Journal of Inequalities in Pure and
Applied Mathematics
CONVOLUTION OPERATORS WITH HOMOGENEOUS SINGULAR
MEASURES ON R3 OF POLYNOMIAL TYPE.
THE REMAINDER CASE.
volume 7, issue 3, article 89,
2006.
MARTA URCIUOLO
Famaf-Ciem, Universidad Nacional de Córdoba-Conicet.
Medina Allende s/n Ciudad Universitaria
5000, Córdoba, Argentina.
Received 22 December, 2005;
accepted 23 September, 2006.
Communicated by: A. Fiorenza
EMail: urciuolo@mate.uncor.edu
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
374-05
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Abstract
Let ϕ (y1 , y2 ) = y2l P (y1 , y2 ) where P is a polynomial function of degree l such
3
that
R P (1, 0) 6= 0. Let µδ be the Borel measure on R defined by µδ (E) =
Vδ χE (x, ϕ (x)) dx where
Vδ = x = (x1 , x2 ) ∈ R2 : |x1 | ≤ 1, and |x1 | ≤ δ |x2 |
and let Tµδ be the convolution operator with the measure µδ . In this paper we
explicitely describe the type set
1 1
Eµδ :=
,
∈ [0, 1] × [0, 1] : kTµδ kp,q < ∞ ,
p q
Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
Marta Urciuolo
for δ small enough.
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2000 Mathematics Subject Classification: 42B20, 26B10.
Key words: Convolution operators, Singular measures.
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Partially supported by Conicet, Agencia Córdoba Ciencia, Agencia Nacional de Promoción Científica y Tecnológica y Secyt-UNC.
The author is deeply indebted to Prof. F. Ricci for his useful suggestions.
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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1.
Introduction
Let ϕ : R2 → R be a homogeneous polynomial function of degree m ≥ 2 and
let D = {y ∈ R2 : |y| ≤ 1} . Let µ be the Borel measure on R3 given by
Z
(1.1)
µ (E) =
χE (y, ϕ (y)) dy
D
and let Tµ be the operator
for f ∈ S (R3 ) , by Tµ f = µ ∗ f. Let Eµ
defined,
be the set of the pairs p1 , 1q ∈ [0, 1] × [0, 1] such that there exists a positive
constant c satisfying kT f kq ≤ c kf kp for all f ∈ S (R3 ) , where the Lp spaces
1 1
3
are taken with respect to the Lebesgue measure on R . For p , q ∈ Eµ , T
can be extended to a bounded operator, still denoted by T, from Lp (R3 ) into
Lq (R3 ) .
Let ϕ = ϕe11 ...ϕenn be a decomposition of ϕ in irreducible factors with ϕi - ϕj
for i 6= j. In [3] we could give a complete description of the set Eµ under the
assumption that ei 6= m2 for each ϕi of degree 1. If det ϕ00 (y) is not identically
zero and if it vanishes somewhere on R2 − {0}, the set of the points y where
det ϕ00 (y) vanishes is a finite union of lines L1 , ..., Lk through the origin. So,
after a possibly linear change of variables, we localized the problem to the x
axes and we studied the type set corresponding to measures µδ defined by
Z
µδ (E) =
χE (y, ϕ (y)) dy,
Vδ
where Vδ = D ∩ {(y1 , y2 ) ∈ R2 : |y2 | ≤ δ |y1 |} and δ is small enough such that
det ϕ00 (y) only vanishes, on Vδ , along the x axes. The only case left was the one
Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
Marta Urciuolo
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corresponding to functions ϕ of the form ϕ (y1 , y2 ) = y2l P (y1 , y2 ) with l = m2 ,
P being a homogeneous polynomial function of degree l such that P (1, 0) 6= 0.
In this paper we characterize Eµδ in this remainder case.
Lp improving properties of convolution operators with singular measures
supported on hypersurfaces in Rn have been widely studied in [2], [5], [6]. In
particular, in [5], the type set was studied under our actual hypothesis, but the
endpoint problem was left open there. Our proof of the main result involves a
biparametric family of dilations and will be based on a suitable adaptation of
arguments due to M. Christ, developed in [1], where the author studied the type
set associated to the two dimensional measure supported on the parabola.
Also, oscillatory integral estimates are involved. A very careful study of this
kind of estimate can be found in [4] where the authors study the boundedness
of maximal operators associated to mixed homogeneous hypersurfaces.
