Muckenhoupt–Wheeden theorem for generalized f-Riesz-type potentials
We obtain the Muckenhoupt-Wheeden theorem for some class of potentials. As a consequence, we describe the equivalent norm in the generalized Bessel potential space of negative order.
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irk-123456789-1647732020-02-11T01:28:43Z Muckenhoupt–Wheeden theorem for generalized f-Riesz-type potentials Knopova, V. Статті We obtain the Muckenhoupt-Wheeden theorem for some class of potentials. As a consequence, we describe the equivalent norm in the generalized Bessel potential space of negative order. Одержано теорему Макенхаупта–Відена для одного класу потенціалів. Як наслідок, описано еквівалентну норму в просторі узагальнених потенціалів Весселя від'ємного порядку. 2008 Article Muckenhoupt–Wheeden theorem for generalized f-Riesz-type potentials / V. Knopova // Український математичний журнал. — 2008. — Т. 60, № 11. — С. 1520–1528. — Бібліогр.: 11 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164773 519.21 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Knopova, V. Muckenhoupt–Wheeden theorem for generalized f-Riesz-type potentials Український математичний журнал |
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We obtain the Muckenhoupt-Wheeden theorem for some class of potentials. As a consequence, we
describe the equivalent norm in the generalized Bessel potential space of negative order. |
| format |
Article |
| author |
Knopova, V. |
| author_facet |
Knopova, V. |
| author_sort |
Knopova, V. |
| title |
Muckenhoupt–Wheeden theorem for generalized f-Riesz-type potentials |
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Muckenhoupt–Wheeden theorem for generalized f-Riesz-type potentials |
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Muckenhoupt–Wheeden theorem for generalized f-Riesz-type potentials |
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Muckenhoupt–Wheeden theorem for generalized f-Riesz-type potentials |
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Muckenhoupt–Wheeden theorem for generalized f-Riesz-type potentials |
| title_sort |
muckenhoupt–wheeden theorem for generalized f-riesz-type potentials |
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Інститут математики НАН України |
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2008 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/164773 |
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Muckenhoupt–Wheeden theorem for generalized f-Riesz-type potentials / V. Knopova // Український математичний журнал. — 2008. — Т. 60, № 11. — С. 1520–1528. — Бібліогр.: 11 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT knopovav muckenhouptwheedentheoremforgeneralizedfriesztypepotentials |
| first_indexed |
2025-07-14T17:21:35Z |
| last_indexed |
2025-07-14T17:21:35Z |
| _version_ |
1837643800872222720 |
| fulltext |
UDC 519.21
V. Knopova (Inst. Cybern. Nat. Acad. Sci., Ukraine, Kyiv)
MUCKENHOUPT – WHEEDEN THEOREM
FOR GENERALIZED f -RIESZ TYPE POTENTIALS
TEOREMA MAKENXAUPTA – VIDENA DLQ UZAHAL|NENYX
POTENCIALIV f -RISIVS|KOHO TYPU
We obtain the Muckenhoupt-Wheeden theorem for some class of potentials. As a consequence, we
describe the equivalent norm in the generalized Bessel potential space of negative order.
OderΩano teoremu Makenxaupta – Videna dlq odnoho klasu potencialiv. Qk naslidok, opysano
ekvivalentnu normu v prostori uzahal\nenyx potencialiv Besselq vid’[mnoho porqdku.
Introduction. The paper is devoted to the generalization of Muckenhoupt – Wheeden
theorem (see [1], Theorem 3.6.1, also [2]) to the case of potentials
I xf µ( ) =
µ ( )dy
x y f x yn
n − −( )−∫ 2
R
, (1)
where µ is any positive measure on Rn , and f is a Bernstein function, which means
that f is a real-valued function defined on ( 0, ∞ ) , satisfying the following conditions:
1) f C∈ ∞∞( , )0 ,
2) f ( x ) ≥ 0,
3) ( ) ( )( )−1 k kf x ≤ 0 for all k ≥ 1.
