Directory UMM :Journals:Journal_of_mathematics:GMJ:
GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 5, 1996, 423-446
CRITERIA OF STRONG TYPE TWO-WEIGHTED
INEQUALITIES FOR FRACTIONAL MAXIMAL
FUNCTIONS
A. GOGATISHVILI AND V. KOKILASHVILI
Abstract. A strong type two-weight problem is solved for fractional
maximal functions defined in homogeneous type general spaces. A
similar problem is also solved for one-sided fractional maximal functions.
1. Introduction
The solution of a two-weight problem for maximal functions in Euclidean
spaces was obtained for the first time in [1]. This paper gives a new complete
description of pairs of weight functions which provides the validity of strong
type two-weighted inequalities for fractional maximal functions defined in
homogeneous type general spaces. A similar result was obtained in [2] for
homogeneous type spaces having group structure. It should be noted that
condition (1.2) below, which has turned out to be a criterion of strong type
two-weight estimates for fractional maximal functions, appeared for the first
time in [3], [4].
In this paper we also solve a strong type two-weight problem for one-sided
maximal functions of fractional order on the real axis.
The homogeneous type space (X, d, µ) is a topological space with a complete measure µ such that compactly supported functions are dense in the
space L1 (X, µ). Moreover, it is assumed that there is a nonnegative realvalued function d : X × X → R1 satisfying the following conditions:
(i) d(x, x) = 0 for all x ∈ X;
(ii) d(x, y) > 0 for all x =
6 y in X;
(iii) there is a constant a0 such that d(x, y) ≤ a0 d(y, x) for all x, y in X;
(iv) there is a constant a1 such that d(x, y) ≤ a1 (d(x, z) + d(z, y)) for all
x, y and z in X;
1991 Mathematics Subject Classification. 42B20, 42B25.
Key words and phrases. Homogeneous type space, fractional maximal function, strong
type two-weighted inequality, one-sided fractional maximal functions.
423
c 1997 Plenum Publishing Corporation
1072-947X/96/0900-0423$12.50/0
424
A. GOGATISHVILI AND V. KOKILASHVILI
(v) for each neighborhood V of x in X there is r > 0 such that the ball
B(x, r) = {y ∈ X : d(x, y) < r} is contained in X.
The balls B(x, r) are measurable for all x and r > 0.
There is a constant b such that µB(x, 2r) ≤ bµB(x, r) for all nonempty
B(x, r) (see [5]).
For a locally summable function f : X → R1 the fractional maximal
function is defined as follows:
Z
(1.0)
Mγ (f )(x) = sup(µB)γ−1 |f (y)|dµ, 0 < γ < 1,
B
where the supremum is taken with respect to all balls B of positive measure
containing the point x.
A measurable function w : X → R1 , which is positive almost everywhere and locally summable, is called a weight function (weight). For a
µ-measurable set E we put
Z
wE = w(x)dµ.
E
Main Theorem. Let 1 < p < q < ∞, 0 < γ < 1, v and w be the weight
functions. For a constant c > 0 to exist so that the inequality
q1
Z
p1
Z
q
p
Mγ (f )(x) v(x)dµ
|f (x)| w(x)dµ
≤c
(1.1)
X
X
would hold, it is necessary and sufficient that the condition
1
′
sup w1−p B(x, 2N0 r) p′ ×
x∈X
r>0
×
Z
X\B(x,r)
(γ−1)q
v(y) µB(x, d(x, y))
dµ
q1
0 such that
Z
X
q1
Z
p1
q
p
Mγ (f )(x) v(x)dµ
≤ c1
|f (x)| w(x)dµ
(2.1)
X
Lpw (X, µ);
for any f ∈
(ii) there exists a constant c2 > 0 such that
p1
q
Z
Z
q1
1−p′
1−p′
(x)dµ
Mγ (χB w
)(x) v(x)dµ
≤ c2
w
B
(2.2)
B
for any ball B ⊂ X.
′
Proof. On substituting f = χB w1−p in (2.1), we obtain (2.2). Therefore
the implication (i) ⇒ (ii) is fulfilled. We will prove (ii) ⇒ (i). Let N =
a1 (1 + 2a1 (1 + a0 )) and N1 = a1 (1 + a12 (1 + a0 )(N + a0 )). Assume b > 1 to
be a constant such that
µ(N1 B) ≤ bµB
for an arbitrary ball B.
Let further B0 be an arbitrarily fixed ball and f be an arbitrary integrable
function which is positive almost everywhere and satisfies the condition
supp f ⊂ B0 .
We set
Ωk = x ∈ X : Mγ (f )(x) > bk , k ∈ Z.
Obviously, for each x ∈ Ωk there exists a ball B(y, r) ∋ x such that
Z
1
f (z)dµ > bk .
(µB(y, r))1−γ
B(y,r)
The set of radius lengths of such balls will be bounded by virtue of the fact
that supp f ⊂ B0 .
426
A. GOGATISHVILI AND V. KOKILASHVILI
Consider the values
n
Rkx = sup r : ∃B(y, r) ∋ x,
Z
1
(µB(y, r))1−γ
B(y,r)
o
f (z)dµ > bk .
Obviously, for arbitrary x ∈ Ωk there exist yx ∈ X and rx >
Z
1
f (z)dµ > bk .
(µB(yx , rx ))1−γ
Rk
x
2
such that
B(yx ,rx )
Along with this, for each ball B ′ which contains the point x and for which
rad B ′ ≥ 2rx we have
Z
1
f (z)dµ ≤ bk .
(2.3)
(µB ′ )1−γ
B′
By Lemma A for each k there exists a sequence (Bjk )j≥1 of nonintersecting
balls such that ∪j≥1 N Bjk ⊃ Ωk and
Z
1
f (z)dµ > bk .
(µBjk )1−γ
Bjk
Let us show that if Bjk ∩ Bin 6= ∅ and n > k, then
N Bin ⊂ N Bjk .
(2.4)
To this end it will first be shown that if ri = rad Bi , rj = rad Bj , and
n > k, then
rjk
.
rin <
a1 (a0 + N )
Assume the opposite, i.e., rjk ≤ a1 (a0 + N )rin . Then for y ∈ Bjk and x ∈
Bjk ∩ Bin we will have
d(xni , y) ≤ a1 d(xin , x) + d(x, y) ≤
≤ a1 rin + a1 a0 d(xkj , x) + d(xjk , y) < a1 rin + a1 (1 + a0 )rjk ≤
≤ a1 1 + a12 (1 + a0 )(N + a0 ) rin = N1 rin .
where xjk are the centers of Bjk . Therefore Bjk ⊂ N1 Bin . Along with this,
2rjk < N1 rin . Therefore by virtue of (2.3) we obtain
Z
Z
1
b
bn <
(z)dµ
≤
f
f (z)dµ ≤ bk+1 .
(µBin )1−γ
µ(N1 Bin )1−γ
Bin
N1 Bin
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
427
Thus n ≤ k, which leads to a contradiction. Therefore
rjk > a1 (a0 + N )rin .
Now for x ∈ N Bin and y ∈ Bin ∩ Bjk we derive
d(xkj , x) ≤ a1 d(xjk , y) + a1 a0 d(xni , y) + d(xin , x) ≤
≤ a1 rjk + a1 (a0 + N )rin ≤ 2a1 rjk < N rjk .
Thus N Bin ⊂ N Bjk provided that Bjk ∩ Bin 6= ∅ and n > k.
We introduce the sets
j−1
[
N Bik ∩ Ωk \Ωk+1 , k ∈ Z, j ∈ N.
Ejk = N Bjk \
i=1
As is easy to verify,
∞
[
6 j.
Ejk = Ωk \Ωk+1 and Ejk ∩ Ein = ∅ for k 6= n or i =
j=1
Therefore we have
Z
∞ X
∞
X
q
Mγ (f )(x) v(x)dµ ≤
b(k+1)q vEjk ≤
X
≤ bq
X
j,k
=c
k=−∞ j=1
1
k
(µBj )1−γ
X σ(N Bjk ) q
j,k
(µBjk )1−γ
vEjk
Z
f (z)dµ
q
Bjk
1
σ(N Bjk )
Z
vEjk =
q
f (z)
σ(z)dµ ,
σ(z)
(2.5)
Bjk
′
where σ = w1−p .
Let now ν be a discrete measure given on Z2 by
σ(N Bjk ) q k
vEj .
ν (j, k) =
(µBjk )1−γ
Define the operator T acting from L1loc (X, dµ) into the set of ν-summable
functions defined on Z2 as follows:
Z
1
|g(z)|σ(z)dµ.
T (g)(j, k) =
σ(N Bjk )
Bjk
428
A. GOGATISHVILI AND V. KOKILASHVILI
Equality (2.5) can now be rewritten as
Z q
Z
q
f
T
(z) dν.
