Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 2, Issue 3, Article 37, 2001
Lp −IMPROVING PROPERTIES FOR MEASURES ON R4 SUPPORTED ON
HOMOGENEOUS SURFACES IN SOME NON ELLIPTIC CASES
E. FERREYRA, T. GODOY, AND M. URCIUOLO
FACULTAD DE M ATEMATICA , A STRONOMIA Y F ISICA -CIEM, U NIVERSIDAD NACIONAL DE C ORDOBA ,
C IUDAD U NIVERSITARIA , 5000 C ORDOBA , A RGENTINA
eferrey@mate.uncor.edu
godoy@mate.uncor.edu
urciuolo@mate.uncor.edu
Received 8 January, 2001; accepted 5 June, 2001
Communicated by L. Pick
A BSTRACT. In Rthis paper we study convolution operators Tµ with measures µ in R4 of the
form µ (E) = B χE (x, ϕ (x)) dx,
B isthe unit ball of R2 , and ϕ is a homogeneous
where
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
certain explicitly described trapezoidal region.
1 1
p, q
belongs to a
Key words and phrases: Singular measures, Lp −improving, Convolution Ooperators.
2000 Mathematics Subject Classification. 42B20, 42B10.
1. I NTRODUCTION
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
ISSN (electronic): 1443-5756
c 2001 Victoria University. All rights reserved.
Partially supported by Agencia Cordoba Ciencia, Secyt-UNC and Conicet.
052-00
2
E. F ERREYRA , T. G ODOY ,
AND
M. U RCIUOLO
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 ϕ00j (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} .
If every point x ∈ B\ {0} is elliptic for ϕ, it is proved in [4] that for m ≥ 3,Eµ is the closed
m
2
1
trapezoidal region Σm with vertices (0, 0) , (1, 1) , m+1
, m−1
and m+1
, 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
l
Vδ = 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 3.6) that for δ small enough, there exists αl ∈ N
and positive constants c and c0 such that
αl
αl
c πL⊥l (x) ≤ inf1 |det (ϕ00 (x) h)| ≤ c0 πL⊥l (x)
h∈S
for all x ∈ Vδl . Following the approach developed in [3], we prove, in Theorem 3.7, that if
α = max1≤l≤k αl and if 7α ≤ m + 1, then the interior of Eµ agrees with the interior of Σm .
◦
◦
Moreover in Theorem 3.8 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.
2. P RELIMINARIES
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)
det (ϕ00 (x) h) = hP (x) h, hi .
J. Inequal. Pure and Appl. Math., 2(3) Art. 37, 2001
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Lp
IMPROVING PROPERTIES FOR SOME MEASURES ON
R4
3
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
inf |det (ϕ00 (x) h)| = |x2 |α |g (x)|
(2.3)
|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.
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 (x
2 ) for
some real analytical
function G = G (x2 ) with G (0) 6= 0 and so λ1 (x1 , x2 ) =
d−α α
x2
d
G xx21 the lemma follows.
x1 λ1 1, x1 = x1 x2 G xx21 . Taking g (x1 , x2 ) = xd−α
1
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.
For r > 0 and w ∈ R2 , 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))| ≥ 21 |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 ) .
2
2
6
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
Z 1
(1 − t)2 3
0
0
00
det (ϕ (x + h) − ϕ (x)) = det (ϕ (x) h) +
dth F (h, h, h) dt.
2
0
Let H (x) = det (ϕ00 (x) h) . The above equation gives
0
0
00
Z
det (ϕ (x + h) − ϕ (x)) = det (ϕ (w) h) +
1
dw+t(x−w) H (h) dt
0
Z
+
0
1
(1 − t)2 3
dth F (h, h, h) dt.
2
Then, for x, x + h ∈ Br (w) we have
|det (ϕ0 (x + h) − ϕ0 (x)) − det (ϕ00 (w) h)| ≤ M |h|3 ≤ 2M r |h|2
J. Inequal. Pure and Appl. Math., 2(3) Art. 37, 2001
http://jipam.vu.edu.au/
4
E. F ERREYRA , T. G ODOY ,
AND
M. U RCIUOLO
with M depending only kϕ1 kC 3 (B) and kϕ2 kC 3 (B) . If we choose c ≤
c inf |h|=1 |det (ϕ00 (w) h)| that
|det (ϕ0 (x + h) − ϕ0 (x))| ≥
and that
JBr (w) (h) ≥
1
,
4M
we get, for 0 < r <
1
|det (ϕ00 (w) h)|
2
1
1
|det (ϕ00 (w) h)| ≥ r |h|2
2
2c
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 Riesz potential
2
6
2
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
1
0
Z
1
d2x+th+s(y−x) ϕ (y − x, h) dsdt.