For a positive measure µ, we define the f -maximal function M f µ
M xf µ( ) = sup
( ( , ))
( )/
r
n
n
n
B x r
f r r>
−
0
2
µ
ω
, (2)
where ωn = dx
Sn−∫ 1 is the volume of a unit ball in Rn . For f ( x ) = x
α, this maxi-
mal function is called a fractional maximal function of a measure µ and is denoted
by Mαµ , see [2], for example. We show that the Lp -norm of M f µ , 1 < p < ∞ ,
is equivalent to the Lp -norm of I f µ . Such an equivalence gives us the description of
an equivalent norm in the generalized Bessel potential space Hp
f n( ),| | ( )⋅ −2 2
R , which is
the closure of the Schwartz space S n( )R under the norm
u Hp
f n( ),| | ( )⋅ −2 2
R
: = F f Fu
Lp
n
− −+ ⋅ ⋅1 2 11( ( ) )( ) ( )
( )R
, 1 < p < ∞ ,
see [3, 4] for more information about the construction of such spaces. Here F , F−1
are respectively the Fourier and the inverse Fourier transforms. Besides others the
generalized Bessel potential spaces are interesting from the analytical point of view as
they are the particular cases of the spaces of generalized smoothness, and appear as
domains of generators of Lp -sub-Markovian semigroups: if f is a Bernstein function,
then − −f ( )∆ is the generator of an Lp -sub-Markovian semigroup, corresponding to a
Lévy process ( )Xt t ≥0 with Lévy exponent f ( )ξ 2 ( i.e., Eei Xt〈 〉ξ, = e t f− ( )ξ 2
) .
The domain of − −f ( )∆ is Hp
f n( ),| | ( )⋅ 2 2
R , which we can identify with the dual of
Hp
f n( ),| | ( )⋅ −2 2
R . In the case where f is a Bernstein function satisfying some growth
© V. KNOPOVA, 2008
1520 1520ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
MUCKENHOUPT – WHEEDEN THEOREM … 1521
restrictions and such that the convolution semigroup associated with it has monotone
0-potencial density, it was proved in [5], Theorem 1.1.2, that the kernel of the resolvent
associated with − −f ( )∆ is equivalent to the kernel of I f .
For u Lp
n∈ ( )R , 1 ≤ p < ∞ , we can also define the potential as
I u xf ( ) : = u y
x y f x y
dyn
n
( )
− −( )−∫ 2
R
. (3)
Therefore in the case µ ( )dy = u y dy( ) , u Lp
n∈ ( )R positive, the generalization of the
Muckenhoupt – Wheeden theorem gives us the equivalence of norms:
u Hp
f n( ),| | ( )⋅ −2 2
R
∼ I u uf
p p+ ∼ M u uf
p p+ ,
where
M u xf ( ) : = sup
( )
( )/
( , )r
n
n
n
B x rf r r
u y dy
>
− ∫
0
2
1
ω
. (4)
Here and below the relation ⋅ ∼ ⋅1 2 means that there exist positive constants c1
and c2 such that c1 1⋅ ≤ ⋅ 2 ≤ c2 2⋅ .
The “classical” Muckenhoupt – Wheeden theorem, i.e., the equivalence of
Lp
n( )R -norms of Riesz potentials Iαµ of a positive measure µ, 0 < α < n, and of
the fractional maximal function Mα , is a useful tool in the theory of function spaces.
This theorem plays an important role in the proof of such a remarkable fact that the
positive cone of Triebel – Lizorkin spaces Fpq
nα ( )R , 1 < p < ∞ , 1 < q ≤ ∞ , α <
< 0, is independent of q, see Corollary 4.3.9 from [2], also [6] for the original result.
Further, the Muckenhoupt – Wheeden theorem is useful for getting estimates for non-
linear potentials, in particular, it is employed to show the equivalence of different
definitions of capacities, see § 4.4 – 4.5 [2] and the reference therein. Also, the
weighted Muckenhoupt – Wheeden inequality applied to I1 allows to obtain some
norm inequalities for the Schrödinger operator L = − −∆ v for v of some type,
which can be used for getting the eigenvalue estimates for L, see [7, 8]. Therefore the
generalized version of the Muckenhoupt – Wheeden theorem may give rise to new
results in the theory of function spaces and applications.
The main result of the paper is formulated in the following theorem.
Theorem 1. Let 1 < p < ∞ , n ≥ 2, and assume that the Bernstein function f
satisfies (6) and (7). Then there exists a constant c such that for any positive
measure µ
c M f
p
−1 µ ≤ I f
p
µ ≤ c M f
p
µ . (5)
Since the left-hand side inequality is trivial, it remains to prove the right-hand side
part. The proof is based on Lemma 1 and Lemma 2 below, see also [2, p. 73 – 74].
