) (β ∈ (−Q, 0)),
then T is bounded on Lp (H n , ω(|x|H )).
Proof of Theorem 3. Let f ∈ Lp (H n , ω(|x|H )) and ω, ω1 be positive increasing functions on (0, ∞). We will prove that T f (x) exists for almost all
x ∈ H n . We take any fixed τ > 0 and represent the function f in the norm
of the sum f1 + f2 , where
(
f (x), if |x|H > τ /2
f1 (x) =
, f2 (x) = f (x) − f1 (x).
0,
if |x|H ≤ τ /2
Let ω(t) be a positive increasing function on (0, ∞) and f ∈ Lp (H n , ω(|x|H )).
Then f1 ∈ Lp (H n ) and therefore T f1 (x) exists for almost all x ∈ H n . Now
we will show that T f2 converges absolutely for all x : |x|H ≥ τ . Note that
C(K) = supx∈SH |K(x)| < ∞. Hence
Z
|T f2 (x)| ≤ C(K)
|f (y)|
|y|H ≤τ /2
2 Qp
≤
τ
Z
|xy−1 |Q
H
dy ≤
(2)
1
|f (y)|ω(|y|H ) p
1
ω(|y|H ) p
|y|H ≤τ /2
dy,
since |xy −1 |H ≥ |x|H − |y|H ≥ τ /2. Thus, by the H¨older inequality we can
estimate (2) as
|T f2 (x)| ≤ Cτ
−Q/p
kf kLp (H n ,ω(|x|H ))
Z
τ /2
′
ω(t)1−p tQ−1 dt
0
1/p′
.
Therefore T f2 (x) converges absolutely for all x : |x|H ≥ τ and thus T f (x)
exists for almost all x ∈ H n . Assume ω
¯ 1 (t) to be an arbitrary continuous
¯ 1 (t) ≤ ω1 (t), ω
increasing function on (0, ∞) such that ω
¯ 1 (0) = ω1 (0+) and
Rt
ω
¯ 1 (t) = 0 ϕ(τ )dτ + ω
¯ 1 (t) exists;
¯ 1 (0), t ∈ (0, ∞) (it is obvious that such ω
Rt
for example, ω
¯ 1 (t) = 0 ω1′ (τ )dτ + ω1 (t)).
We observe that the condition a) implies
∃C1 > 0, ∀t > 0, ω1 (t) ≤ C1 ω(t/2).
(3)
Indeed, from
Z
∃C2 > 0, ∀t > 0,
p−1
Z t/2
′
ω(τ )1−p τ Q−1 dτ
≤ C2
ϕ(τ )τ −Q(p−1) dτ
∞
t
0
(4)
372
V. S. GULIEV
we obtain (3), since
Z ∞
Z
ω1 (τ )τ −1−Q(p−1) dτ ≥ Cω1 (t)t−Q(p−1) ,
t
t/2
′
ω(τ )1−p τ Q−1 dτ
0
and, besides,
p−1
≤ Cω(t/2)−1 tQ(p−1)
Z ∞
Z ∞
Z ∞
1
−Q(p−1)
dτ =
ϕ(τ )dτ
λ−1−Q(p−1) dλ =
ϕ(τ )τ
Q(p − 1) t
t
τ
Z λ
Z ∞
Z ∞
λ−1−Q(p−1) dλ
ϕ(τ )dτ ≤
ω1 (τ )τ −1−Q(p−1) dτ.
=
t
t
t
We have
kT f kLp ,¯ω1 (H n ) ≤
¯ 1 (0)
+ ω
Z
Z
|T f (x)|p dx
Hn
|T f (x)|p dx
Hn
Z
1/p
|x|H
ϕ(t)dt
0
1/p
+
= A1 + A2 .
If ω(0+) > 0, then Lp (H n , ω(|x|H )) ⊂ Lp (H n ), and if ω(0+) = 0, then
¯ (t) ≤ ω1 (t) ≤ Cω(t/2) implies ω
ω
¯ 1 (0) = 0. Therefore in the case ω(0+) = 0
we have A2 = 0.
If ω(0) > 0, then f ∈Lp (Rn ) and we have
Z
1/p
Z
1/p
A2 ≤ C ω
≤
¯ 1 (0)
|f (x)|p ω1 (|x|H )dx
≤C
|f (x)|p dx
Hn
Hn
≤ Ckf kLp (H n ,ω(|x|H )) .
Now we can write
Z
A1 ≤
Z
∞
ϕ(t)dt
0
|x|H >t
1/p
≤ A11 + A12 .
|T f (x)p dx
where
p
A11
=
Ap12 =
Z
Z
∞
ϕ(t)dt
0
Z
Z
|x|H >t |y|H >t/2
∞
ϕ(t)dt
0
Z
Z
|x|H >t |y|H t/2
|y|H >t/2
implies f ∈ Lp ({y ∈ H n : |y|H > t}) for any t > 0.
Hence, on account of (3), we have
Z
1/p
Z ∞
|f (x)|p dx
ϕ(t)dt
=
A11 ≤ C
0
=C
≤C
Z
Z
|x|H >t/2
|f (x)|p dx
Hn
Z
2|x|H
ϕ(t)dt
0
|f (x)|p ω1 (2|x|H )dx
Hn
1/p
1/p
≤
≤ Ckf kLp,ω(|x|H ) (H n ).
Obviously, if |x|H > t, |y|H < t/2, then 21 |x|H ≤ |y −1 x|H ≤
Therefore
Z Z
p
K(xy −1 )f (y)dy dx ≤
3
2 |x|H .
|x|H >t |y|H t
≤2
Qp
C(K)
Z
|xy −1 |−Q
H |f (y)|dy
|y|H Q(1 +
Z
0
SH
|x|H >t
1
p′ ),
by virtue of the H¨older inequality, we have
|f (y)|dy = α
Z
dσ(¯
y)
SH
|y|H