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# Convolution estimates related to space curves

## Abstract

Based on a uniform estimate of convolution operators with measures on a family of plane curves, we obtain optimal Lp -Lq boundedness 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.

## 1 Introduction

Let I be an open interval and ψ : I be a C3 function. Let γ : I3 be the curve given by γ(t) = (t, t2/2,ψ(t)), t I. Associated to γ is the affine arclength measure γ on 3 determined by

$∫ ℝ 3 f d σ γ = ∫ I f ( γ ( t ) ) λ ( t ) d t , f ∈ C 0 ∞ ( ℝ 3 )$

with

$λ ( t ) = ψ ( 3 ) ( t ) 1 6 , t ∈ I .$

The Lp - Lq mapping properties of the corresponding convolution operator $T σ γ$ given by

$T σ γ f ( x ) = f * σ γ ( x ) = ∫ I f ( x - γ ( t ) ) λ ( t ) d t$
(1.1)

have been studied by many authors [18]. 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 by Δ 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 by $S$. It is well known that the typeset of $T σ γ$ is contained in Δ and that under suitable conditions $T σ γ$ is bounded from Lp (3) to Lq (3) with uniform 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 on $T σ γ$ for a wider class of curves γ.

Before we state our main result, we introduce certain conditions on functions defined on intervals. For an interval J1 in , a locally integrable function Φ : J1+, and a positive real number A, we let

and

$E 1 ( A ) : = { Φ : J → ℝ + ∣ Φ ∈ ℭ ( Φ , A ) } .$

An interesting subclass of $E 1 ( 2 A )$ is the collection $E 2 ( A )$, introduced in [9], of functions Φ : J+ such that

1. 1.

Φ is monotone; and

2. 2.

whenever s 1 < s 2 and [s 1, s 2] J,

$Φ ( s 1 ) Φ ( s 2 ) ≤ A Φ ( ( s 1 + s 2 ) ∕ 2 )$

Our main theorem is the following:

Theorem 1.1. Let I = (a, b) be an open interval and let ψ : I be a C3function such that

1. ψ(3)(t) ≥ 0, whenever t I;

2. there exists A (0, ) such that, for each u (0, b - a), $F u : ( a , b − u ) → ℝ +$ given by$F u ( s ) := ψ ( 3 ) ( s + u ) ψ ( 3 ) ( s )$satisfies

$F u ∈ E 1 ( A ) .$
(1.2)

Then, the operator$T σ γ$defined by (1.1) is a bounded operator from Lp (3) to Lq (3) whenever $( p − 1 , q − 1 ) ∈ S$, and the operator norm$T σ γ L p → L q$is dominated by a constant that depends only on A.

The case when $ψ ( 3 ) ∈ E 2 ( A )$ was considered by Oberlin [5]. One can easily see that $ψ ( 3 ) ∈ E 2 ( A ∕ 2 )$ implies (1.2) uniformly in u (0, b - a). The theorem generalizes many results previously known for convolution estimates related to space curves, namely [16].

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 Theorem 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 , and ϕ : J be a C2function such that ϕ″ ≥ 0. Let ω : J be a nonnegative measurable function. Suppose that there exists a positive constant A such that $ω∈G( ϕ ″ ,A)$, i.e.

$ω ( s 1 ) 1 ∕ 2 ω s 2 1 ∕ 2 ≤ A s 2 - s 1 ∫ s 1 s 2 ϕ ″ ( v ) d v$

holds whenever s1< s2and [s1, s2] J. Let S be the operator given by

$S g x 2 , x 3 = ∫ J g ( x 2 - s , x 3 - ϕ ( s ) ) ω 1 ∕ 3 ( s ) d s$

for$g∈ C 0 ∞ ( ℝ 2 )$. Then, there exists a constant C that depends only on A such that

$∥ S g ∥ L 3 ( ℝ 2 ) ≤ C ∥ g ∥ L 3 ∕ 2 ( ℝ 2 )$

holds uniformly in$g∈ C 0 ∞ ( ℝ 2 )$.

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

$∥ S g ∥ 3 3 = ∫ ℝ ∫ ℝ ∫ J ∫ J ∫ J ∏ j = 1 3 g x 2 - s j , x 3 - ϕ s j ω 1 ∕ 3 s j d s 1 d s 2 d s 3 d x 2 d x 3 = ∫ ℝ ∫ ℝ ∫ ℝ G g z 1 , ⋅ , g z 2 , ⋅ , g z 3 , ⋅ z 1 , z 2 , z 3 d z 1 d z 2 d z 3 ,$

where for z1, z2, z3 and suitable functions h1, h2, h3 defined on ,

$[ G ( h 1 , h 2 , h 3 ) ( z 1 , z 2 , z 3 ) : = ∫ ℝ ∫ J ( z 1 , z 2 , z 3 ) ∏ j = 1 3 [ h j ( x 3 − ϕ ( x 2 − z j ) ) ω 1 / 3 ( x 2 − z j ) ] d x 2 d x 3 ,$

and

$J ( z 1 , z 2 , z 3 ) : = ( J + z 1 ) ∩ ( J + z 2 ) ∩ ( J + z 3 ) .$

We will prove that the estimate

$∣ [ G ( h 1 , h 2 , h 3 ) ] ( z 1 , z 2 , z 3 ) ∣ ≤ C ∥ h 1 ∥ L 3 ∕ 2 ℝ ∥ h 2 ∥ L 3 ∕ 2 ℝ ∥ h 3 ∥ L 3 ∕ 2 ℝ ∣ ( z 1 - z 2 ) ( z 1 - z 3 ) ( z 2 - z 3 ) ∣ 1 ∕ 3$
(2.1)

holds uniformly in h1, h2, h3, z1, z2, and z3.

