GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 1, 1999, 65-82
TWO-WEIGHTED INEQUALITIES FOR INTEGRAL
OPERATORS IN LORENTZ SPACES DEFINED ON
HOMOGENEOUS GROUPS
V. KOKILASHVILI AND A. MESKHI
Abstract. The optimal sufficient conditions are found for weights,
which guarantee the validity of two-weighted inequalities for singular
integrals in the Lorentz spaces defined on homogeneous groups. In
some particular case the found conditions are necessary for the corresponding inequalities to be valid. Also, the necessary and sufficient
conditions are found for pairs of weights, which provide the validity
of two-weighted inequalities for the generalized Hardy operator in the
Lorentz spaces defined on homogeneous groups.
Introduction
In this paper, the optimal sufficient conditions are found for pairs of
weights, which provide the validity of two-weighted inequalities for singular
integrals in the Lorentz spaces defined on homogeneous groups. In [1–7],
analogous problems were studied in the Lebesgue spaces for the Hilbert
transform and singular integrals, while in [8] the sufficient conditions are
found for pairs of weights, which guarantee the fulfilment of two-weighted
inequalities for singular integrals in the Lorentz spaces defined on Euclidean
spaces. In this paper, the necessary and sufficient conditions are also found
for pairs of weights, which provide the boundedness of the generalized Hardy
operators in the weighted Lorentz spaces. Analogous problems were considered in the Lebesgue spaces in [9–14] and in the Lorentz spaces in [8], [15]
and [16] (see also [17]).
Finally, we would like to note that most of the results obtained in this
paper are new for the classical singular integrals as well.
1991 Mathematics Subject Classification. 42B20, 43A80, 46E40, 47G10.
Key words and phrases. Hilbert transform, singular integrals, Hardy operator,
Lorentz spaces, two-weighted inequality, homogeneous groups.
65
c 1999 Plenum Publishing Corporation
1072-947X/99/0100-0065$15.00/0 
66
V. KOKILASHVILI AND A. MESKHI
Homogeneous Groups and the Spaces of Functions Defined on
Them. Some Known Results
In this section, we give the notion of homogeneous groups and define the
Lorentz spaces. Some known results on the Lorentz spaces and singular
integrals are also presented.
Definition 1 (see [18], p. 5). A homogeneous group is a connected,
simply connected nilpotent Lie group G on whose Lie algebra g a oneparametric group of extensions δt = exp(A ln t), t > 0, is given, where A is
the diagonizable operator on g whose eigenvalues are positive.
For the homogeneous group G the mappings exp ◦δt ◦ exp−1 , t > 0, are
the automorphisms on G which will again be denoted by δt .
The number Q = trA is called a homogeneous dimension of the group G.
Homogeneous groups can be exemplified by an n-dimensional Euclidean
space, Heisenberg groups and so on.
Onto the homogeneous group G we introduce a homogeneous norm, i.e.,
a continuous function r : G → [0, ∞) which is smooth on G\{e} and satisfies
the following conditions:
1. r(x) = r(x−1 ) for any x ∈ G;
2. r(δt x) = tr(x) for arbitrary x ∈ G and t > 0;
3. r(x) = 0 ⇔ x = e;
4. there exists a constant c0 > 0 such that r(xy) ≤ c0 (r(x) + r(y)) for
arbitrary x and y from G.
For x ∈ G and ρ > 0 we set B(x, ρ) = {y ∈ G : r(xy −1 ) < ρ}. Further,
let S(x, ρ) = {y ∈ G : r(xy −1 ) = ρ}. Note that δρ B(e, 1) = B(e, ρ).
Onto the group G we fixed the normed Haar measure in such a manner
that the measure of the unit ball B(e, 1) be equal to 1. The Haar measure
of any measurable set E ⊂ G willR be denoted by |E|, and the integral on E
with respect to this measure by f (x)dx.
E
It readily follows that
|δt E| = tQ |E|,
d(δt x) = tQ dx.
In particular, for arbitrary x ∈ G and ρ > 0 we have |B(e, ρ)| = ρQ .
Definition 2. An almost everywhere positive, locally integrable function
w : G → R1 will be called a weight.
In what follows, we shall denote by Lpq
w (G) the Lorentz space with weight
w which is a class of all measurable functions f : G → R1 for which
kf kLpq
w (G)
’ Z∞ ’
= q
0
Z
{x∈G}:|f (x)|>λ}
w(x) dx
“ pq
q−1
λ
dλ
“ q1
<∞
TWO-WEIGHTED INEQUALITIES FOR INTEGRAL OPERATORS
when 1 ≤ p ≤ ∞, 1 ≤ q < ∞, and
’
λ
kf kLp∞
=
sup
w (G)
λ>0
Z
w(x) dx
{x∈G:|f (x)|>λ}
“ p1
67
<∞
when 1 ≤ p < ∞.
For p = q, Lpq
w (G) is the Lebesgue space which it is commonly accepted
to denote by Lpw (G).
Theorem A (see [18], p. 14). Let G be a homogeneous group, SG =
{x ∈ G : r(x) = 1}. There exists a unique countably additive measure σ
defined on SG such that the equality
“
’Z
Z
Z∞
Q−1
f (δt ξ) dσ(ξ) dt
f (x) dx = t
0
G
SG
1
holds for any f ∈ L (G).
Let k : G → R1 be a measurable function such that:
c
6 e;
(1) |k(x)| ≤ r(x)
Q for an arbitrary element x =
(2) there exists a positive constant c1 such that for arbitrary x, y ∈ G
with the condition r(xy −1 ) < 12 r(x) we have the inequality

