Directory UMM :Journals:Journal_of_mathematics:GMJ:
Georgian Mathematical Journal
1(1994), No. 5, 495-503
TWO-WEIGHTED ESTIMATES FOR SOME INTEGRAL
TRANSFORMS IN THE LEBESGUE SPACES WITH
MIXED NORM AND IMBEDDING THEOREMS
V.KOKILASHVILI
Abstract. Two-weighted inequalities are proved for anisotropic potentials. These estimates are used to obtain the refinements of the
well-known imbedding theorems in the scale of weighted Lebesgue
spaces.
Two-weighted inequalities are obtained for anisotropic potentials in Lebesgue spaces with mixed norm. These estimates are used to prove imbedding theorems for different metrics and different dimensions for weighted
spaces of anisotropic Bessel potentials.
Nonweighted cases were previously treated in [1–3]. One-weighted estimates for isotropic Bessel potentials can be found in [4].
1. A measurable almost everywhere positive function ̺ : R1 → R1 will
be called a weight function. Let w = (w1 , w2 , . . . , wn ) be a vector-function
where wi (i = 1, 2, . . . , n) is a weight function. By definition a measurable
function f (x) = f (x1 , x2 , . . . , xn ) given on the n-dimensional space Rn
belongs to Lpw , p = (p1 , p2 , . . ., pn), 1 < pi < ∞ (1 = 1, 2, . . . , n), if the
norm
kf kLpw (Rn ) =
Z∞
=
−∞
w1p1 (x1 )dx1
Z∞
−∞
w2p2 (x2 )dx2 . . .
Z∞
pn−1
pp1
pn
2
...
f (x)wnpn (xn )dxn
−∞
! p1
1
is finite.
We shall introduce a class of pairs of weight functions.
r
.
For a given number r, 1 < r < ∞, we write r′ = r−1
1991 Mathematics Subject Classification. 42B25, 46E30.
495
c 1994 Plenum Publishing Corporation
1072-947X/1994/0900-0495$07.00/0
496
V.KOKILASHVILI
Definition 1. A pair of weight functions (̺, σ) given on R1 will be said
to belong to the class Gs,r
β , 0 < β < 1, 1 < r < s < ∞, if the conditions
Z
1s Z
′
1′
σ −r (t)
r
< ∞,
sup
̺ (t)dt
dt
′
(|I| + |tI − t|)βr
1
I
R
Z
1′ Z
1s
̺s (t)
r
−r ′
sup
σ (t)dt
< ∞,
dt
sβ
(|I| + |tI − t|)
s
(1.1)
(1.2)
R1
I
are fulfilled, where the supremum is taken over all bounded one-dimensional
intervals I, with centre and length, tI and |I| respectively.
In the sequel we shall proceed from
Theorem A [5–7]. The fractional integral
Iγ (f )(x) =
Z∞
f (τ )
dτ,
|t − τ |1−γ
0 < γ < 1,
−∞
generates a continuous operator from Lrσ (R1 ) into Ls̺ (R1 ) if and only if
s,r
.
(̺, σ) ∈ G1−γ
Let numbers aj > 0 (j = 1, 2, . . . , n) be given. For x=(x1 , x2 , . . . , xn) we
set
n
1
X
2 2
|x|a =
|x| aj .
i=1
It is obvious that for aj = 1 (j = 1, 2, . . . , n) we obtain an usual Euclidian
distance.
Theorem 1. Let w = (w1 , w2 , . . . , wn ), v = (v1 , v2 , . . . , vn ), where wi
and vi (i = 1, 2, . . . , n) are weight functions given on Rn . We set
Z
f (y)
dy,
Kf (x) =
|x − y|µa
Rn
where
µ=
n
X
aj (1 − γj ),
0 < γj < 1 (j = 1, 2, . . . , n).
j=1
i ,pi
If 1 < pi < qi < ∞, (vi , wi ) ∈ Gq1−γ
(i = 1, 2, . . . , n), then there exists a
i
positive number c such that the inequality kKf kLqv ≤ ckf kLpw holds for any
f ∈ Lpw .
The proof of Theorem 1 will be based on several lemmas. The first lemma
is a weighted analogue of the well–known Hardy–Littlewood inequality (see
[8], Theorem 382).
TWO-WEIGHTED ESTIMATES FOR INTEGRAL TRANSFORMS
497
Lemma 1. Let (̺, σ) ∈ Gs,r
1−γ , 1 < r < s < ∞, 0 < γ < 1. Then there
exists a constant c > 0, such that the inequality
Z∞ Z∞ ϕ(x)ψ(y)
dx
dy
≤ ckϕkLrσ kψkLs′
1/̺
|x − y|1−γ
(1.3)
−∞ −∞
′
s
holds for any arbitrary ϕ ∈ Lrσ (R1 ) and ψ ∈ L1/̺
(R1 ).