0
Now, for z ∈ Br (w),
Z
2
dz ϕ − d2w ϕ (y − x, h) = 1
d3z+u(w−z) ϕ (w
0
− z, y − x, h) du
≤ M r |y − x| |h|
then
Z
1
1
Z
d2x+th+s(y−x) ϕ (y − x, h) dsdt
Z 1Z 1
2
2
= dw ϕ (y − x, h) +
dx+th+s(y−x) ϕ − d2w ϕ (y − x, h) dsdt.
0 =
0
0
0
0
|d2w ϕ (y
So
− 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 and kϕ2 k 2 . Then, for |v| = 1 and x = Av, we have
where M
C (B)
C (B)
f
|Av| ≥ |det A| /M . Thus
2
dw ϕ (y − x, h) ≥ |y − x| |h| inf d2w ϕ (u, v)
|u|=1,|v|=1
= |y − x| |h|
≥
J. Inequal. Pure and Appl. Math., 2(3) Art. 37, 2001
inf
|u|=1,|v|=1
|hϕ00 (w) u, vi|
1
|y − x| |h| inf |det ϕ00 (w) u| .
f
|u|=1
M
http://jipam.vu.edu.au/
Lp
If we choose r <
is proved.
1
f
MM
IMPROVING PROPERTIES FOR SOME MEASURES ON
R4
5
inf |u|=1 | det ϕ00 (w) u| the above inequality implies x = y and the lemma
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 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
RB1 r (w) Fm (x) dx
2
1≤m≤3,m6=j
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 |
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
(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
6
2
Proof. We fix i and j ≥ j0 . For x ∈ Ua,j,i we have
0
0
Z
det (ϕ (x + h) − ϕ (x)) = det
1
00
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 . We consider the first case. Let
J. Inequal. Pure and Appl. Math., 2(3) Art. 37, 2001
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6
E. F ERREYRA , T. G ODOY ,
F (t) = det
R
AND
M. U RCIUOLO
ϕ00 (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.
t
0
0
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
Z
m2
(1)
(2)
f y−ϕ x−x
f.
,y − ϕ x − x
dxdy ≤
x(1) ,x(2)
JUa,j,i (x(2) − x(1) ) R4
Ba,j,i
×R2
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)
is a finite set with at most m2 elements. Indeed, if z = y−ϕ x − x(1) and w = y−ϕ x − x(2)
(1)
(2)
x ,x
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)
x ∈ Ba,j,i
: ϕ x − x(2) − ϕ x − x(1) = z − w
is a finite set with at most m2 points. Since x determines y, the assertion
follows.
Q
4
For a fixed η > 0 and for k = (k1 , ..., k4 ) ∈ Z , let Qk = 1≤n≤4 [kn η, (1 + kn ) η] . Let
(1) (2)
x ,x
2
Φk,j,i : Ba,j,i
× R ∩ Qk → R2 × R2 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) . Thus
det Φ0k,j,i (x, y) ≥ JU
(2.4)
x(2) − x(1)
a,j,i
J. Inequal. Pure and Appl. Math., 2(3) Art. 37, 2001
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Lp
IMPROVING PROPERTIES FOR SOME MEASURES ON
R4
7
(1) (2)
x ,x
for (x, y) ∈ Ba,j,i
× R2 ∩ Qk .
