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ac.kr Department of Mathematics, Ajou University, Suwon 443-749, South Korea Abstract Based on a uniform estimate of convolution operators with measures on a family of plane curves, we o

R E S E A R C H Open Access

Convolution estimates related to space curves

Youngwoo Choi

Correspondence: youngwoo@ajou.

ac.kr

Department of Mathematics, Ajou

University, Suwon 443-749, South

Korea

Abstract

Based on a uniform estimate of convolution operators with measures on a family of plane curves, we obtain optimal Lp-Lqboundedness of convolution operators with affine arclength measures supported on space curves satisfying a suitable condition The result generalizes the previously known estimates

2000 Mathematics Subject Classifications: Primary 42B15; Secondary 42B20

Keywords: affine arclength, convolution operators

1 Introduction

Let I⊂ ℝ be an open interval and ψ : I ® ℝ be a C3

function Let g : I ® ℝ3

be the curve given by g(t) = (t, t2/2,ψ(t)), t Î I Associated to g is the affine arclength measure

dsgonℝ3

determined by



R3

f d σ γ =



I

0 (R3)

with

λ(t) =ψ(3)(t)1

6 , t ∈ I.

The Lp- Lq mapping properties of the corresponding convolution operatorT σ γgiven

by

T σ γ f (x) = f ∗ σ γ (x) =



I

have been studied by many authors [1-8] The use of the affine arclength measure was suggested by Drury [2] to mitigate the effect of degeneracy and has been helpful

to obtain uniform estimates

We denote byΔ the closed convex hull of {(0, 0), (1, 1), (p0-1, q0-1) (p1-1, q1-1)} in the plane, where p0 = 3/2, q0= 2, p1= 2 and q1 = 3 The line segment joining (p0-1, q0-1) and (p1-1, q1-1) is denoted byS It is well known that the typeset ofT σ γis contained in

Δ and that under suitable conditionsT σ γis bounded from Lp(ℝ3

) to Lq (ℝ3

) with uni-form bounds whenever(p−1, q−1)∈ S The most general result currently available was obtained by Oberlin [5] In this article, we establish uniform endpoint estimates onT σ γ

for a wider class of curves g

Before we state our main result, we introduce certain conditions on functions defined on intervals For an interval J1in ℝ, a locally integrable function F : J1 ® ℝ+

,

© 2011 Choi; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,

and a positive real number A, we let

G(, A) :=ω : J1→R+|ω(s1)ω(s2) ≤ A

s2− s1

 s2

s1

(s)ds

whenever s1< s2and [s, s2]⊂ J1



and

E1(A) := { : J →R+| ∈ G(, A)}.

An interesting subclass ofE1(2A)is the collectionE2(A), introduced in [9], of func-tionsF : J ® ℝ+

such that

1.F is monotone; and

2 whenever s1< s2and [s1, s2]⊂ J,



(s1)(s2)≤ A((s1+ s2)/2)

Our main theorem is the following:

Theorem 1.1 Let I = (a, b) ⊂ ℝ be an open interval and let ψ : I ® ℝ be a C3

func-tion such that

1.ψ(3) (t)≥ 0, whenever t Î I;

2 there exists A Î (0, ∞) such that, for each u Î (0, b - a),

Fu : (a, b − u) →R+

given byFu (s) :=

ψ(3)(s + u) ψ(3)(s)satisfies

Then, the operatorT σ γdefined by(1.1) is a bounded operator from Lp(ℝ3

) to Lq(ℝ3

) whenever(p−1, q−1)∈ S, and the operator normT σ γ

L p →L qis dominated by a constant that depends only on A

The case whenψ(3)∈E2(A)was considered by Oberlin [5] One can easily see that

ψ(3)∈E2(A/2)implies (1.2) uniformly in uÎ (0, b - a) The theorem generalizes many

results previously known for convolution estimates related to space curves, namely [1-6]

This article is organized as follows: in the following section, a uniform estimate for convolution operators with measures supported on plane curves The proof of

Theo-rem 1.1 based on a T*T method is given in Section 3

2 Uniform estimates on the plane

The following theorem motivated by Oberlin [10] which is interesting in itself will be

useful:

Theorem 2.1 Let J be an open interval in ℝ, and j : J ® ℝ be a C2

function such that j″ ≥ 0 Let ω : J ® ℝ be a nonnegative measurable function Suppose that there

exists a positive constant A such thatω ∈ G(φ, A), i.e.