Assumptions and auxiliary results. In what follows we will assume that our
Bernstein function satisfies the following assumptions:
1. There exists β > 0 such that for all λ ≥ 1
c1λβ ≤ f x
f x
( )
( )
λ , x > 0; (6)
2. There exists 0 < σ < n p/ 2 such that for all λ ≤ 1
c2λσ ≤ f x
f x
( )
( )
λ , x > 0. (7)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1522 V. KNOPOVA
Here c1, c2 are some positive constants, independent on x and λ .
Consider some examples.
Examples. 1. f x x( ) = α , 0 ≤ α ≤ 1.
2. f x x x( ) ln( )= +α α1 , 0 ≤ α < 1 / 2.
Indeed, (6) is satisfied with β ≤ α due to monotonicity of f x( ); to prove (7) con-
sider the function g x x c x( ) ln( ( ) ) ln( )= + − +1 1λ λα ε α , 0 < α ≤ 1.
′g x( ) = α λ
λ
λα
α
α
ε
αx
x
c
x
−
+
−
+
1
1 1( )
> α λ λ
α
α
α εx
x
c
−
+
−[ ]
1
1
> 0 if α < ε .
Since g( )0 = 0, then g x( ) > 0 for all x > 0, and hence we have (7) with α < σ <
< n p/ 2 .
3. f x x e x( ) = −( )−1 4 . Again, we have (6) with β ≤ 1 / 2 due to monotonici-
ty. To show (7) consider the function g x( ) = 1 14 4− − −( )− −e c ex xλ ελ . Then for
suitable c > 0
′g x( ) = 2
x
cλ λε−( ) > 0 if ε < 1
2
.
Since g( )0 = 0, we get (7) with 1 / 2 < σ < n p/ 2 .
4. f x( ) = x
I x
I x
ν
ν
+1( )
( )
, see [9]. Here Iν is the modified Bessel function of the
first kind, see [10]. Sinse
I xν( ) ∼ 1
1 2Γ ( )ν
ν
+
x as x → 0 ,
I xν( ) ∼ 1
2π x
ex as x → ∞ ,
we have f x x( ) /∼ ν 2 as x → 0 , and f x x( ) ∼ as x → ∞ , hence we can
choose constants c1 and c2 such that (6) and (7) are satisfied.
5. f x( ) = x
K x
K x
ν
ν
−1( )
( )
, see [9]. Here Kν is the modified Bessel function of
the third kind, see [10]. Since
K xν( ) ∼ Γν ν
2
2
x
as x → 0 ,
K xν( ) ∼ π
2x
e x− as x → ∞ ,
we have f x( ) ∼
xΓ
Γ
( )
( )
ν
ν
− 1
2
=
x
2 1( )ν −
, ν > 1, as x → 0 , and f x x( ) ∼ as x →
→ ∞ , hence as above we can choose constants c1 and c2 such that (6) and (7) are
satisfied with β ≥ 1 / 2 and σ > 1.
6. By the same arguments, (6) and (7) are satisfied for Bernstein functions f x( ) =
=
xI x
I x
ν
ν
β
α
( )
( )
and f x( ) =
xK x
K x
ν
ν
α
β
( )
( )
, ν > 0, α > β > 0 (see [9]).
Below we will use the estimates for derivatives of a Bernstein function, see [4]:
f xk( )( ) ≤ k f x
xk
! ( ) , k ≥ 1 , x > 0 . (8)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
MUCKENHOUPT – WHEEDEN THEOREM … 1523
For a positive measure µ define the Hardy – Littlewood maximal finction
M xµ( ) : = sup ( )
( , )r n
n
B x r
w r
dy
>
∫
0
1 µ . (9)
In case µ ( )dy = u y dy( ) for some Lp -function u, 1 ≤ p ≤ ∞ , we will use the
notation Mu .
We will need the Hardy – Littlewood – Wiener theorem, see, for example, [2],
Theorem 1.1.1. Denote by “Vol” the volume of a set.
Theorem 2 (Hardy – Littlewood – Wiener [2]). Let u Lp
n∈ ( )R , 1 ≤ p ≤ ∞ .
