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volume 7, issue 3, article 89, 2006.

Received 22 December, 2005;

accepted 23 September, 2006.

Communicated by:A. Fiorenza

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Journal of Inequalities in Pure and Applied Mathematics

CONVOLUTION OPERATORS WITH HOMOGENEOUS SINGULAR MEASURES ONR³ OF POLYNOMIAL TYPE.

THE REMAINDER CASE.

MARTA URCIUOLO

Famaf-Ciem, Universidad Nacional de Córdoba-Conicet.

Medina Allende s/n Ciudad Universitaria 5000, Córdoba, Argentina.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 374-05

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Convolution Operators with Homogeneous Singular Measures onR³of Polynomial

Type. The Remainder Case Marta Urciuolo

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J. Ineq. Pure and Appl. Math. 7(3) Art. 89, 2006

http://jipam.vu.edu.au

Abstract

Letϕ(y₁, y₂) =y₂^lP(y₁, y₂)whereP is a polynomial function of degreelsuch thatP(1,0) 6= 0. Let µδ be the Borel measure onR³ defined by µδ(E) = R

VδχE(x, ϕ(x))dxwhere V_δ=

x= (x₁, x₂)∈R²:|x₁| ≤1,and|x₁| ≤δ|x₂|

and letT_µ_δ be the convolution operator with the measureµ_δ.In this paper we explicitely describe the type set

Eµ_δ:=

1 p,1

q

∈[0,1]×[0,1] :kT_µ_δk_p,q<∞

, forδsmall enough.

2000 Mathematics Subject Classification:42B20, 26B10.

Key words: Convolution operators, Singular measures.

Partially supported by Conicet, Agencia Córdoba Ciencia, Agencia Nacional de Pro- moción Científica y Tecnológica y Secyt-UNC.

The author is deeply indebted to Prof. F. Ricci for his useful suggestions.

1. Introduction

Letϕ : R² → Rbe a homogeneous polynomial function of degreem ≥ 2and letD={y ∈R² :|y| ≤1}.Letµbe the Borel measure onR³ given by

(1.1) µ(E) =

Z

D

χ_E(y, ϕ(y))dy

and let T_µ be the operator defined, for f ∈ S(R³),by T_µf = µ∗f. Let E_µ be the set of the pairs

1 p,¹_q

∈ [0,1]×[0,1]such that there exists a positive constantcsatisfyingkT fk_q ≤ ckfk_p for allf ∈ S(R³),where theL^p spaces are taken with respect to the Lebesgue measure on R³. For

1 p,¹_q

∈ E_µ, T can be extended to a bounded operator, still denoted by T,from L^p(R³)into L^q(R³).

Letϕ=ϕ^e₁¹...ϕ^e_nⁿbe a decomposition ofϕin irreducible factors withϕ_i -ϕ_j for i 6= j. In [3] we could give a complete description of the set Eµunder the assumption that e_i 6= ^m₂ for eachϕ_i of degree1. Ifdetϕ⁰⁰(y)is not identically zero and if it vanishes somewhere on R² − {0}, the set of the points y where detϕ⁰⁰(y)vanishes is a finite union of lines L1, ..., Lk through the origin. So, after a possibly linear change of variables, we localized the problem to the x axes and we studied the type set corresponding to measuresµ_δdefined by

µ_δ(E) = Z

Vδ

χ_E(y, ϕ(y))dy,

whereV_δ=D∩ {(y₁, y₂)∈R² :|y₂| ≤δ|y₁|}andδis small enough such that detϕ⁰⁰(y)only vanishes, onVδ,along thexaxes. The only case left was the one

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corresponding to functionsϕof the formϕ(y₁, y₂) =y^l₂P (y₁, y₂)withl = ^m₂, P being a homogeneous polynomial function of degreelsuch thatP(1,0)6= 0.

In this paper we characterizeE_µ_δ in this remainder case.

L^p improving properties of convolution operators with singular measures supported on hypersurfaces inRⁿ have been widely studied in [2], [5], [6]. In particular, in [5], the type set was studied under our actual hypothesis, but the endpoint problem was left open there. Our proof of the main result involves a biparametric family of dilations and will be based on a suitable adaptation of arguments due to M. Christ, developed in [1], where the author studied the type set associated to the two dimensional measure supported on the parabola.

