In this paper we study convolution operators Tµ with measures µin R4 of the formµ(E

http://jipam.vu.edu.au/

Volume 2, Issue 3, Article 37, 2001

L^p−IMPROVING PROPERTIES FOR MEASURES ON R⁴ SUPPORTED ON HOMOGENEOUS SURFACES IN SOME NON ELLIPTIC CASES

E. FERREYRA, T. GODOY, AND M. URCIUOLO

FACULTAD DEMATEMATICA, ASTRONOMIA YFISICA-CIEM, UNIVERSIDADNACIONAL DECORDOBA, CIUDADUNIVERSITARIA, 5000 CORDOBA, ARGENTINA

eferrey@mate.uncor.edu godoy@mate.uncor.edu urciuolo@mate.uncor.edu

Received 8 January, 2001; accepted 5 June, 2001 Communicated by L. Pick

ABSTRACT. In this paper we study convolution operators Tµ with measures µin R⁴ of the formµ(E) = R

Bχ_E(x, ϕ(x))dx,whereB is the unit ball ofR², 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, thatTµ is bounded fromL^p intoL^q if

1 p,¹_q

belongs to a certain explicitly described trapezoidal region.

Key words and phrases: Singular measures,L^p−improving, Convolution Ooperators.

2000 Mathematics Subject Classification. 42B20, 42B10.

1. INTRODUCTION

It is well known that a complex measure µ on Rⁿ acts as a convolution operator on the Lebesgue spacesL^p(Rⁿ) : µ∗L^p ⊂ L^pfor1 ≤p ≤ ∞.If for somepthere exists q > psuch thatµ∗L^p ⊂ L^q, µis calledL^p−improving. It is known that singular measures supported on smooth submanifolds ofRⁿmay beL^p−improving. See, for example, [2], [5], [8], [9], [7] and [4].

Letϕ₁, ϕ₂ be two homogeneous polynomial functions onR² of degreem ≥ 2and let ϕ = (ϕ₁, ϕ₂).Letµbe the Borel measure onR⁴ given by

(1.1) µ(E) =

Z

B

χ_E(x, ϕ(x))dx,

ISSN (electronic): 1443-5756

c 2001 Victoria University. All rights reserved.

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

052-00

whereB denotes the closed unit ball around the origin inR² anddxis the Lebesgue measure onR².LetT_µbe the convolution operator given byT_µf =µ∗f, f ∈S(R⁴)and letE_µbe the type set corresponding to the measureµdefined by

Eµ= 1

p,1 q

:kTµk_p,q <∞,1≤p, q ≤ ∞

,

where kTµk_p,q denotes the operator norm of Tµ from L^p(R⁴) into L^q(R⁴) and where theL^p spaces are taken with respect to the Lebesgue measure onR⁴.

Forx, h∈R²,letϕ⁰⁰(x)hbe the2×2matrix whosej−thcolumn isϕ⁰⁰_j (x)h,whereϕ⁰⁰_j (x) denotes the Hessian matrix ofϕ_j atx.Following [3, p. 152], we say thatx ∈ R² is an elliptic point forϕifdet (ϕ⁰⁰(x)h)6= 0for allh ∈R²\ {0}.ForA ⊂R²,we will say thatϕis strongly elliptic onAifdet (ϕ⁰⁰₁(x)h, ϕ⁰⁰₂(y)h)6= 0for allx, y ∈Aandh∈R²\ {0}.

If every pointx∈B\ {0}is elliptic forϕ,it is proved in [4] that form≥3, E_µis the closed trapezoidal regionΣ_mwith 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 ofh.For eachl= 1,2, ..., k,letπ_Lland π_L^⊥

l be the orthogonal projections fromR² ontoL_landL^⊥_l respectively. Forδ >0,1≤l ≤k, let

V_δ^l=n

x∈B : 1/2≤ |π_Ll(x)| ≤1and π_L^⊥

l (x)

≤δ|π_Ll(x)|o .

