Journal of Inequalities in Pure and
Applied Mathematics
Lp IMPROVING PROPERTIES FOR MEASURES ON R4 SUPPORTED
ON HOMOGENEOUS SURFACES IN SOME NON ELLIPTIC CASES
volume 2, issue 3, article 37,
2001.
E. FERREYRA, T. GODOY AND M. URCIUOLO
Facultad de Matematica,
Astronomia y Fisica-CIEM,
Universidad Nacional de Cordoba,
Ciudad Universitaria,
5000 Cordoba, Argentina
EMail: eferrey@mate.uncor.edu
Received 08 January, 2001;
accepted 05 June, 2001.
Communicated by: L. Pick
EMail: godoy@mate.uncor.edu
Abstract
EMail: urciuolo@mate.uncor.edu
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c
2000
School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
052-00
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Abstract
4
In this paper we
R study convolution operators Tµ with measures µ2 in R of the
form µ (E) = B χE (x, ϕ (x)) dx, where B is the unit ballof R , and ϕ is a
homogeneous polynomial function. If inf h∈S 1 det d2x ϕ (h, .) vanishes only on
a finite union of lines,
we prove, under suitable hypothesis, that Tµ is bounded
from Lp into Lq if
region.
1 1
p, q
belongs to a certain explicitly described trapezoidal
2000 Mathematics Subject Classification: 42B20, 42B10.
Key words: Singular measures, Lp −improving, convolution operators.
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
E. Ferreyra, T. Godoy and
M. Urciuolo
Partially supported by Agencia Cordoba Ciencia, Secyt-UNC and Conicet
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
About the Type Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
References
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1.
Introduction
It is well known that a complex measure µ on Rn acts as a convolution operator
on the Lebesgue spaces Lp (Rn ) : µ ∗ Lp ⊂ Lp for 1 ≤ p ≤ ∞. If for some
p there exists q > p such that µ ∗ Lp ⊂ Lq , µ is called Lp − improving. It is
known that singular measures supported on smooth submanifolds of Rn may be
Lp − improving. See, for example, [2], [5], [8], [9], [7] and [4].
Let ϕ1 , ϕ2 be two homogeneous polynomial functions on R2 of degree m ≥
2 and let ϕ = (ϕ1 , ϕ2 ) . Let µ be the Borel measure on R4 given by
Z
(1.1)
µ (E) =
χE (x, ϕ (x)) dx,
B
where B denotes the closed unit ball around the origin in R2 and dx is the
Lebesgue measure on R2 . Let Tµ be the convolution operator given by Tµ f =
µ ∗ f, f ∈ S (R4 ) and let Eµ be the type set corresponding to the measure µ
defined by
1 1
Eµ =
,
: kTµ kp,q < ∞, 1 ≤ p, q ≤ ∞ ,
p q
where kTµ kp,q denotes the operator norm of Tµ from Lp (R4 ) into Lq (R4 ) and
where the Lp spaces are taken with respect to the Lebesgue measure on R4 .
For x, h ∈ R2 , let ϕ00 (x) h be the 2 × 2 matrix whose j − th column is
00
ϕj (x) h, where ϕ00j (x) denotes the Hessian matrix of ϕj at x. Following [3, p.
152], we say that x ∈ R2 is an elliptic point for ϕ if det (ϕ00 (x) h) 6= 0 for
all h ∈ R2 \ {0} . For A ⊂ R2 , we will say that ϕ is strongly elliptic on A if
det (ϕ001 (x) h, ϕ002 (y) h) 6= 0 for all x, y ∈ A and h ∈ R2 \ {0} .
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
E. Ferreyra, T. Godoy and
M. Urciuolo
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If every point x ∈ B\ {0} is elliptic for ϕ, it is proved in [4] that for m ≥ 3,
m
Eµ is the closed trapezoidal region Σm with vertices (0, 0) , (1, 1) , m+1
, m−1
m+1
2
1
and m+1
, m+1
.
Our aim in this paper is to study the case where the set of non elliptic points
consists of a finite union of lines through the origin, L1 , ..., Lk . We assume
from now on, that for x ∈ R2 − {0}, det (ϕ00 (x) h) does not vanish identically,
as a function of h. For each l = 1, 2, ..., k, let πLl and πL⊥l be the orthogonal
projections from R2 onto Ll and L⊥
l respectively. For δ > 0, 1 ≤ l ≤ k, let
n
o
Vδl = x ∈ B : 1/2 ≤ |πLl (x)| ≤ 1 and πL⊥l (x) ≤ δ |πLl (x)| .
It is easy to see (see Lemma 2.1 and Remark 3.2) that for δ small enough, there
exists αl ∈ N and positive constants c and c0 such that
αl
αl
00
0
c πL⊥l (x) ≤ inf1 |det (ϕ (x) h)| ≤ c πL⊥l (x)
h∈S
Vδl .
for all x ∈
Following the approach developed in [3], we prove, in Theorem
3.5, that if α = max1≤l≤k αl and if 7α ≤ m + 1, then the interior of Eµ agrees
with the interior of Σm .
◦
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
E. Ferreyra, T. Godoy and
M. Urciuolo
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◦
Moreover in Theorem 3.6 we obtain that E µ = Σm still holds in some cases
where 7α > m + 1, if we require a suitable hypothesis on the behavior, near the
lines L1 , ..., Lk , of the map (x, y) → inf h∈S 1 |det (ϕ001 (x) h, ϕ002 (y) h)| .
In any case, even though we can not give a complete description of the interior of Eµ , we obtain a polygonal region contained in it.
Throughout the paper c will denote a positive constant not necessarily the
same at each occurrence.
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2.
