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volume 2, issue 3, article 37, 2001.

Received 08 January, 2001;

accepted 05 June, 2001.

Communicated by:L. Pick

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Journal of Inequalities in Pure and Applied Mathematics

L^p IMPROVING PROPERTIES FOR MEASURES ONR⁴ SUPPORTED ON HOMOGENEOUS SURFACES IN SOME NON ELLIPTIC CASES

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 EMail:godoy@mate.uncor.edu EMail:urciuolo@mate.uncor.edu

c

2000School of Communications and Informatics,Victoria University of Technology ISSN (electronic): 1443-5756

052-00

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Improvement of An Ostrowski Type Inequality for Monotonic Mappings and Its Application

for Some Special Means

E. Ferreyra,T. Godoyand M. Urciuolo

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J. Ineq. Pure and Appl. Math. 2(3) Art. 37, 2001

http://jipam.vu.edu.au

Abstract

In this paper we study convolution operatorsTµwith measuresµinR⁴of the form µ(E) = R

Bχ_E(x, ϕ(x))dx, whereB is the unit ball of R², andϕis a homogeneous polynomial function. Ifinf_h∈S1

det d²_xϕ(h, .)

vanishes only on a finite union of lines, we prove, under suitable hypothesis, thatTµis bounded fromL^p intoL^qif

1 p,¹_q

belongs to a certain explicitly described trapezoidal region.

2000 Mathematics Subject Classification:42B20, 42B10.

Key words: Singular measures,L^p−improving, convolution operators.

Partially supported by Agencia Cordoba Ciencia, Secyt-UNC and Conicet

1. Introduction

It is well known that a complex measureµonRⁿacts as a convolution operator on the Lebesgue spaces L^p(Rⁿ) : µ∗L^p ⊂ L^p for 1 ≤ p ≤ ∞.If for some p there exists q > psuch that µ∗L^p ⊂ L^q, µis called L^p− improving. It is known that singular measures supported on smooth submanifolds ofRⁿmay be L^p−improving. See, for example, [2], [5], [8], [9], [7] and [4].

Letϕ₁, ϕ₂ be two homogeneous polynomial functions onR² of degreem≥ 2and letϕ = (ϕ₁, ϕ₂).Letµbe the Borel measure onR⁴given by

(1.1) µ(E) =

Z

B

χ_E(x, ϕ(x))dx,

where B denotes the closed unit ball around the origin in R² and dx is the Lebesgue measure on R².LetT_µ be the convolution operator given byT_µf = µ∗f, f ∈ S(R⁴)and letEµ be the type set corresponding to the measure µ defined by

Eµ= 1

p,1 q

:kTµk_p,q <∞,1≤p, q ≤ ∞

,

wherekT_µk_p,q denotes the operator norm ofT_µ fromL^p(R⁴)intoL^q(R⁴)and where theL^p spaces are taken with respect to the Lebesgue measure onR⁴.

For x, h ∈ R², let ϕ⁰⁰(x)h be the 2 ×2 matrix whose j −th column is ϕ⁰⁰_j (x)h,whereϕ⁰⁰_j(x)denotes the Hessian matrix ofϕj atx.Following [3, p.

152], we say that x ∈ R² is an elliptic point for ϕ if det (ϕ⁰⁰(x)h) 6= 0for all h ∈ R²\ {0}.For A ⊂ R², we will say that ϕ is strongly elliptic on Aif det (ϕ⁰⁰₁(x)h, ϕ⁰⁰₂(y)h)6= 0for allx, y ∈Aandh∈R²\ {0}.

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If every pointx∈B\ {0}is elliptic forϕ,it is proved in [4] that form ≥3, Eµis the closed trapezoidal regionΣmwith vertices(0,0),(1,1), _m+1^m ,^m−1_m+1 and _m+1² ,_m+1¹

.