−1 ‘
Œ
Œ
1
Œk(x) − k(y)Œ ≤ c1 ω r(xy )
,
r(x)
r(x)Q
where ω : [0, 1] → R1 is a nondecreasing function such that ω(0) = 0,
R1
ω(2t) ≤ c2 ω(t) for any t > 0 and ω(t)
t dt < ∞.
0
It will be assumed that the kernel k together with the above-given conditions satisfy the condition: the singular integral
Z
Z
def
k(xy −1 )f (y) dy
T f (x) = p.v. k(xy −1 )f (y) dy = lim
ε→0
G\B(x,ε)
G
defines the bounded operator in the space L2 (G).
The lemmas below are valid.
Lemma A ([19]). Let E ⊂ G be an arbitrary measurable set, w a weight
function on G and f , f1 , f2 measurable functions on G. Then we have:
’Z
“ p1
(1) kχE (·)kLpq
=
w(x)
;
dx
w (G)
E
(2)
kf kLpq
≤ kf kLpq
,
1
2
w (G)
w (G)
for fixed p and q2 ≤ q1 ;
68
V. KOKILASHVILI AND A. MESKHI
(3)
where
p2 q2
≤ ckf1 kLpw1 q1 (G) kf2 kLw
kf1 f2 kLpq
(G) ,
w (G)
1
p
=
1
p1
+
1
1
p2 , q
=
1
q1
+
1
q2 .
Ek be measurable sets of the homogeneous group G such
Lemma B. Let P
that the inequality
χEk ≤ cχ∪Ek with constant c is fulfilled. Then:
k
k
(1) for any f we have the inequality
X


λ
f (·)χE (·)λ rs
≤ c1 f (·)χ∪Ek (·)Lrs (G) ,
L (G)
k
w
k
k
w
where the constant c1 does not depend on f and max(r, s) ≤ λ;
(2) for any f we have the inequality

X
X

γ

f (·)χE (·)γ pq ,


f (·)χEk (·)
≤ c2

L (G)
k
w
pq
Lw
(G)
k
k
where the constant c2 does not depend on f and 0 < γ ≤ min(p, q).
Lemma B is proved as Lemma 2 from [8]. For the case c = 1 analogous
result was obtained in [20], [15] (see also [16]).
Definition 3. A function v : G → R1+ is called radial if there exists a
function β : R1+ → R1+ such that the equality v(x) = β(r(x)) holds for any
x ∈ G. In what follows, instead of β we shall use the notation v.
Definition 4. Let 1 < p < ∞. A weight function w belongs to Ap (G) if
’
“’
“p−1
Z
Z
1
1
1−p0
sup
w(x) dx
w
(x) dx
< ∞,
|B|
|B|
B
B
where the least upper bound is taken with respect to all balls B, B ⊂ G.
Theorem B ([21]). Let 1 < p < ∞, w ∈ Ap (G). Then the operator T
is bounded in Lpw (G).
Theorem C (see [22], p. 207). Let 1 < p, q < ∞. If w ∈ Ap (G),
then the operator T is bounded in Lpq
w (G). If the Hilbert transform acts
continuously in Lpq
w (R), then w ∈ Ap (R).
We shall need
Lemma C ([4], [7]). Let 1 < p < ∞, ρ ∈ Ap (G), 0 ≤ c1 < c2 ≤ c3 <
c4 < ∞. Then there exists a positive constant c such that the inequality
Z
Z
ρ(x) dx ≤ c
ρ(x) dx
{x∈G:c3 t<r(x)<c4 t}
holds for any t, t > 0.
{x∈G:c1 t<r(x)<c2 t}
TWO-WEIGHTED INEQUALITIES FOR INTEGRAL OPERATORS
69
1. Two-Weighted Inequalities for the Hardy Operator in the
Lorentz Spaces Defined on Homogeneous Groups
In this paragraph, the necessary and sufficient conditions are found for
the boundedness of the operators
Z
b(y)f (y)w(y) dy,
Hf (x) = a(x)
r(y)≤r(x)
Z
∗
H f (x) = b(x)
a(y)f (y)w(y) dy
r(y)≥r(x)
Lrs
w (G)
from
into
functions on G.
Lpq
v (G),
where a, b, w and v are measurable non-negative
Theorem 1.1. Let r = s = 1 or r = s = ∞ or r ∈ (1, ∞) and s ∈
(1, ∞), p = q = 1 or p = q = ∞ or p ∈ (1, ∞) and q ∈ (1, ∞); max(r, s) ≤
min(p, q). For the inequality
kHf (·)kLvpq (G) ≤ ckf (·)kLrs
,
w (G)
(1.1)
where the constant c does not depend on f , to be valid it is necessary and
sufficient that the condition