It is easy to see (see Lemma 2.1 and Remark 3.6) that forδsmall enough, there existsαl ∈ N and positive constantscandc⁰ such that

c π_L^⊥

l (x)

α_l

≤ inf

h∈S¹|det (ϕ⁰⁰(x)h)| ≤c⁰ π_L^⊥

l (x)

α_l

for all x ∈ V_δ^l. Following the approach developed in [3], we prove, in Theorem 3.7, that if α= max1≤l≤kα_l and if7α≤m+ 1,then the interior ofE_µagrees with the interior ofΣ_m.

Moreover in Theorem 3.8 we obtain that

◦

E_µ =

◦

Σ_m 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 ofEµ, we obtain a polygonal region contained in it.

Throughout the papercwill denote a positive constant not necessarily the same at each oc- currence.

2. PRELIMINARIES

Letϕ₁, ϕ₂ : R² → Rbe two homogeneous polynomials functions of degree m ≥ 2and let ϕ= (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 forϕ inV_δ0 is contained in thex₁ axis.

Forx ∈ 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.

Lemma 2.1. There existδ ∈ (0, δ₀), α ∈ Nand a real analytic function g = g(x₁, x₂)onV_δ withg(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. SinceP(x) is real analytic onV_δ andP (x) 6= 0forx 6= 0,it follows that, forδsmall enough, there exists two real analytic functionsλ₁(x) andλ₂(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 forϕ and that P(x) 6= 0for x 6= 0, diminishingδ if necessary, we can assume thatλ₁(1,0) = 0and that|λ₁(1, x₂)| ≤ |λ₂(1, x₂)|for|x₂| ≤ δ.

SinceP (x)is homogeneous in x,we have that λ₁(x)and λ₂(x)are homogeneous in xwith the same homogeneity degree d. Thus |λ₁(x)| ≤ |λ₂(x)| for allx ∈ V_δ.Now, λ₁(1, x₂) = x^α₂G(x₂)for some real analytical function G = G(x₂) withG(0) 6= 0and soλ₁(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

J_U(x−y)^−1+αf(y)dy.

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

We have the following

Lemma 2.2. Let wbe an elliptic point for ϕ. Then there exist positive constants candc⁰ de- pending only onkϕ₁k_C3(B)andkϕ₂k_C3(B)such that if0< 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).

Proof. LetF (h) = det (ϕ⁰(x+h)−ϕ⁰(x))and letd^j_xF denotes thej−th differential ofF at x.Applying the Taylor formula toF (h)aroundh= 0and taking into account thatF (0) = 0, d₀F (h) = 0and thatd²₀F(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

withM depending onlykϕ₁k_C3(B)andkϕ₂k_C3(B).If we choose c≤ _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_Br(w)(h)≥ 1

2|det (ϕ⁰⁰(w)h)| ≥ 1 2cr|h|² Thus

R^B1^r(w)

2

(f) 6

≤ c⁰r⁻¹² kI₂(f)k₆ ≤ c⁰r⁻¹² kfk3

2 , where I_α denotes the Riesz potential onR⁴,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. Letwbe an elliptic point forϕ.Then there exists a positive constantcdepending only onkϕ₁k_C3(B) andkϕ₂k_C3(B) such that if 0 < r ≤ cinf_|h|=1|det (ϕ⁰⁰(w)h)| then for all h6= 0the 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 toBr(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

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.

So|d²_wϕ(y−x, h)| ≤M r|y−x| |h|withM depending only onkϕ₁k_C3(B)andkϕ₂k_C3(B). On the other hand,wis an elliptic point forϕand so, for|u|= 1,the matrixA:= ϕ⁰⁰(w)u is invertible. AlsoA⁻¹ = (detA)⁻¹Ad(A),then

A⁻¹x

=|detA|⁻¹|Ad(A)x| ≤ Mf

|detA||x|,

whereMfdepends only onkϕ₁k_C2(B) andkϕ₂k_C2(B). Then, for |v| = 1andx = Av,we have

|Av| ≥ |detA|/fM .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=yand 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_µA be the convolution operator given byT_µAf =µ_A∗f.

Proposition 2.4. Letw be an elliptic point for ϕ. Then there exist positive constantscand c⁰ depending only onkϕ₁k_C3(B)andkϕ₂k_C3(B)such that if0< r < cinf|h|=1|detϕ⁰⁰(w)h|then

T_µBr(w)f

3 ≤c⁰r⁻¹³ kfk³

2 .