Preliminaries
Let ϕ1 , ϕ2 : R2 → R be two homogeneous polynomials functions of degree
m ≥ 2 and let ϕ = (ϕ1 , ϕ2 ). For δ > 0 let
1
(2.1)
Vδ = (x1 , x2 ) ∈ B : ≤ |x1 | ≤ 1 and |x2 | ≤ δ |x1 | .
2
We assume in this section that, for some δ0 > 0, the set of the non elliptic points
for ϕ in Vδ0 is contained in the x1 axis.
For x ∈ R2 , let P = P (x) be the symmetric matrix that realizes the
quadratic form h → det (ϕ00 (x) h) , so
(2.2)
E. Ferreyra, T. Godoy and
M. Urciuolo
det (ϕ00 (x) h) = hP (x) h, hi .
Lemma 2.1. There exist δ ∈ (0, δ0 ) , α ∈ N and a real analytic function g =
g (x1 , x2 )on Vδ with g (x1 , 0) 6= 0 for x1 6= 0 such that
(2.3)
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
inf |det (ϕ00 (x) h)| = |x2 |α |g (x)|
|h|=1
for all x ∈ Vδ .
Proof. Since P (x) is real analytic on Vδ and P (x) 6= 0 for x 6= 0, it follows that, for δ small enough, there exists two real analytic functions λ1 (x)
and λ2 (x) wich are the eigenvalues of P (x) . Also, inf |h|=1 |det (ϕ00 (x) h)| =
min {|λ1 (x)| , |λ2 (x)|} for x ∈ Vδ . Since we have assumed that (1, 0) is not an
elliptic point for ϕ and that P (x) 6= 0 for x 6= 0, diminishing δ if necessary, we
can assume that λ1 (1, 0) = 0 and that |λ1 (1, x2 )| ≤ |λ2 (1, x2 )| for |x2 | ≤ δ.
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Since P (x) is homogeneous in x, we have that λ1 (x) and λ2 (x) are homogeneous in x with the same homogeneity degree d. Thus |λ1 (x)| ≤ |λ2 (x)| for
all x ∈ Vδ . Now, λ1 (1, x2 ) = xα2 G (x2 ) for some real
function G =
analytical
d−α
G (x2 ) with G (0) 6= 0 and so λ1 (x1 , x2 ) = xd1 λ1 1, xx21 = x1 xα2 G xx21 .
x2
Taking g (x1 , x2 ) = xd−α
G
the lemma follows.
1
x1
Following [3], for U ⊂ R2 let JU : R2 → R ∪ {∞} given by
JU (h) =
inf
x, x+h∈U
|det (ϕ0 (x + h) − ϕ0 (x))| ,
where the infimum of the empty set is understood to be ∞. We also set, as there,
for 0 < α < 1
Z
U
Rα (f ) (x) = JU (x − y)−1+α f (y) dy.
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
E. Ferreyra, T. Godoy and
M. Urciuolo
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2
For r > 0 and w ∈ R , let Br (w) denotes the open ball centered at w with
radius r.
We have the following
Lemma 2.2. Let w be an elliptic point for ϕ. Then there exist positive constants
c and c0 depending only on kϕ1 kC 3 (B) and kϕ2 kC 3 (B) such that if 0 < r ≤
c inf |h|=1 |det (ϕ00 (w) h)| then
(1) |det (ϕ0 (x + h) − ϕ0 (x))| ≥ |det (ϕ00 (w) h)| if x, x + h ∈ Br (w) .
1
B (w)
(2) R 1 r (f ) ≤ c0 r− 2 kf k 3 , f ∈ S (R4 ) .
1
2
2
6
2
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Proof. Let F (h) = det (ϕ0 (x + h) − ϕ0 (x)) and let djx F denotes the j−th
differential of F at x. Applying the Taylor formula to F (h) around h = 0
and taking into account that F (0) = 0, d0 F (h) = 0 and that d20 F (h, h) ≡
2 det (ϕ00 (x) h) we obtain
0
0
00
Z
1
det (ϕ (x + h) − ϕ (x)) = det (ϕ (x) h) +
0
(1 − t)2 3
dth F (h, h, h) dt.
2
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
Let H (x) = det (ϕ00 (x) h) . The above equation gives
0
0
00
Z
1
det (ϕ (x + h) − ϕ (x)) = det (ϕ (w) h) +
dw+t(x−w) H (h) dt
0
Z 1
(1 − t)2 3
dth F (h, h, h) dt.
+
2
0
|det (ϕ0 (x + h) − ϕ0 (x)) − det (ϕ00 (w) h)| ≤ M |h|3 ≤ 2M r |h|2
1
|det (ϕ (x + h) − ϕ (x))| ≥ |det (ϕ00 (w) h)|
2
0
0
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Then, for x, x + h ∈ Br (w) we have
with M depending only kϕ1 kC 3 (B) and kϕ2 kC 3 (B) . If we choose c ≤
get, for 0 < r < c inf |h|=1 |det (ϕ00 (w) h)| that
E. Ferreyra, T. Godoy and
M. Urciuolo
1
,
4M
we
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and that
JBr (w) (h) ≥
1
1
|det (ϕ00 (w) h)| ≥ r |h|2
2
2c
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1
1
B (w)
Thus R 1 r (f ) ≤ c0 r− 2 kI2 (f )k6 ≤ c0 r− 2 kf k 3 , where Iα denotes the
2
6
2
Riesz potential on R4 , defined as in [10, p. 117]. So the lemma follows from
the Hardy–Littlewood–Sobolev theorem of fractional integration as stated e.g.
in [10, p. 119].