(2)
(1)
(1)
−
ϕ
x
−
x
=
ϕ
x
−
x
−
Since Φk,j,i
(x,
y)
=
Φ
k,j,i (x, y) implies that ϕ x − x
(2)
ϕ x−x
, taking into account Lemma 2.1, from Lemma 2.3 it follows the existence of
e
j ∈ N with e
j independent of x(1) , x(2) such that for j ≥ e
j there exists ηe = ηe (j) > 0
satisfyingthat for 0 < η < ηe (j) the map Φk,j,i is injective for all k ∈ Z 4 . Let Ψk,j,i :
x(1) ,x(2)
Wk,j,i → Ba,j,i
× R2 ∩ Qk its inverse. Lemma 2.5 says that det Φ0k,j,i ≥ c2−jβ |h|2 on
(1) (2)
x ,x
Ba,j,i
× R2 ∩ Qk . We have
Z
x(1) ,x(2)
Ba,j,i
×R2
f y − ϕ x − x(1) , y − ϕ x − x(2) dxdy
=
XZ
k∈Z 4
=
≤
x(1) ,x(2)
×R2
Ba,j,i
XZ
f y − ϕ x − x(1) , y − ϕ x − x(2) dxdy
∩Qk
f (z, w) Φ0k,j,i
det
Z X
k∈Z 4
Wk,j,i
JUa,j,i
(x(2)
1
− x(1) )
m2
≤
JUa,j,i (x(2) − x(1) )
1
dzdw
(Ψk,j,i (z, w))
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 h ∈ R2 , x, y ∈ Ua,j,i , j ≥ j0 ,
0
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
µU ∗ f 3 =
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
with
Z
Aj =
Fj (x)
R2
Y
1≤m≤3,m6=j
U
R 1 a,j,i Fm (x) dx
2
and Fj (x) = kf (x, .)k 3 . Now the proof follows as in Proposition 2.4.
2
J. Inequal. Pure and Appl. Math., 2(3) Art. 37, 2001
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8
E. F ERREYRA , T. G ODOY ,
3. A BOUT
AND
M. U RCIUOLO
T YPE S ET
THE
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 con
2
1
tains the closed trapezoidal region with vertices (0, 0) , (1, 1) , 7α−1
, 7α−2
, 7α
, 7α
, except
7α
7α
perhaps the closed edge parallel to the 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
If w = (w1 , w2 ) ∈ U 1 ,j then 2
≤ |w2 | ≤ 2
. Thus, from Lemmas 2.2, 2.3 and Propo2
sition 2.7, follows the existence of j0 ∈ N and of a positive constant c = c kϕ1 kC 3 (B) , kϕ2 kC 3 (B)
such that if rj = c2−jα , then
TµBr
j
jα
f
≤ c0 2 3 kf k 3
(w)
2
3
0
4
for some c > 0 and all j ≥ j0 , w ∈ U 1 ,j , f ∈ S (R ) . For 0 ≤ t ≤ 1 let pt , qt be defined
2
1 1
2 1
by pt , qt = t 3 , 3 + (1 − t) (1, 1) . We have also TµBr (w) f ≤ πc2 2−2jα kf k1 , thus, the
j
1
Riesz-Thorin theorem gives
tα
TµBr (w) f ≤ c2j ( 3 −(1−t)2α) kf kpt .
qt
Since U 1 ,j can be covered with N of such balls Br (w) with N ' 2j(2α−1) we get that
2
j ( 73 αt−1)
Tµ
≤
c2
.
U
1 2 ,j
pt ,qt
Let U = ∪j≥j0 U 1 ,j . We have that kTµU kpt ,qt ≤
2
we have
1
pt
< ∞, for t <
pt ,qt
3
.
7α
Since for
◦
2
and q1t = 1 − 7α
and since every point in Vδ \U is an elliptic point
(and so, from Theorem 3 in [3], TµVδ \U 3 < ∞), we get that (1 − θ) (1, 1)+θ 7α−1
, 7α−2
∈
7α
7α
t=
3
7α
2 ,j
j≥j0 TµU 1
P
=1−
1
7α
2
,3
EµVδ for 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 g : R4 → C, g ∨ (x) = g (−x) .
δ
δ
Thus EµVδ is symmetric with respect to the nonprincipal diagonal. Finally, after an application
of the Riesz-Thorin interpolation theorem, the proposition follows.
For δ > 0, let Aδ = {(x1 , x2 ) ∈ B : |x2 | ≤ δ |x1 |}.
Remark 3.2. 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
4
4
for all f ∈ S (R ) , x ∈ R .
From this it follows easily that
Tµ2−j V δ
= 2−j (
2(m+1)
2(m+1)
− p +2
q
p,q
)
T
µVδ .
p,q
This fact implies that
(3.2)
Eµ ⊂
J. Inequal. Pure and Appl. Math., 2(3) Art. 37, 2001
1 1
,
p q
1
1
1
: ≥ −
q
p m+1
http://jipam.vu.edu.au/
Lp
and that if
1
q
>
1
p
−
1
m+1
then
IMPROVING PROPERTIES FOR SOME MEASURES ON
1 1
,
p q
∈ EµAδ if and only if
1 1
,
p q
R4
9
∈ EµVδ .