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, ..., L_k. We assume from now on, that forx∈R²− {0},det (ϕ⁰⁰(x)h)does not vanish identically, as a function of h. For eachl = 1,2, ..., k,let π_L_l and π_L^⊥

l be the orthogonal projections fromR²ontoL_landL^⊥_l respectively. Forδ >0,1≤l ≤k,let

V_δ^l =n

x∈B : 1/2≤ |π_L_l(x)| ≤1and π_L^⊥

l (x)

≤δ|π_L_l(x)|o . It is easy to see (see Lemma2.1and Remark3.2) that forδsmall enough, there existsα_l∈Nand positive constantscandc⁰such that

c π_L^⊥

l (x)

αl

≤ inf

h∈S¹|det (ϕ⁰⁰(x)h)| ≤c⁰ π_L^⊥

l (x)

αl

for allx∈V_δ^l. Following the approach developed in [3], we prove, in Theorem 3.5, that ifα = max1≤l≤kα_land if7α ≤m+ 1,then the interior ofE_µ agrees with the interior ofΣm.

Moreover in Theorem3.6we obtain that

◦

Eµ =

◦

Σm still holds in some cases where7α > m+ 1,if we require a suitable hypothesis on the behavior, near the linesL₁, ..., L_k,of the map(x, y)→infh∈S¹|det (ϕ⁰⁰₁(x)h, ϕ⁰⁰₂(y)h)|.

In any case, even though we can not give a complete description of the interior ofE_µ, we obtain a polygonal region contained in it.

Throughout the paperc 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 ≥2and letϕ= (ϕ₁, ϕ₂). Forδ >0let

(2.1) V_δ =

(x₁, x₂)∈B : 1

2 ≤ |x₁| ≤1and |x₂| ≤δ|x₁|

.

We assume in this section that, for someδ₀ >0,the set of the non elliptic points forϕinV_δ₀ is contained in thex₁ axis.

For x ∈ R², let P = P(x) be the symmetric matrix that realizes the quadratic formh→det (ϕ⁰⁰(x)h),so

(2.2) det (ϕ⁰⁰(x)h) =hP (x)h, hi.

Lemma 2.1. There exist δ ∈ (0, δ₀), α ∈ N and a real analytic functiong = g(x₁, x₂)onV_δwithg(x₁,0)6= 0forx₁ 6= 0such that

(2.3) inf

|h|=1|det (ϕ⁰⁰(x)h)|=|x₂|^α|g(x)|

for allx∈V_δ.

Proof. Since P(x) is real analytic on V_δ 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 ofP (x).Also, inf|h|=1|det (ϕ⁰⁰(x)h)| = min{|λ₁(x)|,|λ₂(x)|}forx∈V_δ.Since we have assumed that(1,0)is not an elliptic point forϕand thatP(x)6= 0forx6= 0,diminishingδif necessary, we can assume that λ1(1,0) = 0and that |λ1(1, x2)| ≤ |λ2(1, x2)|for |x2| ≤ δ.

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SinceP (x)is homogeneous inx,we have that λ₁(x)andλ₂(x)are homogeneous in x with the same homogeneity degreed. Thus|λ1(x)| ≤ |λ2(x)|for allx ∈ V_δ.Now,λ₁(1, x₂) =x^α₂G(x₂)for some real analytical functionG = G(x₂) withG(0) 6= 0 and so λ₁(x₁, x₂) = x^d₁λ₁

1,^x_x²

1

= x^d−α₁ x^α₂G

x2

x1

. Takingg(x1, x2) = x^d−α₁ G

x2

x1

the lemma follows.

Following [3], forU ⊂R² letJ_U :R² →R∪ {∞}given by J_U(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, for0< α <1

R^U_α(f) (x) = Z

JU(x−y)^−1+αf(y)dy.

Forr > 0andw ∈ R²,letB_r(w)denotes the open ball centered atw with radiusr.

We have the following

Lemma 2.2. Letwbe an elliptic point forϕ.Then there exist positive constants c and c⁰ depending only on kϕ₁k_C3(B) and kϕ₂k_C3(B) such that if 0 < r ≤ cinf|h|=1|det (ϕ⁰⁰(w)h)|then

(1) |det (ϕ⁰(x+h)−ϕ⁰(x))| ≥ ¹₂|det (ϕ⁰⁰(w)h)|ifx, x+h∈B_r(w). (2)

R^B1^r^(w)

2

(f) 6

≤c⁰r⁻¹² kfk3

2 , f ∈S(R⁴).

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Proof. Let F (h) = det (ϕ⁰(x+h)−ϕ⁰(x)) and let d^j_xF 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, d₀F (h) = 0 and that d²₀F(h, h) ≡ 2 det (ϕ⁰⁰(x)h)we obtain

det (ϕ⁰(x+h)−ϕ⁰(x)) = det (ϕ⁰⁰(x)h) + Z 1

0

(1−t)²

2 d³_thF (h, h, h)dt.