Lemma 2.3. Let w be an elliptic point for ϕ. Then there exists a positive constant c depending only on kϕ1 kC 3 (B) and kϕ2 kC 3 (B) such that if 0 < r ≤
c inf |h|=1 |det (ϕ00 (w) h)| then for all h 6= 0 the map x → ϕ (x + h) − ϕ (x)
is injective on the domain {x ∈ B : x, x + h ∈ Br (w)} .
Proof. Suppose that x, y, x + h and y + h belong to Br (w) and that
ϕ (x + h) − ϕ (x) = ϕ (y + h) − ϕ (y) .
From this equation we get
Z 1
Z
0
0
0=
(ϕ (x + th) − ϕ (y + th)) hdt =
0
0
Now, for z ∈ Br (w),
Z
2
2
dz ϕ − dw ϕ (y − x, h) = 0
1
1
Z
1
d2x+th+s(y−x) ϕ (y − x, h) dsdt.
0
3
dz+u(w−z) ϕ (w − z, y − x, h) du
≤ M r |y − x| |h|
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
E. Ferreyra, T. Godoy and
M. Urciuolo
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then
Z
1Z
1
d2x+th+s(y−x) ϕ (y − x, h) dsdt
0
0
Z 1Z 1
2
2
dx+th+s(y−x) ϕ − d2w ϕ (y − x, h) dsdt.
= dw ϕ (y − x, h) +
0 =
0
0
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So |d2w ϕ (y − x, h)| ≤ M r |y − x| |h| with M depending only on kϕ1 kC 3 (B)
and kϕ2 kC 3 (B) .
On the other hand, w is an elliptic point for ϕ and so, for |u| = 1, the matrix
A := ϕ00 (w) u is invertible. Also A−1 = (det A)−1 Ad (A) , then
−1 A x = |det A|−1 |Ad (A) x| ≤
f
M
|x| ,
|det A|
f depends only on kϕ1 k 2
where M
C (B) and kϕ2 kC 2 (B) . Then, for |v| = 1 and
f. Thus
x = Av, we have |Av| ≥ |det A| /M
2
dw ϕ (y − x, h) ≥ |y − x| |h|
= |y − x| |h|
≥
1
f
MM
inf
2
dw ϕ (u, v)
inf
|hϕ00 (w) u, vi|
|u|=1,|v|=1
|u|=1,|v|=1
1
|y − x| |h| inf |det ϕ00 (w) u| .
f
|u|=1
M
00
If we choose r <
inf |u|=1 | det ϕ (w) u| the above inequality implies x = y
and the lemma is proved.
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
E. Ferreyra, T. Godoy and
M. Urciuolo
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For anyR measurable set A ⊂ B, let µA be the Borel measure defined by
µA (E) = A χE (x, ϕ (x)) dx and let TµA be the convolution operator given by
TµA f = µA ∗ f.
Proposition 2.4. Let w be an elliptic point for ϕ. Then there exist positive
constants c and c0 depending only on kϕ1 kC 3 (B) and kϕ2 kC 3 (B) such that if
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0 < r < c inf |h|=1 |det ϕ00 (w) h| then
1
TµBr (w) f ≤ c0 r− 3 kf k 3 .
2
3
Proof. Taking account of Lemma 2.3, we can proceed as in Theorem 0 in [3] to
obtain, as there, that
3
1
µBr (w) ∗ f ≤ (A1 A2 A3 ) 3 ,
3
where
Z
Aj =
Fj (x)
R2
Y
1≤m≤3,m6=j
RB1 r (w) Fm (x) dx
2
and Fj (x) = kf (x, .)k 3
2
Then the proposition follows from Lemma 2.2 and an application of the triple
Hölder inequality.
For 0 < a < 1 and j ∈ N let
Ua,j = (x1 , x2 ) ∈ B : |x1 | ≥ a, 2−j |x1 | ≤ |x2 | ≤ 2−j+1 |x1 |
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
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M. Urciuolo
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and let Ua,j,i , i = 1, 2, 3, 4 the connected components of Ua,j .
We have
Lemma 2.5. Let 0 < a < 1. Suppose that there exist β ∈ N, j0 ∈ N and
a positive constant c such that |det (ϕ001 (x) h, ϕ002 (y) h)| ≥ c2−jβ |h|2 for all
h ∈ R2 , x, y ∈ Ua,j,i , j ≥ j0 and i = 1, 2, 3, 4. Thus
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(1) For all j ≥ j0 , i = 1, 2, 3, 4 if x and x + h belong to Ua,j,i then
|det (ϕ0 (x + h) − ϕ0 (x))| ≥ c2−jβ |h|2 .
(2) There exists a positive constant c0 such that for all j ≥ j0 , i = 1, 2, 3, 4
jβ
Ua,j,i
R
(f
)
1
≤ c0 2 2 kf k 3 .
2
2
6
Proof. We fix i and j ≥ j0 . For x ∈ Ua,j,i we have
Z 1
0
0
00
det (ϕ (x + h) − ϕ (x)) = det
ϕ (x + sh) hds .
0
For each h ∈ R2 \ {0} we have either det (ϕ001 (x) h, ϕ002 (y) h) > c2−jβ |h|2 for
all x, y ∈ Ua,j,i or det (ϕ001 (x) h, ϕ002 (y) h) < −c2−jβ |h|2 for all
x, y ∈ Ua,j,i .
R t 00
We consider the first case. Let F (t) = det 0 ϕ (x + sh) hds . Then
Z
t
0
00
00
F (t) = det
ϕ1 (x + sh) hds, ϕ2 (x + th) h
0
Z t
00
00
+ det ϕ1 (x + th) h,
ϕ2 (x + sh) hds
0
Z t
=
det (ϕ001 (x + sh) h, ϕ002 (x + th) h) ds
0
Z t
+
det (ϕ001 (x + th) h, ϕ002 (x + sh) h) ds ≥ c2−jβ |h|2 t.