Mγ (f )(x) ν(x)dµ ≤ c
σ
(2.6)
Z2
X
Our aim is to prove that the operator T has a strong type (p, q). It is
obvious that the operator T has a strong type (∞, ∞). Now let us show
that T is the operator of a weak type (1, pq ), i.e.,
νΓ(λ) = ν (j, k) ∈ Z2 : T (g)(j, k) > λ ≤
pq
Z
q
cλ− p
|g(z)|p σ(z)dµ
(2.7)
X
for any λ > 0 and g ∈ L1 (X, σdµ).
We fix some k0 ∈ Z and first show that
q
λ p νΓk0 (λ) = ν (j, k) ∈ Z × Zk0 : T (g)(j, k) > λ} ≤
Z
p1
|g(z)|p σ(z)dµ ,
≤
(2.8)
X
where Zk0 = {k0 , k0 + 1, . . . }.
Now fix λ > 0 and consider the system of balls (Bjk )(j,k)∈Γk0 (λ) . Choose
from the latter system a subsystem of nonintersecting balls in the following
manner: take all balls of “rank” k0 , i.e., all balls Bjk0 , j = 1, 2, . . . . Pass
to “rank” k0 + 1. If some Bik0 +1 intersects with none of Bjk0 , then include
it in the subsystem and otherwise discard. Next compare the balls of rank
k0 + 2 with the ones already chosen in the above-described manner and so
on. We obtain thus the sequence of nonintersecting balls {Bi }i . According
to (2.4) if Bjk ∈ {Bi }i , then N Bjk ⊂ N Bi0 for some i0 ≥ 1 and therefore
∞
[
N Bi =
i=1
[
N Bjk .
(j,k)∈Γk0 (λ)
Hence we obtain
q
X
q
λ p νΓk0 (λ) = λ p
(j,k)∈Γk0 (λ)
q
≤ cλ p
∞
X
σ(N B k ) q
j
X
(µBjk )1−γ
i=1 N B k ⊂N Bi
j
vEjk ≤
σ(N Bjk )
(µ(N Bjk ))1−γ
q
vEjk ≤
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
q
≤ cλ p
∞
X
Z
X
i=1 N B k ⊂N Bi k
j
E
j
≤ cλ
∞ Z
X
q
p
i=1N B
≤c
∞
X
i=1
=c
Bi
q
Mγ (χN Bi σ)(z) v(z)dµ ≤
∞
q
q
q X
Mγ (χN Bi σ)(z) v(z)dµ ≤ cλ p
σ(N Bi ) p ≤
i=1
q
σ(N Bi ) p
∞ Z
X
i=1
i
429
Z
1
σ(N Bi )
|g(z)|σ(z)dµ
pq
=
Bi
X
pq
pq
∞ Z
|g(z)|σ(z)dµ
≤c
≤
|g(z)|σ(z)dµ
i=1 B
Z
pq
|g(z)|σ(z)dµ .
≤c
i
X
Thus we have proved (2.8) where the constant does not depend on k0 .
Making now k0 tend to −∞, we obtain (2.7). We have shown that the
operator T has a weak type (1, pq ).
Since the operator T has a weak type (1, pq ) and a strong type (∞, ∞), by
the Marcinkiewicz interpolation theorem we conclude that T has a strong
type (p, q). Then (3.6) implies
Z
Z q
q
f
Mγ (f )(x) v(x)dµ ≤ c
T
(x) dν ≤
σ
Z2
X
pq
pq
Z
Z
f p
p
≤ c1
f (x)w(x)dµ .
(x)σ(x)dµ
= c1
σ
X
X
Let now f be an arbitrary function from Lp (X, wdµ). By virtue of the
foregoing arguments, for an arbitrary ball B0 we will have
Z
≤c
Z
B0
X
p
q
Mγ (χB0 f ) v(x)dµ
f (x)w(x)dµ
p1
≤c
Z
q1
p
≤
f (x)w(x)dµ
p1
.
X
Making rad B0 tend to infinity, by the Fatou lemma we obtain (2.1).
Next we will give two theorems which are proved in [7]. They concern
two-weight estimates of integral transforms with a positive kernel, in par-
430
A. GOGATISHVILI AND V. KOKILASHVILI
ticular, analogs of fractional integrals defined in spaces with a quasi-metric
and measure.
Consider the integral operators
Z
K(f )(x) = k(x, y)f (y)dy
X
∗
K (f )(x) =
Z
k(y, x)f (y)dy.
X
Theorem A. Let 1 < p < q < ∞, k : X × X → R1 be an arbitrary
positive measurable kernel; v and µ be arbitrary finite measures on X so
that µ{x} = 0 for any x ∈ X. If the condition
1
c0 = sup vB(x, 2N0 r) q
x∈X
r>0
Z
′
′
k p (x, y)w1−p (y)dµ
1′
p
< ∞,
X\B(x,r)
where N0 = a1 (1 + 2a0 ) and the constants a0 and a1 are from the definition
of a quasi-metric, is fulfilled, then there exists the constant c > 0 such that
the inequality
Z
pq
v x ∈ X : K(f )(x) > λ ≤ cλ−q
|f (x)|p dµ
X
holds for any µ-measurable nonnegative function f : X → R1 and arbitrary
λ > 0.
Definition 2.1. A positive measurable kernel k : X × X → R1 will be
said to satisfy condition (V ) (k ∈ V ) if there exists a constant c > 0 such
that
k(x, y) < ck(x′ , y)
for arbitrary x, y, and x′ from X which satisfy the condition d(x, x′ ) <
N d(x, y), where N = 2N0 .
Theorem B. Let 1 < p < q < ∞, µ be an arbitrary locally finite measure,
w be a weight, and k ∈ V . Then the following conditiona are equivalent:
(i) there exists a constant c1 > 0 such that the inequality
w
1−p′
x ∈ X : K(f )(x) > λ ≤ c1 λ−q
Z
p
|f (x)| v
1
1−q
X
holds for arbitrary λ > 0 and nonnegative f ∈ Lp (X, wdµ);
p′′
q
(x)dµ
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
431
(ii) there exists a constant c2 > 0 such that
Z
X
∗
K (χB w
1−p′
Z
p1
q1
1−p′
w
(y)dµ
≤ c2
)(x) v(x)dµ
q
B
for an arbitrary ball B ⊂ X;
(iii)
1
′
sup w1−p B(x, 2N0 r) p′
x∈X
r>0
q1
k q (x, y)v(y)dµ
< ∞.
Z
X\B(x,r)
3. Proof of the Main Theorem
Using the results of the preceding sections we will prove the main theorem
of this paper.
Proof. Our aim is to show that the implication (1.1) ⇔ (1.2) is valid. First
we will prove the implication (1.2) ⇒ (1.1). Consider an operator given on
1
′
Lq (X, v 1−q ) in the form
Z
|f (y)|
Tγ (f )(x) =
dµ.
(µB(x, d(x, y)))1−γ
X
The latter operator is an analog of the Riesz potential for homogeneous type
spaces.
Using Theorem A, from condition (1.2) we conclude that the weak type
inequality
′
w1−p x ∈ X : Tγ (f )(x) > λ ≤ c4 λ−p
′
Z
′
|f (x)|q v
1
1−q
p′′
q
(x)dµ
X
(3.1)
with the constant not depending on λ > 0 and f is valid.
Further, by virtue of Theorem B the latter inequality implies that there
exists a constant c2 > 0 such that for any ball B ⊂ X we have
Z
X
q
′
Tγ∗ (χB w1−p )(x) v(x)dµ
where
Tγ∗ (ϕ)(x)
=
Z
X
q1
≤ c2
Z
′
w1−p (x)dµ
B
|ϕ(y)|
dµ.
(µB(y, d(x, y)))1−γ
1′
p
,
(3.2)
432
A. GOGATISHVILI AND V. KOKILASHVILI
On the other hand, there exist constants c3 > 0 and c4 > 0 such that
c3 µB y, d(x, y) ≤ µB x, d(x, y) ≤ c4 µB y, d(x, y) .
(3.3)
The latter follows from the fact that B(x, d(x, y)) ⊂ a1 (a0 + 1)d(x, y). Indeed, let d(x, z) ≤ d(x, y). Then
d(y, z) ≤ a1 d(y, x) + d(x, z) ≤ a1 (a0 + 1)d(x, y).
By virtue of the doubling property for measure we obtain
µB y, a1 (a0 + 1)d(x, y) ≤ c5 µB y, d(x, y) .
Hence we conclude that (3.3) is valid. Next, from (3.3) and (3.2) we derive
Z
X
q
′
Tγ (χB w1−p )(x) v(x)dµ
1q
≤ c6
Z
′
w1−p (x)dµ
p1
.
(3.4)
B
Now we will show that for an arbitrary nonnegative measurable function ϕ
we have
Mγ (ϕ)(x) ≤ c7 Tγ (ϕ)(x),
(3.5)
where the constant c7 does not depend on ϕ and x.