sup a(·)χ{r(y)>t} (·)Lpq (G) b(·)χ{r(y)<t} (·)Lr0 s0 (G) < ∞
(1.2)
v
t>0
(r0 =
r
r−1 ,
s0 =
s
s−1 )
w
be fulfilled.
Proof.
Sufficiency. Assume that condition (1.2) is fulfilled. We take f ≥ 0. If
R
f (x)b(x)w(x) dx < ∞, then it belongs to the interval (2m , 2m+1 ] for some
G
m
integer m. By virtue of Theorem A we can choose a sequence {xk }k=−∞
such that
Z
Z
2k =
b(y)f (y)w(y) dy for k ≤ m − 1 (1.3)
b(y)f (y)w(y) dy =
r(y)≤xk
xk <r(y)≤xk+1
and
2
m
Z
=
b(y)f (y)w(y) dy.
(1.4)
r(y)≤xm
Let Gk = {y : xk < r(y) ≤ xk+1 }, k ≤ m and xm+1 = ∞. Then the sets
Gk do not intersect pairwise and, since we can assume that lim xk = 0,
k→−∞
we obtain
S
k≤m
Gk = G \ {e}.
(1.5)
70
If
V. KOKILASHVILI AND A. MESKHI
R
G
b(y)f (y)w(y) dy = ∞, then we choose a sequence {xk }+∞
k=−∞ such that
(1.3) is fulfilled for any integer k (in this case m = +∞). By (1.3) and (1.4)
we have
Hf (x) ≤ a(x) · 2k+1
for x ∈ Gk ,
k ≤ m.
(1.6)
Choose a number σ such that max(r, s) ≤ σ ≤ min(p, q). Using (1.5),
Lemma B, (1.6), Hölder’s inequality, we obtain
X
σ


σ

(Hf )(·)χGk (·)
≤
kHf (·)kLvpq (G) = 

≤
≤
X
2
k≤m
≤ 4σ
σ(k+1)
X’
k≤m
≤4
X
k≤m
k(Hf )(·)χGk (·)kσLpq
≤
v (G)
ka(·)χGk (·)kσLvpq (G) = 22σ
Z
X
k≤m
X
2σ(k−1) ka(·)χGk (·)kσLpq
≤
v (G)
k≤m
“σ
b(y)f (y)w(y) dy
xk−1 <r(y)≤xk
σ
Lpq
v (G)
k≤m
≤
ka(·)χ{r(y)>xk } (·)kσLpq
v (G)
σ
kf (·)χGk−1 (·)kσLw
×
rs (G) kb(·)χ{r(y)<x } (·)k r 0 s0
L
(G)
k
w
σ
×ka(·)χ{r(y)>xk } (·)kσLvpq (G) ≤ ckf (·)kL
rs
w (G)
where the constant c does not depend on f .
rs (G) ≤ 1 and t ∈
Necessity. Let inequality (1.1) be fulfilled. If kf kLw
(0, ∞), then we have
c ≥ ckf kLrs
≥
≥ kHf kLvpq (G) ≥ k(Hf )(·)χ{r(y)>t} (·)kLpq
v (G)
w (G)
Z
.
≥
b(y)f (y)w(y) dy ka(·)χ{r(y)>t} (·)kLpq
v (G)
r(y)<t
Taking the least upper bound with respect to all such f and t, we obtain
condition (1.2).
If we apply the dual arguments (see also [8]), then we easily obtain
Theorem 1.2. Let the numbers r, s, p, q satisfy the conditions of Theorem 1.1. For the inequality
kH ∗ f (·)kLpq
≤ ckf (·)kLrs
,
v (G)
w (G)
where the constant c does not depend on f , to be valid it is necessary and
sufficient that the condition




sup b(·)χ{r(y)<t} (·)Lpq (G) a(·)χ{r(y)>t} (·)Lr0 s0 (G) < ∞
t>0
v
w
TWO-WEIGHTED INEQUALITIES FOR INTEGRAL OPERATORS
71
be fulfilled.
Rx
Analogous results for the operator Hf (x) = a(x) b(t)f (t)w(t) dt are
0
given in [16]. A certain analog in Rn was obtained in [8].
Corollary 1.1. Let 1 < r, s, p, q < ∞, max(r, s) ≤ min(p, q), γ < − Q
p,
β = γr + Qr( r10 + p1 ). Then the inequality


Z


r(·)γ

≤ ckf (·)kLrs β (G)
dy
f
(y)


r(y)≤r(·)
Lpq (G)
r(·)
holds, where the constant c does not depend on f .
2. Weighted Inequalities for Singular Integrals in the
Lorentz Spaces Defined on Homogeneous Groups
In this paragraph, the sufficient conditions are found for pairs of weights,
which provide the boundedness of a singular integral operator in the weighted Lorentz spaces defined on homogeneous groups.
First we investigate the existence of T f (x).
Lemma 1. Let 1 < s ≤ p < ∞, ρ ∈ Ap (G). If the weight functions w
and w1 satisfy the conditions:
(1) there exists a positive increasing on (0, ∞) function σ such that for
almost all x ∈ G we have the inequality
€

σ r(x) ρ(x) ≤ bw(x)w1p (x),
where the positive constant b does not depend on x;


1


(2) 
< ∞ for any t > 0,
χ{r(y)<t} (·) p0 s0
w(·)w1 (·)
Lw (G)
then for arbitrary ϕ with the condition kϕ(·)w1 (·)kLps
< ∞, T ϕ(x) exists
w (G)
almost everywhere on G.
Proof. Fix the number α, α > 0, and let
n
αo
.
Sα = x ∈ G : r(x) >
2
Take a function ϕ with the condition kϕ(·)w1 (·)kLps
< ∞. We write φ
w (G)
as
ϕ(x) = ϕ1 (x) + ϕ2 (x),
where ϕ1 (x) = ϕ(x) · χSα (x), ϕ2 (x) = ϕ(x) − ϕ1 (x).
72
V. KOKILASHVILI AND A. MESKHI
For ϕ1 we obtain
Z
Z
σ( α2 )
|ϕ(x)|p ρ(x) dx ≤
|ϕ1 (x)|p ρ(x) dx =
σ( α2 )
G
Sα
Z
Z
€