Proof. By virtue of the H¨older inequality we have
Z∞ Z∞ ϕ(x)ψ(y)
dx
dy
≤
|x − y|1−γ |
−∞ −∞
Z Z |ψ(y)|dy r′ 1
1′
r
≤ kϕσkLr
dx
.
|x − y|1−γ
σ r′ (x)
R1
R1
′
′
r ,s
1 1
From the condition (̺, σ) ∈ Gs,r
1−γ readily follows that ( σ , ̺ ) ∈ G1−γ . Using
Theorem A we obtain the estimate
Z∞ Z∞ ϕ(x)ψ(y)
dx
dy
≤ ckϕkLrσ · kψkLs′ .
1/̺
|x − y|1−γ
−∞ −∞
Bellow we shall set ̺ = ( v11 , v12 , . . . , v1n ) for v = (v1 , v2 , . . . , vn ).
i ,pi
Lemma 2. Let 1 < pi < qi < ∞, (vi , wi ) ∈ Gq1−γ
, 0 < γi < 1. Then
i
there exists a positive constant c such that
Z Z ϕ(x)ψ(y)
· kψkLq′
µ dx dy ≤ ckϕkLp
w
̺
|x − y|a
Rn Rn
′
for arbitrary ϕ ∈ Lpw (Rn ) and ψ ∈ Lq̺ (Rn ).
Proof. We shall apply the reduction technique to the one-dimensional case.
Let x′ = (x1 , x2 , . . . , xn−1 ), y ′ = (y1 , y2 , . . . , yn−1 ) and
µ′ =
n−1
X
aj (1 − γj ), a′ = (a1 , a2 , . . . , an−1 ).
j=1
It readily follows that
′
|x − y|aµ = |x − y|µa |x − y|aan (1−γn ) ≥ |x′ − y ′ |aµ′ |xn − yn |1−γn .
′
498
V.KOKILASHVILI
Therefore by Lemma 1 we obtain
Z∞ Z∞
Z Z
Z Z ϕ(x)ψ(y)
dx′ dy ′
|ϕ(x)| |ψ(y)|
dxn dyn ≤
′
µ dx dy ≤
µ
′
′
|x − y|a
|xn − yn |1−γn
|x − y |a
n
n
n−1
n−1
−∞ −∞
R R
R
R
Z Z
F (x′ )H(y ′ ) ′ ′
≤c
′ dx dy ,
|x′ − y ′ |µa′
Rn−1 Rn−1
where
Z
F (x′ ) =
|ϕ(x)|pn wnpn (xn )dxn
R1
′
H(x ) =
Z
′
′
−qn
|ψ(y)|qn vn
R1
p1
(yn )dyn
n
1
′
qn
,
.
Further reduction leads us to the proof of Lemma 2.
Proof of Theorem 1. By the property of the norm and also by Lemma 2 we
have
Z
q
kKf kLv = sup Kf (x)g(x)dx,
Rn
where the least upper bound is taken over all functions g for which
1 1
1
kgkLq′ ≤ 1, ̺ =
, ,... ,
.
̺
v1 v2
vn
Next, by Lemma 2 we have
Z Z f (y)g(x)
≤ ckf kLpw · kgkLq′ ≤ ckf kLpw .
dx
dy
̺
|x − y|aµ
Rn Rn
Theorem 2. Let 1≤m≤n, v+=(v1 , v2 , . . . , vm ), w+=(w1 , w2 , . . . , wm ),
w = (w1 , w2 , . . . , wm , 1, . . . , 1), 1 < pi < qi < ∞ (i = 1, 2, . . . , m), q+ =
(q1 , q2 , . . . , qm ), p+ = (p1 , p2 , . . . , pm ), 1 < pi < ∞ (i = m + 1, . . . , n).
Next we set
µ=
m
X
aj (1 − γj ) +
i=1
n
X
aj
,
p′
j=m+1 j
(1.4)
where aj > 0, 0 < γj < 1 (j = 1, 2, . . . , m)
qi ,pi
(i = 1, 2, . . . , m), then there exists a constant c > 0
If (vi , wi ) ∈ G1−γ
i
0
, . . . , x0n ) the function
such that for any f ∈ Lpw (Rn ) and arbitrary (xm+1
Z
f (y)
dy
h(x) =
|x − y|µa
Rn
TWO-WEIGHTED ESTIMATES FOR INTEGRAL TRANSFORMS
499
+
belongs to the space Lvq + (Rm ) and the inequality holds khkLq+ ≤ ckf kLpw ,
v+
where the constant c is independent of f .
The proof of Theorem 2 is based on the following
Lemma 3. Let the conditions of Theorem 2 be fulfilled. If ̺ =( v11 , v12 , . . . ,
1
′
′
′
′
vn ), q+ = (q1 , q2 , . . . , qm ), then there exists a constant c > 0 such that for
1
n
all f : R → R and g : Rm → R1 we have
Z Z g(x)f (y)
p
′
.
(1.5)
µ ≤ ckf kLw
(Rn ) · kgk q+
|x − y|a
L̺+ (Rm )
Rm Rn
Proof. Let y ′=(y1 , y2 , . . . , ym ), y ′′=(ym+1 , . . . , yn ), p+ = (p1 , p2 , . . . , pm ),
p− = (pm+1 , . . . , pn ), p′− = (p′m+1 , . . . , pn′ ).