LetH(x) = det (ϕ⁰⁰(x)h).The above equation gives det (ϕ⁰(x+h)−ϕ⁰(x)) = det (ϕ⁰⁰(w)h) +

Z 1 0

dw+t(x−w)H(h)dt

+ Z 1

0

(1−t)²

2 d³_thF(h, h, h)dt.

Then, forx, x+h∈B_r(w)we have

|det (ϕ⁰(x+h)−ϕ⁰(x))−det (ϕ⁰⁰(w)h)| ≤M|h|³ ≤2M r|h|² with M depending only kϕ₁k_C3(B) and kϕ₂k_C3(B). If we choosec ≤ _4M¹ ,we get, for0< r < cinf|h|=1|det (ϕ⁰⁰(w)h)|that

|det (ϕ⁰(x+h)−ϕ⁰(x))| ≥ 1

2|det (ϕ⁰⁰(w)h)|

and that

J_B_r_(w)(h)≥ 1

2|det (ϕ⁰⁰(w)h)| ≥ 1 2cr|h|²

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Thus R^B1^r^(w)

2

(f) 6

≤ c⁰r⁻¹² kI2(f)k₆ ≤ c⁰r⁻¹² kfk³

2 , where Iα denotes the 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 wbe an elliptic point forϕ.Then there exists a positive con- stant c depending only on kϕ₁k_C3(B) and kϕ₂k_C3(B) such that if 0 < r ≤ cinf|h|=1|det (ϕ⁰⁰(w)h)| then for all h 6= 0 the mapx → ϕ(x+h)−ϕ(x) is injective on the domain{x∈B :x, x+h∈Br(w)}.

Proof. Suppose thatx, y, x+handy+hbelong toB_r(w)and that ϕ(x+h)−ϕ(x) = ϕ(y+h)−ϕ(y).

From this equation we get 0 =

Z 1 0

(ϕ⁰(x+th)−ϕ⁰(y+th))hdt= Z 1

0

Z 1 0

d²x+th+s(y−x)ϕ(y−x, h)dsdt.

Now, forz ∈B_r(w),

d²_zϕ−d²_wϕ

(y−x, h) =

Z 1 0

d³_z+u(w−z)ϕ(w−z, y−x, h)du

≤ M r|y−x| |h|

then 0 =

Z 1 0

d²x+th+s(y−x)ϕ(y−x, h)dsdt

= d²_wϕ(y−x, h) + Z 1

0

Z 1 0

d²x+th+s(y−x)ϕ−d²_wϕ

(y−x, h)dsdt.

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So |d²_wϕ(y−x, h)| ≤ M r|y−x| |h| with M depending only on kϕ₁k_C3(B)

andkϕ2k_C3(B).

On the other hand,wis an elliptic point forϕand so, for|u|= 1,the matrix A :=ϕ⁰⁰(w)uis invertible. AlsoA⁻¹ = (detA)⁻¹Ad(A),then

A⁻¹x

=|detA|⁻¹|Ad(A)x| ≤ Mf

|detA||x|,

where Mfdepends only on kϕ1k_C2(B) and kϕ2k_C2(B). Then, for |v| = 1 and x=Av,we have|Av| ≥ |detA|/M .f Thus

d²_wϕ(y−x, h)

≥ |y−x| |h| inf

|u|=1,|v|=1

d²_wϕ(u, v)

= |y−x| |h| inf

|u|=1,|v|=1|hϕ⁰⁰(w)u, vi|

≥ 1

Mf|y−x| |h| inf

|u|=1|detϕ⁰⁰(w)u|. If we chooser < ¹

MMfinf|u|=1|detϕ⁰⁰(w)u|the above inequality impliesx=y and the lemma is proved.

For any measurable set A ⊂ B, let µ_A be the Borel measure defined by µ_A(E) =R

Aχ_E(x, ϕ(x))dxand letT_µ_A be the convolution operator given by T_µ_Af =µ_A∗f.