First we will show that for any x there exists a ball Bx = B(x, rx ) such
that
Z
c8
Mγ (ϕ)(x) ≤
ϕ(z)dµ,
(3.6)
(µBx )1−γ
B
where the positive constant c8 does not depend on ϕ and Bx . Indeed, there
exists a ball B(y, r) such that x ∈ B(y, r) and
Z
2
Mγ (ϕ)(x) ≤
ϕ(z)dµ.
(3.7)
(µB(y, r))1−γ
B(y,r)
Assuming now that z ∈ B(y, r), we obtain
d(x, z) ≤ a1 d(x, y) + d(y, z) ≤ a1 (a0 + 1)r.
Therefore B(y, r) ⊂ B(x, a1 (1+a0 )r). On the other hand, we have B(x, a1 (1+
a0 )r) ⊂ B(y, a1 (1 + a1 (1 + a0 ))r) since
d(y, z) ≤ a1 d(y, x) + d(x, z) ≤ a1 r + a1 (1 + a0 r) = a1 1 + a1 (1 + a0 ) r
for any z ∈ B(x, a1 (1 + a0 )r).
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
433
Now applying the doubling condition, from (3.7) we find that
Mγ (ϕ)(x) ≤
Z
2
(µB(y, r))1−γ
ϕ(z)dµ ≤
B(x,a1 (1+a0 )r)
Z
c8
≤
(µB(x, a1 (1 + a0 )r))1−γ
ϕ(z)dµ.
B(x,a1 (1+a0 )r)
Replacing rx by the number a1 (1 + a0 )r, we obtain (3.6).
Now we will prove (3.5). We have
Z
Tγ (ϕ)(x) ≥
ϕ(y)
dµ ≥
(µB(x, d(x, y)))1−γ
B(x,rx )
≥
Z
1
(µB(x, rx ))1−γ
ϕ(z)dµ ≥
1
Mγ (ϕ)(x).
c8
B(x,rx )
From (3.4) and (3.5) we obtain
Z
X
Mγ (χB w
1−p′
q
)(x) v(x)dµ
q1
≤ c9
Z
w
1−p′
(x)dµ
p1
.
B
By Theorem 2.1 we conclude that inequality (1.1) is valid.
Thus we have proved the implication (1.2) ⇒ (1.1).
Let us show the validity of the inverse implication (1.1) ⇒ (1.2). Fix an
arbitrary ball B(x, r) and assume
′
f (y) = χN B(x,r) w1−p (y),
where N is an arbitrary number greater than unity. Obviously, by virtue of
the doubling condition we have
Z
1
Mγ (f )(y) ≥
µB(x, N d(x, y))
′
w1−p (z)dz ≥
B(x,N d(x,y))∩N B(x,r)
c10
≥
µB(x, d(x, y))
Z
N B(x,r)
for any y ∈ X\B(x, r).
′
w1−p (z)dµ
434
A. GOGATISHVILI AND V. KOKILASHVILI
Therefore (1.1) implies
Z
w
1−p′
(z)dµ
Z
v(y)
dµ
(µB(x, d(x, y))(1−γ)q
q1
≤
X\B(x,r)
N B(x,r)
≤c
Z
w
1−p′
(z)dµ
p1
.
N B(x,r)
From the latter inequality we obtain the validity of (1.2).
Definition 3.1. Measure ν satisfies the reverse doubling condition (ν ∈
(RD)) if there exist numbers δ and ε from (0, 1) such that
νB(x, r1 ) ≤ ενB(x, r2 )
for µB(x, r1 ) ≤ δµB(x, r2 ), 0 < r1 < r2 .
′
In the particular case where measure w1−p satisfies the reverse doubling
condition, (1.2) in the main theorem can be replaced by a simpler condition.
′
Theorem 3.1. Let 1 < p < q < ∞, 0 < γ < 1 and measure w1−p
satisfy the reverse doubling condition. Then (1.1) holds iff
γ−1 1−p′
1
1
sup µB(x, r)
B(x, r) p′ vB(x, r) q < ∞.
w
(3.8)
x∈X
r>0
Proof. The implication (1.1) ⇒ (3.8) is obtained immediately by substitut′
ing the function f (y) = χB w1−p (y) into (1.1).
By virtue of the main theorem to prove the implication (3.8) ⇒ (1.1) it
′
is sufficient to show that if w1−p ∈ (RD), then (3.8) ⇒ (1.2). Let x ∈ X
and r > 0 be fixed. Choose numbers rk (k = 0, 1, . . . ) as follows:
r0 = 2N0 r and rk = inf r : µB(x, rk−1 ) < δµB(x, r) .
Obviously,
µB(x, rk−1 ) < δµB(x, rk ) < cµB(x, rk−1 ).
Again applying the condition (RD), we obtain
′
′
w1−p B(x, r0 ) ≤ εk w1−p B(x, rk ) k = 1, 2, . . . .
The latter inequalities imply
1−p′
1
w
B(x, 2N0 r) p′
Z
X\B(x,r)
(γ−1)q
v(y) µB(x, d(x, y))
dµ(y)
q1
≤
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
∞
1 X
′
≤ w1−p B(x, r0 ) p′
k=1
+ w
1−p′
≤
1
B(x, r0 ) p′
∞
X
k=1
Z
(γ−1)q
v(y) µB(x, d(x, y))
dµ(y)
q1
435
+
B(x,rk )\B(x,rk−1 )
Z
B(x,r0 )\B(x,r)
q1
(γ−1)q
≤
v(y) µB(x, d(x, y))
dµ(y)
γ−1
1
1
k
′
+
ε p′ w1−p B(x, rk ) p′ vB(x, rk ) q µB(x, rk−1 )
γ−1
1
1
′
≤
+ w1−p B(x, r0 ) p′ vB(x, r0 ) q µB(x, r0 )
≤c
∞
X
k=0
γ−1 1−p′
1
1
k
B(x, rk ) p′ vB(x, rk ) q ≤
ε p′ µB(x, rk )
w
≤c
∞
X
k
ε p′ < ∞.
k=0
Remark. A similar result was obtained in [2] under the assumption that
′
the measure w1−p satisfies the doubling condition. Theorem 3.1 contains a
stronger result, since condition (RD) is weaker than the doubling condition.
(For instance, the function e|x| ∈ (RD) but it does not satisfy the doubling
condition.) Along with this, the proof in [2] essentially differs from the
above.
The above proofs also remains valid for fractional maximal functions:
Z
1
|f (y)|dσ(y),
(3.9)
Mγ (f dσ)(x) = sup
1−γ
B∋x (µB)
B
where 0 ≤ γ < 1 and σ is a Borel measure while the supremum is taken
over all balls B ⊂ X with a positive measure containing x.
Theorem 3.2. Let X be an arbitrary homogeneous type space, 1 < p <
q < ∞, 0 < γ < 1, ω and σ be positive Borel measures. For a constant
c > 0 to exist such that the inequality
p1
q1
Z
Z
q
p
|f (x)| dω(x)
≤c
(3.10)
Mγ (f dσ)(x) dω(x)
X
X
holds, it is necessary and sufficient that the condition
1
σB(2N0 r) p′ ×
sup
x∈X
r>0
µB(x,r)6=0
436
A. GOGATISHVILI AND V. KOKILASHVILI
×
Z
X\B(x,r)
µB(x, d(x, y))
(γ−1)q
q1
0 such that
Z
X
q1
p1
Z
q
Mγ (f dσ)(x) dω(x)
≤ c1
|f (x)|p dσ(x) ,
(3.12)
X
for any f ∈ Lp (X, dσ);
(ii) there exists a constant c2 > 0 such that
Z
B
q1
q
1
≤ c2 (σB) p
Mγ (χB dσ)(x) dω(x)
(3.13)
for any B ⊂ X.
Theorem 3.4. Let 1 < p < q < ∞, 0 < γ < 1, and the measure σ
satisfy the reverse doubling condition (σ ∈ (RD)). Then (3.10) holds iff
1
1
sup (µB)γ−1 (σB) p′ (ωB) q < ∞.
B
µB6=0
Theorems 3.2 and 3.3 in fact were previously proved in [2] under the
assumption that the space X possesses group structure, while Theorem
3.4 was proved for homogeneous type spaces under a stronger restriction
imposed on the measure σ.
Remark. The above-formulated theorems remain valid also in the case
where X = R1 is an arbitrary positive Borel measure. This follows from the
fact that we proved Theorem A for such measures and from the general conclusion that the well-known technique is applicable for maximal functions
with respect to a Borel measure on the real axis (see, for instance, [9]).
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
437
4. A Strong Type Two-Weight Problem for Fractional
Maximal Functions in a Half-Space
This section deals with “H¨ormander type” fractional maximal functions.
b = X × [0, ∞) and for (x, t) ∈ X
b
Let X
Z
γ−1
f
(4.1)
Mγ (f dσ)(x, t) = sup(µB)
|f (y)|dσ(y),
B
where 0 ≤ γ < 1, σ is a Borel measure in X and the supremum is taken
over all balls B ⊂ X containing x and having a radius greater than 2t .