Obviously
Z
Z Z g(x)f (y)
Z Z |f (y)|dy ′′
dx
dy
dy ′ dx. (1.6)
|g(x)|
≤
|x − y|µa
|x − y|µa
Rm
Rm
Rm Rn
Rn−m
By virtue of the H¨older inequality
Z
|f (y)|dy ′′
≤ kf kLp− (Rn−m ) · k |x − y|a−µ k p′− n−m .
)
L (R
|x − y|µa
(1.7)
Rn−m
We introduce some notation:
ϕ(y ′ ) = kf (y ′ , ·)kLp− (Rn−m ) , ϕ1 (x, y ′ ) = k |x − y|−µ
a k
L
T =
m
X
2
|xj − yj | aj
j=1
21
p′
− (Rn−m )
.
Let us prove that there exists a positive number c1 such that
Pn
aj
j=m+1 p′
j
ϕ1 (x, y ′ ) ≤ c1 T
−µ
.
We have
n
µ
X
2 − 2
ϕ1 (x, y ′ ) = T 2 +
|xj − yj | aj
= T2 +
j=m+1
n
X
j=m+1
p′
− (Rn−m )
L
2
|yj | aj
− µ2
p′
− (Rn−m )
L
,
.
=
(1.8)
500
V.KOKILASHVILI
The change of the variable yj = T aj tj in the latter expression leads to
the equality
Pn
aj
n
µ
X
2 − 2
−µ
p′
′
|tj | aj
ϕ(x, y ) = T j=m+1 j 1 +
.
′
p−
n−m
L
j=m+1
Now it is obvious that
Pn
n
n
µ
1
X
X
2 − 2
2 − 2
k=m+1
aj
aj
= 1+
1+
|tj |
|tj |
ak
p′
k
(R
Pm
+
j=m+1
j=m+1
n
Y
=
j=m+1
1+
n
X
|tj |
j=m+1
2
aj
− 21 ( a′j +ε)
p
j
n
Y
≤
1 + |tj |
2
aj
j=m+1
i=1
)
ai (1−γi )
=
a
− 12 ( p′j +ε)
j
,
where ε > 0.
Therefore
n
µ
X
2 − 2
|tj | aj
1+
p′
− (Rn−m )
L
j=m+1
n
1 aj
2 − ( ′ +ε)
2 p
Y
j
1 + |tj | aj
≤
R1
1 + |tj |
dtj
≤
1 aj
2 ( ′ +ε)p′
2 p
j
a
j
p′
− (Rn−m )
L
j=m+1
On the other hand,
Z
≤
j
Z
R1
dtj
1+
(1 + |tj |)
εp′
j
aj
.
< ∞.
Thus we have proved the estimate (1.8). It implies
Pm
m
m
1
X
Y
aj (1−γj )
2 − 2
j=1
′
aj
ϕ(x, y ) ≤
|xj − yj |
≤ c1
|xj − yj |(γj −1) . (1.9)
j=1
j=1
Using the generalized H¨older inequality and Lemma 1 with (1.6), (1.7),
(1.8) and (1.9) we obtain:
Z Z
Z
Z
|g(x)|ϕ(y ′ )
g(x)f (y)
Qm
dx dy ≤
dx
µ dy ≤ c2
1−γj
|x − y|a
j=1 |xj − yj |
Rm
Rm Rm
Rn
′
≤ c3 kϕ(y )kLpw+ (Rm ) kgkLq′ (Rm ) .
+
̺
This implies that
Z
Z
g(x)f (y)
dx
dy ≤ c3 kf kLpw (Rn ) kgkL̺q+ (Rm ) .
|x − y|µa
Rm
Rn
TWO-WEIGHTED ESTIMATES FOR INTEGRAL TRANSFORMS
501
Proof of Theorem 2. Using the standard arguments, the validity of Theorem 2 readily follows from Lemma 3.
2. In this paragraph we shall prove the imbedding theorems for different
metrics and different dimensions for weighted spaces of anisotropic Bessel
potentials.
Definition 2 (see[2]). Let r = (r1 , r2 , . . . , rn ), p = (p1 , p2 , . . . , pn), rj >
n
0 (j = 1, 2, . . . , n). It will be said that f ∈ Lp,r
w (R ) if
Z
f (x) = Gr (x − y)g(y)dy,
Rn
p
(Rn ). By
where Gr is the anisotropic Bessel–Macdonald kernel and g ∈ Lw
the definition, kf kLp,r
= kgkLpw .
w
The kernel Gr is characterized by its Fourier transform as follows (see[2])
∗
n
b 2 (λ) = [1 + σ 2 (λ)]− r2 where the function σ(λ) is determined by the
(2π) 2 G
equation
n
n
X
λj2
1X 1
r∗
1
=
.
= 1, aj = ,
σ 2aj
rj
r∗
n j=1 rj
j=1
The kernel Gr obeys, along each j – the coordinate direction, the estimates
Pn
1
−r
−1
i=1 ri
(2.1)
|Gr (x)| ≤ c|xj | j
.