0
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
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Since F (0) = 0 we get F (1) =
R1
0
F 0 (t) dt ≥ c2−jβ |h|2 . Thus
det (ϕ0 (x + h) − ϕ0 (x)) = F (1) ≥ c2−jβ |h|2 .
Then JUa,j,i , (h) ≥ c2−jβ |h|2 , and the lemma follows, as in Lemma 2.2, from
the Hardy–Littlewood–Sobolev theorem of fractional integration. The other
case is similar.
For fixed x(1) , x(2) ∈ R2 , let
x(1) ,x(2)
Ba,j,i
= x ∈ R2 : x − x(1) ∈ Ua,j,i and x − x(2) ∈ Ua,j,i , i = 1, 2, 3, 4.
We have
Lemma 2.6. Let 0 < a < 1 and let x(1) , x(2) ∈ R2 . Suppose that there exist
β ∈ N, j0 ∈ N and a positive constant c such that |det (ϕ001 (x) h, ϕ002 (y) h)| ≥
c2−jβ |h|2 for all h ∈ R2 , x, y ∈ Ua,j,i , j ≥ j0 and i = 1, 2, 3, 4. Then there
exists j1 ∈ N independent of x(1) , x(2) such that for all j ≥ j1 , i = 1, 2, 3, 4 and
all nonnegative f ∈ S (R4 ) it holds that
Z
(1)
(2)
dxdy
f
y
−
ϕ
x
−
x
,
y
−
ϕ
x
−
x
(1) (2)
x
,x
Ba,j,i
×R2
m2
≤
JUa,j,i (x(2) − x(1) )
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
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M. Urciuolo
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Z
f.
R4
Proof. We assert that, if j ≥ j0 then for each (z, w) ∈ R2 ×R2 and i = 1, 2, 3, 4,
the set
n
o
x(1) ,x(2)
(x, y) ∈ Ba,j,i
× R2 : z = y − ϕ x − x(1) and w = y − ϕ x − x(2)
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is a finite set with at most m2 elements. Indeed, if z = y − ϕ x − x(1) and
x(1) ,x(2)
w = y − ϕ x − x(2) with x ∈ Ba,j,i
, Lemma 2.5 says that, for j large
enough,
det ϕ0 x − x(1) − ϕ0 x − x(2) ≥ c2−jβ |h|2 .
Thus the Bezout’s Theorem (See e.g.[1, Lemma 11.5.1, p. 281]) implies that
for each (z, w) ∈ R2 × R2 the set
n
o
x(1) ,x(2)
: ϕ x − x(2) − ϕ x − x(1) = z − w
x ∈ Ba,j,i
is a finite set with at most m2 points. Since x determines y, the assertion follows.
For a fixed η > 0 and for k = (k1 , ..., k4 ) ∈ Z 4 , let
Y
[kn η, (1 + kn ) η] .
Qk =
1≤n≤4
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
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M. Urciuolo
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Let Φk,j,i :
x(1) ,x(2)
Ba,j,i
×R
2
2
2
∩ Qk → R × R be the function defined by
Φk,j,i (x, y) = y − ϕ x − x(1) , y − ϕ x − x(2)
and let Wk,j,i its image. Also
det Φ0k,j,i (x, y) = det ϕ0 x − x(2) − ϕ0 x − x(1) .
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Thus
(2.4)
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J
det Φ0k,j,i (x, y) ≥ JU
x(2) − x(1)
a,j,i
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(1) (2)
x ,x
for (x, y) ∈ Ba,j,i
× R2 ∩ Qk .
(1)
(1)
Since Φk,j,i
(x,
y)
=
Φ
−
ϕ
x
−
x
=
k,j,i (x, y) implies that ϕ x − x
(2)
(2)
, taking into account Lemma 2.1, from Lemma 2.3 it
ϕ x − x −ϕ x − x
e
follows the existence of j ∈ N with e
j independent of x(1) , x(2) such that for j ≥
e
j there exists ηe = ηe (j) > 0 satisfying that for
0 < η < ηe (j) the map Φk,j,i is
(1)
(2)
x ,x
× R2 ∩ Qk its inverse.
injective for all k ∈ Z 4 . Let Ψk,j,i : Wk,j,i → Ba,j,i
2
x(1) ,x(2)
0
−jβ
2
Lemma 2.5 says that det Φk,j,i ≥ c2
|h| on Ba,j,i
× R ∩ Qk . We
have
Z
f y − ϕ x − x(1) , y − ϕ x − x(2) dxdy
(1) (2)
x
,x
Ba,j,i
=
×R2
XZ
k∈Z 4
=
XZ
k∈Z 4
≤
(1)
(2)
x
,x
Ba,j,i
Wk,j,i
×R2 ∩Qk
f (z, w) 1
JUa,j,i (x(2) − x(1) )
2
≤
f y − ϕ x − x(1) , y − ϕ x − x(2) dxdy
m
JUa,j,i (x(2) − x(1) )
1
dzdw
(Ψk,j,i (z, w))
det Φ0k,j,i
Z X
χWk,j,i (v) f (v) dv
R4 k∈Z 4
Z
f
R4
where we have used (2.4).
Proposition 2.7. Let 0 < a < 1. Suppose that there exist β ∈ N, j0 ∈ N
and a positive constant c such that |det (ϕ001 (x) h, ϕ002 (y) h)| ≥ c2−jβ |h|2 for all
Improvement of An Ostrowski
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0
h ∈ R2 , x, y ∈ Ua,j,i , j ≥ j0 , i = 1, 2, 3, 4. Then, there exist j1 ∈ N, c > 0
such that for all j ≥ j1 , f ∈ S (R4 )
0 jβ
TµUa,j f ≤ c 2 3 kf k 3 .