Theorem 3.3. 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
2
m
1
the two closed trapezoidal regions with vertices (0, 0) , (1, 1) , m+1
,
and
, m−1
,
m+1
m+1 m+1
2
1
7α−1 7α−2
(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 EµAδ is the open trapezoidal region with vertices
2
m
1
(0, 0) , (1, 1) , m+1
, m−1
and
,
.
m+1
m+1 m+1
Proof. Taking into account Proposition 3.1, the theorem follows from the facts of Remark 3.2.
For 0 < a < 1 and δ > 0 we set Va,δ = {(x1 , x2 ) ∈ B : a ≤ |x1 | ≤ 1 and |x2 | ≤ δ |x1 |} . We
have
Proposition 3.4. 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
β+1
2
1
,
,
,
, except perhaps the closed edge
region with vertices (0, 0) , (1, 1) , β+2
β+3 β+3
β+3 β+3
parallel to the principal diagonal.
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 .
3 2
Also, for some c > 0 and all f ∈ S (R4 ) we have TµUa,j,i f ≤ c2−j kf k1 . Then TµUa,j,i f ≤
1
qt
j (t β3 −(1−t))
kf kpt where pt , qt are defined as in the proof of Proposition 3.1. Let U = ∪j≥j1 Ua,j .
c2
3
Then kTµU f kpt ,qt < ∞ if t < β+3
. Now, the proof follows as in Proposition 3.1.
Theorem 3.5. 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
β+2 β+1
m
m−1
2
1
regions with vertices (0, 0) , (1, 1) , m+1 , m+1 , m+1 , m+1 and (0, 0) , (1, 1) , β+3 , β+3 ,
1
2
,
, respectively, except perhaps the closed edge parallel to the diagonal.
β+3 β+3
Moreover, if β ≤ m −2 then the interior
of Eµ is the open trapezoidal region with vertices
m
m−1
2
1
(0, 0) , (1, 1) , m+1 , m+1 and m+1 , m+1 .
Proof. Follows as in Theorem 3.3 using now Proposition 3.4 instead of Proposition 3.1.
Remark 3.6. 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
the origin,
through
L1 ,...,Lk .
For each l, 1 ≤ l ≤ k, let Alδ = x ∈ R2 : πL⊥l x ≤ δ |πLl x| where πLl and πL⊥l denote the
l
orthogonal projections from R2 into Ll and L⊥
l respectively. Thus each Aδ 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 , c00l such that
α
α
(3.3)
c0l πL⊥ w l ≤ inf |det (ϕ00 (w) h)| ≤ c00l πL⊥ x l
l
|h|=1
l
for all w ∈ Alδ . Indeed, after a rotation the situation reduces to that considered in Remark 3.2.
J. Inequal. Pure and Appl. Math., 2(3) Art. 37, 2001
http://jipam.vu.edu.au/
10
E. F ERREYRA , T. G ODOY ,
AND
M. U RCIUOLO
Theorem 3.7. 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
trapezoidal regions with
the 2intersection
of the two closed 7α−1
vertices (0, 0) , (1, 1) ,
m
2
m−1
1
7α−2
1
,
, m+1 , m+1 and (0, 0) , (1, 1) , 7α , 7α , 7α , 7α , respectively, except perm+1 m+1
haps the closed edge parallel to the diagonal.
Moreover, if 7α ≤ m + 1 then the
of E
interior
µ is the interior of the trapezoidal regions with
2
1
m
vertices (0, 0) , (1, 1) , m+1
, m−1
,
,
.
m+1
m+1 m+1
Proof. For l = 1, 2, ..., k, let Alδ be as above. From Theorem 3.3, we obtain that EµAl contains
δ
m
the intersection of the two closed trapezoidal regions with vertices (0, 0) ,(1, 1) , m+1
,
, m−1
m+1
2
, 1
and (0, 0) , (1, 1), 7α7αl −1
, 7α7αl −2
, 7α2 l , 7α1 l respectively, except perhaps the
m+1 m+1
l
l
closed edge parallel to the diagonal.
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 = x ∈ B\ ∪l Alδ : 12 ≤ |x| . Then (using the
2
symmetry of EµD , the fact of that µD is a finite measure
and the Riesz-Thorin theorem) EµD
2 1
is the closed triangle with vertices (0, 0) , (1, 1) , 3 , 3 . Now, proceeding as in the proof of
1
Theorem 3.3 we get that TµB\∪ Al < ∞ if 1q > p1 − m+1
. Then the first assertion of the
l
δ
p,q
theorem is true. The second one follows also using the facts of Remark 3.2.