Now we shall prove the imbedding theorem of different metrics.
qi ,pi
(i = 1, 2, . . . , n). Put
Theorem 3. Let 1 < pj < qj < ∞, (vi , wi ) ∈ G1−γ
i
κ =1−
n
X
γj
j=1
rj
and ̺ = κr, r = (r1 , r2 , . . . , rn ).
p,r
n
Then each function f ∈ Lw
(Rn ) belongs to the space Lq,̺
v (R ) and the
inequality kf kLq,̺
≤ ckf kLp,r
holds, where the constant c is independant of
v
w
f.
Proof. We have
f (x) =
Z
G̺ (x − y)h(y)dy,
Rn
where
h(x) =
Z
Rn
and g ∈
p
Lw
(Rn ).
Gr(1−κ) (x − y)g(y)dy
502
or
V.KOKILASHVILI
Now it will be shown that h ∈ Lqv (Rn ). Due to (2.1) we have
Pn
1
−1
−rj (1−κ)
ri (1−κ)
i=1
,
|Gr(1−κ) (x)| ≤ c|xi |
−rj
|Gr(1−κ) (x)| ≤ c|xi |
Let aj =
1
rj
and
max |xj |
1
aj
1≤j≤n
Pn
i=1
ai (1−γi )
Pn
= |xj0 |
i=1
1
aj
0
1−γi
ri
Pn
.
i=1
ai (1−γi )
.
As can be easily verified,
n
X
i=1
|xj |
2
aj
12 Pn
i=1
ai (1−γi )
1
≤ c1 |xj0 | aj0
Pn
i=1
ai (1−γi )
.
Therefore
|Gr(1−κ) (x)| ≤ c2 |x|a−µ ,
where
µ=
n
X
(2.2)
aj (1 − γj ).
j=1
Hence
|h(x)| ≤
Z
|g(y)| |x − y|−µ
a dy.
Rn
Applying Theorem 1, we obtain khkLqv ≤ c3 kf kLpw , which implies kf kLq,̺
v
≤ kf kLp,r
.
w
Using Theorem 2 one may prove an imbedding theorem of different dimensions in a similar manner.
Theorem 4. Let 1 < pi < ∞ (i = 1, 2, . . . , n), 1 < pi < qi < ∞
i ,pi
(i = 1, 2, . . . , m), 1 ≤ m ≤ n. It is also assumed that (vi , wi ) ∈ Gq1−γ
,
i
0 < γi < 1 (i = 1, 2, . . . ).
If
κ =1−
m
X
γj
j=1
rj
−
n
X
1
,
r
p
j=m+1 j j
(2.3)
n
then for an arbitrary function f from the space Lp,r
w (R ) the function
0
0
F (x1 , x2 , . . . , xm ) = f (x1 , x2 , . . . , xm , xm+1 , . . . , xn ) belongs to the space
≤ ckf kLp,r
Lvq,̺ (Rm ) and the inequality kF kLq,̺
holds where the
m
n
v (R )
w (R )
constant c is independent of f .
TWO-WEIGHTED ESTIMATES FOR INTEGRAL TRANSFORMS
503
Proof. In the case under consideration the kernel G(1−κ)r admits the estimate |G(1−κ)r (x)| ≤ c|x|a−µ , where a = (a1 , a2 , . . . , an ), ai = r1j and
µ=
m
X
1 − γi
i=1
ri
+
n
X
1
.
r
p′j
j
j=m+1
(2.4)
Hence we can apply Theorem 2. The rest of the proof is as for the
preceding theorem.
References
1. P.I. Lizorkin, Generalized Liouville differentiation and the method of
multipliers in the theory of imbedding of classes of differentiable functions.
(Russian) Trudy Mat. Inst. Steklov. 105(1969), 87–167; English transl.:
Proc. Steklov Inst. Math. 105(1969).
2. —–, Multipliers of Fourier integrals and bounds of convolutions in
spaces with mixed norms. Applications. (Russian) Izv. Akad. Nauk SSSR,
Ser. Mat. 34(1970), 218-247; English transl.: Math. USSR Izv. 4(1970).
3. I.P. Ilin, Properties of some classes of differentiable functions of several
variables. (Russian) Trudy Mat. Inst. Steklov. 66(1962), 227-363; English
transl. Proc. Steclov. Inst. Mat. (2)81(1969).
4. V. Kokilashvili, On weighted Lizorkin-Triebel spaces. Singular integrals, multipliers, imbedding theorems. (Russian) Trudy Math. Inst.
Steklov. 161(1983), 125-149; English transl.: Proc. Steklov Inst. Mat.
3(1984), 135-162.
5. V. Kokilashvili and M. Krbec, Weighted inequalities in Lorentz and
Orlicz spaces. World Scientific, Singapur–New Jersey–London–Hong Kong,
1991.
6. M. Gabidzashvili, I. Genebashvili and V. Kokilashvili, Two weighted
inequalities for generalized potentials. (Russian) Trudy Mat. Inst. Steklov.