Proposition 2.4. Let w be an elliptic point for ϕ. Then there exist positive constants c and c⁰ depending only on kϕ₁k_C3(B) and kϕ₂k_C3(B) such that if

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0< r < cinf|h|=1|detϕ⁰⁰(w)h|then

T_µ_Br(w)f

3 ≤c⁰r⁻¹³ kfk³

2 .

Proof. Taking account of Lemma2.3, we can proceed as in Theorem 0 in [3] to obtain, as there, that

µ_Br(w)∗f

3 3

≤(A₁A₂A₃)¹³ , where

A_j = Z

R²

F_j(x) Y

1≤m≤3,m6=j

R^Br(w)1 2

F_m(x)dx andF_j(x) =kf(x, .)k3

2

Then the proposition follows from Lemma2.2and an application of the triple Hölder inequality.

For0< a <1andj ∈N let U_a,j =

(x₁, x₂)∈B :|x₁| ≥a, 2^−j|x₁| ≤ |x₂| ≤2^−j+1|x₁| and letU_a,j,i, i= 1,2,3,4the connected components ofU_a,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 ∈U_a,j,i, j≥j₀andi= 1,2,3,4.Thus

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(1) For allj ≥j₀, i= 1,2,3,4ifxandx+hbelong toU_a,j,i then

|det (ϕ⁰(x+h)−ϕ⁰(x))| ≥c2^−jβ|h|².

(2) There exists a positive constantc⁰ such that for allj ≥j₀, i = 1,2,3,4

R^U1^a,j,i

2

(f) 6

≤c⁰2^jβ² kfk3 2

.

Proof. We fixiandj ≥j₀.Forx∈U_a,j,i we have det (ϕ⁰(x+h)−ϕ⁰(x)) = det

Z 1 0

ϕ⁰⁰(x+sh)hds

.

For eachh ∈ R²\ {0}we have eitherdet (ϕ⁰⁰₁(x)h, ϕ⁰⁰₂(y)h) > c2^−jβ|h|² for all x, y ∈ U_a,j,i ordet (ϕ⁰⁰₁(x)h, ϕ⁰⁰₂(y)h) < −c2^−jβ|h|² for all x, y ∈ U_a,j,i. We consider the first case. LetF(t) = det

Rt

0ϕ⁰⁰(x+sh)hds

.Then F⁰(t) = det

Z t

0

ϕ⁰⁰₁(x+sh)hds, ϕ⁰⁰₂(x+th)h

+ det

ϕ⁰⁰₁(x+th)h, Z t

0

ϕ⁰⁰₂(x+sh)hds

= Z t

0

det (ϕ⁰⁰₁(x+sh)h, ϕ⁰⁰₂(x+th)h)ds +

Z t

0

det (ϕ⁰⁰₁(x+th)h, ϕ⁰⁰₂(x+sh)h)ds≥c2^−jβ|h|²t.

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SinceF (0) = 0we getF(1) =R1

0 F⁰(t)dt ≥c2^−jβ|h|².Thus det (ϕ⁰(x+h)−ϕ⁰(x)) = F(1) ≥c2^−jβ|h|².

Then J_U_a,j,i_,(h) ≥ c2^−jβ|h|², and the lemma follows, as in Lemma2.2, from the Hardy–Littlewood–Sobolev theorem of fractional integration. The other case is similar.

For fixedx⁽¹⁾, x⁽²⁾ ∈R²,let B_a,j,i^x⁽¹⁾^,x⁽²⁾ =

x∈R² :x−x⁽¹⁾ ∈U_a,j,iandx−x⁽²⁾ ∈U_a,j,i , i= 1,2,3,4.

We have

Lemma 2.6. Let 0 < a < 1and letx⁽¹⁾, x⁽²⁾ ∈ R². Suppose that there exist β ∈ N, j₀ ∈ Nand a positive constantcsuch that|det (ϕ⁰⁰₁(x)h, ϕ⁰⁰₂(y)h)| ≥ c2^−jβ|h|² for allh ∈ R², x, y ∈ U_a,j,i, j ≥ j₀ and i = 1,2,3,4. Then there existsj1 ∈Nindependent ofx⁽¹⁾, x⁽²⁾such that for allj ≥j1, i= 1,2,3,4and all nonnegativef ∈S(R⁴)it holds that

Z

B^x_a,j,i⁽¹⁾^,x⁽²⁾×R²

f y−ϕ x−x⁽¹⁾

, y−ϕ x−x⁽²⁾ dxdy

≤ m²

J_U_a,j,i(x⁽²⁾−x⁽¹⁾) Z

R⁴

f.