Proof. Taking account of Lemma 2.3, 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 Lemma 2.2 and an application of the triple Hölder in-

equality.

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. Let0 < a < 1. Suppose that there existβ ∈ N, j₀ ∈ Nand 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 ≥ j₀ and i= 1,2,3,4.Thus

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

|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∈Ua,j,iwe 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 allx, y ∈ U_a,j,i or det (ϕ⁰⁰₁(x)h, ϕ⁰⁰₂(y)h) < −c2^−jβ|h|² for allx, y ∈ U_a,j,i.We consider the first case. Let

F (t) = det Rt

0 ϕ⁰⁰(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.

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

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

We have

Lemma 2.6. Let0< a <1and letx⁽¹⁾, x⁽²⁾ ∈R².Suppose that there existβ ∈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 exists j₁ ∈ N independent of x⁽¹⁾, x⁽²⁾ such that for all j ≥j₁, i= 1,2,3,4and all nonnegativef ∈S(R⁴)it holds that

Z

B_a,j,i^x(1),x(2)×R²

f y−ϕ x−x⁽¹⁾

, y−ϕ x−x⁽²⁾

dxdy ≤ m²

J_Ua,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 is a finite set with at mostm²elements. Indeed, ifz =y−ϕ x−x⁽¹⁾

andw=y−ϕ x−x⁽²⁾ withx∈B_a,j,i^x(1),x(2),Lemma 2.5 says that, forj 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 mostm² points. Sincexdeterminesy,the assertion follows.

For a fixed η > 0 and fork = (k₁, ..., k₄) ∈ Z⁴, let Q_k = Q

1≤n≤4[k_nη,(1 +k_n)η].Let Φ_k,j,i :

B_a,j,i^x(1),x(2) ×R²

∩Q_k →R²×R² be the function defined by Φk,j,i(x, y) = y−ϕ x−x⁽¹⁾

, y−ϕ x−x⁽²⁾ and letW_k,j,i its image. Alsodet 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)

≥J_Ua,j,i x⁽²⁾−x⁽¹⁾

for(x, y)∈

B_a,j,i^x(1),x(2)×R²

∩Q_k.

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 Lemma 2.1, from Lemma 2.3 it follows the existence of ej ∈ N with ej independent of x⁽¹⁾, x⁽²⁾ such that for j ≥ ej there exists ηe = eη(j) > 0 satisfying that for 0 < η < ηe(j) the map Φ_k,j,i is injective for all k ∈ Z⁴. Let Ψ_k,j,i : Wk,j,i →

B_a,j,i^x(1),x(2) ×R²

∩Qkits inverse. Lemma 2.5 says that

det Φ⁰_k,j,i

≥c2^−jβ|h|²on

B_a,j,i^x(1),x(2)×R²

∩Q_k.We have Z

B_a,j,i^x(1),x(2)×R²

f y−ϕ x−x⁽¹⁾

, y−ϕ x−x⁽²⁾ dxdy

= X

k∈Z⁴

Z

B^x_a,j,i^(1),x(2)×R²

∩Q_k

f y−ϕ x−x⁽¹⁾

, y−ϕ x−x⁽²⁾ dxdy

= 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_Ua,j,i(x⁽²⁾−x⁽¹⁾) Z

R⁴

X

k∈Z⁴

χ_Wk,j,i(v)f(v)dv

≤ m²

J_Ua,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 allh∈R², x, y ∈U_a,j,i, j ≥j₀, i= 1,2,3,4.Then, there existj1 ∈N, c⁰ >0such that for allj ≥j1, f ∈S(R⁴)

T_µUa,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_Ua,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 Lemma 2.6 and reasoning, with the obvious changes, as in [3], Theorem 0, we obtain that

µ_Ua,j,i ∗f

3

3 ≤m²(A₁A₂A₃)¹³ with

Aj = Z

R²

Fj(x) Y

1≤m≤3,m6=j

R^U1^a,j,i 2

Fm(x)dx

andF_j(x) = kf(x, .)k³

2 .Now the proof follows as in Proposition 2.4.