194(1992), 85-96; English transl.: Proc. Steklov Inst. Mat. (to appear).
7. E. Sawyer and R.L. Wheeden, Weighted inequalities for fractional
integrals on euclidian and homogeneous spaces. Amer. J. Math. 114(1992),
813-874.
8. G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities. London,
Cambridge University Press, 1951.
(Received 02.08.1993)
Author’s address:
A. Razmadze Mathematical Institute
Georgian Academy of Sciences
1, Z. Rukhadze St., Tbilisi 380093
Republic of Georgia
1(1994), No. 5, 495-503
TWO-WEIGHTED ESTIMATES FOR SOME INTEGRAL
TRANSFORMS IN THE LEBESGUE SPACES WITH
MIXED NORM AND IMBEDDING THEOREMS
V.KOKILASHVILI
Abstract. Two-weighted inequalities are proved for anisotropic potentials. These estimates are used to obtain the refinements of the
well-known imbedding theorems in the scale of weighted Lebesgue
spaces.
Two-weighted inequalities are obtained for anisotropic potentials in Lebesgue spaces with mixed norm. These estimates are used to prove imbedding theorems for different metrics and different dimensions for weighted
spaces of anisotropic Bessel potentials.
Nonweighted cases were previously treated in [1–3]. One-weighted estimates for isotropic Bessel potentials can be found in [4].
1. A measurable almost everywhere positive function ̺ : R1 → R1 will
be called a weight function. Let w = (w1 , w2 , . . . , wn ) be a vector-function
where wi (i = 1, 2, . . . , n) is a weight function. By definition a measurable
function f (x) = f (x1 , x2 , . . . , xn ) given on the n-dimensional space Rn
belongs to Lpw , p = (p1 , p2 , . . ., pn), 1 < pi < ∞ (1 = 1, 2, . . . , n), if the
norm
kf kLpw (Rn ) =
Z∞
=
−∞
w1p1 (x1 )dx1
Z∞
−∞
w2p2 (x2 )dx2 . . .
Z∞
pn−1
pp1
pn
2
...
f (x)wnpn (xn )dxn
−∞
! p1
1
is finite.
We shall introduce a class of pairs of weight functions.
r
.
For a given number r, 1 < r < ∞, we write r′ = r−1
1991 Mathematics Subject Classification. 42B25, 46E30.
495
c 1994 Plenum Publishing Corporation
1072-947X/1994/0900-0495$07.00/0
496
V.KOKILASHVILI
Definition 1. A pair of weight functions (̺, σ) given on R1 will be said
to belong to the class Gs,r
β , 0 < β < 1, 1 < r < s < ∞, if the conditions
Z
1s Z
′
1′
σ −r (t)
r
< ∞,
sup
̺ (t)dt
dt
′
(|I| + |tI − t|)βr
1
I
R
Z
1′ Z
1s
̺s (t)
r
−r ′
sup
σ (t)dt
< ∞,
dt
sβ
(|I| + |tI − t|)
s
(1.1)
(1.2)
R1
I
are fulfilled, where the supremum is taken over all bounded one-dimensional
intervals I, with centre and length, tI and |I| respectively.
In the sequel we shall proceed from
Theorem A [5–7]. The fractional integral
Iγ (f )(x) =
Z∞
f (τ )
dτ,
|t − τ |1−γ
0 < γ < 1,
−∞
generates a continuous operator from Lrσ (R1 ) into Ls̺ (R1 ) if and only if
s,r
.
(̺, σ) ∈ G1−γ
Let numbers aj > 0 (j = 1, 2, . . . , n) be given. For x=(x1 , x2 , . . . , xn) we
set
n
1
X
2 2
|x|a =
|x| aj .
i=1
It is obvious that for aj = 1 (j = 1, 2, . . . , n) we obtain an usual Euclidian
distance.
Theorem 1. Let w = (w1 , w2 , . . . , wn ), v = (v1 , v2 , . . . , vn ), where wi
and vi (i = 1, 2, . . . , n) are weight functions given on Rn . We set
Z
f (y)
dy,
Kf (x) =
|x − y|µa
Rn
where
µ=
n
X
aj (1 − γj ),
0 < γj < 1 (j = 1, 2, . . . , n).
j=1
i ,pi
If 1 < pi < qi < ∞, (vi , wi ) ∈ Gq1−γ
(i = 1, 2, . . . , n), then there exists a
i
positive number c such that the inequality kKf kLqv ≤ ckf kLpw holds for any
f ∈ Lpw .
The proof of Theorem 1 will be based on several lemmas. The first lemma
is a weighted analogue of the well–known Hardy–Littlewood inequality (see
[8], Theorem 382).
TWO-WEIGHTED ESTIMATES FOR INTEGRAL TRANSFORMS
497
Lemma 1. Let (̺, σ) ∈ Gs,r
1−γ , 1 < r < s < ∞, 0 < γ < 1. Then there
exists a constant c > 0, such that the inequality
Z∞ Z∞ ϕ(x)ψ(y)
dx
dy
≤ ckϕkLrσ kψkLs′
1/̺
|x − y|1−γ
(1.3)
−∞ −∞
′
s
holds for any arbitrary ϕ ∈ Lrσ (R1 ) and ψ ∈ L1/̺
(R1 ).