Proof. We assert that, ifj ≥j0then for each(z, w)∈R²×R²andi= 1,2,3,4, the set

n

(x, y)∈B_a,j,i^x⁽¹⁾^,x⁽²⁾ ×R² :z =y−ϕ x−x⁽¹⁾

andw=y−ϕ x−x⁽²⁾o

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is a finite set with at most m² elements. Indeed, if z = y−ϕ x−x⁽¹⁾ and w = y −ϕ x−x⁽²⁾

with x ∈ B_a,j,i^x⁽¹⁾^,x⁽²⁾, 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

x∈B_a,j,i^x⁽¹⁾^,x⁽²⁾ :ϕ x−x⁽²⁾

−ϕ x−x⁽¹⁾

=z−wo

is a finite set with at mostm²points. Sincexdeterminesy,the assertion follows.

For a fixedη >0and fork = (k₁, ..., k₄)∈Z⁴,let Q_k = Y

1≤n≤4

[k_nη,(1 +k_n)η].

LetΦk,j,i :

B^x_a,j,i⁽¹⁾^,x⁽²⁾ ×R²

∩Qk →R²×R²be the function defined by Φ_k,j,i(x, y) = y−ϕ x−x⁽¹⁾

, y−ϕ x−x⁽²⁾ and letW_k,j,iits image. Also

det Φ⁰_k,j,i

(x, y) = det ϕ⁰ x−x⁽²⁾

−ϕ⁰ x−x⁽¹⁾ .

Thus

(2.4)

det Φ⁰_k,j,i (x, y)

≥JUa,j,i x⁽²⁾−x⁽¹⁾

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for(x, y)∈

B_a,j,i^x⁽¹⁾^,x⁽²⁾ ×R²

∩Qk.

SinceΦk,j,i(x, y) = Φk,j,i(x, y)implies thatϕ x−x⁽¹⁾

−ϕ x−x⁽¹⁾

= ϕ x−x⁽²⁾

−ϕ x−x⁽²⁾

,taking into account Lemma2.1, from Lemma2.3it follows the existence ofej ∈N withejindependent ofx⁽¹⁾, x⁽²⁾such that forj ≥ ej there existsηe=ηe(j) >0satisfying that for0 < η <ηe(j)the mapΦ_k,j,i is injective for allk ∈Z⁴.LetΨ_k,j,i :W_k,j,i →

B_a,j,i^x⁽¹⁾^,x⁽²⁾ ×R²

∩Q_kits inverse.

Lemma2.5says that

det Φ⁰_k,j,i

≥c2^−jβ|h|² on

B_a,j,i^x⁽¹⁾^,x⁽²⁾ ×R²

∩Q_k.We have

Z

B^x_a,j,i⁽¹⁾^,x⁽²⁾×R²

= X

k∈Z⁴

Z

B^x_a,j,i⁽¹⁾^,x⁽²⁾×R²

∩Q_k

= X

k∈Z⁴

Z

Wk,j,i

f(z, w) 1

det Φ⁰_k,j,i

(Ψ_k,j,i(z, w))

dzdw

≤ 1

J_U_a,j,i(x⁽²⁾−x⁽¹⁾) Z

R⁴

X

k∈Z⁴

χ_W_k,j,i(v)f(v)dv

≤ m²

J_U_a,j,i(x⁽²⁾−x⁽¹⁾) Z

R⁴

f where we have used (2.4).

Proposition 2.7. Let 0 < a < 1. Suppose that there exist β ∈ N, j₀ ∈ N and a positive constantcsuch that|det (ϕ⁰⁰₁(x)h, ϕ⁰⁰₂(y)h)| ≥c2^−jβ|h|² for all

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h ∈ R², x, y ∈ Ua,j,i, j ≥ j0, i = 1,2,3,4.Then, there exist j1 ∈ N, c⁰ > 0 such that for allj ≥j₁, f ∈S(R⁴)

T_µ_Ua,jf

3 ≤c⁰2^jβ³ kfk³

2 . Proof. Fori= 1,2,3,4,let

K_a,j,i =

x, y, x⁽¹⁾, x⁽²⁾, x⁽³⁾

∈R²×R²×R²×R²×R² :x−x^(s) ∈U_a,j,i, s= 1,2,3 . We can proceed as in Theorem 0 in [3] to obtain, as there, that