For 0 < a < 1, we set
l
Ua,j
= x ∈ R2 : a ≤ |πLl (x)| ≤ 1 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 3.8. 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)| ≥ c2−jβj |h|2 for
l
all x, y ∈ Ua,j,i
, j ≥ j0 and i = 1, 2, 3, 4. Let β = max1≤j≤k βj . Then Eµ contains the intersec
m
2
1
tion of the two closed trapezoidal regions with vertices (0, 0) , (1, 1) , m+1
, m−1
, m+1
, m+1
m+1
β+2 β+1
2
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
of E
interior
µ is the interior of the trapezoidal region with
m
m−1
2
1
vertices (0, 0) , (1, 1) , m+1 , m+1 , m+1 , m+1 .
Proof. Follows as in Theorem 3.7, using now Theorem 3.5 instead of Theorem 3.3.
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 |xp
1 | with δ small enough, its eigenvalue of lower absolute value
2
2
is λ1 (x1 , x2 ) = 4x1 + 4x2 − 4 (x42 − 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 3.7, we obtain
α = 2 and so
13 6
1
Eµ contains the closed trapezoidal region with vertices (0, 0) , (1, 1) , 14 , 7 and 17 , 14
except
J. Inequal. Pure and Appl. Math., 2(3) Art. 37, 2001
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Lp
IMPROVING PROPERTIES FOR SOME MEASURES ON
R4
11
perhaps the closed edge parallel to the principal diagonal. Observe that, in this case, Theorem
3.8 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 ) + 4h1 h2 (x1 x
e2 + x
e1 x2 ) + 4h22 (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 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 3.8 characterizes E µ . Let
ϕ (x1 , x2 ) = x31 x2 − 3x1 x32 , 3x21 x22 − x42 .
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 .
In order to apply Theorem 3.8, we consider the quadratic form in h = (h1 , h2 )
det (ϕ001 (x1 , x2 ) h, ϕ002 (e
x1 , x
e2 ) h) .
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
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
c1 |det A| ≤ |λ1 (x, x
e)| ≤ c2 |det A|
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 21 ≤ t ≤ 2. Then
2
det A = 324x22 −x21 x
e21 − 9t2 x42 − 12tx1 x22 x
e1 + 2t2 x22 x21 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
1 a negative maximum on {1} × {1} × 2 , 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 3.8 are satisfied with β = 2 and
such a. Moreover, we have β = m − 2, then we conclude
2 1 that the interior of Eµ is the open
3 4
trapezoidal region with vertices (0, 0) , (1, 1) , 5 , 5 , 5 , 5 .
On the other hand, in a similar way than in Example 3.1 we can see that α = 2 (in fact
2
det A (x, x) = 648 (x21 + 9x22 ) (x21 + x22 ) x22 ), so in this case Theorem 3.8 gives a better result
◦
◦
(a precise description of E µ ) than that given by Theorem 3.7, that asserts
only that E µ contains
1 1
13 6
the trapezoidal region with vertices (0, 0) , (1, 1) , 14 , 7 and 7 , 14 .
J. Inequal. Pure and Appl. Math., 2(3) Art. 37, 2001
http://jipam.vu.edu.au/
12
E. F ERREYRA , T. G ODOY ,
AND
M. U RCIUOLO
◦
Example 3.3. The following is an example where Theorem 3.7 characterizes E µ . Let
ϕ (x1 , x2 ) = x2 Re (x1 + ix2 )12 , x2 Im (x1 + ix2 )12 .
A computation gives that for x = (x1 , x2 ) and h = (h1 , h2 )
10
det (ϕ00 (x) h) = 288 x21 + x22
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
x1 x2
66x22
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
Since det A (1, x2 ) = (288)2 (1 + x22 ) 143
x22 + 5148x42 , we have that α = 2. So 7α =
4
m + 1 and, from Theorem 3.7, we conclude that the interior of Eµ is the open trapezoidal region
13 6
1
with vertices (0, 0) , (1, 1) , 14
, 7 , 17 , 14
.
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J. Inequal. Pure and Appl. Math., 2(3) Art. 37, 2001
http://jipam.vu.edu.au/