Proof. By virtue of the H¨older inequality we have
Z∞ Z∞ ϕ(x)ψ(y)
dx
dy
≤
|x − y|1−γ |
−∞ −∞
Z Z |ψ(y)|dy r′ 1
1′
r
≤ kϕσkLr
dx
.
|x − y|1−γ
σ r′ (x)
R1
R1
′
′
r ,s
1 1
From the condition (̺, σ) ∈ Gs,r
1−γ readily follows that ( σ , ̺ ) ∈ G1−γ . Using
Theorem A we obtain the estimate
Z∞ Z∞ ϕ(x)ψ(y)
dx
dy
≤ ckϕkLrσ · kψkLs′ .
1/̺
|x − y|1−γ
−∞ −∞
Bellow we shall set ̺ = ( v11 , v12 , . . . , v1n ) for v = (v1 , v2 , . . . , vn ).
i ,pi
Lemma 2. Let 1 < pi < qi < ∞, (vi , wi ) ∈ Gq1−γ
, 0 < γi < 1. Then
i
there exists a positive constant c such that
Z Z ϕ(x)ψ(y)
· kψkLq′
µ dx dy ≤ ckϕkLp
w
̺
|x − y|a
Rn Rn
′
for arbitrary ϕ ∈ Lpw (Rn ) and ψ ∈ Lq̺ (Rn ).
Proof. We shall apply the reduction technique to the one-dimensional case.
Let x′ = (x1 , x2 , . . . , xn−1 ), y ′ = (y1 , y2 , . . . , yn−1 ) and
µ′ =
n−1
X
aj (1 − γj ), a′ = (a1 , a2 , . . . , an−1 ).
j=1
It readily follows that
′
|x − y|aµ = |x − y|µa |x − y|aan (1−γn ) ≥ |x′ − y ′ |aµ′ |xn − yn |1−γn .
′
498
V.KOKILASHVILI
Therefore by Lemma 1 we obtain
Z∞ Z∞
Z Z
Z Z ϕ(x)ψ(y)
dx′ dy ′
|ϕ(x)| |ψ(y)|
dxn dyn ≤
′
µ dx dy ≤
µ
′
′
|x − y|a
|xn − yn |1−γn
|x − y |a
n
n
n−1
n−1
−∞ −∞
R R
R
R
Z Z
F (x′ )H(y ′ ) ′ ′
≤c
′ dx dy ,
|x′ − y ′ |µa′
Rn−1 Rn−1
where
Z
F (x′ ) =
|ϕ(x)|pn wnpn (xn )dxn
R1
′
H(x ) =
Z
′
′
−qn
|ψ(y)|qn vn
R1
p1
(yn )dyn
n
1
′
qn
,
.
Further reduction leads us to the proof of Lemma 2.
Proof of Theorem 1. By the property of the norm and also by Lemma 2 we
have
Z
q
kKf kLv = sup Kf (x)g(x)dx,
Rn
where the least upper bound is taken over all functions g for which
1 1
1
kgkLq′ ≤ 1, ̺ =
, ,... ,
.
̺
v1 v2
vn
Next, by Lemma 2 we have
Z Z f (y)g(x)
≤ ckf kLpw · kgkLq′ ≤ ckf kLpw .
dx
dy
̺
|x − y|aµ
Rn Rn
Theorem 2. Let 1≤m≤n, v+=(v1 , v2 , . . . , vm ), w+=(w1 , w2 , . . . , wm ),
w = (w1 , w2 , . . . , wm , 1, . . . , 1), 1 < pi < qi < ∞ (i = 1, 2, . . . , m), q+ =
(q1 , q2 , . . . , qm ), p+ = (p1 , p2 , . . . , pm ), 1 < pi < ∞ (i = m + 1, . . . , n).
Next we set
µ=
m
X
aj (1 − γj ) +
i=1
n
X
aj
,
p′
j=m+1 j
(1.4)
where aj > 0, 0 < γj < 1 (j = 1, 2, . . . , m)
qi ,pi
(i = 1, 2, . . . , m), then there exists a constant c > 0
If (vi , wi ) ∈ G1−γ
i
0
, . . . , x0n ) the function
such that for any f ∈ Lpw (Rn ) and arbitrary (xm+1
Z
f (y)
dy
h(x) =
|x − y|µa
Rn
TWO-WEIGHTED ESTIMATES FOR INTEGRAL TRANSFORMS
499
+
belongs to the space Lvq + (Rm ) and the inequality holds khkLq+ ≤ ckf kLpw ,
v+
where the constant c is independent of f .
The proof of Theorem 2 is based on the following
Lemma 3. Let the conditions of Theorem 2 be fulfilled. If ̺ =( v11 , v12 , . . . ,
1
′
′
′
′
vn ), q+ = (q1 , q2 , . . . , qm ), then there exists a constant c > 0 such that for
1
n
all f : R → R and g : Rm → R1 we have
Z Z g(x)f (y)
p
′
.