Throughout this paper c will denote a positive constant, not the same at each
occurrence.
Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
Marta Urciuolo
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2.
The Main Result
We assume ϕ (y1 , y2 ) = y2l P (y1 , y2 ) , where l = m2 and P is a homogeneous
polynomial function of degree l such that P (1, 0) 6= 0. We take δ1 > 0 such
that, for y ∈ Vδ1 such that y2 6= 0, det ϕ00 (y) 6= 0. Moreover, since P (1, 0) 6=
0 we can assume that P (y) 6= 0 and P1 (y) 6=
0 for all y ∈ Vδ1 . Now, if
maxVδ1 |P2 (y1 , y2 )| =
6 0, we choose δ < min
l minVδ |P (y1 ,y2 )|
1
2 maxVδ |P2 (y1 ,y2 )|
, δ1 . In the
1
other case we take δ = δ1 .
The main result we prove is the following.
Theorem 2.1. Let ϕ (y1 , y2 ) = y2l P (y1 , y2 ) where l = m2 and P is a homogeneous polynomial function of degree l such that P (1, 0) 6= 0 and y2 - P (y1 , y2 ) .
Let Vδ be defined as above and let EVδ be the corresponding type set. Then
EVδ is the closed polygonal region with vertices (0, 0) , (1, 1) , 2l+1
, 2l−1 and
2l+2 2l+2
3
, 1 .
2l+2 2l+2
Standard arguments (see, for example Lemma 2 and Lemma 3 in [3]) imply
the following result.
1
Lemma 2.2. If p1 , 1q ∈ Eµδ then 1q ≤ p1 , 1q ≥ p3 − 2 and 1q ≥ p1 − l+1
.
So, since kTµδ k1,1 < ∞, by duality arguments it only remains to prove that
(2.1)
kTµδ k 2l+2
, 2l+2
2l+1 2l−1
< ∞.
δ δ
∞
2
We set Q0 = 41 , 2 × 64
, 8 . We take a truncation
1 function
δ δ θ ∈ C (R ),
θ (y1 , y2 ) ≥ 0, supp θ ⊂ Q0 and θ (y1 , y2 ) = 1 on 2 , 1 × 32 , 16 . We define,
Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
Marta Urciuolo
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for ε, γ > 0, the biparametric family of dilations on R2 and R3 given
by (ε, γ) ◦
l l
(y1 , y2 ) = (εy1 , γy2 ) and (ε, γ) ◦ (y1 , y2 , y3 ) = εy1 , γy2 , ε γ y3 repectively.
Also, for j, k ≥ 0, we set Qj,k = 2−j , 2−k ◦ Q0 .
For f ∈ S (R3 ) , we define
(2.2) Tj,k f (x1 , x2 , x3 )
Z
= f (x1 − y1 , x2 − y2 , x3 − ϕ (y1 , y2 )) θ 2j y1 , 2k y2 dy1 dy2
so for f ≥ 0,
Tµ δ f ≤ c
(2.3)
8
X
Tj,k f.
Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
0≤j≤k
Marta Urciuolo
To study
P
Tj,k f, we will adapt the argument developed by M. Christ (see
0≤j≤k
[1]) to the setting of biparametric dilations. First of all, we prove the following
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Proposition 2.3. There exists a positive constant c > 0 such that for 0 ≤ j ≤ k,
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kTj,k k 2l+2 , 2l+2 ≤ c.
2l+1 2l−1
Proof.
Tj,k f (x1 , x2 , x3 )
Z
= f (x1 − y1 , x2 − y2 , x3 − ϕ (y1 , y2 )) θ 2j y1 , 2k y2 dy1 dy2
Z
−(j+k)
=2
f x1 − 2−j y1 , x2 − 2−k y2 , x3 − ϕ 2−j y1 , 2−k y2 θ (y1 , y2 ) dy1 dy2
= 2−(j+k) T (j−k) fj,k 2j x1 , 2k x2 , 2(j+k)l x3 ,
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where we denote
T
(j)
Z
f (x1 , x2 , x3 ) =
f x1 − y1 , x2 − y2 , x3 − y2l P y1 , 2j y2
θ (y1 , y2 ) dy1 dy2
and
fj,k (x1 , x2 , x3 ) = f
2−j , 2−k ◦ (x1 , x2 , x3 ) .