Also, oscillatory integral estimates are involved. A very careful study of this kind of estimate can be found in [4] where the authors study the boundedness of maximal operators associated to mixed homogeneous hypersurfaces.

Throughout this papercwill denote a positive constant, not the same at each occurrence.

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2. The Main Result

We assume ϕ(y₁, y₂) = y^l₂P (y₁, y₂),where l = ^m₂ andP is a homogeneous polynomial function of degreel such thatP (1,0) 6= 0. We take δ₁ > 0 such that, for y ∈ V_δ₁ such thaty₂ 6= 0,detϕ⁰⁰(y)6= 0.Moreover, sinceP (1,0) 6=

0 we can assume that P (y) 6= 0 and P₁(y) 6= 0 for all y ∈ V_δ₁. Now, if max_V_δ

1|P₂(y₁, y₂)| 6= 0, we choose δ < min

lmin_Vδ

1|P(y1,y2)|

2 max_Vδ

1|P2(y1,y2)|, δ₁

. In the other case we takeδ=δ₁.

The main result we prove is the following.

Theorem 2.1. Let ϕ(y1, y2) = y₂^lP (y1, y2)where l = ^m₂ andP is a homoge- neous polynomial function of degreelsuch thatP (1,0)6= 0andy₂ -P (y₁, y₂). Let V_δ be defined as above and let E_V_δ be the corresponding type set. Then EV_δ is the closed polygonal region with vertices(0,0),(1,1), ^2l+1_2l+2,^2l−1_2l+2

and

3

2l+2,_2l+2¹ .

Standard arguments (see, for example Lemma 2 and Lemma 3 in [3]) imply the following result.

Lemma 2.2. If 1

p,¹_q

∈Eµ_δ then ¹_q ≤ ¹_p, ¹_q ≥ ³_p −2and ¹_q ≥ ¹_p − _l+1¹ . So, sincekT_µ_δk

1,1 <∞,by duality arguments it only remains to prove that

(2.1) kT_µ_δk

2l+2 2l+1,2l+2

2l−1

<∞.

We setQ₀ = ₁

4,2

×_δ

64,^δ₈

.We take a truncation functionθ ∈ C^∞(R²), θ(y1, y2)≥0,suppθ⊂Q0 andθ(y1, y2) = 1on₁

2,1

× _δ

32,₁₆^δ

.We define,

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forε, γ >0,the biparametric family of dilations onR²andR³given by(ε, γ)◦ (y1, y2) = (εy1, γy2) and (ε, γ)◦(y1, y2, y3) = εy1, γy2, ε^lγ^ly3

repectively.

Also, forj, k ≥0,we setQ_j,k = 2^−j,2^−k

◦Q₀. Forf ∈S(R³),we define

(2.2) T_j,kf(x₁, x₂, x₃)

= Z

f(x₁−y₁, x₂−y₂, x₃−ϕ(y₁, y₂))θ 2^jy₁,2^ky₂

dy₁dy₂

so forf ≥0,

(2.3) T_µ_δ

8

f ≤c X

0≤j≤k

T_j,kf.

To study P

0≤j≤k

T_j,kf, we will adapt the argument developed by M. Christ (see [1]) to the setting of biparametric dilations. First of all, we prove the following Proposition 2.3. There exists a positive constantc >0such that for0≤j ≤k,

kT_j,kk2l+2

2l+1,^2l+2_2l−1 ≤c.

Proof.

T_j,kf(x₁, x₂, x₃)

= Z

f(x₁−y₁, x₂−y₂, x₃−ϕ(y₁, y₂))θ 2^jy₁,2^ky₂

dy₁dy₂

= 2^−(j+k) Z

f x₁−2^−jy₁, x₂−2^−ky₂, x₃−ϕ 2^−jy₁,2^−ky₂

θ(y₁, y₂)dy₁dy₂

= 2^−(j+k)T^(j−k)fj,k 2^jx1,2^kx2,2^(j+k)lx3

,

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where we denote T^(j)f(x₁, x₂, x₃) =

Z

f x₁−y₁, x₂−y₂, x₃−y^l₂P y₁,2^jy₂

θ(y₁, y₂)dy₁dy₂

and

f_j,k(x₁, x₂, x₃) =f 2^−j,2^−k

◦(x₁, x₂, x₃) . So

(2.4) kTj,kf(x1, x2, x3)k_q = 2^(j+k)(^1+lp −^1+l_q −1)

T^(j−k)

p,qkfk_p. Now,

det y₂^lP y1,2^j−ky2

⁰⁰

= 2(2−2l)(j−k)

det (ϕ)⁰⁰ y1,2^j−ky2

.

so as in the proof of Lemma 4 in [3] we obtain that there existsc > 0such that T^(j−k)

2l+2

2l+1,^2l+2_2l−1 ≤cfor0≤j ≤k,and the proposition follows.