2
3
Proof. For i = 1, 2, 3, 4, let
Ka,j,i =
x, y, x(1) , x(2) , x(3)
∈ R2 × R2 × R2 × R2 × R2 : x − x(s)
∈ Ua,j,i , s = 1, 2, 3 .
We can proceed as in Theorem 0 in [3] to obtain, as there, that
Z
Y
3
µU ∗ f =
f (xj , y − ϕ (x − xj )) dxdydx(1) dx(2) dx(3)
a,j,i
3
Ka,j,i 1≤j≤3
taking into account of Lemma 2.6 and reasoning, with the obvious changes, as
in [3], Theorem 0, we obtain that
1
µU ∗ f 3 ≤ m2 (A1 A2 A3 ) 3
a,j,i
3
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with
Z
Aj =
Fj (x)
R2
Y
1≤m≤3,m6=j
U
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R 1 a,j,i Fm (x) dx
and Fj (x) = kf (x, .)k 3 . Now the proof follows as in Proposition 2.4.
Page 15 of 27
2
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3.
About the Type Set
Proposition 3.1. For δ > 0 let Vδ be defined by (2.1). Suppose that the set of the
non elliptic points for ϕ in Vδ are those lying in the x1 axis and let α be defined
by (2.3).Then EµVδ contains the closed trapezoidal region with vertices (0, 0) ,
2
1
(1, 1) , 7α−1
, 7α−2
, 7α
, 7α
, except perhaps the closed edge parallel to the
7α
7α
principal diagonal.
7α−2
Proof. We first show that (1 − θ) (1, 1) + θ 7α−1
,
∈ EµVδ if 0 ≤ θ < 1.
7α
7α
−j−1
−j+1
≤ |w2 | ≤ 2
. Thus, from Lemmas
If w = (w1 , w2 ) ∈ U 1 ,j then 2
2
2.2, 2.3 and Proposition
2.7,
follows
the
existence
of
j
∈
N and of a positive
0
constant c = c kϕ1 kC 3 (B) , kϕ2 kC 3 (B) such that if rj = c2−jα , then
TµBr
j
0 jα
f
≤
c
2 3 kf k 3
(w)
3
for some c0 > 0 and all j ≥ j0 , w ∈ U 1 ,j , f ∈ S (R4 ) . For 0 ≤ t ≤ 1 let pt , qt
2
1 1
2 1
be defined by pt , qt = t 3 , 3 +(1 − t) (1, 1) . We have also TµBr (w) f ≤
j
1
qt
Since U 1 ,j can be covered with N of such balls Br (w) with N ' 2j(2α−1) we
2
get that
7
Tµ
≤ c2j ( 3 αt−1) .
U 12 ,j pt ,qt
E. Ferreyra, T. Godoy and
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2
πc2 2−2jα kf k1 , thus, the Riesz-Thorin theorem gives
tα
TµBr (w) f ≤ c2j ( 3 −(1−t)2α) kf kpt .
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Let U = ∪j≥j0 U 1 ,j . We have that kTµU kpt ,qt ≤
2
for t <
3
.
7α
Since for t =
◦
3
7α
we have
1
pt
2 ,j
j≥j0 TµU 1
P
= 1−
1
7α
and
1
qt
pt ,qt
= 1−
< ∞,
2
7α
and
since every
3 in [3],
point in Vδ \U is an elliptic point (and so, from Theorem
, 7α−2
∈ EµVδ for
TµVδ \U 3 < ∞), we get that (1 − θ) (1, 1) + θ 7α−1
7α
7α
2
,3
0 ≤ θ < 1. On the other hand, a standard
computation
∨ shows that the adjoint operator Tµ∗V is given by Tµ∗V f = TµVδ (f ∨ ) , where we write, for
δ
δ
g : R4 → C, g ∨ (x) = g (−x) . Thus EµVδ is symmetric with respect to the
nonprincipal diagonal. Finally, after an application of the Riesz-Thorin interpolation theorem, the proposition follows.
Improvement of An Ostrowski
Type Inequality for Monotonic
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E. Ferreyra, T. Godoy and
M. Urciuolo
For δ > 0, let Aδ = {(x1 , x2 ) ∈ B : |x2 | ≤ δ |x1 |}.
Remark 3.1. For s > 0, x = (x1 , ..., x4 ) ∈ R4 we set s•x = (sx1 , sx2 , sm x3 , sm x4 ) .
If E ⊂ R2 , F ⊂ R4 we set sE = {sx : x ∈ E} and s • F = {s • x : x ∈ F } .
For f : R4 → C, s > 0, let fs denotes the function given by fs (x) = f (s • x) .
A computation shows that
(3.1)
Tµ2−j V f 2−j • x = 2−2j TµVδ f2−j (x)
δ
for all f ∈ S (R4 ) , x ∈ R4 .
From this it follows easily that
2(m+1)
2(m+1)
−j
− p +2) Tµ2−j V = 2 ( q
TµVδ δ
p,q
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.
p,q
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This fact implies that
(3.2)
and that if
1
q
>
1
p
1 1
1
1
1
Eµ ⊂
,
: ≥ −
p q
q
p m+1
1
− m+1
then p1 , 1q ∈ EµAδ if and only if p1 , 1q ∈ EµVδ .