We have
Theorem 4.1. If 1 < p < q < ∞, 0 < γ < 1, and X is an arbitrary
homogeneous type space, then the inequality
Z
b
X
q1
Z
p1
q
p
f
Mγ (f dσ))(x, t) dω(x, t)
≤c
|f (x)| dσ(x)
(4.2)
X
holds for all f with c independent of f iff
1
sup σB(x, 2N0 r) p′ ×
x∈X
r>0
×
Z
b B(x,r)
b
X\
q1
(γ−1)q
µB(x, d(x, y) + t)
dω(x, t)
< ∞,
(4.3)
b r) = B(x, r) × [0, r).
where B(x,
Proof. The validity of the theorem follows from the main theorem using the
following arguments. A similar idea is used in [18].
On the product space X × R we define the quasi-metric as
db (x, t), (y, s) = max d(x, y), |s − t|
b = µ ⊕ δ0 , where δ0 is the Dirac measure concentrated at
and the measure µ
zero.
b with respect to db lies in X
b and is
Note that the center of a ball from X
b Since
obtained by the intersection of the respective ball in X × R with X.
b
b we have
B((x,
t), r) = B(x, r) × (t − r, t + r) ∩ X,
b (x, t), r = µB(x, r) for r ≥ t,
µ
bB
b (x, t), r = 0 for r < t.
bB
µ
438
A. GOGATISHVILI AND V. KOKILASHVILI
bµ
b d,
Therefore (X,
b) is a homogeneous type space. It can be easily verified
c is actually a function of the form
that the maximal function M
Z
γ−1
b
Mγ (f dσ)(x, t) = sup µB((z,
t), r)
|f (y)|db
σ,
b
),r)
B((z,τ
where σ
b = σ ⊕ δ0 .
Hence to derive Theorem 4.2 from the main theorem it is sufficient to
show that in the considered case condition (2.11) can be reduced to condition
b
b
bB((x,
τ ), r) =
τ ), r) 6= 0 for τ < r and B((x,
(4.3). For this, in turn, since µ
B(x, r) × [0, r + τ ) it is sufficient to show that
Indeed,
b
b (x, τ ), d((x,
µ
bB
τ ), (y, t)) ∼ µB x, d(x, y) + t
for τ < r < db (x, τ ), (y, t) = max d(x, y), |τ − t| .
d(x, y) + t ≤ d(x, y) + |τ − t| + τ ≤ d(x, y) + |τ − t| + r ≤ 3db (x, τ ), (y, t) .
Since 0 < τ < max{d(x, y), |τ − t|}, we have {d(x, y), |τ − t|} < d(x, y) + t,
i.e.,
b
τ ), (y, t)).
d(x, y) + t ∼ d((x,
Therefore by virtue of the doubling condition
b
b (x, t), d((x,
t), (y, s)) =
µ
bB
b
= µB x, d((x,
τ ), (y, t)) ∼ µB x, d(x, y) + t .
Theorem 4.2. Let X be an arbitrary homogeneous type space, 1 < p ≤
q < ∞, 0 ≤ γ < 1. Then (4.2) holds for all f with c independent of f iff
Z
bb
B
q
Mγ (χB dσ)(x, t) dω(x, t)
q1
1
≤ c1 (σB) P ,
(4.4)
bb
= B × [0, 2 rad B).
where B
Theorem 4.3. If 1 < p < q < ∞, 0 < γ < 1, σ ∈ (RD), then (4.2)
holds iff
1
bb q1
sup(µB)γ−1 (σB) p′ ω B
< ∞,
where the supremum is taken with respect to all balls B.
(4.5)
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
439
It should be said that theorems of the type of Theorems 4.1 and 4.2 have
previously been known only for homogeneous type spaces with a group
structure [10], [11]. In the latter papers instead of condition (4.3) there
figures a condition which actually implies the simultaneous fulfillment of
conditions (4.3) and (4.5). As for Theorem 4.3, it is proved in [10], [11] but
under a more restrictive doubling condition for the measure σ.
We should also note papers [12], [13], where some more particular results
are obtained.
5. A Strong Type Two-Weight Problem for One-Sided
Fractional Maximal Functions
The method developed in the preceding sections enables one to solve the
following strong type two-weight problem:
Mγ+ (f )(x)
= sup h
γ−1
h>0
= sup h
h>0
|f (t)|dt,
x
and
Mγ− (f )(x)
x+h
Z
γ−1
Zx
|f (t)|dt,
x−h
where x > 0 and 0 < γ < 1.
To this end it is sufficient to follow the proof of the main theorem and
apply an analog of Theorem 2.1 for one-sided maximal functions (see, for
instance, [17, Theorem 2]), Theorem 1.6 from [7], and also the fact that
the Riemann-Liouville operator Rγ and the Weyl operator Wγ control the
one-sided maximal functions
Mγ− (f )(x) ≤ Rγ (|f |)(x)
and
Mγ+ (f )(x) ≤ Wγ (|f |)(x),
where
Rγ (f )(x) =
Zx
(x − t)γ−1 f (t)dt
0
and
Z∞
Wγ (f )(x) = (t − x)γ−1 f (t)dt,
x
where 0 < γ < 1, x > 0.
We thus come to the validity of the following two statements:
440
A. GOGATISHVILI AND V. KOKILASHVILI
Theorem 5.1. Let 1 < p < q < ∞, 0 < γ < 1. For the inequality
Z∞
0
q
Mγ+ (f )(x) v(x)dx
q1
Z∞
p1
p
≤c
|f (x)| w(x)dx
0
with the positive constant c independent of f to hold, it is necessary and
sufficient that the condition
1′ Z∞
a+h
Z
p
1−p′
w
(y)dy
sup
a,h
0 0 such that
Z
B
p′
K (χB dv) dµ
∗
1′
p
1
≤ c2 (vB) q′ .
(6.2)
The theorem also hold in the case p = 1, 1 < q < ∞ provided that
condition (6.2) is replaced by the condition
1
expB K ∗ χB dv (x) ≤ c2 (vB) q′
for each ball B.
Proof. Let f be a bounded nonnegative function with a compact support.
For given λ > 0 we set
Ωλ = x ∈ X : K(f )(x) > λ .
Let Bj = B(xj , rj ) be the sequence of balls from Lemma B corresponding
to the number c = 2a1 . Then there exists a constant c3 > 0 such that (see,
for instance, [17])
K χX\cBj f (x) ≤ c3 λ
(6.3)
for any x ∈ Bj , where cBj = B(xj , crj ). Hence K(χX\cBj f )(x) > c3 λ for
any x ∈ Bj ∩ Ω2c3 λ .
442
A. GOGATISHVILI AND V. KOKILASHVILI
Next,
∞
X
v Bj ∩ Ω2c3 λ =
vΩ2c3 λ ≤
j=1
X
=
j∈F
where
+
X
v Bj ∩ Ω2c3 λ = I1 + I2 ,
j∈G
F = j : v Bj ∩ Ω2c3 λ > βv(cBj )
and
G = v Bj ∩ Ω2c3 λ ≤ βv(cBj ) ,
while the number β will be chosen below so that 0 < β < 1.
Applying the H¨older inequality and condition (6.2) we obtain
c3 λv(cBj ) ≤ c3 β −1 λv Bj ∩ Ω2c3 λ ≤
Z Z
≤ β −1
k(x, y)f (y)dµ(y) dv(x) ≤
Bj
cBj
cBj
Bj
≤ β −1
≤ β −1
Z Z
Z Z
cBj
k(x, y)dv(x) f (y)dµ(y) ≤
Bj
p1
1′ Z
p′
p
k(x, y)dv(x) dµ(y)
≤
f p (y)dµ(y)
Bj
1
≤ β −1 c2 v(cBj ) q′
Z
f p (y)dµ
p1
.
cBj
Therefore
q
λ v(cBj ) ≤
−1
(c−1
c2 )q
3 β
Z
p
|f (y)| dµ(y)
pq
.
cBj
The summation of the latter inequality gives
pq
Z
q
−q −1 −1
p
I1 ≤ λ (c3 β c2 )
|f (y)| dµ .
X
P∞
Here we have used the fact that > 1 and j=1 χcBj ≤ M . Next for
j ∈ G we obtain
v Bj ∩ Ω2c3 λ ≤ βv(cBj ).
q
p
Therefore
I2 ≤ βM vΩλ .
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
443
Finally, we find that
vΩ2c3 λ ≤ c4 λ
−q
Z
p
|f (y)| dµ
pq
+ βM vΩλ .