We take0 ≤ j ≤ k,and denote byµj,k andµ^(j) the measures associated to T_j,k andT^(j) respectively. Forξ = (ξ_1,ξ₂, ξ₃),

µ\^(j−k)(ξ) = Z

e⁻ⁱ(^ξ¹^y¹^+ξ²^y²^+ξ³^y2^lP(^y¹^,2^j−k^y²))θ(y₁, y₂)dy₁dy₂.

If for someξon the unit sphere,Ω^(j−k)_ξ (y₁, y₂) = ξ₁y₁+ξ₂y₂+ξ₃y₂^lP y₁,2^j−ky₂ has a critical point belonging to thesuppθ,then

ξ1+ξ3y₂^lP1 y1,2^j−ky2

= 0

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and

ξ2+ξ3 2^j−ky^l₂P2 y1,2^j−ky2

+ly^l−1₂ P y1,2^j−ky2

= 0,

but then, since P1(y) 6= 0for y ∈ Vδ1,from the first equation we obtain that there exist constants a, b ∈ Z with a < bsuch 2^a|ξ₃| ≤ |ξ₁| ≤ 2^b|ξ₃|, and, from the second one and the choice of δ we obtain constants c, d ∈ Z² with c < dsuch that2^c|ξ3| ≤ |ξ2| ≤2^d|ξ3|. Soξbelongs to the cone

C₀ =

ξ∈R³ : 2^a|ξ₃|<|ξ₁|<2^b|ξ₃|, 2^c|ξ₃|<|ξ₂|<2^d|ξ₃| . Lemma 2.4. SupposeC₀is as above. Then the family of cones

2^j,2^k

◦C₀ _j,k∈

Z

has finite overlapping (i.e.,#

(j, k)∈Z² :C₀∩ 2^j,2^k

◦C₀

6=∅ <∞).

Proof. We supposeξ ∈C₀ and 2^j,2^k

◦ξ ∈C₀,then

2^a|ξ₃|<|ξ₁|<2^b|ξ₃|, 2^c|ξ₃|<|ξ₂|<2^d|ξ₃| and

2^(j+k)l+a|ξ₃|<2^j|ξ₁|<2^(j+k)l+b|ξ₃|, 2^(j+k)l+c|ξ₃|<2^k|ξ₂|<2^(j+k)l+d|ξ₃| so

2^j|ξ₁|<2^(j+k)l+b|ξ₃|<2^(j+k)l+b−a|ξ₁| and

2^b|ξ₃|>|ξ₁|>2^−j2^(j+k)l+a|ξ₃|, so

a−b−kl < j(l−1)< b−a−kl,

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analogously we obtain

c−d−jl < k(l−1)< d−c−jl, thus

(c−d) (l−1) + (a−b)l

l²−(l−1)² < k < (d−c) (l−1) + (b−a)l l²−(l−1)² and so

a−b

l−1 −l(d−c) (l−1) + (b−a)l l²−(l−1)²

(l−1)

< j < (b−a)

l−1 +l(d−c) (l−1) + (b−a)l l²−(l−1)²

(l−1) .