3. ABOUT THE TYPE SET

Proposition 3.1. Forδ > 0 letVδ be defined by (2.1). Suppose that the set of the non elliptic points for ϕ inVδ are those lying in the x1 axis and let α be defined by (2.3).ThenEµ_Vδ con- tains 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_µVδ if0≤θ <1.

Ifw= (w₁, w₂)∈U¹

2,jthen2^−j−1 ≤ |w₂| ≤2^−j+1.Thus, from Lemmas 2.2, 2.3 and Propo- sition 2.7, follows the existence ofj₀ ∈N and of a positive constantc=c

kϕ₁k_C3(B),kϕ₂k_C3(B)

such that ifr_j =c2^−jα,then T_µ

Brj(w)f 3

≤c⁰2^jα3 kfk3 2

for somec⁰ > 0and all j ≥ j₀, w ∈ U¹

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

1 pt,_q¹

t

=t ²₃,¹₃

+ (1−t) (1,1).We have also T_µ

Brj(w)f 1

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

T_µBr(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_µU

1 2,j

pt,qt

≤c2^j(⁷3αt−1).

LetU =∪_j≥j0U¹

2,j.We have thatkT_µUk_p

t,qt ≤P

j≥j0

T_µU

1 2,j

pt,qt

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

t = 1− _7α¹ and _q¹

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

Tµ_Vδ\U

3

2,3 <∞), we get that(1−θ) (1,1)+θ ^7α−1_7α ,^7α−2_7α

∈ E_µVδ for0≤θ < 1.On the other hand, a standard computation shows that the adjoint operator T_µ^∗

Vδ is given by T_µ^∗

Vδf =

Tµ_Vδ(f^∨) ∨

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

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

Remark 3.2. For s > 0, x = (x₁, ..., x₄) ∈ R⁴ we set s• x = (sx₁, sx₂, s^mx₃, s^mx₄). If E ⊂ R², F ⊂ R⁴ we setsE ={sx:x∈E}ands•F = {s•x:x∈F}.Forf : R⁴ → C, s >0, letfsdenotes the function given byfs(x) =f(s•x).A computation shows that (3.1)

Tµ₂−j Vδf

2^−j •x

= 2^−2j

Tµ_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_µVδ

p,q

.

This fact implies that

(3.2) E_µ ⊂

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_µAδ if and only if

1 p,¹_q

∈E_µVδ.

Theorem 3.3. Suppose that for someδ > 0 the set of the non elliptic points forϕ inA_δ are those lying on thex₁ axis and letαbe defined by (2.3). ThenE_µAδ 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_µAδ 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 Proposition 3.1, the theorem follows from the facts of Remark 3.2.

For0< a < 1andδ >0we setV_a,δ ={(x₁, x₂)∈B :a≤ |x₁| ≤1and |x₂| ≤δ|x₁|}.We have

Proposition 3.4. Let0< a < 1.Suppose that for some0< a < 1, j0, β∈N and some positive constantcwe 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, forδpositive and small enough,E_µVa,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.

Proof. Proposition 2.7 says that there existj1 ∈N and a positive constantcsuch that forj ≥j1

andf ∈S(R⁴)

T_µUa,j,if

3 ≤c2^jβ3 kfk³

2 . Also, for somec >0and allf ∈S(R⁴)we have

T_µUa,j,if 1

≤c2^−jkfk₁.Then

T_µUa,j,if qt

≤ c2^j(^tβ₃^−(1−t))kfk_ptwherept, qtare defined as in the proof of Proposition 3.1. LetU =∪j≥j1Ua,j. ThenkT_µUfk_p

t,qt <∞ift < _β+3³ .Now, the proof follows as in Proposition 3.1.

Theorem 3.5. Suppose that for some0< 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 ≥j₀ andi = 1,2,3,4.Then for δ positive and small enough,E_µAδ 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),

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_µ is 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 Theorem 3.3 using now Proposition 3.4 instead of Proposition 3.1.

Remark 3.6. 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.