1
b
p
≤
|ϕ(x)| ρ(x)σ r(x) dx ≤
|ϕ(x)|p w1p (x)w(x) dx ≤
σ( α2 )
σ( α2 )
Sα
≤
Sα
b
b
kϕ(·)w1 (·)kpLp (G) ≤
kϕ(·)w1 (·)kpLps (G) .
w
w
σ( α2 )
σ( α2 )
We find by Theorem B that T ϕ1 ∈ Lpρ (G) and T ϕ1 (x) exists almost everywhere on G.
Now we shall show that T ϕ2 (x) converges absolutely for r(x) > αc0 . Note
that when r(x) > αc0 and r(y) < α2 , we have r(x) ≤ c0 (r(xy −1 ) + r(y)) ≤
r(x)
α
−1
c0 (r(xy −1 ) + α2 ) ≤ c0 (r(xy −1 ) + r(x)
).
2c0 ) and 2 < 2c0 ≤ r(xy
We obtain
Z
Z
c2
ϕ(y)
≤
dy
|ϕ(y)| dy =
|T ϕ2 (x)| ≤ c1
r(xy −1 )Q
αQ
{y:r(y)< α
2}
=
≤
c2
αQ
Z
{y:r(y)< α
2}
{y:r(y)< α
2}
ϕ(y)
1
w1 (y)w(y)dy ≤
w1 (y)w(y)


c2
1


< ∞.
kϕ(·)w1 (·)kLps
 p0 s0
χ{y:r(y)< α } (·)
w (G)
Q
2
α
w1 (·)w(·) Lw (G)
Since we can take α arbitrarily small, T ϕ(x) exists almost everywhere on
G.
The next lemma is proved in a similar manner.
Lemma 2. Let 1 < p, s < ∞, s ≤ p, ρ ∈ Ap (G). If the weight functions
w and w1 satisfy the conditions:
(1) there exists a positive decreasing on (0, ∞) function σ such that for
almost all x ∈ G we have the inequality
€

σ r(x) ρ(x) ≤ bw(x)w1p (x),
where the positive constant b does not depend on x;

 r(·)−Q


(2) 
χ{r(y)>t} (·) p0 s0
< ∞ for any t > 0,
w(·)w1 (·)
Lw (G)
then for arbitrary ϕ with the condition kϕ(·)w1 (·)kLps
< ∞, T ϕ(x) exists
w (G)
almost everywhere on G.
Lemma 1 readily implies
TWO-WEIGHTED INEQUALITIES FOR INTEGRAL OPERATORS
73
Lemma 3. Let 1 < p, s, < ∞, s ≤ p. If u and u1 are positive increasing
functions on (0, ∞) and




1

<∞
χ{r(y)<t} (·) p0 s0
u1 (r(·))u(r(·))
Lu(r(·)) (G)
for arbitrary t > 0, then for any ϕ with the condition kϕ(·)u1 (r(·))kLps (G)
u(r(·))
< ∞, T ϕ(x) exists almost everywhere on G.
Analogously, Lemma 2 implies
Lemma 4. Let 1 < p, s < ∞, s ≤ p. If u and u1 are positive decreasing
functions on (0, ∞) and




r(·)−Q

<∞
χ{r(y)>t} (·) p0 s0
u(r(·))u1 (r(·))
Lu(r(·)) (G)
for arbitrary t > 0, then for ϕ with the condition ϕ(·)u1 (r(·)) ∈ Lps
u(r(·)) (G)
< ∞, T ϕ(x) exists almost everywhere on G.
Now we shall formulate and prove our basic theorems.
Theorem 2.1. Let 1 < s ≤ p ≤ q < ∞, σ be a positive increasing
function on (0, ∞), the function ρ ∈ Ap (G), w a weight function on G,
v(x) = σ(r(x))ρ(x). Let the following conditions be fullfilled:
(1) there exists a positive constant b such that almost for all x ∈ G
€

σ 2c0 r(x) ρ(x) ≤ bw(x);
(2)
 1



kr(·)−Q χ{r(y)>t} (·)kLpq
χ
(·)
≤ c < ∞.