There exists a constant A depending only on p and n such that
a) if p = 1, then
Vol x Mu x: ( ) >{ }λ ≤ A u
λ 1 for all λ > 0;
b) if 1 < p ≤ ∞ , then
Mu p ≤ A u p. (10)
Lemma 1. Let f be a Bernstein function satisfying (6) and (7), I f be as in
(3), and 1 ≤ p < ∞ . Then
I u xf ( ) ≤ cMu x
f Mu x
u p
p n
( )
( )
/
2 . (11)
Proof. Take 0 < δ < 1, split the integral:
I u xf ( ) = u y
x y f x y
dy
u y
x y f x y
dyn
x y
n
x y
( ) ( )
− −( ) +
− −( )−
− <
−
− ≥
∫ ∫2 2
δ δ
= I I1 2+ ,
and consider the terms I1 and I2 separately.
Changing the variables y = r ζ , r ∈ +R , ζ ∈ −Sn 1, we obtain
I1 =
0
2
1
δ ζ σ ζ∫ ∫ −
−
−
u x r
r f r
d drn n
Sn
( )
( )
( ) ,
where σ ζn d( ) is the surface measure on Sn−1. Let
ρ( , )x dr = u x r d drn
Sn
( ) ( )−
−
∫ ζ σ ζ
1
and
ρ( , )x r =
0 1
r
n
S
u x d d
n
∫ ∫ −
−
( ) ( )τζ σ ζ τ = u y dy
B x r
( )
( , )
∫ .
Using (8), we get
I1 =
0
2
δ ρ∫ −
( , )
( )
x dr
r f rn =
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1524 V. KNOPOVA
=
ρ ρ
δ δ
( , )
( )
( , )
( )
( )
( )
( )
x r
r f r
x r
n
r f r
n
f r
r f r
drn n n− + −
−
+ −− − + +
′
∫2
0 0
1 2
2
3 2 22 ≤
≤
ρ δ
δ δ
ρδ
( , )
( )
( , )
( )
x
f
n
x r
r f r
drn n− + −+ ∫2
0
1 2 ≤
≤ ω
δ
n Mu x
f r
n dr
r f r
( )
( ) ( )
1
2
0
2− −+
∫ .
Since f satisfies (6), we have that the last integral is less than
1
2 1 20
1
f
dx
x( )δ β− −∫ and
thus
I1 ≤ c
M u x
f
( )
( )δ−2 .
Further, by Hölder’s inequality, we get
I2 ≤ u
x y f x y
dyp n
p
x y
p
1
2
1
− −( )
−
′
− ≥
′
∫
δ
/
=
= u
c
r f r
r drp n
p
n
p
δ
∞
−
′
−
′
∫
1
2
1
1
( )
/
=
= u
c
f
dp n
p
n n
p
1
1
2 2
1
1∞
− −
′
−
′
∫
( ) ( )
/
δτ δ τ
δ τ τ ≤
≤ c
u
f
dp
n p p
np n p
p
1
1
2
1
1 2
1
δ
δ
τ
τ σ
( )/ /
( )
− ′ ′
−
∞
′− + − ′
′
∫
≤
c u
f
p
n p
2
2
δ
δ
−
−
( / )
( )
,
where (7) is used in the third line.
For δ > 1 the estimates are the same due to the restricttions on σ and β (we
need not to pose the restriction β < n / p , since β can be arbitrary small in (6)).
Combining the estimates for I1 and I2 and choosing δ =
u
Mu
p
p n
/
, we arrive
at (11).
Remark 1. Let g xp( ) = x
f x p n( )/2 , x > 0. The function gp
−1 is convex, mono-
tone increasing for 2p n/ ≤ 1 and monotone decreasing for 2p n/ > 1. Then we
can get the inequalities analogous to the Sobolev inequality, but under some restrictions
on the norm u p .
1. If 2p n/ ≤ 1, then cg xp
−1( ) ≤ g cxp
−1( ) if c ≥ 1 and then for u , u p ≤ 1,
we have, due to the Hardy – Littlewood – Wiener theorem,
1 1
u
g I u
p
p p
f
p
p− ( ) ≤ g
I u x
up
f
p p
p
−
1 ( ) ≤
Mu
u
p
p
p
p ≤ C,
or
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
MUCKENHOUPT – WHEEDEN THEOREM … 1525
g I up
f
p
−1( ) ≤ u p . (12)
2. If 2p n/ > 1, then we have g cxp
−1( ) ≤ g xp
−1( ) for c ≥ 1, and in this case if
u p > 1,
g I up
f−1( ) ≤ g g Mu
u
up p
p
p
−
1 ≤ Mu
u p
,
whence (12) is satisfied if u p > 1.