µ_U_a,j,i ∗f

3 3 =

Z

Ka,j,i

Y

1≤j≤3

f(x_j, y−ϕ(x−x_j))dxdydx⁽¹⁾dx⁽²⁾dx⁽³⁾

taking into account of Lemma2.6and reasoning, with the obvious changes, as in [3], Theorem 0, we obtain that

µ_U_a,j,i ∗f

3

3 ≤m²(A₁A₂A₃)¹³ with

A_j = Z

R²

F_j(x) Y

1≤m≤3,m6=j

R^U1^a,j,i 2

F_m(x)dx andFj(x) =kf(x, .)k³

2 .Now the proof follows as in Proposition2.4.

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3. About the Type Set

Proposition 3.1. Forδ >0letVδbe defined by (2.1). Suppose that the set of the non elliptic points forϕinV_δare those lying in thex₁ axis and letαbe defined by (2.3).ThenE_µ_Vδ contains the closed trapezoidal region with vertices(0,0), (1,1), ^7α−1_7α ,^7α−2_7α

, _7α² ,_7α¹

, except perhaps the closed edge parallel to the principal diagonal.

Proof. We first show that(1−θ) (1,1) +θ ^7α−1_7α ,^7α−2_7α

∈E_µ_Vδ if0≤θ < 1.

Ifw = (w₁, w₂) ∈ U¹

2,j then 2^−j−1 ≤ |w₂| ≤ 2^−j+1.Thus, from Lemmas 2.2, 2.3and Proposition2.7, follows the existence ofj₀ ∈ N and of a positive constantc=c

kϕ₁k_C3(B),kϕ₂k_C3(B)

such that ifr_j =c2^−jα,then

Tµ_Br

j(w)f 3

≤c⁰2^jα³ kfk³

2

for somec⁰ > 0and allj ≥j₀, w ∈ U¹

2,j, f ∈S(R⁴).For0 ≤t ≤ 1letp_t, q_t be defined by

1 pt,_q¹

t

=t ²₃,¹₃

+ (1−t) (1,1).We have also Tµ_Br

j(w)f 1

≤ πc²2^−2jαkfk₁,thus, the Riesz-Thorin theorem gives

T_µ_Br(w)f qt

≤c2^j(^tα₃^{−(1−t)2α})kfk_p

t. SinceU¹

2,j can be covered withN of such ballsB_r(w)withN ' 2^j(2α−1) we get that

T_µ_U

12,j

pt,qt

≤c2^j(⁷₃^αt−1).

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Let U = ∪j≥j₀U¹

2,j. We have that kT_µ_Uk_p

t,qt ≤ P

j≥j0

T_µ_U

12,j

pt,qt

< ∞, for t < _7α³ . Since for t = _7α³ we have _p¹

t = 1 − _7α¹ and _q¹

t = 1 − _7α² and since every point in V_δ\U^◦ is an elliptic point (and so, from Theorem 3 in [3],

T_µ

Vδ\U

3

2,3

< ∞), we get that (1−θ) (1,1) + θ ^7α−1_7α ,^7α−2_7α

∈ E_µ_Vδ for 0 ≤ θ < 1. On the other hand, a standard computation shows that the ad- joint operator T_µ^∗

Vδ is given by T_µ^∗

Vδf =

T_µ_Vδ(f^∨)∨

, where we write, for g : R⁴ → C, g^∨(x) = g(−x). Thus E_µ_Vδ is symmetric with respect to the nonprincipal diagonal. Finally, after an application of the Riesz-Thorin interpo- lation theorem, the proposition follows.

Forδ >0,letA_δ ={(x₁, x₂)∈B :|x₂| ≤δ|x₁|}.

Remark 3.1. Fors >0, x = (x₁, ..., x₄)∈R⁴we sets•x= (sx₁, sx₂, s^mx₃, s^mx₄). IfE ⊂ R², F ⊂ R⁴ we setsE = {sx:x∈E}ands•F ={s•x:x∈F}. Forf :R⁴ →C, s >0, letf_sdenotes the function given byf_s(x) =f(s•x). A computation shows that

(3.1)

T_µ

2−j Vδ

f

2^−j•x

= 2^−2j

T_µ_Vδf₂^−j (x) for allf ∈S(R⁴), x ∈R⁴.