For eachl,1≤l ≤ k,letA^l_δ =

x∈R² : π_L^⊥

lx

≤δ|π_Llx| whereπ_Ll andπ_L^⊥

l denote the orthogonal projections fromR² intoL_l andL^⊥_l respectively. Thus eachA^l_δ is a closed conical sector aroundL_l.We chooseδsmall 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

α_l

≤ inf

|h|=1|det (ϕ⁰⁰(w)h)| ≤c⁰⁰_l π_L^⊥

lx

α_l

for allw∈A^l_δ.Indeed, after a rotation the situation reduces to that considered in Remark 3.2.

Theorem 3.7. Suppose that the set of non elliptic points is a finite union of lines through the origin, L₁,...,L_k.Forl = 1,2, ..., k,letα_l be defined by (3.3), and let α = max1≤l≤kα_l.Then E_µ contains the intersection of the two closed trapezoidal regions with vertices(0,0),(1,1),

m

m+1,^m−1_m+1

, _m+1² ,_m+1¹

and (0,0), (1,1), ^7α−1_7α ,^7α−2_7α

, _7α² ,_7α¹

, respectively, except per- haps the closed edge parallel to the diagonal.

Moreover, if7α ≤m+ 1then the interior ofE_µis the interior of the trapezoidal 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_δ be as above. From Theorem 3.3, we obtain thatE_µ

Alδ

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

,

2

m+1,_m+1¹

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

7αl−1 7α_l ,^7α_7α^l−2

l

,

2 7α_l,_7α¹

l

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

Since every x ∈ B\ ∪_l A^l_δ is an elliptic point for ϕ, Theorem 0 in [3] and a compactness argument give that kTµ_Dk3

2,3 < ∞ where D =

x∈B\ ∪lA^l_δ : ¹₂ ≤ |x| . Then (using the symmetry of E_µD, the fact of that µ_D is a finite measure and the Riesz-Thorin theorem) E_µD is the closed triangle with vertices (0,0), (1,1), ²₃,¹₃

. Now, proceeding as in the proof of Theorem 3.3 we 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 3.2.

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

x∈R² :a≤ |π_Ll(x)| ≤1and2^−j|π_Ll(x)| ≤

π_L^⊥l(x)

≤2^−j+1|π_Ll(x)|

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

Theorem 3.8. Suppose that the set of non elliptic points forϕis a finite union of lines through the origin,L1,...,Lk.Let0< a < 1and letj0 ∈N such that

Forl = 1,2, ..., k, there existsβ_l ∈ N satisfying|det (ϕ⁰⁰₁(x)h, ϕ⁰⁰₁(y)h)| ≥ c2^−jβj|h|² for allx, y ∈U_a,j,i^l , j ≥j₀andi= 1,2,3,4.Letβ = max1≤j≤kβ_j.ThenE_µcontains the intersec- tion 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 par- allel to the diagonal.

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

, _m+1² ,_m+1¹ .

Proof. Follows as in Theorem 3.7, using now Theorem 3.5 instead of Theorem 3.3.

Example 3.1. ϕ(x₁, x₂) = (x²₁x₂−x₁x²₂, x²₁x₂+x₁x²₂)

It is easy to check that the set of non elliptic points is the union of the coordinate axes. Indeed, forh = (h₁, h₂)we havedetϕ⁰⁰(x₁, x₂)h = 8x²₂h²₁ + 8x₁x₂h₁h₂ + 8x²₁h²₂ and this quadratic form in(h₁, h₂)has non trivial zeros only ifx₁ = 0orx₂ = 0.The associated symmetric matrix to the quadratic form is

8x²₂ 4x₁x₂ 4x₁x₂ 8x²₁

and for x1 6= 0 and|x2| ≤ δ|x1| withδ small enough, its eigenvalue of lower absolute value isλ₁(x₁, x₂) = 4x²₁+ 4x²₂ −4p

(x⁴₂−x²₁x²₂+x⁴₁).Thusλ₁(x₁, x₂) ' 6x²₂ for such(x₁, x₂). Similarly, for x₂ 6= 0and|x₁| ≤ δ|x₂|withδ small enough, the eigenvalue of lower absolute value is comparable with6x²₁. Then, in the notation of Theorem 3.7, we obtainα = 2and so E_µcontains the closed trapezoidal region with vertices(0,0),(1,1), ¹³₁₄,⁶₇

and ¹₇,₁₄¹

except

perhaps the closed edge parallel to the principal diagonal. Observe that, in this case, Theorem 3.8 does not apply. In fact, forx= (x₁, x₂),xe= (ex₁,ex₂)andh= (h₁, h₂)we have

det (ϕ⁰⁰₁(x)h, , ϕ⁰⁰₂(ex)h)