ω(s1)1/2ω(s2)1/2≤ A

s2− s1

 s2

s1

φ(v)dv

holds whenever s1 < s2 and[s1, s2]⊂ J Let S be the operator given by

Sg (x2, x3) =



J g(x2− s, x3− φ(s))ω1/3(s)ds

for g ∈ C∞

0(R2) Then, there exists a constant C that depends only on A such that

||Sg|| L3 (R2 )≤ C||g|| L3/2 (R 2 )

holds uniformly in g ∈ C∞

0(R2) Proof of Theorem 2.1 Our proof is based on the method introduced by Drury and Guo [11], which was later refined by Oberlin [10]

We have

||Sg||3

3=



R



R



J



J



J

3



j=1

g x2− s j , x3− φ s j

ω1/3 s j

ds1ds2ds3dx2dx3

=



R



R



R G g (z1,·) , g (z2,·) , g (z3,·)(z1, z2, z3) dz1dz2dz3,

where for z1, z2, z3Î ℝ and suitable functions h1, h2, h3defined on ℝ,

[G(h1, h2, h3)(z1, z2, z3) :=



R



J (z1,z2,z3)

3



j=1 [h j (x3− φ(x2− z j))ω1/3(x2− z j)]

dx2dx3,

and

J(z1, z2, z3) := (J + z1)∩ (J + z2)∩ (J + z3)

We will prove that the estimate

|[G(h1, h2, h3)](z1, z2, z3)| ≤ C ||h1||L3/2(R) ||h2||L3/2(R) ||h3||L3/2(R)

|(z1− z2)(z1− z3)(z2− z3)|1/3 (2:1) holds uniformly in h1, h2, h3, z1, z2, and z3

To establish (2.1) we let

[G k (h1, h2, h3)](z1, z2, z3) :=



R



J (z1,z2,z3) h k (x3− φ(x2− z k))



1≤j≤3 j [h j (x3− φ(x2− z j))ω1/2(x2− z j )]dx2dx3

for k = 1, 2, 3 Then, we have

|[ 1(h1, h2, h3)](z1, z2, z3 )| ≤ ||h1 || ∞



R



J(z1,z2,z3)

3



j=2

|h j (x3− φ(x2− z j))|ω1/2(x2− z j)

dx2dx3

For z2, z3Î ℝ and x2 Î J (z1, z2, z3), we have

φ)(x

2− z2)− φ(x

2− z3)=

 x2−z3

x2−z2

φ(s)ds



≥ A−1|z2− z3|ω1/2(x2− z2)ω1/2(x2− z3)

by hypothesis Hence,

|[ 1(h1, h2, h3)](z1, z2, z3 ) | ≤A ||h1 || ∞

|z2− z3 |



R



J(z1,z2,z3)

3



j=2

|h j (x3− φ(x2− z j)) |

|φ(x − z )− φ(x − z )|dx dx .

A change of variables gives

|[ 1(h1, h2, h3)](z1, z2, z3)| ≤ A ||h1||∞

|z2− z3|



R



R|h2(z2)| |h3(z3)|dz2dz3

Thus, we obtain

|[ 1(h1, h2, h3)](z1, z2, z3)| ≤ A ||h1||∞||h2||1||h3||1

Similarly, we get

|[ 2(h1, h2, h3)](z1, z2, z3)| ≤ A ||h1||1||h2||∞||h3||1

and

|[ 2(h1, h2, h3)](z1, z2, z3)| ≤ A ||h1||1||h2||1||h3||∞

Interpolating (2.2), (2.3) and (2.4) provides (2.1) Combining this with Proposition 2.2

in Christ [12] finishes the proof

The special case in which ω = j″ provides a uniform estimate for the convolution operators with affine arclength measure on plane curves

Corollary 2.2 Let J be an open interval in ℝ, and j : J ® ℝ be a C2

function such that j″ ≥ 0 Suppose that there exists a constant A such thatφ∈E1(A), i.e

φ(s

1)1/2φ(s

2)1/2≤ A

s2− s1

 s2

s1

φ(v)dv

holds whenever s1 < s2 and[s1, s2]⊂ J LetSbe the operator given by

Sg(x2, x3) =



J g(x2− s, x3− φ(s))φ(s)1/3ds

for g ∈ C∞

0(R2) Then, there exists a constant C that depends only on A such that

||Sg|| L3 (R 2 )≤ C||g|| L3/2 (R2 )

holds uniformly in g ∈ C∞

0(R2)