We definem₀(ξ) =n(ξ₁, ξ₃)r(ξ₂, ξ₃)wherenandrbelong toC^∞(R²− {0}), are homogeneous of degree zero with respect to the isotropic dilations,

suppn ⊂

(ξ₁, ξ₃) : 2^a−1|ξ₃|<|ξ₁|<2^b+1|ξ₃| n ≥0andn ≡1on

(ξ₁, ξ₃) : 2^a|ξ₃|<|ξ₁|<2^b|ξ₃| , suppr⊂

(ξ₂, ξ₃) : 2^c−1|ξ₃|<|ξ₂|<2^d+1|ξ₃| ,

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r ≥0andr ≡1on

(ξ₂, ξ₃) : 2^c|ξ₃|<|ξ₂|<2^d|ξ₃| ,som₀is homogeneous of degree zero with respect to the isotropic dilations, it belongs to C^∞ on each octant ofR³, m0 ≥0, m0 ≡1onC0and

suppm0 ⊂Cf0

=

ξ ∈R³ : 2^a−1|ξ₃|<|ξ₁|<2^b+1|ξ₃|, 2^c−1|ξ₃|<|ξ₂|<2^d+1|ξ₃| . For(j, k)∈Z²,we definem_j,k(ξ) =m₀ 2^−j,2^−k

◦ξ

andQ_j,kthe operator with multiplier mj,k. If ξ belongs to an open octant of R³ then ξ belongs to

2^j,2^k

◦C0 for some(j, k)∈ Z²(indeed2^−k ∼ ^|ξ_|ξ¹^|

3| and2^−j ∼ ^|ξ_|ξ²^|

3|) and from the previous lemma, it belongs to a finite number of them (independent of ξ).

So P

(j,k)∈Z²

m_j,k(ξ) ≤ c. Now it is easy to check that, for 1 < p < ∞, there existsA_p >0such that forf ∈L²∩L^p and any choice ofε_j,k =±1,

(2.5)

X

(j,k)∈Z²

ε_j,kQ_j,kf p

≤A_pkfk_p.

Indeed, we now show that

m(ξ) = X

(j,k)∈Z²

ε_j,km_j,k(ξ)

satisfies the hypothesis of the Marcinkiewicz Theorem, as stated in Theorem 6’

in [7].

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We have just observed that

|m(ξ)| ≤ X

(j,k)∈Z²

m_j,k(ξ)≤c.

Now we want to estimate

∂

∂ξ1m(ξ)

. We recall that _∂ξ^∂

1m₀ is homogeneous of degree−1.We pickξ in an open octant. In a small neighborhood ofξ only finitely many(j, k)∈Z²(independent ofξ) are involved. For each one of them,

∂

∂ξ₁mj,k(ξ) = 2^−j ∂

∂ξ₁m0 2^−jξ1,2^−kξ2,2^−(j+k)lξ3

≤c2^−j

2^−jξ1,2^−kξ2,2^−(j+k)lξ3

−1 ≤c2^−j 2^−jξ1

−1,

so

sup

ξ2,ξ3

Z 2^s+1

2^s

∂

∂ξ₁m(ξ)

dξ₁ ≤c,

and in a similar way (using the homogeneity of the derivatives of m_j,k) we obtain that for each0< k≤3,

sup

ξk+1,...,ξ3

Z

ρ

∂^k

∂ξ₁...∂ξ_km(ξ)

dξ₁ ≤c,

asρranges over dyadic rectangles ofR^kand that this inequality holds for every one of the six pemutations of the variablesξ₁, ξ₂, ξ₃.

We now define h(ξ) ∈ C^∞(R³), h ≥ 0, h ≡ 1 on the unit ball of R³, hj,k(ξ) = h 2^−j,2^−k

◦ξ

andRj,k the operators with multipliershj,k.

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Lemma 2.5. There exists a constantC > 0,independent ofK,such that

X

0≤j≤k≤K

T_j,kR_j,k

2l+2 2l+1,2l+2

2l−1

≤C.

Proof. LetK_j,k be the kernel ofT_j,kR_j,k.A computation shows that, K_j,k(x) = 2^(j+k)l µ^(j−k)∗hˆ^∨

2^j,2^k

◦x .

Thus

X

0≤j≤k≤K

|K_j,k(ξ)| ≤ X

0≤j≤k

2^(j+k)l

G^(j,k) 2^j,2^k

◦ξ

withG^(j,k)defined by G^(j,k)∧

= µ^(j−k)∧

h.Sincej−k ≤0, as in Lemma 7 in [3] we obtain that G^(j,k)∧

∈ S(R³)with each seminorm bounded on j, k, it follows that the same holds forG^(j,k).Now