So
(2.4)
kTj,k f (x1 , x2 , x3 )kq = 2(j+k)(
1+l
− 1+l
−1
p
q
)
T (j−k) p,q
Convolution Operators with
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Measures on R3 of Polynomial
Type. The Remainder Case
kf kp .
Now,
det y2l P y1 , 2j−k y2
00
Marta Urciuolo
= 2(2−2l)(j−k) det (ϕ)00 y1 , 2j−k y2 .
so
in the proof of Lemma 4 in [3] we obtain that there exists c > 0 such that
as
T (j−k) 2l+2 2l+2 ≤ c for 0 ≤ j ≤ k, and the proposition follows.
,
2l+1 2l−1
We take 0 ≤ j ≤ k, and denote by µj,k and µ(j) the measures associated to
Tj,k and T (j) respectively. For ξ = (ξ1, ξ2 , ξ3 ) ,
Z
l
j−k
\
(j−k)
µ
(ξ) = e−i(ξ1 y1 +ξ2 y2 +ξ3 y2 P (y1 ,2 y2 )) θ (y1 , y2 ) dy1 dy2 .
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(j−k)
If for some ξ on the unit sphere, Ωξ
(y1 , y2 ) = ξ1 y1 +ξ2 y2 +ξ3 y2l P y1 , 2j−k y2
has a critical point belonging to the supp θ, then
ξ1 + ξ3 y2l P1 y1 , 2j−k y2 = 0
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and
ξ2 + ξ3 2j−k y2l P2 y1 , 2j−k y2 + ly2l−1 P y1 , 2j−k y2 = 0,
but then, since P1 (y) 6= 0 for y ∈ Vδ1 , from the first equation we obtain that
there exist constants a, b ∈ Z with a < b such 2a |ξ3 | ≤ |ξ1 | ≤ 2b |ξ3 | , and,
from the second one and the choice of δ we obtain constants c, d ∈ Z2 with
c < d such that 2c |ξ3 | ≤ |ξ2 | ≤ 2d |ξ3 |. So ξ belongs to the cone
C0 = ξ ∈ R3 : 2a |ξ3 | < |ξ1 | < 2b |ξ3 | , 2c |ξ3 | < |ξ2 | < 2d |ξ3 | .
Lemma 2.4. Suppose C0 is as above. Then the family of cones 2j , 2k ◦ C0 j,k∈Z
has finite overlapping (i.e., # (j, k) ∈ Z2 : C0 ∩ 2j , 2k ◦ C0 6= ∅ < ∞).
Proof. We suppose ξ ∈ C0 and 2j , 2k ◦ ξ ∈ C0 , then
2a |ξ3 | < |ξ1 | < 2b |ξ3 | ,
2c |ξ3 | < |ξ2 | < 2d |ξ3 |
Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
Marta Urciuolo
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and
(j+k)l+a
2
j
(j+k)l+b
|ξ3 | < 2 |ξ1 | < 2
|ξ3 | ,
2(j+k)l+c |ξ3 | < 2k |ξ2 | < 2(j+k)l+d |ξ3 |
so
2j |ξ1 | < 2(j+k)l+b |ξ3 | < 2(j+k)l+b−a |ξ1 |
and
2b |ξ3 | > |ξ1 | > 2−j 2(j+k)l+a |ξ3 | ,
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so
a − b − kl < j (l − 1) < b − a − kl,
J. Ineq. Pure and Appl. Math. 7(3) Art. 89, 2006
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analogously we obtain
c − d − jl < k (l − 1) < d − c − jl,
thus
(c − d) (l − 1) + (a − b) l
(d − c) (l − 1) + (b − a) l
<k<
2
2
l − (l − 1)
l2 − (l − 1)2
and so
a−b
(d − c) (l − 1) + (b − a) l
−l
l−1
l2 − (l − 1)2 (l − 1)
(b − a)
(d − c) (l − 1) + (b − a) l
+l
.