Theorem 3.2. Suppose that for some δ > 0 the set of the non elliptic points
for ϕ in Aδ are those lying on the x1 axis and let α be defined by (2.3). Then
EµAδ contains the intersection of the two closed trapezoidal regions with ver
2
7α−1 7α−2
m
1
,
and
(0,
0)
,
(1,
1)
,
,
tices (0, 0) , (1, 1) , m+1
, m−1
,
,
m+1
m+1 m+1
7α
7α
1
2
respectively, except perhaps the closed edge parallel to the diagonal.
,
7α 7α
Moreover, if 7α ≤ m + 1 then the interior of EµAδ is the open trapezoidal
2
m
1
and
.
region with vertices (0, 0) , (1, 1) , m+1
, m−1
,
m+1
m+1 m+1
Proof. Taking into account Proposition 3.1, the theorem follows from the facts
of Remark 3.1.
For 0 < a < 1 and δ > 0 we set Va,δ = {(x1 , x2 ) ∈ B : a ≤ |x1 | ≤ 1 and
|x2 | ≤ δ |x1 |} . We have
Proposition 3.3. Let 0 < a < 1. Suppose that for some 0 < a < 1, j0 , β ∈ N
and some positive constant c we have |det (ϕ001 (x) h, ϕ001 (y) h)| ≥ c2−jβ |h|2
for all h ∈ R2 , x, y ∈ Ua,j,i , j ≥ j0 and i = 1, 2, 3, 4. Then, for δ positive and
small enough, EµVa,δ contains the closed trapezoidal region with vertices (0, 0) ,
β+2 β+1
2
1
(1, 1) , β+3
, β+3 , β+3
, β+3
, except perhaps the closed edge parallel to the
principal diagonal.
Improvement of An Ostrowski
Type Inequality for Monotonic
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M. Urciuolo
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Proof. Proposition 2.7 says that there exist j1 ∈ N and a positive constant c
such that for j ≥ j1 and f ∈ S (R4 )
jβ
TµUa,j,i f ≤ c2 3 kf k 3 .
2
3
Also, for some c > 0 and all f ∈ S (R ) we have TµUa,j,i f ≤ c2−j kf k1 .
1
j (t β3 −(1−t))
Then TµUa,j,i f ≤ c2
kf kpt where pt , qt are defined as in the proof
4
qt
of Proposition 3.1. Let U = ∪j≥j1 Ua,j . Then kTµU f kpt ,qt < ∞ if t <
Now, the proof follows as in Proposition 3.1.
3
.
β+3
Theorem 3.4. Suppose that for some 0 < a < 1, j0 , β ∈ N and for some
positive constant c we have |det (ϕ001 (x) h, ϕ001 (y) h)| ≥ c2−jβ |h|2 for all x, y ∈
Ua,j,i , j ≥ j0 and i = 1, 2, 3, 4. Then for δ positive and small enough, EµAδ contains the intersection of the two closed trapezoidal regions
withvertices
(0, 0),
β+2 β+1
m−1
2
1
2
1
m
,
(1, 1) , m+1 , m+1 , m+1 , m+1 and (0, 0) , (1, 1) , β+3 , β+3 , β+3 , β+3
respectively, except perhaps the closed edge parallel to the diagonal.
Moreover, if β ≤ m − 2 then the interior
of Eµ is the open
trapezoidal region
m
m−1
2
1
with vertices (0, 0) , (1, 1) , m+1 , m+1 and m+1 , m+1 .
Proof. Follows as in Theorem 3.2 using now Proposition 3.3 instead of Proposition 3.1.
Improvement of An Ostrowski
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Remark 3.2. We now turn out to the case when ϕ is a homogeneous polynomial
function whose set of non elliptic points is a finite union of lines through the
origin, L1 ,...,Lk .
Page 19 of 27
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For each l, 1 ≤ l ≤ k, let Alδ = x ∈ R2 : πL⊥l x ≤ δ |πLl x| where πLl
and πL⊥l denote the orthogonal projections from R2 into Ll and L⊥
l respectively.
Thus each Alδ is a closed conical sector around Ll . We choose δ small enough
such that Alδ ∩ Aiδ = ∅ for l 6= i.
It is easy to see that there exists (a unique) αl ∈ N and positive constants c0l ,
00
cl such that
α
α
(3.3)
c0l πL⊥l w l ≤ inf |det (ϕ00 (w) h)| ≤ c00l πL⊥l x l
|h|=1
for all w ∈ Alδ . Indeed, after a rotation the situation reduces to that considered
in Remark 3.1.
Theorem 3.5. Suppose that the set of non elliptic points is a finite union of
lines through the origin, L1 ,...,Lk . For l = 1, 2, ..., k, let αl be defined by (3.3),
and let α = max1≤l≤k αl . Then Eµ contains the intersection
two closed
of the
m
2
m−1
1
trapezoidal regions with vertices (0, 0) , (1, 1) , m+1 , m+1 , m+1 , m+1
and
7α−1 7α−2
2
1
(0, 0) , (1, 1) , 7α , 7α , 7α , 7α , respectively, except perhaps the closed
edge parallel to the diagonal.
Moreover, if 7α ≤ m + 1 then the interior of Eµ is the interior of
the trapem
m−1
2
1
zoidal regions with vertices (0, 0) , (1, 1) , m+1 , m+1 , m+1 , m+1 .
Proof. For l = 1, 2, ..., k, let Alδ be as above. From Theorem 3.2, we obtain that
EµAl contains the intersection of the two closed trapezoidal regions with verδ
7αl −1 7αl −2
m
2
1
tices (0, 0) ,(1, 1) , m+1
, m−1
,
,
and
(0,
0)
,
(1,
1),
,
,
m+1
m+1 m+1
7αl
7αl
2
, 1 respectively, except perhaps the closed edge parallel to the diagonal.