From this it follows easily that

T_µ

2−j Vδ

p,q

= 2^−j(^2(m+1)_q ⁻^2(m+1)_p ⁺²) T_µ_Vδ

p,q

.

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This fact implies that

(3.2) Eµ⊂

1 p,1

q

: 1 q ≥ 1

p− 1 m+ 1

and that if ¹_q > ¹_p − _m+1¹ then

1 p,¹_q

∈E_µ_Aδ if and only if

1 p,¹_q

∈E_µ_Vδ. Theorem 3.2. Suppose that for some δ > 0 the set of the non elliptic points for ϕ in A_δ are those lying on the x₁ axis and let α be defined by (2.3). Then E_µ_Aδ contains the intersection of the two closed trapezoidal regions with ver- tices(0,0),(1,1), _m+1^m ,^m−1_m+1

, _m+1² ,_m+1¹

and(0,0),(1,1), ^7α−1_7α ,^7α−2_7α ,

2 7α,_7α¹

respectively, except perhaps the closed edge parallel to the diagonal.

Moreover, if7α ≤ m + 1then the interior ofEµ_Aδ is the open trapezoidal region with vertices(0,0),(1,1), _m+1^m ,^m−1_m+1

and _m+1² ,_m+1¹ .

Proof. Taking into account Proposition3.1, the theorem follows from the facts of Remark3.1.

For0 < a < 1andδ > 0we set V_a,δ = {(x₁, x₂)∈B :a ≤ |x₁| ≤1 and

|x₂| ≤δ|x₁|}.We have

Proposition 3.3. Let0 < a <1.Suppose that for some0 < a <1, j0, β ∈N and some positive constant c we have |det (ϕ⁰⁰₁(x)h, ϕ⁰⁰₁(y)h)| ≥ c2^−jβ|h|² for allh ∈ R², x, y ∈U_a,j,i, j ≥ j₀ andi = 1,2,3,4.Then, forδ positive and small enough,Eµ_Va,δ contains the closed trapezoidal region with vertices(0,0), (1,1),

β+2 β+3,^β+1_β+3

,

2

β+3,_β+3¹

, except perhaps the closed edge parallel to the principal diagonal.

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Proof. Proposition 2.7 says that there exist j₁ ∈ N and a positive constant c such that forj ≥j1 andf ∈S(R⁴)

T_µ_Ua,j,if 3

≤c2^jβ³ kfk3 2 . Also, for some c > 0 and allf ∈ S(R⁴) we have

Tµ_Ua,j,if

1 ≤ c2^−jkfk₁. Then

T_µ_Ua,j,if qt

≤c2^j(^t^β3−(1−t))kfk_p

t wherep_t, q_tare defined as in the proof of Proposition 3.1. Let U = ∪j≥j₁U_a,j. Then kT_µ_Ufk_p

t,qt < ∞ if t < _β+3³ . Now, the proof follows as in Proposition3.1.

Theorem 3.4. Suppose that for some 0 < a < 1, j₀, β ∈ N and for some positive constantcwe have|det (ϕ⁰⁰₁(x)h, ϕ⁰⁰₁(y)h)| ≥c2^−jβ|h|² for allx, y ∈ U_a,j,i, j ≥j₀andi= 1,2,3,4.Then forδpositive and small enough,E_µ_Aδ con- tains the intersection of the two closed trapezoidal regions with vertices(0,0), (1,1), _m+1^m ,^m−1_m+1

, _m+1² ,_m+1¹

and(0,0), (1,1),

β+2 β+3,^β+1_β+3

,

2

β+3,_β+3¹ , respectively, except perhaps the closed edge parallel to the diagonal.

Moreover, ifβ ≤m−2then the interior ofE_µis the open trapezoidal region with vertices(0,0),(1,1), _m+1^m ,^m−1_m+1

and _m+1² ,_m+1¹ .

Proof. Follows as in Theorem3.2using now Proposition3.3instead of Propo- sition3.1.

Remark 3.2. We now turn out to the case whenϕis a homogeneous polynomial function whose set of non elliptic points is a finite union of lines through the origin,L₁,...,L_k.