<j<
l−1
l2 − (l − 1)2 (l − 1)
Convolution Operators with
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Measures on R3 of Polynomial
Type. The Remainder Case
Marta Urciuolo
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We define m0 (ξ) = n (ξ1 , ξ3 ) r (ξ2 , ξ3 ) where n and r belong to C ∞ (R2 − {0}) ,
are homogeneous of degree zero with respect to the isotropic dilations,
supp n ⊂ (ξ1 , ξ3 ) : 2a−1 |ξ3 | < |ξ1 | < 2b+1 |ξ3 |
n ≥ 0 and n ≡ 1 on (ξ1 , ξ3 ) : 2a |ξ3 | < |ξ1 | < 2b |ξ3 | ,
supp r ⊂ (ξ2 , ξ3 ) : 2c−1 |ξ3 | < |ξ2 | < 2d+1 |ξ3 | ,
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r ≥ 0 and r ≡ 1 on (ξ2 , ξ3 ) : 2c |ξ3 | < |ξ2 | < 2d |ξ3 | , so m0 is homogeneous
of degree zero with respect to the isotropic dilations, it belongs to C ∞ on each
octant of R3 , m0 ≥ 0, m0 ≡ 1 on C0 and
f0
supp m0 ⊂ C
= ξ ∈ R3 : 2a−1 |ξ3 | < |ξ1 | < 2b+1 |ξ3 | , 2c−1 |ξ3 | < |ξ2 | < 2d+1 |ξ3 | .
For (j, k) ∈ Z2 , we define mj,k (ξ) = m0 2−j , 2−k ◦ ξ and Qj,k the operator
with multiplier mj,k . If ξ belongs to an open octant of R3 then ξ belongs to
|ξ1 |
2|
2j , 2k ◦ C0 for some (j, k) ∈ Z2 (indeed 2−k ∼ |ξ
and 2−j ∼ |ξ
) and from
|ξ3 |
3|
the previous
lemma,
it
belongs
to
a
finite
number
of
them
(independent
of ξ).
P
So
mj,k (ξ) ≤ c. Now it is easy to check that, for 1 < p < ∞, there
Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
Marta Urciuolo
(j,k)∈Z2
exists Ap > 0 such that for f ∈ L2 ∩ Lp and any choice of εj,k = ±1,
X
≤ Ap kf k .
(2.5)
ε
Q
f
j,k
j,k
p
(j,k)∈Z2
p
Indeed, we now show that
m (ξ) =
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X
εj,k mj,k (ξ)
(j,k)∈Z2
satisfies the hypothesis of the Marcinkiewicz Theorem, as stated in Theorem 6’
in [7].
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We have just observed that
|m (ξ)| ≤
X
mj,k (ξ) ≤ c.
(j,k)∈Z2
Now we want to estimate ∂ξ∂1 m (ξ) . We recall that ∂ξ∂1 m0 is homogeneous
of degree −1. We pick ξ in an open octant. In a small neighborhood of ξ only
finitely many (j, k) ∈ Z2 (independent of ξ) are involved. For each one of them,
∂
∂
mj,k (ξ) = 2−j
m0 2−j ξ1 , 2−k ξ2 , 2−(j+k)l ξ3
∂ξ1
∂ξ1
−1
−1
−j −j
≤ c2 2 ξ1 , 2−k ξ2 , 2−(j+k)l ξ3 ≤ c2−j 2−j ξ1 ,
so
Z
2s+1
sup
ξ2 ,ξ3
2s
∂
∂ξ1 m (ξ) dξ1 ≤ c,
and in a similar way (using the homogeneity of the derivatives of mj,k ) we
obtain that for each 0 < k ≤ 3,
Z ∂k
m (ξ) dξ1 ≤ c,
sup
ξk+1 ,...,ξ3 ρ ∂ξ1 ...∂ξk
k
as ρ ranges over dyadic rectangles of R and that this inequality holds for every
one of the six pemutations of the variables ξ1 , ξ2 , ξ3 .
We now define h (ξ)
∈ C ∞ (R3 ) , h ≥ 0, h ≡ 1 on the unit ball of R3 ,
hj,k (ξ) = h 2−j , 2−k ◦ ξ and Rj,k the operators with multipliers hj,k .
Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
Marta Urciuolo
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Lemma 2.5. There exists a constant C > 0, independent of K, such that
X
Tj,k Rj,k ≤ C.
0≤j≤k≤K
2l+2 , 2l+2
2l+1 2l−1
Proof. Let Kj,k be the kernel of Tj,k Rj,k . A computation shows that,
Kj,k (x) = 2(j+k)l µ(j−k) ∗ hˆ∨
2j , 2k ◦ x .