7αl 7αl
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Since every x ∈ B\ ∪l Alδ is an elliptic point for ϕ, Theorem 0 in [3] and a
compactness argument give that kTµD k 3 ,3
<
∞ where D
=
2
1
l
x ∈ B\ ∪l Aδ : 2 ≤ |x| . Then (using the symmetry of EµD , the fact of that
µD is a finite measure and the Riesz-Thorin
theorem) EµD is the closed triangle
with vertices (0, 0) ,(1, 1) , 23 , 13 . Now, proceeding as in the proof of Theo
1
rem 3.2 we get that TµB\∪ Al < ∞ if 1q > p1 − m+1
. Then the first assertion
l
δ
p,q
of the theorem is true. The second one follows also using the facts of Remark
3.1.
For 0 < a < 1, we set
l
Ua,j
= x ∈ R2 : a ≤ |πLl (x)| ≤ 1
Improvement of An Ostrowski
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and 2−j |πLl (x)| ≤ πL⊥l (x) ≤ 2−j+1 |πLl (x)|
l
l
let Ua,j,i
, i = 1, 2, 3, 4 be the connected components of Ua,j
.
Theorem 3.6. Suppose that the set of non elliptic points for ϕ is a finite union
of lines through the origin, L1 ,...,Lk . Let 0 < a < 1 and let j0 ∈ N such that
For l = 1, 2, ..., k, there exists βl ∈ N satisfying |det (ϕ001 (x) h, ϕ001 (y) h)| ≥
l
c2−jβj |h|2 for all x, y ∈ Ua,j,i
, j ≥ j0 and i = 1, 2, 3, 4. Let β = max1≤j≤k βj .
Then Eµ contains the intersection of the two closed trapezoidal regions
with ver
β+1
m
m−1
2
1
tices (0, 0) , (1, 1) , m+1 , m+1 , m+1 , m+1 and (0, 0) , (1, 1) , β+2
,
,
β+3 β+3
2
, 1 , respectively, except perhaps the closed edge parallel to the diagoβ+3 β+3
nal.
Moreover, if β ≤ m − 2 then the interior of Eµ is
the 2interior
of the trapem
1
, m−1
,
,
.
zoidal region with vertices (0, 0) , (1, 1) , m+1
m+1
m+1 m+1
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Proof. Follows as in Theorem 3.5, using now Theorem 3.4 instead of Theorem
3.2.
Example 3.1. ϕ (x1 , x2 ) = (x21 x2 − x1 x22 , x21 x2 + x1 x22 )
It is easy to check that the set of non elliptic points is the union of the coordinate axes. Indeed, for h = (h1 , h2 ) we have det ϕ00 (x1 , x2 ) h = 8x22 h21 +
8x1 x2 h1 h2 + 8x21 h22 and this quadratic form in (h1 , h2 ) has non trivial zeros only
if x1 = 0 or x2 = 0. The associated symmetric matrix to the quadratic form is
8x22 4x1 x2
4x1 x2 8x21
and for x1 6= 0 and |x2 | ≤ δ |x1 | with δ small enough,
its eigenvalue of
p
2
2
4
lower absolute value is λ1 (x1 , x2 ) = 4x1 + 4x2 − 4 (x2 − x21 x22 + x41 ). Thus
λ1 (x1 , x2 ) ' 6x22 for such (x1 , x2 ) . Similarly, for x2 6= 0 and |x1 | ≤ δ |x2 | with
δ small enough, the eigenvalue of lower absolute value is comparable with 6x21 .
Then, in the notation of Theorem 3.5, we obtain α = 2 and so Eµ contains
the
13 6
1 1
closed trapezoidal region with vertices (0, 0) , (1, 1) , 14 , 7 and 7 , 14 except
perhaps the closed edge parallel to the principal diagonal. Observe that, in this
case, Theorem 3.6 does not apply. In fact, for x = (x1 , x2 ) , x
e = (e
x1 , x
e2 ) and
h = (h1 , h2 ) we have
det (ϕ001 (x) h, ϕ002 (e
x) h) = 4h21 (x2 x
e1 − x
e2 x1 + 2x2 x
e2 )
2
+ 4h1 h2 (x1 x
e2 + x
e1 x2 ) + 4h2 (x1 x
e2 − x2 x
e1 + 2x2 x
e1 ) .
Take x1 = x
e1 = 1 and let A = A (x2 , x
e2 ) the matrix of the above quadratic
form in (h1 , h2 ) . For x2 = 2−j , x
e2 = 2−j+1 we have det A < 0 for j large
Improvement of An Ostrowski
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enough but if we take x2 = 2−j+1 and x
e2 = 2−j , we get det A > 0 for j large
enough, so, for all j large enough, det A = 0 for some 2−j ≤ x2 , x
e2 ≤ 2−j+1 .
Thus, for such x2 , x
e2 ,
inf
|(h1 ,h2 )|=1
det (ϕ001 (1, x2 ) (h1 , h2 ) , ϕ002 (1, x
e2 ) (h1 , h2 )) = 0.
◦
Example 3.2. Let us show an example where Theorem 3.6 characterizes E µ .
Let
ϕ (x1 , x2 ) = x31 x2 − 3x1 x32 , 3x21 x22 − x42 .
Improvement of An Ostrowski
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In this case the set of non elliptic points for ϕ is the x1 axis. Indeed,
det (ϕ00 (x1 , x2 ) (h1 , h2 )) = 18 x21 + x22 (h2 x1 + x2 h1 )2 + 2x22 h21 + 6h22 x22 .
E. Ferreyra, T. Godoy and
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In order to apply Theorem 3.6, we consider the quadratic form in h = (h1 , h2 )
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det (ϕ001 (x1 , x2 ) h, ϕ002 (e
x1 , x
e2 ) h) .