(1.5)
µ ≤ ckf kLw
(Rn ) · kgk q+
|x − y|a
L̺+ (Rm )
Rm Rn
Proof. Let y ′=(y1 , y2 , . . . , ym ), y ′′=(ym+1 , . . . , yn ), p+ = (p1 , p2 , . . . , pm ),
p− = (pm+1 , . . . , pn ), p′− = (p′m+1 , . . . , pn′ ).
Obviously
Z
Z Z g(x)f (y)
Z Z |f (y)|dy ′′
dx
dy
dy ′ dx. (1.6)
|g(x)|
≤
|x − y|µa
|x − y|µa
Rm
Rm
Rm Rn
Rn−m
By virtue of the H¨older inequality
Z
|f (y)|dy ′′
≤ kf kLp− (Rn−m ) · k |x − y|a−µ k p′− n−m .
)
L (R
|x − y|µa
(1.7)
Rn−m
We introduce some notation:
ϕ(y ′ ) = kf (y ′ , ·)kLp− (Rn−m ) , ϕ1 (x, y ′ ) = k |x − y|−µ
a k
L
T =
m
X
2
|xj − yj | aj
j=1
21
p′
− (Rn−m )
.
Let us prove that there exists a positive number c1 such that
Pn
aj
j=m+1 p′
j
ϕ1 (x, y ′ ) ≤ c1 T
−µ
.
We have
n
µ
X
2 − 2
ϕ1 (x, y ′ ) = T 2 +
|xj − yj | aj
= T2 +
j=m+1
n
X
j=m+1
p′
− (Rn−m )
L
2
|yj | aj
− µ2
p′
− (Rn−m )
L
,
.
=
(1.8)
500
V.KOKILASHVILI
The change of the variable yj = T aj tj in the latter expression leads to
the equality
Pn
aj
n
µ
X
2 − 2
−µ
p′
′
|tj | aj
ϕ(x, y ) = T j=m+1 j 1 +
.
′
p−
n−m
L
j=m+1
Now it is obvious that
Pn
n
n
µ
1
X
X
2 − 2
2 − 2
k=m+1
aj
aj
= 1+
1+
|tj |
|tj |
ak
p′
k
(R
Pm
+
j=m+1
j=m+1
n
Y
=
j=m+1
1+
n
X
|tj |
j=m+1
2
aj
− 21 ( a′j +ε)
p
j
n
Y
≤
1 + |tj |
2
aj
j=m+1
i=1
)
ai (1−γi )
=
a
− 12 ( p′j +ε)
j
,
where ε > 0.
Therefore
n
µ
X
2 − 2
|tj | aj
1+
p′
− (Rn−m )
L
j=m+1
n
1 aj
2 − ( ′ +ε)
2 p
Y
j
1 + |tj | aj
≤
R1
1 + |tj |
dtj
≤
1 aj
2 ( ′ +ε)p′
2 p
j
a
j
p′
− (Rn−m )
L
j=m+1
On the other hand,
Z
≤
j
Z
R1
dtj
1+
(1 + |tj |)
εp′
j
aj
.
< ∞.
Thus we have proved the estimate (1.8). It implies
Pm
m
m
1
X
Y
aj (1−γj )
2 − 2
j=1
′
aj
ϕ(x, y ) ≤
|xj − yj |
≤ c1
|xj − yj |(γj −1) . (1.9)
j=1
j=1
Using the generalized H¨older inequality and Lemma 1 with (1.6), (1.7),
(1.8) and (1.9) we obtain:
Z Z
Z
Z
|g(x)|ϕ(y ′ )
g(x)f (y)
Qm
dx dy ≤
dx
µ dy ≤ c2
1−γj
|x − y|a
j=1 |xj − yj |
Rm
Rm Rm
Rn
′
≤ c3 kϕ(y )kLpw+ (Rm ) kgkLq′ (Rm ) .
+
̺
This implies that
Z
Z
g(x)f (y)
dx
dy ≤ c3 kf kLpw (Rn ) kgkL̺q+ (Rm ) .
|x − y|µa
Rm
Rn
TWO-WEIGHTED ESTIMATES FOR INTEGRAL TRANSFORMS
501
Proof of Theorem 2. Using the standard arguments, the validity of Theorem 2 readily follows from Lemma 3.
2. In this paragraph we shall prove the imbedding theorems for different
metrics and different dimensions for weighted spaces of anisotropic Bessel
potentials.
Definition 2 (see[2]). Let r = (r1 , r2 , . . . , rn ), p = (p1 , p2 , . . . , pn), rj >
n
0 (j = 1, 2, . . . , n). It will be said that f ∈ Lp,r
w (R ) if
Z
f (x) = Gr (x − y)g(y)dy,
Rn
p
(Rn ). By
where Gr is the anisotropic Bessel–Macdonald kernel and g ∈ Lw
the definition, kf kLp,r
= kgkLpw .
w
The kernel Gr is characterized by its Fourier transform as follows (see[2])
∗
n
b 2 (λ) = [1 + σ 2 (λ)]− r2 where the function σ(λ) is determined by the
(2π) 2 G
equation
n
n
X
λj2
1X 1
r∗
1
=
.