 p 0 s0
v (G)
w(·) {r(y)<t}
Lw (G)
Then there exists a positive constant c such that the inequality
kT f (·)kLpq
≤ ckf (·)kLps
w (G)
v (G)
holds for any f ∈ Lps
w (G).
Proof. Without loss of generality we can write the function σ as
σ(t) = σ(0) +
Zt
0
ϕ(τ ) dτ,
(2.1)
74
V. KOKILASHVILI AND A. MESKHI
where σ(0) = lim σ(t) and ϕ(τ ) ≥ 0 on (0, ∞). We have
t→0
kT f (·)k
Lpq
v (G)
’
’ Z∞
q−1
≤ c1 q λ
0
’ Z∞
’
q−1
+c1 q λ
0
“ pq “ q1
ρ(x)σ(0) dx dλ
+
Z
{x:|T f (x)|>λ}
’ r(x)
“ “ pq “ q1
Z
=
ρ(x)
ϕ(t) dt dx dλ
Z
{x:|T f (x)|>λ}
0
= I1 + I 2 .
If σ(0) = 0, then I1 = 0, and if σ(0) 6= 0, then by Theorem C and Lemma
A we obtain
1
1
≤ c2 σ p (0)kf (·)kLpq
≤
I1 = c1 σ p (0)kT f (·)kLpq
ρ (G)
ρ (G)
1
ps
≤ c3 kf (·)kLw
≤ c2 σ p (0)kf (·)kLps
(G) .
ρ (G)
Now we shall estimate I2 . Let f1t (x) = f (x) · χ{r(x)>
f (x) − f1t (x). We have
’ Z∞
’ Z∞
’
q−1
I2 = c1 q λ
ϕ(t)
0
0
Z
t }
2c0
(x), f2t (x) =
“ “ pq “ q1
≤
ρ(x) dx dt dλ
{x:r(x)>t, |T f (x)|>λ}
’ Z∞
’ Z∞
’
q−1
ϕ(t)
≤ c4 q λ
Z
“ “ pq “ q1
n
λo
ρ(x)dx dt dλ +
χ x : |T f1t (x)| >
2
’ Z∞
’ Z∞
’
q−1
+c4 q λ
ϕ(t)
Z
“ “ pq “ q1
n
λo
ρ(x)dx dt dλ =
χ x : |T f2t (x)| >
2
0
0
0
0
{x:r(x)>t}
{x:r(x)>t}
= I21 + I22 .
Applying Minkowski’s inequality twice ( pq ≥ 1,
obtain
I21 ≤ c5
≤ c6
’ Z∞
0
’ Z∞
0
’ Z∞
’
ϕ(t)
λq−1
0
Z
p
s
≥ 1) and Theorem C, we
ρ(x) dx
{x:|T f1t (x)|>λ}
“ pq
dλ
“ pq
dt
“ p1
≤
“ p1
’ Z∞
“ p1
p
p
ϕ(t)kf1t (·)kLpq (G) dt
≤ c6
ϕ(t)kf1t (·)kLps (G) dt
≤
ρ
ρ
0
TWO-WEIGHTED INEQUALITIES FOR INTEGRAL OPERATORS
≤ c6
= c6
’ Z∞
λ
s−1
ϕ(t)
0
0
’ Z∞
’ Z∞
s−1
λ
0
’
’
Z
ρ(x)χ{r(y)>
t
2c0
{x:|f (x)|>λ}
“ “ ps “ 1s
=
(x) dx dt dλ
}
’ 2cZ0 r(x)
“ “ ps “ 1s
≤ c7 kf kLps
.
ρ(x)
ϕ(t) dt dx dλ
w (G)
Z
0
{x:|f (x)|>λ}
Next, we shall estimate I22 . Note that if r(x) > t and r(y) <
r(x) ≤ 2c0 r(xy −1 ). By Theorem 1.1 we obtain
I22
’ Z∞
’ Z∞
’
q−1
ϕ(t)
= c4 q λ
0
0

 1

= c8 
r(·)Q
75
{x:
Z
{r(y)<r(·)}
(2c0 )Q
r(x)Q
t
2c0 ,
then
“ “ pq “ q1
ρ(x)χ{r(y)>t} (x) dx dt dλ ≤
Z
R
f (y)dy> λ
2}
{y:r(y)<r(x)}


|f (y)| dy 

Lpq
v (G)
ps
≤ c9 kf (·)kLw
(G) .
If we write the function σ as
σ(t) = σ(+∞) +
Z∞
ψ(τ )dτ,
t
where σ(+∞) = lim σ(t), ψ(τ ) ≥ 0,
t→+∞
on (0, ∞), again apply Theorem C, Lemma A and Theorem 1.2, then we
shall have
Theorem 2.2. Let 1 < s ≤ p ≤ q < ∞, σ be a positive decreasing function on (0, ∞), w a weight function on G, ρ ∈ Ap (G), v(x) = σ(r(x))ρ(x).
Let the following conditions be fulfilled:
(1) there exists a positive constant b such that the inequality
ρ(x)σ
 r(x) ‘
2c0
≤ bw(x)
holds for almost all x ∈ G;
(2)

 r(·)−Q


χ
< ∞.
(·)
sup kχ{r(y)<t} (·)kLpq
 p0 s0

v (G)
w(·) {r(y)>t}
Lw (G)
t>0
Then inequality (2.1) is valid.
It is proved in [8] that if inequality (2.1) is fulfilled for the Hilbert transform, then the inequality v(x) ≤ b1 w(x) holds almost everywhere on R1 .
76
V. KOKILASHVILI AND A. MESKHI
Corollary 2.1. Let 1 < s ≤ p ≤ q < ∞, σ1 and σ2 be positive increasing
functions on (0, ∞), ρ ∈ Ap (G), v(x) = σ2 (r(x))ρ(x), w(x) = σ1 (r(x))ρ(x).
If the conditions
(1) there exists a positive constant b such that the inequality
σ2 (2c0 t) ≤ bσ1 (t)
holds for any t > 0;
(2)

 1


<∞
χ{r(y)<t} (·) p0 s0
sup kr(·)−Q χ{r(y)>t} (·)kLpq

v (G)
w(·)
Lw (G)
t>0
are fulfilled, then inequality (2.1) is valid.
Corollary 2.2. Let 1 < s ≤ p ≤ q < ∞, σ1 and σ2 be positive decreasing
functions on (0, ∞), ρ ∈ Ap (G), v(x) = σ2 (r(x))ρ(x), w(x) = σ1 (r(x))ρ(x).
If the conditions
 t ‘
(1) σ2
≤ bσ1 (t) for any t > 0;
2c0

 r(·)−Q


<∞
χ
(·)
(2) sup kχ{r(y)<t} (·)kLpq
 p0 s0

v (G)
w(·) {r(y)>t}
Lw (G)
t>0
are fulfilled, then inequality (2.1) is valid.
Theorem 2.3. Let 1 < p ≤ q < ∞, σ1 and σ2 be positive increasing
functions on (0, ∞), ρ ∈ Ap (G), v(x) = σ2 (r(x))ρ(x), w(x) = σ1 (r(x))ρ(x).
If v and w satisfy the condition
 1