X
Multiplying the latter inequality by λq and taking the exact upper bound
with respect to λ, 0 < λ < 2cδ3 , we obtain
q
sup λ vΩλ ≤ c4
0
CRITERIA OF STRONG TYPE TWO-WEIGHTED
INEQUALITIES FOR FRACTIONAL MAXIMAL
FUNCTIONS
A. GOGATISHVILI AND V. KOKILASHVILI
Abstract. A strong type two-weight problem is solved for fractional
maximal functions defined in homogeneous type general spaces. A
similar problem is also solved for one-sided fractional maximal functions.
1. Introduction
The solution of a two-weight problem for maximal functions in Euclidean
spaces was obtained for the first time in [1]. This paper gives a new complete
description of pairs of weight functions which provides the validity of strong
type two-weighted inequalities for fractional maximal functions defined in
homogeneous type general spaces. A similar result was obtained in [2] for
homogeneous type spaces having group structure. It should be noted that
condition (1.2) below, which has turned out to be a criterion of strong type
two-weight estimates for fractional maximal functions, appeared for the first
time in [3], [4].
In this paper we also solve a strong type two-weight problem for one-sided
maximal functions of fractional order on the real axis.
The homogeneous type space (X, d, µ) is a topological space with a complete measure µ such that compactly supported functions are dense in the
space L1 (X, µ). Moreover, it is assumed that there is a nonnegative realvalued function d : X × X → R1 satisfying the following conditions:
(i) d(x, x) = 0 for all x ∈ X;
(ii) d(x, y) > 0 for all x =
6 y in X;
(iii) there is a constant a0 such that d(x, y) ≤ a0 d(y, x) for all x, y in X;
(iv) there is a constant a1 such that d(x, y) ≤ a1 (d(x, z) + d(z, y)) for all
x, y and z in X;
1991 Mathematics Subject Classification. 42B20, 42B25.
Key words and phrases. Homogeneous type space, fractional maximal function, strong
type two-weighted inequality, one-sided fractional maximal functions.
423
c 1997 Plenum Publishing Corporation
1072-947X/96/0900-0423$12.50/0
424
A. GOGATISHVILI AND V. KOKILASHVILI
(v) for each neighborhood V of x in X there is r > 0 such that the ball
B(x, r) = {y ∈ X : d(x, y) < r} is contained in X.
The balls B(x, r) are measurable for all x and r > 0.
There is a constant b such that µB(x, 2r) ≤ bµB(x, r) for all nonempty
B(x, r) (see [5]).
For a locally summable function f : X → R1 the fractional maximal
function is defined as follows:
Z
(1.0)
Mγ (f )(x) = sup(µB)γ−1 |f (y)|dµ, 0 < γ < 1,
B
where the supremum is taken with respect to all balls B of positive measure
containing the point x.
A measurable function w : X → R1 , which is positive almost everywhere and locally summable, is called a weight function (weight). For a
µ-measurable set E we put
Z
wE = w(x)dµ.
E
Main Theorem. Let 1 < p < q < ∞, 0 < γ < 1, v and w be the weight
functions. For a constant c > 0 to exist so that the inequality
q1
Z
p1
Z
q
p
Mγ (f )(x) v(x)dµ
|f (x)| w(x)dµ
≤c
(1.1)
X
X
would hold, it is necessary and sufficient that the condition
1
′
sup w1−p B(x, 2N0 r) p′ ×
x∈X
r>0
×
Z
X\B(x,r)
(γ−1)q
v(y) µB(x, d(x, y))
dµ
q1
0 such that
Z
X
q1
Z
p1
q
p
Mγ (f )(x) v(x)dµ
≤ c1
|f (x)| w(x)dµ
(2.1)
X
Lpw (X, µ);
for any f ∈
(ii) there exists a constant c2 > 0 such that
p1
q
Z
Z
q1
1−p′
1−p′
(x)dµ
Mγ (χB w
)(x) v(x)dµ
≤ c2
w
B
(2.2)
B
for any ball B ⊂ X.
′
Proof. On substituting f = χB w1−p in (2.1), we obtain (2.2). Therefore
the implication (i) ⇒ (ii) is fulfilled. We will prove (ii) ⇒ (i). Let N =
a1 (1 + 2a1 (1 + a0 )) and N1 = a1 (1 + a12 (1 + a0 )(N + a0 )). Assume b > 1 to
be a constant such that
µ(N1 B) ≤ bµB
for an arbitrary ball B.
Let further B0 be an arbitrarily fixed ball and f be an arbitrary integrable
function which is positive almost everywhere and satisfies the condition
supp f ⊂ B0 .
We set
Ωk = x ∈ X : Mγ (f )(x) > bk , k ∈ Z.
Obviously, for each x ∈ Ωk there exists a ball B(y, r) ∋ x such that
Z
1
f (z)dµ > bk .
(µB(y, r))1−γ
B(y,r)
The set of radius lengths of such balls will be bounded by virtue of the fact
that supp f ⊂ B0 .
426
A. GOGATISHVILI AND V. KOKILASHVILI
Consider the values
n
Rkx = sup r : ∃B(y, r) ∋ x,
Z
1
(µB(y, r))1−γ
B(y,r)
o
f (z)dµ > bk .
Obviously, for arbitrary x ∈ Ωk there exist yx ∈ X and rx >
Z
1
f (z)dµ > bk .
(µB(yx , rx ))1−γ
Rk
x
2
such that
B(yx ,rx )
Along with this, for each ball B ′ which contains the point x and for which
rad B ′ ≥ 2rx we have
Z
1
f (z)dµ ≤ bk .
(2.3)
(µB ′ )1−γ
B′
By Lemma A for each k there exists a sequence (Bjk )j≥1 of nonintersecting
balls such that ∪j≥1 N Bjk ⊃ Ωk and
Z
1
f (z)dµ > bk .
(µBjk )1−γ
Bjk
Let us show that if Bjk ∩ Bin 6= ∅ and n > k, then
N Bin ⊂ N Bjk .
(2.4)
To this end it will first be shown that if ri = rad Bi , rj = rad Bj , and
n > k, then
rjk
.
rin <
a1 (a0 + N )
Assume the opposite, i.e., rjk ≤ a1 (a0 + N )rin . Then for y ∈ Bjk and x ∈
Bjk ∩ Bin we will have
d(xni , y) ≤ a1 d(xin , x) + d(x, y) ≤
≤ a1 rin + a1 a0 d(xkj , x) + d(xjk , y) < a1 rin + a1 (1 + a0 )rjk ≤
≤ a1 1 + a12 (1 + a0 )(N + a0 ) rin = N1 rin .
where xjk are the centers of Bjk . Therefore Bjk ⊂ N1 Bin . Along with this,
2rjk < N1 rin . Therefore by virtue of (2.3) we obtain
Z
Z
1
b
bn <
(z)dµ
≤
f
f (z)dµ ≤ bk+1 .
(µBin )1−γ
µ(N1 Bin )1−γ
Bin
N1 Bin
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
427
Thus n ≤ k, which leads to a contradiction. Therefore
rjk > a1 (a0 + N )rin .
Now for x ∈ N Bin and y ∈ Bin ∩ Bjk we derive
d(xkj , x) ≤ a1 d(xjk , y) + a1 a0 d(xni , y) + d(xin , x) ≤
≤ a1 rjk + a1 (a0 + N )rin ≤ 2a1 rjk < N rjk .
Thus N Bin ⊂ N Bjk provided that Bjk ∩ Bin 6= ∅ and n > k.
We introduce the sets
j−1
[
N Bik ∩ Ωk \Ωk+1 , k ∈ Z, j ∈ N.
Ejk = N Bjk \
i=1
As is easy to verify,
∞
[
6 j.
Ejk = Ωk \Ωk+1 and Ejk ∩ Ein = ∅ for k 6= n or i =
j=1
Therefore we have
Z
∞ X
∞
X
q
Mγ (f )(x) v(x)dµ ≤
b(k+1)q vEjk ≤
X
≤ bq
X
j,k
=c
k=−∞ j=1
1
k
(µBj )1−γ
X σ(N Bjk ) q
j,k
(µBjk )1−γ
vEjk
Z
f (z)dµ
q
Bjk
1
σ(N Bjk )
Z
vEjk =
q
f (z)
σ(z)dµ ,
σ(z)
(2.5)
Bjk
′
where σ = w1−p .
Let now ν be a discrete measure given on Z2 by
σ(N Bjk ) q k
vEj .
ν (j, k) =
(µBjk )1−γ
Define the operator T acting from L1loc (X, dµ) into the set of ν-summable
functions defined on Z2 as follows:
Z
1
|g(z)|σ(z)dµ.
T (g)(j, k) =
σ(N Bjk )
Bjk
428
A. GOGATISHVILI AND V. KOKILASHVILI
Equality (2.5) can now be rewritten as
Z q
Z
q
f
T
(z) dν.