Remark 2. Lemma 1 and Theorem 2a) imply weak type estimates:
Vol x I u xf: ( ) >{ }λ ≤ c
g u− ( )1
1λ /
, (13)
where g x( ) = g x1( ) = x
f x n( )/2 . Indeed,
x I u xf: ( ) >{ }λ ⊂ x Mu x u g
u
: ( ) >
−
1
1
1
λ .
Applying Theorem 2a), we get (13). Moreover, (13) is valid for finite measures on Rn ,
u 1 = µ ( )R
n .
Remark 3. Note that the statements of Lemma 1 and Remark 2 can be naturally
generalized to the case of finite measures on Rn .
Remark 4. For the case of Riezs potentials, i.e. when f ( x ) = x
α, 0 < α < 1,
see [2], Proposition 3.1.2 and Theorem 3.1.4.
Lemma 2. There exists a > 1, b > 0, such that for all λ > 0, for all ε , 0 <
< ε < 1,
Vol x I x af: ( )µ λ>{ } ≤
≤ b
g
x I x x M xf f
− − >{ } + >{ }1 1( )
: ( ) : ( )
ε
µ λ µ ελVol Vol . (14)
Proof. Since µ is a positive measure, by Fatou’s Lemma the potential (1) is lo-
wer semicontinuous. Then the set x I f: µ λ>{ } is open. By Whitney decompositi-
on theorem there exists a set of dyadic cubes Qi{ } with disjoint interior such that for
all Qi there exists
x : dist ( x , Qi ) ≤ 4 diam Q
i
. (15)
For such x we have I xf µ( ) ≤ λ . Assume that Q Qi∈{ }, a > 1 and consider the
set x Q I x af∈ >{ }: ( )µ λ .
1. Suppose Q x M xf∩ : ( )µ ελ≤{ } ≠ ∅. Let P be a ball concentric with Q,
with radius 6 diam Q . Let µ1 : = µ P , µ2 : = µ µ− 1. By Remark 2,
Vol x I x
af: ( )µ λ
1 2
>{ } ≤ c
g a dn
− −
∫( )
1
1
1
λ µ
R
.
Let x Q0 ∈ be such that M xf µ( )0 ≤ λε and let B x( )0 = B x Q( , )0 8diam ( then P ⊂
⊂ B x( )0 ) . Then
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1526 V. KNOPOVA
d
n
µ1
R
∫ ≤ d
P
µ∫ ≤ d
B x
µ
( )0
∫ ≤ M x g B xf µ( ) ( ( ))0 0Vol( ) ≤ λε g B xVol( ( ))0( ).
By definition, f is increasing. Then for δ > 0 there exists cδ > 0 such that for all
δ ≤ λ ≤ 1, x ≥ 1,
f x( )λ ≤ c f x fδ λ( ) ( ).
Since for n ≥ 2, g = g2 is also increasing, for such x and λ we get g g x( ) ( )λ ≤
≤ c g xδ λ( ) , and hence due to monotonocity ( take λ = g
−1( )v , x = g w−1( ) ) , we have
g w−1( )v ≤
g c g g g w− − −( )1 1 1
δ ( )( ) ( )v ≤
≤
g g c g g w− − −( )1 1 1( )( ) ( )δ v ≤ g c g w− −1 1( ) ( )δ v .
Further, for small diameters of Q we have dn µ1
R∫ ≤ 1, and g B xVol( ( ))0( ) < 1.
Then from
1
ε
<
g B x
dn
Vol( ( ))0
1
( )
∫ µ
R
,
for a > 1 such that λ a > 0, we get
g a−
1
ε
≤ g
a g B x
dn
− ( )
∫
1 0
1
λ
µ
Vol( ( ))
R
≤ g
a
d
c B x
n
−
∫
1
1
0
λ
µ δ
R
Vol( ( )),
where the constant cδ depends on the size of the cubes Qi{ } but is independent of
the choice of Qi (i.e., we may assume that the size of Qi{ } is bounded from below
by 2−M with M fixed and large enough).