sup kr(·)−Q χ{r(y)>t} (·)kLpq
(·)
χ
<∞

 p0
v (G)
w(·) {r(y)<t}
Lw (G)
t>0
then the inequality
≤ ckf (·)kLpw (G)
kT f (·)kLpq
v (G)
holds, where the positive constant c does not depend on f .
Proof. By Corollary 2.1. it is sufficient to show that the inequality
σ2 (2c0 t) ≤ bσ1 (t)
holds for any t > 0.
0
Using Hölder’s inequality and Lemma C (ρ1−p ∈ Ap0 (G)), we obtain
σ2 (2c0 t) −Qp
t
≤ c1
σ1 (t)
’
σ2 (2c0 t)
≤
σ1 (t)
“’
Z
ρ(x) dx
{x:2c0 t<r(x)<4c0 t}
Z
1−p0
ρ
“p−1
≤
(x) dx
{x:2c0 t<r(x)<4c0 t}
TWO-WEIGHTED INEQUALITIES FOR INTEGRAL OPERATORS
≤ c2
σ2 (2c0 t) −Qp
t
σ1 (t)
≤ c2 t
−Qp
’
Z
ρ(x) dx
{x:2c0 t<r(x)<4c0 t}
’
Z
Z
0
ρ1−p (x) dx
{x:r(x)<t}
“’
v(x) dx
{x:2c0 t<r(x)<4c0 t}
=
“’
Z
w
1−p0
{x:r(x)<t}

 1
c2 t−Qp kχ{x:2c0 t<r(x)<4c0 t} (·)kpLpq (G) 
v
w(·)
“p−1
77
≤
“p−1
(x) dx
=
p

χ{r(y)<t} (·) p0
Lw (G)
 1
p


≤ c3 kr(·)−Q χ{r(y)>t} (·)kpLpq (G) 
< ∞.
(·)
χ
 p0
v
w(·) {r(y)<t}
Lw (G)
≤
Theorem 2.4. Let 1 < s ≤ p ≤ q < ∞; σ1 , σ2 , u1 and u2 be weight
functions on G, ρ ∈ Ap (G), v = σ2 ρ, w = σ1 ρ. Let the following three
conditions be fulfilled:
(1) there exists a positive constant b such that for any t > 0 we have the
inequality
1
1
sup σ2p (x) sup u2 (x) ≤ b inf σ1p (x) inf u2 (x),
Ft
where Ft = {x ∈ G :
Ft
Ft
t
c0
Ft
< r(x) < 8c0 t};

1

χ{r(y)<t} (·) p0 s0
< ∞;
u1 (·)w(·)
Lw (G)
t>0


1


r(·)−Q χ{r(y)>t}(·) p0 s0
(3) sup ku2 (·)χ{r(y)<t} (·)kLvpq (G) 
< ∞.
u1 (·)w(·)
Lw (G)
t>0


(2) sup ku2 (·)r(·)−Q χ{r(y)>t} (·)kLpq

v (G)
Then the inequality
ps
≤ cku1 (·)f (·)kLw
ku2 (·)T f (·)kLpq
(G)
v (G)
(2.2)
is valid, where the positive constant c does not depend on f .
Proof. Let Ek = {x ∈ G : 2k < r(x) ≤ 2k+1 }, Gk1 = {x ∈ G : r(x) ≤
k−1
2k−1
c0 },
k+2
Gk2 = {x ∈ G : 2 c0 < r(x) ≤ c0 2k+2 }, Gk3 = {x ∈ G : r(x) > c0 2
We estimate the left-hand side of inequality (2.2) as follows:
p
X


p


T f (·)u2 (·)χEk (·)
≤
ku2 (·)T f (·)kLpq (G) = 
v
k∈Z
Lpq
v (G)
X
p



≤ c1 
u2 (·)T (f · χGk1 )(·)χEk (·)
+
 pq
Lv (G)
k∈Z

X
p


(f
(·)
u
(·)T
·
χ
)(·)χ
+c1 
+
2
Gk2
Ek


k∈Z
Lpq
v (G)
}.
78
V. KOKILASHVILI AND A. MESKHI
X
p




+c1 
u2 (·)T (f · χGk3 )(·)χEk (·)
pq
= c1 (S1p + S2p + S3p ).
Lv (G)
k∈Z
Next, we shall estimate S1p . Note that for x ∈ Ek and y ∈ Gk1 we have
k−1
2k
−1
≤ r(x)
) ≥ r(x)
r(y) ≤ 2 c0 = 2c
2c0 < r(x). Moreover, r(xy
2c0 and we obtain
0
ŒZ
Œ
Œ
Œ
−1
Œ
|T (f · χGk1 )(x)| = Œ k(xy )f (y)χGk1 (y) dy ŒŒ ≤
≤ c2
Z
G
|f (y)|χGk1 (y)
r(xy −1 )Q
G
Z
1
dy ≤ c3
r(x)Q
|f (y)| dy
r(y)<r(x)
for any x ∈ Ek . By Theorem 1.1 we find that

’
“p
Z


p
p
−Q
S1 ≤ c3 u2 (·)r(·)
f (y) dy 

Lpq
v (G)
{r(y)≤r(x)}
≤ c4 ku1 (·)f (·)kpLps (G) .
w
Let us estimate S3p . As is easy to verify, for x ∈ Ek and y ∈ Gk3 we have
r(y) ≥ r(x) and r(xy −1 ) ≥ r(y)
2c0 . For x ∈ Ek we obtain
Z
|f (y)|
|T (f · χGk3 )(x)| ≤ c5
dy.
r(y)Q
{r(y)≥r(x)}
By Theorem 1.2 we find that