Mγ (f )(x) ν(x)dµ ≤ c
σ
(2.6)
Z2
X
Our aim is to prove that the operator T has a strong type (p, q). It is
obvious that the operator T has a strong type (∞, ∞). Now let us show
that T is the operator of a weak type (1, pq ), i.e.,
νΓ(λ) = ν (j, k) ∈ Z2 : T (g)(j, k) > λ ≤
pq
Z
q
cλ− p
|g(z)|p σ(z)dµ
(2.7)
X
for any λ > 0 and g ∈ L1 (X, σdµ).
We fix some k0 ∈ Z and first show that
q
λ p νΓk0 (λ) = ν (j, k) ∈ Z × Zk0 : T (g)(j, k) > λ} ≤
Z
p1
|g(z)|p σ(z)dµ ,
≤
(2.8)
X
where Zk0 = {k0 , k0 + 1, . . . }.
Now fix λ > 0 and consider the system of balls (Bjk )(j,k)∈Γk0 (λ) . Choose
from the latter system a subsystem of nonintersecting balls in the following
manner: take all balls of “rank” k0 , i.e., all balls Bjk0 , j = 1, 2, . . . . Pass
to “rank” k0 + 1. If some Bik0 +1 intersects with none of Bjk0 , then include
it in the subsystem and otherwise discard. Next compare the balls of rank
k0 + 2 with the ones already chosen in the above-described manner and so
on. We obtain thus the sequence of nonintersecting balls {Bi }i . According
to (2.4) if Bjk ∈ {Bi }i , then N Bjk ⊂ N Bi0 for some i0 ≥ 1 and therefore
∞
[
N Bi =
i=1
[
N Bjk .
(j,k)∈Γk0 (λ)
Hence we obtain
q
X
q
λ p νΓk0 (λ) = λ p
(j,k)∈Γk0 (λ)
q
≤ cλ p
∞
X
σ(N B k ) q
j
X
(µBjk )1−γ
i=1 N B k ⊂N Bi
j
vEjk ≤
σ(N Bjk )
(µ(N Bjk ))1−γ
q
vEjk ≤
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
q
≤ cλ p
∞
X
Z
X
i=1 N B k ⊂N Bi k
j
E
j
≤ cλ
∞ Z
X
q
p
i=1N B
≤c
∞
X
i=1
=c
Bi
q
Mγ (χN Bi σ)(z) v(z)dµ ≤
∞
q
q
q X
Mγ (χN Bi σ)(z) v(z)dµ ≤ cλ p
σ(N Bi ) p ≤
i=1
q
σ(N Bi ) p
∞ Z
X
i=1
i
429
Z
1
σ(N Bi )
|g(z)|σ(z)dµ
pq
=
Bi
X
pq
pq
∞ Z
|g(z)|σ(z)dµ
≤c
≤
|g(z)|σ(z)dµ
i=1 B
Z
pq
|g(z)|σ(z)dµ .
≤c
i
X
Thus we have proved (2.8) where the constant does not depend on k0 .
Making now k0 tend to −∞, we obtain (2.7). We have shown that the
operator T has a weak type (1, pq ).
Since the operator T has a weak type (1, pq ) and a strong type (∞, ∞), by
the Marcinkiewicz interpolation theorem we conclude that T has a strong
type (p, q). Then (3.6) implies
Z
Z q
q
f
Mγ (f )(x) v(x)dµ ≤ c
T
(x) dν ≤
σ
Z2
X
pq
pq
Z
Z
f p
p
≤ c1
f (x)w(x)dµ .
(x)σ(x)dµ
= c1
σ
X
X
Let now f be an arbitrary function from Lp (X, wdµ). By virtue of the
foregoing arguments, for an arbitrary ball B0 we will have
Z
≤c
Z
B0
X
p
q
Mγ (χB0 f ) v(x)dµ
f (x)w(x)dµ
p1
≤c
Z
q1
p
≤
f (x)w(x)dµ
p1
.
X
Making rad B0 tend to infinity, by the Fatou lemma we obtain (2.1).
Next we will give two theorems which are proved in [7]. They concern
two-weight estimates of integral transforms with a positive kernel, in par-
430
A. GOGATISHVILI AND V. KOKILASHVILI
ticular, analogs of fractional integrals defined in spaces with a quasi-metric
and measure.
Consider the integral operators
Z
K(f )(x) = k(x, y)f (y)dy
X
∗
K (f )(x) =
Z
k(y, x)f (y)dy.
X
Theorem A. Let 1 < p < q < ∞, k : X × X → R1 be an arbitrary
positive measurable kernel; v and µ be arbitrary finite measures on X so
that µ{x} = 0 for any x ∈ X. If the condition
1
c0 = sup vB(x, 2N0 r) q
x∈X
r>0
Z
′
′
k p (x, y)w1−p (y)dµ
1′
p
< ∞,
X\B(x,r)
where N0 = a1 (1 + 2a0 ) and the constants a0 and a1 are from the definition
of a quasi-metric, is fulfilled, then there exists the constant c > 0 such that
the inequality
Z
pq
v x ∈ X : K(f )(x) > λ ≤ cλ−q
|f (x)|p dµ
X
holds for any µ-measurable nonnegative function f : X → R1 and arbitrary
λ > 0.
Definition 2.1. A positive measurable kernel k : X × X → R1 will be
said to satisfy condition (V ) (k ∈ V ) if there exists a constant c > 0 such
that
k(x, y) < ck(x′ , y)
for arbitrary x, y, and x′ from X which satisfy the condition d(x, x′ ) <
N d(x, y), where N = 2N0 .
Theorem B. Let 1 < p < q < ∞, µ be an arbitrary locally finite measure,
w be a weight, and k ∈ V . Then the following conditiona are equivalent:
(i) there exists a constant c1 > 0 such that the inequality
w
1−p′
x ∈ X : K(f )(x) > λ ≤ c1 λ−q
Z
p
|f (x)| v
1
1−q
X
holds for arbitrary λ > 0 and nonnegative f ∈ Lp (X, wdµ);
p′′
q
(x)dµ
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
431
(ii) there exists a constant c2 > 0 such that
Z
X
∗
K (χB w
1−p′
Z
p1
q1
1−p′
w
(y)dµ
≤ c2
)(x) v(x)dµ
q
B
for an arbitrary ball B ⊂ X;
(iii)
1
′
sup w1−p B(x, 2N0 r) p′
x∈X
r>0
q1
k q (x, y)v(y)dµ
< ∞.
Z
X\B(x,r)
3. Proof of the Main Theorem
Using the results of the preceding sections we will prove the main theorem
of this paper.
Proof. Our aim is to show that the implication (1.1) ⇔ (1.2) is valid. First
we will prove the implication (1.2) ⇒ (1.1). Consider an operator given on
1
′
Lq (X, v 1−q ) in the form
Z
|f (y)|
Tγ (f )(x) =
dµ.
(µB(x, d(x, y)))1−γ
X
The latter operator is an analog of the Riesz potential for homogeneous type
spaces.
Using Theorem A, from condition (1.2) we conclude that the weak type
inequality
′
w1−p x ∈ X : Tγ (f )(x) > λ ≤ c4 λ−p
′
Z
′
|f (x)|q v
1
1−q
p′′
q
(x)dµ
X
(3.1)
with the constant not depending on λ > 0 and f is valid.
Further, by virtue of Theorem B the latter inequality implies that there
exists a constant c2 > 0 such that for any ball B ⊂ X we have
Z
X
q
′
Tγ∗ (χB w1−p )(x) v(x)dµ
where
Tγ∗ (ϕ)(x)
=
Z
X
q1
≤ c2
Z
′
w1−p (x)dµ
B
|ϕ(y)|
dµ.
(µB(y, d(x, y)))1−γ
1′
p
,
(3.2)
432
A. GOGATISHVILI AND V. KOKILASHVILI
On the other hand, there exist constants c3 > 0 and c4 > 0 such that
c3 µB y, d(x, y) ≤ µB x, d(x, y) ≤ c4 µB y, d(x, y) .
(3.3)
The latter follows from the fact that B(x, d(x, y)) ⊂ a1 (a0 + 1)d(x, y). Indeed, let d(x, z) ≤ d(x, y). Then
d(y, z) ≤ a1 d(y, x) + d(x, z) ≤ a1 (a0 + 1)d(x, y).
By virtue of the doubling property for measure we obtain
µB y, a1 (a0 + 1)d(x, y) ≤ c5 µB y, d(x, y) .
Hence we conclude that (3.3) is valid. Next, from (3.3) and (3.2) we derive
Z
X
q
′
Tγ (χB w1−p )(x) v(x)dµ
1q
≤ c6
Z
′
w1−p (x)dµ
p1
.
(3.4)
B
Now we will show that for an arbitrary nonnegative measurable function ϕ
we have
Mγ (ϕ)(x) ≤ c7 Tγ (ϕ)(x),
(3.5)
where the constant c7 does not depend on ϕ and x.