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For each l,1 ≤ l ≤ k, letA^l_δ =

x∈R² : π_L^⊥

lx

≤δ|π_L_lx| where π_L_l andπ^⊥_L

l denote the orthogonal projections fromR²intoL_landL^⊥_l respectively.

Thus eachA^l_δ is a closed conical sector aroundLl.We chooseδ small enough such thatA^l_δ∩Aⁱ_δ =∅forl 6=i.

It is easy to see that there exists (a unique)α_l∈N and positive constantsc⁰_l, c⁰⁰_l such that

(3.3) c⁰_l π^⊥_L_lw

αl

≤ inf

|h|=1|det (ϕ⁰⁰(w)h)| ≤c⁰⁰_l π_L^⊥_lx

αl

for allw ∈ A^l_δ.Indeed, after a rotation the situation reduces to that considered in Remark3.1.

Theorem 3.5. Suppose that the set of non elliptic points is a finite union of lines through the origin,L1,...,Lk.Forl = 1,2, ..., k,letαl be defined by (3.3), and let α = max1≤l≤kα_l.ThenE_µ contains the intersection of the two closed trapezoidal regions with vertices (0,0), (1,1), _m+1^m ,^m−1_m+1

, _m+1² ,_m+1¹ and (0,0), (1,1), ^7α−1_7α ,^7α−2_7α

, _7α² ,_7α¹

, respectively, except perhaps the closed edge parallel to the diagonal.

Moreover, if7α ≤m+ 1then the interior ofE_µis the interior of the trape- zoidal regions with vertices(0,0),(1,1), _m+1^m ,^m−1_m+1

, _m+1² ,_m+1¹ .

Proof. Forl = 1,2, ..., k,letA^l_δbe as above. From Theorem3.2, we obtain that E_µ

Alδ

contains the intersection of the two closed trapezoidal regions with vertices(0,0),(1,1), _m+1^m ,^m−1_m+1

, _m+1² ,_m+1¹

and(0,0),(1,1),

7αl−1 7αl ,^7α_7α^l⁻²

l

, 2

7α_l,_7α¹

l

respectively, except perhaps the closed edge parallel to the diagonal.

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Since everyx∈ B\ ∪_lA^l_δis an elliptic point forϕ, Theorem 0 in [3] and a compactness argument give that kTµDk3

2,3 < ∞ where D = x∈B\ ∪_lA^l_δ : ¹₂ ≤ |x| .Then (using the symmetry of E_µ_D,the fact of that µ_D is a finite measure and the Riesz-Thorin theorem)E_µ_D is the closed triangle with vertices (0,0), (1,1), ²₃,¹₃

.Now, proceeding as in the proof of Theo- rem3.2we get that

T_µ

B\∪lAl δ

p,q

<∞if ¹_q > ¹_p −_m+1¹ .Then the first assertion of the theorem is true. The second one follows also using the facts of Remark 3.1.

For0< a <1,we set U_a,j^l =

x∈R² :a≤ |π_L^l(x)| ≤1

and2^−j|π_L^l(x)| ≤

π_L^⊥l(x)

≤2^−j+1|π_L^l(x)|

letU_a,j,i^l , i= 1,2,3,4be the connected components ofU_a,j^l .

Theorem 3.6. Suppose that the set of non elliptic points for ϕis a finite union of lines through the origin,L₁,...,L_k.Let0< a <1and letj₀ ∈N such that

Forl = 1,2, ..., k,there existsβ_l ∈N satisfying|det (ϕ⁰⁰₁(x)h, ϕ⁰⁰₁(y)h)| ≥ c2^−jβ^j|h|² for allx, y ∈U_a,j,i^l , j ≥j₀ andi = 1,2,3,4.Letβ = max1≤j≤kβ_j. ThenEµcontains the intersection of the two closed trapezoidal regions with ver- tices (0,0), (1,1), _m+1^m ,^m−1_m+1

, _m+1² ,_m+1¹

and (0,0), (1,1),

β+2 β+3,^β+1_β+3

, 2

β+3,_β+3¹

, respectively, except perhaps the closed edge parallel to the diago- nal.

Moreover, ifβ ≤ m−2then the interior ofE_µ is the interior of the trape- zoidal region with vertices(0,0),(1,1), _m+1^m ,^m−1_m+1

, _m+1² ,_m+1¹ .