Z
p

f (y)

u2 (·)
S3p ≤ cp5 
dy
≤ c6 ku1 (·)f (·)kpLps (G) .
w
r(y)Q Lpq
v (G)
{r(y)≥r(x)}
Now we shall estimate S2p . By Lemma B (second part) we obtain
X p
X
Sk2 .
ku2 (·)T (f · χGk2 )(·)χEk (·)kpLpq (G) =
S2p ≤
v
k
k∈Z
Introducing the notation
u2k = sup u2 (x), σ2k = sup σ2 (x), u1k =
x∈Ek
x∈Ek
inf u1 (x), σ1k = inf σ1 (x), by Lemma A and Theorem C we obtain
x∈Ek
x∈Ek
1
1
p
p
Sk2 ≤ u2k σ2k
kT (f · χGk2 )(·)kLpq
kf (·)χGk2 (·)kLpq
≤ c7 u2k σ2k
≤
ρ (G)
ρ (G)
1
1
p
p
kf (·)χGk2 (·)kLps
≤ c8 u1k σ1k
kf (·)χGk2 (·)kLps
≤
≤ c7 u2k σ2k
ρ (G)
ρ (G)
s
1
”X ’
“
•
Z
p
s
up1k σ1k ρ(x) dx
≤ c9
2js
≤
j
Gk2 ∩{|f |>2j }
TWO-WEIGHTED INEQUALITIES FOR INTEGRAL OPERATORS
≤ c9
”X
(u1k 2j )s
j
’
Z
w(x) dx
Gk2 ∩{u1 f >u1k 2j }
“ ps • 1s
79
ps
≤ c10 ku1 (·)f (·)χGk2 (·)kLw
(G) .
By Lemma B (first part) we have
p
S2p ≤ c11 ku1 (·)f (·)kL
.
ps
(G)
w
In [8] (see also [1], [4]) it is shown that for the Hilbert transform conditions
(2) and (3) of Theorem 2.4 are necessary for inequality (2.2) to be fulfilled.
Remark 1. One can easily verify that Theorem 2.4 remains in force when
condition (1) is replaced by
€ the condition

σ2 (y)up2 (y) ≤ b1 σ1 (x)u1p (x) for a.a. x ∈ G.
(10 )
sup
r(x)
4c0 <r(y)≤4c0 r(x)
Now let us consider the case with radial weight functions.
Corollary 2.3. Let 1 < s ≤ p ≤ q < ∞; u1 , u2 and v be positive
increasing functions on (0, ∞) which satisfy the condition


1


sup ku2 (·)r(·)−Q χ{r(y)>t} (·)kLpq (G) 
=
χ{r(y)<t} (·) p0 s0
v(r(·))
u1 (r(·))
L
(G)
t>0
= sup B(t) < ∞.
t>0
Then the inequality
kT f (·)u2 (r(·))kLpq
v(r(·))
(G)
≤ ckf (·)u1 (r(·))kLps (G)
(2.3)
holds, where the positive constant c does not depend on f .
Proof. First we shall show that the inequality
1
u2 (8c0 t)v p (8c0 t) ≤ bu1
t‘
c0
holds, where the positive constant b does not depend on t. Indeed, using
Lemma A and Theorem A, we obtain
B(t) ≥ ku2 (r(·))r(·)−Q χ{t<r(y)<2t} (·)kLpq
v(r(·))


׍
(G)
×
“ p1
’ Z2t

1

−Q
Q−1
χ
≥ c1 t u2 (t)
v(τ )τ
×
dτ

u1 (r(·)) {r(y)< 8ct02 } Lp0 s0 (G)
t
Œn
1
1
1
t oŒŒ p10
Œ
×
:
.
y
r(y)
<
Œ
Œ ≥ c2 u2 (t)v p (t)
8c20
u1 ( 8ct 2 )
u1 ( 8ct 2 )
0
Moreover,
0
B1 (t) = ku2 (r(·))χ{r(y)<t} (·)kLpq
v(r(·))
(G)
×
80
V. KOKILASHVILI AND A. MESKHI

1

r(·)−Q χ{r(y)>t}  p0 s0
≤
u1 (r(·))
L
(G)
Q
1
1
kr(·)−Q χ{r(y)> t } (·)kLp0 s0 (G) .
≤ c3 u2 (t)v p (t)t p
u1 ( 8ct 2 )
8c2
0


׍
0
It is easy to verify that
kr(·)−Q χ{r(y)>
Q
t }
8c2
0
(·)kLp0 s0 (G) ≤ c4 t− p ,
where c4 does not depend on t > 0. We have sup B1 (t) ≤ c5 . Using
t>0
Theorem 2.4, we obtain inequality (2.3).
An analogous reasoning is used to prove
Corollary 2.4. Let 1 < s ≤ p ≤ q < ∞; u1 , u2 and v be positive
decreasing functions on (0, ∞) and


1


< ∞.
r(·)−Q χ{r(y)>t} (·) p0 s0
sup ku2 (r(·))χ{r(y)<t} (·)kLpq (G) 
v(r(·))
u1 (r(·))
L
(G)
t>0
Then inequality (2.3) holds.
Now we shall give examples illustrating pairs of weights.
Example 1. Let 1 < p < q < ∞, v(t) = tp−1 , w(t) = tp−1 lnγ 2π
t , where
p − 1 < γ < p and γ = pq + p − 1. Then the pair (v, w) satisfies the condition



v(|·|)
sup k| · |−1 χ{t<|y|<π} (·)kLpq
0<t<π
and therefore the inequality
kfe(·)kLpq
v(|·|)
1
1
p
w (| · |)