First we will show that for any x there exists a ball Bx = B(x, rx ) such
that
Z
c8
Mγ (ϕ)(x) ≤
ϕ(z)dµ,
(3.6)
(µBx )1−γ
B
where the positive constant c8 does not depend on ϕ and Bx . Indeed, there
exists a ball B(y, r) such that x ∈ B(y, r) and
Z
2
Mγ (ϕ)(x) ≤
ϕ(z)dµ.
(3.7)
(µB(y, r))1−γ
B(y,r)
Assuming now that z ∈ B(y, r), we obtain
d(x, z) ≤ a1 d(x, y) + d(y, z) ≤ a1 (a0 + 1)r.
Therefore B(y, r) ⊂ B(x, a1 (1+a0 )r). On the other hand, we have B(x, a1 (1+
a0 )r) ⊂ B(y, a1 (1 + a1 (1 + a0 ))r) since
d(y, z) ≤ a1 d(y, x) + d(x, z) ≤ a1 r + a1 (1 + a0 r) = a1 1 + a1 (1 + a0 ) r
for any z ∈ B(x, a1 (1 + a0 )r).
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
433
Now applying the doubling condition, from (3.7) we find that
Mγ (ϕ)(x) ≤
Z
2
(µB(y, r))1−γ
ϕ(z)dµ ≤
B(x,a1 (1+a0 )r)
Z
c8
≤
(µB(x, a1 (1 + a0 )r))1−γ
ϕ(z)dµ.
B(x,a1 (1+a0 )r)
Replacing rx by the number a1 (1 + a0 )r, we obtain (3.6).
Now we will prove (3.5). We have
Z
Tγ (ϕ)(x) ≥
ϕ(y)
dµ ≥
(µB(x, d(x, y)))1−γ
B(x,rx )
≥
Z
1
(µB(x, rx ))1−γ
ϕ(z)dµ ≥
1
Mγ (ϕ)(x).
c8
B(x,rx )
From (3.4) and (3.5) we obtain
Z
X
Mγ (χB w
1−p′
q
)(x) v(x)dµ
q1
≤ c9
Z
w
1−p′
(x)dµ
p1
.
B
By Theorem 2.1 we conclude that inequality (1.1) is valid.
Thus we have proved the implication (1.2) ⇒ (1.1).
Let us show the validity of the inverse implication (1.1) ⇒ (1.2). Fix an
arbitrary ball B(x, r) and assume
′
f (y) = χN B(x,r) w1−p (y),
where N is an arbitrary number greater than unity. Obviously, by virtue of
the doubling condition we have
Z
1
Mγ (f )(y) ≥
µB(x, N d(x, y))
′
w1−p (z)dz ≥
B(x,N d(x,y))∩N B(x,r)
c10
≥
µB(x, d(x, y))
Z
N B(x,r)
for any y ∈ X\B(x, r).
′
w1−p (z)dµ
434
A. GOGATISHVILI AND V. KOKILASHVILI
Therefore (1.1) implies
Z
w
1−p′
(z)dµ
Z
v(y)
dµ
(µB(x, d(x, y))(1−γ)q
q1
≤
X\B(x,r)
N B(x,r)
≤c
Z
w
1−p′
(z)dµ
p1
.
N B(x,r)
From the latter inequality we obtain the validity of (1.2).
Definition 3.1. Measure ν satisfies the reverse doubling condition (ν ∈
(RD)) if there exist numbers δ and ε from (0, 1) such that
νB(x, r1 ) ≤ ενB(x, r2 )
for µB(x, r1 ) ≤ δµB(x, r2 ), 0 < r1 < r2 .
′
In the particular case where measure w1−p satisfies the reverse doubling
condition, (1.2) in the main theorem can be replaced by a simpler condition.
′
Theorem 3.1. Let 1 < p < q < ∞, 0 < γ < 1 and measure w1−p
satisfy the reverse doubling condition. Then (1.1) holds iff
γ−1 1−p′
1
1
sup µB(x, r)
B(x, r) p′ vB(x, r) q < ∞.
w
(3.8)
x∈X
r>0
Proof. The implication (1.1) ⇒ (3.8) is obtained immediately by substitut′
ing the function f (y) = χB w1−p (y) into (1.1).
By virtue of the main theorem to prove the implication (3.8) ⇒ (1.1) it
′
is sufficient to show that if w1−p ∈ (RD), then (3.8) ⇒ (1.2). Let x ∈ X
and r > 0 be fixed. Choose numbers rk (k = 0, 1, . . . ) as follows:
r0 = 2N0 r and rk = inf r : µB(x, rk−1 ) < δµB(x, r) .
Obviously,
µB(x, rk−1 ) < δµB(x, rk ) < cµB(x, rk−1 ).
Again applying the condition (RD), we obtain
′
′
w1−p B(x, r0 ) ≤ εk w1−p B(x, rk ) k = 1, 2, . . . .
The latter inequalities imply
1−p′
1
w
B(x, 2N0 r) p′
Z
X\B(x,r)
(γ−1)q
v(y) µB(x, d(x, y))
dµ(y)
q1
≤
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
∞
1 X
′
≤ w1−p B(x, r0 ) p′
k=1
+ w
1−p′
≤
1
B(x, r0 ) p′
∞
X
k=1
Z
(γ−1)q
v(y) µB(x, d(x, y))
dµ(y)
q1
435
+
B(x,rk )\B(x,rk−1 )
Z
B(x,r0 )\B(x,r)
q1
(γ−1)q
≤
v(y) µB(x, d(x, y))
dµ(y)
γ−1
1
1
k
′
+
ε p′ w1−p B(x, rk ) p′ vB(x, rk ) q µB(x, rk−1 )
γ−1
1
1
′
≤
+ w1−p B(x, r0 ) p′ vB(x, r0 ) q µB(x, r0 )
≤c
∞
X
k=0
γ−1 1−p′
1
1
k
B(x, rk ) p′ vB(x, rk ) q ≤
ε p′ µB(x, rk )
w
≤c
∞
X
k
ε p′ < ∞.
k=0
Remark. A similar result was obtained in [2] under the assumption that
′
the measure w1−p satisfies the doubling condition. Theorem 3.1 contains a
stronger result, since condition (RD) is weaker than the doubling condition.
(For instance, the function e|x| ∈ (RD) but it does not satisfy the doubling
condition.) Along with this, the proof in [2] essentially differs from the
above.
The above proofs also remains valid for fractional maximal functions:
Z
1
|f (y)|dσ(y),
(3.9)
Mγ (f dσ)(x) = sup
1−γ
B∋x (µB)
B
where 0 ≤ γ < 1 and σ is a Borel measure while the supremum is taken
over all balls B ⊂ X with a positive measure containing x.
Theorem 3.2. Let X be an arbitrary homogeneous type space, 1 < p <
q < ∞, 0 < γ < 1, ω and σ be positive Borel measures. For a constant
c > 0 to exist such that the inequality
p1
q1
Z
Z
q
p
|f (x)| dω(x)
≤c
(3.10)
Mγ (f dσ)(x) dω(x)
X
X
holds, it is necessary and sufficient that the condition
1
σB(2N0 r) p′ ×
sup
x∈X
r>0
µB(x,r)6=0
436
A. GOGATISHVILI AND V. KOKILASHVILI
×
Z
X\B(x,r)
µB(x, d(x, y))
(γ−1)q
q1
0 such that
Z
X
q1
p1
Z
q
Mγ (f dσ)(x) dω(x)
≤ c1
|f (x)|p dσ(x) ,
(3.12)
X
for any f ∈ Lp (X, dσ);
(ii) there exists a constant c2 > 0 such that
Z
B
q1
q
1
≤ c2 (σB) p
Mγ (χB dσ)(x) dω(x)
(3.13)
for any B ⊂ X.
Theorem 3.4. Let 1 < p < q < ∞, 0 < γ < 1, and the measure σ
satisfy the reverse doubling condition (σ ∈ (RD)). Then (3.10) holds iff
1
1
sup (µB)γ−1 (σB) p′ (ωB) q < ∞.
B
µB6=0
Theorems 3.2 and 3.3 in fact were previously proved in [2] under the
assumption that the space X possesses group structure, while Theorem
3.4 was proved for homogeneous type spaces under a stronger restriction
imposed on the measure σ.
Remark. The above-formulated theorems remain valid also in the case
where X = R1 is an arbitrary positive Borel measure. This follows from the
fact that we proved Theorem A for such measures and from the general conclusion that the well-known technique is applicable for maximal functions
with respect to a Borel measure on the real axis (see, for instance, [9]).
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
437
4. A Strong Type Two-Weight Problem for Fractional
Maximal Functions in a Half-Space
This section deals with “H¨ormander type” fractional maximal functions.
b = X × [0, ∞) and for (x, t) ∈ X
b
Let X
Z
γ−1
f
(4.1)
Mγ (f dσ)(x, t) = sup(µB)
|f (y)|dσ(y),
B
where 0 ≤ γ < 1, σ is a Borel measure in X and the supremum is taken
over all balls B ⊂ X containing x and having a radius greater than 2t .