= 4h²₁(x₂ex₁−ex₂x₁+ 2x₂xe₂) + 4h₁h₂(x₁ex₂+xe₁x₂) + 4h²₂(x₁ex₂−x₂xe₁+ 2x₂ex₁). Takex₁ = ex₁ = 1and let A = A(x₂,xe₂)the matrix of the above quadratic form in (h₁, h₂). Forx₂ = 2^−j,xe₂ = 2^−j+1 we havedetA <0forj large enough but if we takex₂ = 2^−j+1and ex₂ = 2^−j,we getdetA > 0forj large enough, so, for all j large enough,detA = 0for some 2^−j ≤x₂,xe₂ ≤2^−j+1.Thus, for suchx₂,ex₂,

|(h₁inf,h2)|=1det (ϕ⁰⁰₁(1, x₂) (h₁, h₂), ϕ⁰⁰₂(1,ex₂) (h₁, h₂)) = 0.

Example 3.2. Let us show an example where Theorem 3.8 characterizes

◦

E_µ.Let ϕ(x₁, x₂) = x³₁x₂−3x₁x³₂,3x²₁x²₂ −x⁴₂

. In this case the set of non elliptic points forϕis thex₁axis. Indeed,

det (ϕ⁰⁰(x₁, x₂) (h₁, h₂)) = 18 x²₁+x²₂

(h₂x₁+x₂h₁)² + 2x²₂h²₁+ 6h²₂x²₂ . In order to apply Theorem 3.8, we consider the quadratic form inh= (h₁, h₂)

det (ϕ⁰⁰₁(x₁, x₂)h, ϕ⁰⁰₂(ex₁,ex₂)h).

Ifx= (x₁, x₂)andex= (xe₁,xe₂),letA=A(x,x)e its associated symmetric matrix. An explicit computation ofAshows that, for a given0< a <1and for alljlarge enough andi= 1,2,3,4, ifxandxebelong toU_a,j,i,then

a² ≤tr(A)≤20

thus, ifλ₁(x,ex)denotes the eigenvalue of lower absolute value ofA(x,ex),we have, forx,xe∈ W_athat

c₁|detA| ≤ |λ₁(x,x)| ≤e c₂|detA|

wherec₁, c₂ are positive constants independent ofj.Now, a computation gives detA= 324 −x²₁ex²₁−9x²₂ex²₂ −12x₁x₂xe₁xe₂+ 2x²₁ex²₂

× x²₂ex²₁−2x²₂xe²₂−4x₁x₂xe₁xe₂+x²₁xe²₂ . Now we writeex2 =tx2,with ¹₂ ≤t ≤2.Then

detA= 324x²₂

−x²₁ex²₁−9t²x⁴₂−12tx1x²₂ex1+ 2t²x²₂x²₁ ex²₁−2t²x²₂−4tx1xe1+t²x²₁ . Note that the the first bracket is negative forx,ex ∈ Wa ifj is large enough. To study the sign of the second one, we consider the functionF (t, x₁,xe₁) = xe²₁ −4tx₁ex₁ +t²x²₁.SinceF has a negative maximum on{1} × {1} ×₁

2,2

,it follows easily that we can choosea such that forx,xe ∈ Wa andj large enough, the same assertion holds for the second bracket. So detA is comparable with2^−2j,thus the hypothesis of the Theorem 3.8 are satisfied withβ = 2and such a.Moreover, we have β = m −2, then we conclude that the interior ofE_µ is the open trapezoidal region with vertices(0,0),(1,1), ³₅,⁴₅

, ²₅,¹₅ .

On the other hand, in a similar way than in Example 3.1 we can see that α = 2 (in fact detA(x, x) = 648 (x²₁+ 9x²₂) (x²₁+x²₂)²x²₂), 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_µcontains the trapezoidal region with vertices(0,0),(1,1), ¹³₁₄,⁶₇

and ¹₇,₁₄¹ .