Then by covering
Vol x Q I x
af∈ >{ }: ( )µ λ
1 2
≤ b
g
Q− −1 1( )
( )
ε
Vol , (16)
or, covering the whole set x I x
af: ( )µ λ
1 2
>{ } ⊂ x I xf: ( )µ λ>{ }, a > 1,
Vol x I x
af: ( )µ λ
1 2
>{ } ≤ b
g
x I xf
− − >{ }1 1( )
: ( )
ε
µ λVol . (17)
2. Take x Q1 ∉ ; then we have (15). Because of the choice of P, there exists a
constant L depending only on n and such that for all y Pc∈ , ∀ ∈x Q : x y1 − ≤
≤ L x y− ( we may assume here L ≥ 1) . By the property of Bernstein functions
f f x y−( )2 ≤ f L x y1
2−( ) ≤ L f x y1
2−( ) ,
and since I xf µ( )1 ≤ λ (due to the conditions of Whitney decomposition, we have
(15)), we get
I xf µ2( ) ≤ L I xn f+1
2 1µ ( ) ≤ λLn+1.
Choose a > 2 1Ln+ ; then if I xf µ2( ) ≤ aλ / 2 and I xf µ( ) > aλ , we obtain
I xf µ1( ) > aλ / 2. Thus, for all x Q∈ such that Q x M f∩ : µ ελ≤{ } ≠ ∅, we
can write the inclusion
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
MUCKENHOUPT – WHEEDEN THEOREM … 1527
x Q I x af∈ >{ }: ( )µ λ ⊂ x Q I x
af∈ >
: ( )µ λ
1 2
.
Let us summarize the statements proved above. The set x I x af: ( )µ λ>{ } can be
covered by cubes of two types:
1) Q : Q x M f∩ : µ ελ≤{ } ≠ ∅. In the case of such Q, for all x Q∈ we have
x I x af: ( )µ λ>{ } ⊂ I x
af µ λ
1 2
( ) >{ };
2) Q̃ : Q̃ ⊂ x M f: µ ελ>{ } .
Covering x I x af: ( )µ λ>{ } by such cubes, in view of (17), we get the esti-
mate (14).
Lemma 2 is proved.
Proof of Theorem 1. For any r > 0
I xf µ( ) ≥
µ ( )dy
x y f x yn
x y r
− −( )−
− ≤
∫ 2 ≥ 1
2r f r
dyn
x y r
( )
( )−
− ≤
∫ µ ,
and by the definition of M
f we get the lower bound.
Let us show the upper bound. Integrate (14):
Vol x I x a df p
R
: ( )µ λ λ λ>{ } −∫ 1
0
≤
≤ b
g
x I x df p
R
− −
−>{ }∫1 1
1
0
( )
: ( )
ε
µ λ λ λVol +
+ Vol x M x df p
R
: ( )µ ελ λ λ>{ } −∫ 1
0
.
Changing the variables, we get
a x I x dp f p
aR
− −>{ }∫ Vol : ( )µ λ λ λ1
0
≤
≤ b
g
x I x df p
R
− −
−>{ }∫1 1
1
0
( )
: ( )
ε
µ λ λ λVol +
+ ε µ λ λ λ
ε
− −>{ }∫p f p
R
x M x dVol : ( ) 1
0
.
If µ is compactly supported, then all the integrals are finite. Choose ε so small that
b
g− −1 1( )ε
≤ a p−
2
.
Then
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1528 V. KNOPOVA
a x I x dp f p
aR
− −>{ }∫ Vol : ( )µ λ λ λ1
0
≤
≤ 2 1
0
ε µ ελ λ λ
ε
− −>{ }∫p f p
R
x M x dVol : ( ) ,
and letting R → ∞ we get
a I x dxp f p
n
− ∫ µ( )
R
≤ 2ε µ λ− ∫p f p
M x d
n
( )
R
.
If µ has no compact support, approximate with µ µn B n= ( , )0 , n = 1, … . Then
I f
n p
µ ≤ C M f
p
µ ,
and we get the statement of Theorem 1 by letting n → ∞ .
Acknowledgment. I am very grateful to Niels Jacob, University of Swansea, for
useful remarks. I also thank the referee for useful suggestions. The INTAS Grant YSF
06-1000019-6024 is gratefully acknowledged.
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Received 26.09.07,
after revision — 12.02.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
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