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If x = (x1 , x2 ) and x
e = (e
x1 , x
e2 ) , let A = A (x, x
e) its associated symmetric
matrix. An explicit computation of A shows that, for a given 0 < a < 1 and for
all j large enough and i = 1, 2, 3, 4, if x and x
e belong to Ua,j,i , then
a2 ≤ tr (A) ≤ 20
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thus, if λ1 (x, x
e) denotes the eigenvalue of lower absolute value of A (x, x
e) , we
have, for x, x
e ∈ Wa that
Page 23 of 27
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e)| ≤ c2 |det A|
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where c1 , c2 are positive constants independent of j. Now, a computation gives
det A = 324 −x21 x
e21 − 9x22 x
e22 − 12x1 x2 x
e1 x
e2 + 2x21 x
e22
× x22 x
e21 − 2x22 x
e22 − 4x1 x2 x
e1 x
e2 + x21 x
e22 .
Now we write x
e2 = tx2 , with 12 ≤ t ≤ 2. Then
e21 − 9t2 x42 − 12tx1 x22 x
e1 + 2t2 x22 x21
det A = 324x22 −x21 x
2
× x
e1 − 2t2 x22 − 4tx1 x
e1 + t2 x21 .
Note that the the first bracket is negative for x, x
e ∈ Wa if j is large enough.
To study the sign of the second one, we consider the function F (t, x1 , x
e1) =
x
e21 − 4tx1 x
e1 + t2 x21 . Since F has a negative maximum on {1} × {1} × 12 , 2 , it
follows easily that we can choose a such that for x, x
e ∈ Wa and j large enough,
the same assertion holds for the second bracket. So det A is comparable with
2−2j , thus the hypothesis of the Theorem 3.6 are satisfied with β = 2 and such
a. Moreover, we have β = m − 2, then we conclude that the
interior
of Eµ is
the open trapezoidal region with vertices (0, 0) , (1, 1) , 53 , 45 , 25 , 15 .
On the other hand, in a similar way than in Example 3.1 we can see that
2
α = 2 (in fact det A (x, x) = 648 (x21 + 9x22 ) (x21 + x22 ) x22 ), so in this case
◦
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Theorem 3.6 gives a better result (a precise description of E µ ) than that given
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by Theorem 3.5, that assertsonly that Eµ contains the trapezoidal region with
1
vertices (0, 0) , (1, 1) , 13
, 6 and 17 , 14
.
14 7
Example 3.3. The following is an example where Theorem 3.5 characterizes
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E µ . Let
ϕ (x1 , x2 ) = x2 Re (x1 + ix2 )12 , x2 Im (x1 + ix2 )12 .
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A computation gives that for x = (x1 , x2 ) and h = (h1 , h2 )
det (ϕ00 (x) h) = 288 x21 + x22
10
66x22 h21 + 11x1 x2 h1 h2 + x21 + 78x22 h22
and this quadratic form in (h1 , h2 ) does not vanish for h 6= 0 unless x2 = 0.
So the set of non elliptic points for ϕ is the x1 axis. Moreover, its associate
symmetric matrix
11
66x22
x1 x2
2 10
2
2
A = A (x) = 288 x1 + x2
11
x x x21 + 78x22
2 1 2
satisfies c1 ≤ trA (x) ≤ c2 for x ∈ B, 12 ≤ |x1 | , and |x2 | ≤ δ |x1 | , δ > 0 small
enough.
Thus if λ1 = λ1 (x) denotes the eigenvalue of lower absolute value of A (x) ,
we have, for x in this region, that
k1 |det A| ≤ |λ1 | ≤ k2 |det A| ,
where k1 and k2 are positive constants.
20
2
4
Since det A (1, x2 ) = (288)2 (1 + x22 ) 143
x
+
5148x
2
2 , we have that
4
α = 2. So 7α = m + 1 and, from Theorem 3.5, we conclude that the interiorof
1
Eµ is the open trapezoidal region with vertices (0, 0) , (1, 1) , 13
, 6 , 17 , 14
.
14 7
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
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M. Urciuolo
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References
[1] J. BOCHNAK, M. COSTE
Springer, 1998.
AND
M. F. ROY, Real Algebraic Geometry,
[2] M. CHRIST. Endpoint bounds for singular fractional integral operators,
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[3] S. W. DRURY AND K. GUO, Convolution estimates related to surfaces
of half the ambient dimension. Math. Proc. Camb. Phil. Soc., 110 (1991),
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[4] E. FERREYRA, T. GODOY AND M. URICUOLO. The type set for some
measures on R2n with n dimensional support, Czech. Math. J., (to appear).
[5] A. IOSEVICH AND E. SAWYER, Sharp Lp − Lq estimates for a class of
averaging operators, Ann Inst. Fourier, 46(5) (1996), 359–1384.
[6] T. KATO, Perturbation Theory for Linear Operators, Second edition.
Springer Verlag, Berlin Heidelberg- New York, 1976.
[7] D. OBERLIN, Convolution estimates for some measures on curves, Proc.
Amer. Math. Soc., 99(1) (1987), 56–60.
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
E. Ferreyra, T. Godoy and
M. Urciuolo
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[8] F. RICCI, Limitatezza Lp − Lq per operatori di convoluzione definiti da
misure singolari in Rn , Bollettino U.M.I., (77) 11-A (1997), 237–252.
Close
[9] F. RICCI AND E. M. STEIN, Harmonic analysis on nilpotent groups and
singular integrals III, fractional integration along manifolds, J. Funct.
Anal., 86 (1989), 360–389.
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[10] E. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970.
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
E. Ferreyra, T. Godoy and
M. Urciuolo
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