= 1, aj = ,
σ 2aj
rj
r∗
n j=1 rj
j=1
The kernel Gr obeys, along each j – the coordinate direction, the estimates
Pn
1
−r
−1
i=1 ri
(2.1)
|Gr (x)| ≤ c|xj | j
.
Now we shall prove the imbedding theorem of different metrics.
qi ,pi
(i = 1, 2, . . . , n). Put
Theorem 3. Let 1 < pj < qj < ∞, (vi , wi ) ∈ G1−γ
i
κ =1−
n
X
γj
j=1
rj
and ̺ = κr, r = (r1 , r2 , . . . , rn ).
p,r
n
Then each function f ∈ Lw
(Rn ) belongs to the space Lq,̺
v (R ) and the
inequality kf kLq,̺
≤ ckf kLp,r
holds, where the constant c is independant of
v
w
f.
Proof. We have
f (x) =
Z
G̺ (x − y)h(y)dy,
Rn
where
h(x) =
Z
Rn
and g ∈
p
Lw
(Rn ).
Gr(1−κ) (x − y)g(y)dy
502
or
V.KOKILASHVILI
Now it will be shown that h ∈ Lqv (Rn ). Due to (2.1) we have
Pn
1
−1
−rj (1−κ)
ri (1−κ)
i=1
,
|Gr(1−κ) (x)| ≤ c|xi |
−rj
|Gr(1−κ) (x)| ≤ c|xi |
Let aj =
1
rj
and
max |xj |
1
aj
1≤j≤n
Pn
i=1
ai (1−γi )
Pn
= |xj0 |
i=1
1
aj
0
1−γi
ri
Pn
.
i=1
ai (1−γi )
.
As can be easily verified,
n
X
i=1
|xj |
2
aj
12 Pn
i=1
ai (1−γi )
1
≤ c1 |xj0 | aj0
Pn
i=1
ai (1−γi )
.
Therefore
|Gr(1−κ) (x)| ≤ c2 |x|a−µ ,
where
µ=
n
X
(2.2)
aj (1 − γj ).
j=1
Hence
|h(x)| ≤
Z
|g(y)| |x − y|−µ
a dy.
Rn
Applying Theorem 1, we obtain khkLqv ≤ c3 kf kLpw , which implies kf kLq,̺
v
≤ kf kLp,r
.
w
Using Theorem 2 one may prove an imbedding theorem of different dimensions in a similar manner.
Theorem 4. Let 1 < pi < ∞ (i = 1, 2, . . . , n), 1 < pi < qi < ∞
i ,pi
(i = 1, 2, . . . , m), 1 ≤ m ≤ n. It is also assumed that (vi , wi ) ∈ Gq1−γ
,
i
0 < γi < 1 (i = 1, 2, . . . ).
If
κ =1−
m
X
γj
j=1
rj
−
n
X
1
,
r
p
j=m+1 j j
(2.3)
n
then for an arbitrary function f from the space Lp,r
w (R ) the function
0
0
F (x1 , x2 , . . . , xm ) = f (x1 , x2 , . . . , xm , xm+1 , . . . , xn ) belongs to the space
≤ ckf kLp,r
Lvq,̺ (Rm ) and the inequality kF kLq,̺
holds where the
m
n
v (R )
w (R )
constant c is independent of f .
TWO-WEIGHTED ESTIMATES FOR INTEGRAL TRANSFORMS
503
Proof. In the case under consideration the kernel G(1−κ)r admits the estimate |G(1−κ)r (x)| ≤ c|x|a−µ , where a = (a1 , a2 , . . . , an ), ai = r1j and
µ=
m
X
1 − γi
i=1
ri
+
n
X
1
.
r
p′j
j
j=m+1
(2.4)
Hence we can apply Theorem 2. The rest of the proof is as for the
preceding theorem.
References
1. P.I. Lizorkin, Generalized Liouville differentiation and the method of
multipliers in the theory of imbedding of classes of differentiable functions.
(Russian) Trudy Mat. Inst. Steklov. 105(1969), 87–167; English transl.:
Proc. Steklov Inst. Math. 105(1969).
2. —–, Multipliers of Fourier integrals and bounds of convolutions in
spaces with mixed norms. Applications. (Russian) Izv. Akad. Nauk SSSR,
Ser. Mat. 34(1970), 218-247; English transl.: Math. USSR Izv. 4(1970).
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Steklov. 161(1983), 125-149; English transl.: Proc. Steklov Inst. Mat.
3(1984), 135-162.
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inequalities for generalized potentials. (Russian) Trudy Mat. Inst. Steklov.
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Cambridge University Press, 1951.
(Received 02.08.1993)
Author’s address:
A. Razmadze Mathematical Institute
Georgian Academy of Sciences
1, Z. Rukhadze St., Tbilisi 380093
Republic of Georgia