We have
Theorem 4.1. If 1 < p < q < ∞, 0 < γ < 1, and X is an arbitrary
homogeneous type space, then the inequality
Z
b
X
q1
Z
p1
q
p
f
Mγ (f dσ))(x, t) dω(x, t)
≤c
|f (x)| dσ(x)
(4.2)
X
holds for all f with c independent of f iff
1
sup σB(x, 2N0 r) p′ ×
x∈X
r>0
×
Z
b B(x,r)
b
X\
q1
(γ−1)q
µB(x, d(x, y) + t)
dω(x, t)
< ∞,
(4.3)
b r) = B(x, r) × [0, r).
where B(x,
Proof. The validity of the theorem follows from the main theorem using the
following arguments. A similar idea is used in [18].
On the product space X × R we define the quasi-metric as
db (x, t), (y, s) = max d(x, y), |s − t|
b = µ ⊕ δ0 , where δ0 is the Dirac measure concentrated at
and the measure µ
zero.
b with respect to db lies in X
b and is
Note that the center of a ball from X
b Since
obtained by the intersection of the respective ball in X × R with X.
b
b we have
B((x,
t), r) = B(x, r) × (t − r, t + r) ∩ X,
b (x, t), r = µB(x, r) for r ≥ t,
µ
bB
b (x, t), r = 0 for r < t.
bB
µ
438
A. GOGATISHVILI AND V. KOKILASHVILI
bµ
b d,
Therefore (X,
b) is a homogeneous type space. It can be easily verified
c is actually a function of the form
that the maximal function M
Z
γ−1
b
Mγ (f dσ)(x, t) = sup µB((z,
t), r)
|f (y)|db
σ,
b
),r)
B((z,τ
where σ
b = σ ⊕ δ0 .
Hence to derive Theorem 4.2 from the main theorem it is sufficient to
show that in the considered case condition (2.11) can be reduced to condition
b
b
bB((x,
τ ), r) =
τ ), r) 6= 0 for τ < r and B((x,
(4.3). For this, in turn, since µ
B(x, r) × [0, r + τ ) it is sufficient to show that
Indeed,
b
b (x, τ ), d((x,
µ
bB
τ ), (y, t)) ∼ µB x, d(x, y) + t
for τ < r < db (x, τ ), (y, t) = max d(x, y), |τ − t| .
d(x, y) + t ≤ d(x, y) + |τ − t| + τ ≤ d(x, y) + |τ − t| + r ≤ 3db (x, τ ), (y, t) .
Since 0 < τ < max{d(x, y), |τ − t|}, we have {d(x, y), |τ − t|} < d(x, y) + t,
i.e.,
b
τ ), (y, t)).
d(x, y) + t ∼ d((x,
Therefore by virtue of the doubling condition
b
b (x, t), d((x,
t), (y, s)) =
µ
bB
b
= µB x, d((x,
τ ), (y, t)) ∼ µB x, d(x, y) + t .
Theorem 4.2. Let X be an arbitrary homogeneous type space, 1 < p ≤
q < ∞, 0 ≤ γ < 1. Then (4.2) holds for all f with c independent of f iff
Z
bb
B
q
Mγ (χB dσ)(x, t) dω(x, t)
q1
1
≤ c1 (σB) P ,
(4.4)
bb
= B × [0, 2 rad B).
where B
Theorem 4.3. If 1 < p < q < ∞, 0 < γ < 1, σ ∈ (RD), then (4.2)
holds iff
1
bb q1
sup(µB)γ−1 (σB) p′ ω B
< ∞,
where the supremum is taken with respect to all balls B.
(4.5)
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
439
It should be said that theorems of the type of Theorems 4.1 and 4.2 have
previously been known only for homogeneous type spaces with a group
structure [10], [11]. In the latter papers instead of condition (4.3) there
figures a condition which actually implies the simultaneous fulfillment of
conditions (4.3) and (4.5). As for Theorem 4.3, it is proved in [10], [11] but
under a more restrictive doubling condition for the measure σ.
We should also note papers [12], [13], where some more particular results
are obtained.
5. A Strong Type Two-Weight Problem for One-Sided
Fractional Maximal Functions
The method developed in the preceding sections enables one to solve the
following strong type two-weight problem:
Mγ+ (f )(x)
= sup h
γ−1
h>0
= sup h
h>0
|f (t)|dt,
x
and
Mγ− (f )(x)
x+h
Z
γ−1
Zx
|f (t)|dt,
x−h
where x > 0 and 0 < γ < 1.
To this end it is sufficient to follow the proof of the main theorem and
apply an analog of Theorem 2.1 for one-sided maximal functions (see, for
instance, [17, Theorem 2]), Theorem 1.6 from [7], and also the fact that
the Riemann-Liouville operator Rγ and the Weyl operator Wγ control the
one-sided maximal functions
Mγ− (f )(x) ≤ Rγ (|f |)(x)
and
Mγ+ (f )(x) ≤ Wγ (|f |)(x),
where
Rγ (f )(x) =
Zx
(x − t)γ−1 f (t)dt
0
and
Z∞
Wγ (f )(x) = (t − x)γ−1 f (t)dt,
x
where 0 < γ < 1, x > 0.
We thus come to the validity of the following two statements:
440
A. GOGATISHVILI AND V. KOKILASHVILI
Theorem 5.1. Let 1 < p < q < ∞, 0 < γ < 1. For the inequality
Z∞
0
q
Mγ+ (f )(x) v(x)dx
q1
Z∞
p1
p
≤c
|f (x)| w(x)dx
0
with the positive constant c independent of f to hold, it is necessary and
sufficient that the condition
1′ Z∞
a+h
Z
p
1−p′
w
(y)dy
sup
a,h
0 0 such that
Z
B
p′
K (χB dv) dµ
∗
1′
p
1
≤ c2 (vB) q′ .
(6.2)
The theorem also hold in the case p = 1, 1 < q < ∞ provided that
condition (6.2) is replaced by the condition
1
expB K ∗ χB dv (x) ≤ c2 (vB) q′
for each ball B.
Proof. Let f be a bounded nonnegative function with a compact support.
For given λ > 0 we set
Ωλ = x ∈ X : K(f )(x) > λ .
Let Bj = B(xj , rj ) be the sequence of balls from Lemma B corresponding
to the number c = 2a1 . Then there exists a constant c3 > 0 such that (see,
for instance, [17])
K χX\cBj f (x) ≤ c3 λ
(6.3)
for any x ∈ Bj , where cBj = B(xj , crj ). Hence K(χX\cBj f )(x) > c3 λ for
any x ∈ Bj ∩ Ω2c3 λ .
442
A. GOGATISHVILI AND V. KOKILASHVILI
Next,
∞
X
v Bj ∩ Ω2c3 λ =
vΩ2c3 λ ≤
j=1
X
=
j∈F
where
+
X
v Bj ∩ Ω2c3 λ = I1 + I2 ,
j∈G
F = j : v Bj ∩ Ω2c3 λ > βv(cBj )
and
G = v Bj ∩ Ω2c3 λ ≤ βv(cBj ) ,
while the number β will be chosen below so that 0 < β < 1.
Applying the H¨older inequality and condition (6.2) we obtain
c3 λv(cBj ) ≤ c3 β −1 λv Bj ∩ Ω2c3 λ ≤
Z Z
≤ β −1
k(x, y)f (y)dµ(y) dv(x) ≤
Bj
cBj
cBj
Bj
≤ β −1
≤ β −1
Z Z
Z Z
cBj
k(x, y)dv(x) f (y)dµ(y) ≤
Bj
p1
1′ Z
p′
p
k(x, y)dv(x) dµ(y)
≤
f p (y)dµ(y)
Bj
1
≤ β −1 c2 v(cBj ) q′
Z
f p (y)dµ
p1
.
cBj
Therefore
q
λ v(cBj ) ≤
−1
(c−1
c2 )q
3 β
Z
p
|f (y)| dµ(y)
pq
.
cBj
The summation of the latter inequality gives
pq
Z
q
−q −1 −1
p
I1 ≤ λ (c3 β c2 )
|f (y)| dµ .
X
P∞
Here we have used the fact that > 1 and j=1 χcBj ≤ M . Next for
j ∈ G we obtain
v Bj ∩ Ω2c3 λ ≤ βv(cBj ).
q
p
Therefore
I2 ≤ βM vΩλ .
CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES
443
Finally, we find that
vΩ2c3 λ ≤ c4 λ
−q
Z
p
|f (y)| dµ
pq
+ βM vΩλ .
X
Multiplying the latter inequality by λq and taking the exact upper bound
with respect to λ, 0 < λ < 2cδ3 , we obtain
q
sup λ vΩλ ≤ c4
0