χ{|y|<t} (·)
Lp0
<∞
≤ ckf (·)kLp
w(|·|)
holds, where fe is the conjugate function and the constant c does not depend
on f .
p
Example 2. Let 1 < s ≤ p < ∞, sp0 ≤ γ < p, s = p+1−γ
(γ =
Then the inequality


γ 2π 
p−1
p−1

kfe(·)| · | p kLp ≤ cf (·)| · | p ln p
 ps
|·| L
holds, where the positive constant c does not depend on f .
p
s0
+ 1).
TWO-WEIGHTED INEQUALITIES FOR INTEGRAL OPERATORS
81
Example 3. Let 1 < s ≤ p ≤ q < ∞, γ = p( 1q + 1 − 1s ). Then the
inequality


γ 2π 
p−1
p−1

kfe(·)| · | p kLpq ≤ cf (·)| · | p ln p
(2.4)

| · | Lps
holds, where the constant c > 0 does not depend on f . Inequality (2.4)
remains in force for γ > p( 1q + 1 − 1s ) too.
Acnowledgement
The first author was partially supported by the Georgian Academy of
Sciences under Grant No. 1.7 and the second by the grant of the Soros
International Educational Program.
References
1. B. Muckenhoupt and R. L. Wheeden, Two-weight function norm inequalities for the Hardy–Littlewood maximal function and the Hilbert transform. Studia Math. 55(1976), No. 3, 279–294.
2. E. G. Gusseinov, Singular integrals in the spaces of functions summable
with monotone weight. (Russian) Mat. Sb. 132(174)(1977), No. 1, 28–44.
3. V. S. Guliev, Two-weighted Lp inequalities for singular integral operators on Heisenberg groups. Georgian Math. J. 1(1994), No. 4, 367–376.
4. D. E. Edmunds and V. M. Kokilashvili, Two-weighted inequalities for
singular integrals. Canad. Math. Bull. 38(1995), No. 3, 295–303.
5. A. Meskhi, Two-weight inequalities for singular integrals defined on
Heisenberg groups. Bull. Acad. Sci. Georgia 153(1996), No. 1, 34–36.
6. V. Kokilashvili, New aspects in weight theory and applications. Function spaces, differential operators and nonlinear analysis (Paseky nad Jizerou, 1995), 1–16, Prometheus Publishing House, Prague, 1996.
7. V. Kokilashvili and A. Meskhi, Two-weight inequalities for singular
integrals defined on homogeneous groups. Proc. A. Razmadze Math. Inst.
Georgian Acad. Sci. 112(1997), 57–91.
8. Y. Rakotondratsimba, On the boundedness of classical operators on
weighted Lorentz spaces. Georgian Math. J. 5(1998), No. 2, 177–200.
9. V. M. Kokilashvili, On Hardy’s inequalities in weighted spaces. (Russian) Soobshch. Akad. Nauk Gruzin. SSR 96(1979), 37–40.
10. J. S. Bradley, Hardy’s inequalities with mixed norms. Canad. Math.
Bull. (1) 21(1978), 405–408.
11. V. G. Maz’ya, Sobolev spaces. Springer, Berlin, 1985.
12. B. Muckenhoupt, Hardy’s inequality with weight. Studia Math.
44(1972), No. 1, 31–38.
82
V. KOKILASHVILI AND A. MESKHI
13. G. Tomaselli, A class of inequalities. Bull. Un. Mat. Ital. 21(1969),
622–631.
14. G. Talenti, Osservazioni sopra una classe di disuguaglianze. Rend.
Sem. Mat. Fis. Milano 36(1969), 171–185.
15. E. Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the
Hardy operator. Trans. Amer. Math. Soc. 281(1984), 329–337.
16. D. Edmunds, P. Gurka, and L. Pick, Compactness of Hardy-type integral operators in weighted Banach function spaces. Studia Math. 109(1994),
No. 1, 73–90.
17. M. Krbec, B. Opic, L. Pick, and J. Rakosnik, Some recent results on
Hardy type operators in weighted function spaces and related topics. Function Spaces, differential operators and nonlinear analysis (Friedrichroda,
1992), 158–184, Teubner-Texte Math., 133, Teubner, Stuttgart, 1993.
18. G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups.
Princeton University Press and University of Tokyo Press, Princeton, New
Jersey, 1982.
19. A. Hunt, On L(p, q) spaces. Enseign. Math. 12(1966), 249–276.
20. H.-M. Chung, R. A. Hunt, and P. S. Kurtz, The Hardy–Littlewood
maximal function on L(p, q) spaces with weights. Indiana Univ. Math. J.
31(1982) 109–120.
21. S. Hoffman, Weighted norm inequalities and vector-valued inequalities for certain rough operators. Indiana Univ. Math. J. 42(1993), No. 1,
1–14.
22. V. Kokilashvili and M. Krbec, Weighted inequalities in Lorentz and
Orlicz spaces. World Scientific, Singapore, New Jersey, London, Hong
Kong, 1991.
(Received 21.11.1996)
Authors’ address:
A. Razmadze Mathematical Institute
Georgian Academy of Sciences
1, M. Aleksidze St., Tbilisi 380093
Georgia