X

0≤j≤k

2^(j+k)l

G^(j,k) 2^j,2^k

◦ξ

≤ X

j,k,h≥0

2^ja+ka+ha

G^(j,k,h) 2^jξ₁,2^kξ₂,2^hξ₃

witha= _l+1^l , G^(j,k,h)=G^(j,k)forh=l(j+k)andG^(j,k,h) = 0otherwise. It is well known that from the uniform boundedness properties of G^(j,k,h)it follows that

X

j,k,h≥0

2^ja+ka+ha

G^(j,k,h) 2^jξ₁,2^kξ₂,2^hξ₃

≤ c

|ξ₁|â|ξ₂|â|ξ₃|â,

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so

X

0≤j≤k≤K

|Kj,k(ξ)| ≤ c

|ξ₁|^l+1^l |ξ₂|^l+1^l |ξ₃|^l+1^l ,

so P

0≤j≤k≤K

T_j,kR_j,kconvolvesL^p(R³)intoL^q(R³)for¹_q = _p¹−_l+1¹ with bounds independent ofK.

Lemma 2.6. There exists a constantC > 0,independent ofK,such that

X

1≤j≤k≤K

Tj,k(I−Pj,k) (I−Qj,k)

2l+2 2l+1,2l+2

2l−1

≤C.

Proof. The kernelH_j,k of X

1≤j≤k≤K

Tj,k(I−Pj,k) (I−Qj,k)

satisfies

X

1≤j≤k≤K

|H_j,k(ξ)| ≤ X

0≤j≤k

2^(j+k)l

g^(j,k) 2^j,2^k

◦ξ

withg^(j,k)defined by g^(j,k)^∧

= µ^(j−k)^∧

(1−h) (1−m₀). Observe that, from Lemma 7 in [3], we have µ^(j−k)^∧

(1−h) (1−m₀) ∈ S(R³)with each seminorm bounded onj, k. From this fact the proof follows as in the previous lemma.

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Proof of the theorem. We have just observed that it is enough to prove (2.1).

Since we can suppose f ≥ 0, by (2.3), we need only check that there exists C > 0, independent ofK such that

X

0≤j≤k≤K

T_j,k

2l+2 2l+1,2l+2

2l−1

≤C,

where T_j,k are defined by (2.2). For a constant c₀ > 0, we define Q⁰_j,k = P

|i−j|≤c0

Q_i,k. SoQ⁰_j,k have the same properties asQ_j,k andQ⁰_j,k ◦Q_j,k = Q_j,k thus we have that (2.5) holds forQ⁰_j,k.Then, for1< p <∞and

F ={f_j,k}_j,k≥0 ∈L^p l² ,

X

j,k≥0

Q⁰_j,kf_j,k p

≤c_pkFk_Lp(l²).

We decompose X

0≤j≤k≤K

Tj,kf

= X

0≤j≤k≤K

Tj,k(I−Pj,k) I−Q⁰_j,k

f+ X

0≤j≤k≤K

Tj,kPj,kf

+ X

0≤j≤k≤K

T_j,kQ⁰_j,k(I −P_j,k)f.

Now, proceeding as in [1], the theorem follows from Proposition2.3, Lemmas 2.5 and2.6and the remarks in [8, p. 85] concerning the multiparameter maximal function.

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References

[1] M. CHRIST, Endpoint bounds for singular fractional integral operators, UCLA, preprint, (1988).

[2] E. FERREYRA, T. GODOY AND M. URCIUOLO, Endpoint bounds for convolution operators with singular measures, Coll. Math., 76(1) (1998), 35–47.

[3] E. FERREYRA, T. GODOY AND M. URCIUOLO, The type set for homogeneous singular measures onR³ of polynomial type, to appear in Coll.

Math.

[4] I. IKROMOV, M. KEMPEANDD. MÜLLER, Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces, Duke Math. J., 126(3) (2005), 471–490.

[5] A. IOSEVICHANDE. SAWYER, SharpL^p−L^qestimates for a class of av- eraging operators, Annales de l’institut Fourier, 46(5) (1996), 1359–1384.

[6] F. RICCIANDE.M. STEIN, Harmonic analysis on nilpotent groups and sin- gular integrals. III. Fractional integration along manifolds, J. Funct. Anal., 86 (1989), 360–389.

[7] E.M. STEIN, Singular Integrals and Differentiability Properties of Func- tions, Princeton University Press (1970).

[8] E.M. STEIN, Harmonic Analysis. Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press (1993).