3 Proof of the main theorem

Before we proceed the proof of Theorem 1.1, we note that the uniform estimate (1.2)

in uÎ (0, b - a) implies

by continuity ofψ(3)

By duality and interpolation, it suffices to show that

holds uniformly for f Î L3/2(ℝ3

)

Recall the following lemma observed by Oberlin [3]:

Lemma 3.1 Suppose there exists a constant C1such that

||T∗

holds uniformly in fÎ L3/2

(R3) Then, (3.2) holds for each fÎ L3/2

(ℝ3 )

To establish (3.3), we write

T∗σ γ T σ γ f (x) =



I



I

f (x − γ (t) + γ (s))λ(t)λ(s) dtds

equivT(1)f (x) + T(2)f (x),

where

T(1)f (x) =



t,s ∈I

t >s

f (x − γ (t) + γ (s))λ(t)λ(s) dtds,

T(2)f (x) =



t,s ∈I

t <s

f (x − γ (t) + γ (s))λ(t)λ(s) dtds.

By symmetry, it suffices to prove

|| (1)f||L3 (R3 )≤ C1||f || L3/2 (R3 )

Next we make a change of variables, u = t - s and write for uÎ (0, b - a)

I u={s ∈ R : a < s < b − u},

u (s) = ψ(s + u) − ψ(s).

Then, we obtain:

T(1)f (x) =



I

 b −s

0

f (x1− u, x2− u(s + u/2), x3− u (s)) λ(s + u)λ(s) duds

=

 b −a 0



I u

f (x1− u, x2− u(s + u/2), x3− u (s)) λ(s + u)λ(s) dsdu,

and so

T(1)f (x1, x2, x3) =

 b −a

0

T u [f u (x1− u, ·, ··)]((x2− u2/2)/u, x3) du

u2/3,

where

f u (x1, x2, x3) := u1/3f (x1, ux2, x3)

T u g(x2, x3) :=



I u

g(x2− s, x3− u (s)) 1/3u (s) ds

u (s) := u λ3(s + u) λ3(s)

= u



ψ(3)(s + u) ψ(3)(s)

for x1, x2, x3Î ℝ, u Î (0, b - a), s Î Iu Notice that for u Î (0, b - a) and [s1, s2]⊂ Iu, we have

1/2

u (s1) 1/2u (s2)≤ Au

s2− s1

 s2

s1



ψ(3)(s + u) ψ(3)(s)ds

s2− s1

 s2

s1

1

u

 s+u

s

ψ(3)(v)dvds

2

s2− s1

 s2

s1

(ψ(s + u) − ψ(s))ds

2

s2− s1

 s2

s1



u (s)ds

by (1.2) and (3.1) By Theorem 2.1,|| u||L3/2 (R 2 )→L 3 (R 2 )is uniformly bounded Hence,

we obtain

|| (1)f|| 3 ≤

⎛

⎜

⎝



R

⎡

⎣

R2

b −a

0



T u f u (x1− u, ·, ··)



x2− u2 /2



u du2/3

 3

dx2dx3

⎤

⎦

1

3 ·3

dx1

⎞

⎟

⎠

1 3

≤

⎛

⎜

R

⎡

⎢ b −a

0



R2



T u f u (x1− u, ·, ··)



x2− u2 /2



3dx2dx3

 1

du

u2/3

⎤

⎥

3

dx1

⎞

⎟

1

≤ C(A)

⎛

⎜

R

⎡

⎣ b −a

0

u

1

3||f u (x1− u, ·, ··) || L3/2 (R2 )

du

u2/3

⎤

⎦

3

dx1

⎞

⎟

1

≤ C(A)

⎛

⎝

R

b −a

0 ||f (x1− u, ·, ··)|| L3/2 (R2 )

du

u2/3

! 3

dx1

⎞

⎠

1 3

.

By Hardy-Littlewood-Sobolev theorem on fractional integration, we obtain

|| (1)f||3≤ C1(A) ||f ||3/2

This finishes the proof of Theorem 1.1

Competing interests

The author declares that they have no competing interests.

Received: 27 April 2011 Accepted: 25 October 2011 Published: 25 October 2011

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2011 2011:91.

(ℝ3 )

To establish (3.3), we write

T∗σ... )|dx dx .