Thus
X
X
|Kj,k (ξ)| ≤
0≤j≤k≤K
2(j+k)l G(j,k)
2j , 2k ◦ ξ 0≤j≤k
∧
∧
with G(j,k) defined by G(j,k) = µ(j−k) h. Since j − k ≤ 0, as in Lemma 7
∧
in [3] we obtain that G(j,k) ∈ S (R3 ) with each seminorm bounded on j, k,
it follows that the same holds for G(j,k) . Now
X
2(j+k)l G(j,k) 2j , 2k ◦ ξ 0≤j≤k
X
≤
2ja+ka+ha G(j,k,h) 2j ξ1 , 2k ξ2 , 2h ξ3 j,k,h≥0
l
,
l+1
(j,k,h)
(j,k)
Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
Marta Urciuolo
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(j,k,h)
with a =
G
=G
for h = l (j + k) and G
= 0 otherwise. It is
well known that from the uniform boundedness properties of G(j,k,h) it follows
that
X
c
2ja+ka+ha G(j,k,h) 2j ξ1 , 2k ξ2 , 2h ξ3 ≤
a
a
a,
|ξ
1 | |ξ2 | |ξ3 |
j,k,h≥0
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so
X
|Kj,k (ξ)| ≤
0≤j≤k≤K
so
P
c
|ξ1 |
l
l+1
l
l
,
|ξ2 | l+1 |ξ3 | l+1
Tj,k Rj,k convolves Lp (R3 ) into Lq (R3 ) for
0≤j≤k≤K
1
q
1
= p1 − l+1
with bounds
independent of K.
Lemma 2.6. There exists a constant C > 0, independent of K, such that
X
≤ C.
T
(I
−
P
)
(I
−
Q
)
j,k
j,k
j,k 1≤j≤k≤K
2l+2 , 2l+2
2l+1 2l−1
Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
Marta Urciuolo
Proof. The kernel Hj,k of
X
Tj,k (I − Pj,k ) (I − Qj,k )
Contents
1≤j≤k≤K
satisfies
X
1≤j≤k≤K
|Hj,k (ξ)| ≤
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X
2(j+k)l g (j,k)
2j , 2k ◦ ξ 0≤j≤k
∧
∧
with g (j,k) defined by g (j,k) = µ(j−k) (1 − h) (1 − m0 ) .
∧
Observe that, from Lemma 7 in [3], we have µ(j−k) (1 − h) (1 − m0 ) ∈
S(R3 ) with each seminorm bounded on j, k. From this fact the proof follows as
in the previous lemma.
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Proof of the theorem. We have just observed that it is enough to prove (2.1).
Since we can suppose f ≥ 0, by (2.3), we need only check that there exists
C > 0, independent of K such that
X
Tj,k ≤ C,
0≤j≤k≤K
2l+2 , 2l+2
2l+1 2l−1
where
by (2.2). For a constant c0 > 0, we define Q0j,k =
P Tj,k are defined
Qi,k . So Q0j,k have the same properties as Qj,k and Q0j,k ◦ Qj,k = Qj,k
|i−j|≤c0
thus we have that (2.5) holds for Q0j,k . Then, for 1 < p < ∞ and
X
p 2
0
F = {fj,k }j,k≥0 ∈ L l ,
Qj,k fj,k ≤ cp kF kLp (l2 ) .
j,k≥0
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0≤j≤k≤K
=
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p
We decompose
X
Tj,k f
X
Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
X
Tj,k (I − Pj,k ) I − Q0j,k f +
0≤j≤k≤K
Tj,k Pj,k f
0≤j≤k≤K
+
X
Tj,k Q0j,k (I − Pj,k ) f.
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0≤j≤k≤K
Now, proceeding as in [1], the theorem follows from Proposition 2.3, Lemmas
2.5 and 2.6 and the remarks in [8, p. 85] concerning the multiparameter maximal function.
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References
[1] M. CHRIST, Endpoint bounds for singular fractional integral operators,
UCLA, preprint, (1988).
[2] E. FERREYRA, T. GODOY AND M. URCIUOLO, Endpoint bounds for
convolution operators with singular measures, Coll. Math., 76(1) (1998),
35–47.
[3] E. FERREYRA, T. GODOY AND M. URCIUOLO, The type set for homogeneous singular measures on R3 of polynomial type, to appear in Coll.
Math.
[4] I. IKROMOV, M. KEMPE AND D. MÜLLER, Damped oscillatory integrals
and boundedness of maximal operators associated to mixed homogeneous
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Convolution Operators with
Homogeneous Singular
Measures on R3 of Polynomial
Type. The Remainder Case
Marta Urciuolo
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J. Ineq. Pure and Appl. Math. 7(3) Art. 89, 2006
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