To establish (2.1) we let

$[ G k ( h 1 , h 2 , h 3 ) ] ( z 1 , z 2 , z 3 ) : = ∫ ℝ ∫ J z 1 , z 2 , z 3 h k ( x 3 - ϕ ( x 2 - z k ) ) ∏ 1 ≤ j ≤ 3 j ≠ k [ h j ( x 3 - ϕ ( x 2 - z j ) ) ω 1 ∕ 2 ( x 2 - z j ) ] d x 2 d x 3$

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

$∣ [ G 1 ( h 1 , h 2 , h 3 ) ] ( z 1 , z 2 , z 3 ) ∣ ≤ ∥ h 1 ∥ ∞ ∫ ℝ ∫ J z 1 , z 2 , z 3 ∏ j = 2 3 ∣ h j ( x 3 - ϕ ( x 2 - z j ) ) ∣ ω 1 ∕ 2 ( x 2 - z j ) d x 2 d x 3 .$

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

$| ϕ ′ ) ( x 2 − z 2 ) − ϕ ′ ( x 2 − z 3 ) | = | ∫ x 2 − z 2 x 2 − z 3 ϕ ″ ( s ) d s | ≥ A − 1 ∣ z 2 − z 3 ∣ ω 1 / 2 ( x 2 − z 2 ) ω 1 / 2 ( x 2 − z 3 )$

by hypothesis. Hence,

$∣ [ G 1 ( h 1 , h 2 , h 3 ) ] ( z 1 , z 2 , z 3 ) ∣ ≤ A ∥ h 1 ∥ ∞ z 2 - z 3 ∫ ℝ ∫ J z 1 , z 2 , z 3 ∏ j = 2 3 ∣ h j ( x 3 - ϕ ( x 2 - z j ) ) ∣ ∣ ϕ ′ ( x 2 - z 2 ) - ϕ ′ ( x 2 - z 3 ) ∣ d x 2 d x 3 .$

A change of variables gives

$∣ [ G 1 ( h 1 , h 2 , h 3 ) ] ( z 1 , z 2 , z 3 ) ∣ ≤ A ∥ h 1 ∥ ∞ ∣ z 2 - z 3 ∣ ∫ ℝ ∫ ℝ ∣ h 2 ( z 2 ) ∣ ∣ h 3 ( z 3 ) ∣ d z 2 d z 3 .$

Thus, we obtain

$∣ [ G 1 ( h 1 , h 2 , h 3 ) ] ( z 1 , z 2 , z 3 ) ∣ ≤ A ∥ h 1 ∥ ∞ ∥ h 2 ∥ 1 ∥ h 3 ∥ 1 ∣ z 2 - z 3 ∣ .$
(2.2)

Similarly, we get

$∣ [ G 2 ( h 1 , h 2 , h 3 ) ] ( z 1 , z 2 , z 3 ) ∣ ≤ A ∥ h 1 ∥ 1 ∥ h 2 ∥ ∞ ∥ h 3 ∥ 1 ∣ z 1 - z 3 ∣$
(2.3)

and

$∣ [ G 2 ( h 1 , h 2 , h 3 ) ] ( z 1 , z 2 , z 3 ) ∣ ≤ A ∥ h 1 ∥ 1 ∥ h 2 ∥ 1 ∥ h 3 ∥ ∞ ∣ z 1 - z 2 ∣ .$
(2.4)

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 = ϕ″ 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 , and ϕ : J be a C2function such that ϕ″ ≥ 0. Suppose that there exists a constant A such that$ϕ ″ ∈ E 1 ( A )$, i.e.

$ϕ ″ ( s 1 ) 1 ∕ 2 ϕ ″ ( s 2 ) 1 ∕ 2 ≤ A s 2 - s 1 ∫ s 1 s 2 ϕ ″ ( v ) d v$

holds whenever s1< s2and [s1, s2] J. Let$S$be the operator given by

$S g ( x 2 , x 3 ) = ∫ J g ( x 2 - s , x 3 - ϕ ( s ) ) ϕ ″ ( s ) 1 ∕ 3 d s$

for$g∈ C 0 ∞ ( ℝ 2 )$. Then, there exists a constant C that depends only on A such that

$∥ S g ∥ L 3 ( ℝ 2 ) ≤ C ∥ g ∥ L 3 ∕ 2 ( ℝ 2 )$

holds uniformly in$g∈ C 0 ∞ ( ℝ 2 )$.

## 3 Proof of the main theorem

Before we proceed the proof of Theorem 1.1, we note that the uniform estimate (1.2) in u (0, b - a) implies

$ψ ( 3 ) ∈ E 1 ( A )$
(3.1)

by continuity of ψ(3).

By duality and interpolation, it suffices to show that

$∥ T σ γ f ∥ L 2 ( ℝ 3 ) ≤ C ∥ f ∥ L 3 ∕ 2 ( ℝ 3 )$
(3.2)

holds uniformly for f L3/2 (3).

Recall the following lemma observed by Oberlin [3]:

Lemma 3.1. Suppose there exists a constant C1such that

$∥ T σ γ * T σ γ f ∥ L 3 ( ℝ 3 ) ≤ C 1 ∥ f ∥ L 3 ∕ 2 ( ℝ 3 )$
(3.3)

holds uniformly in f L3/2 (R3). Then, (3.2) holds for each f L3/2 (3).

To establish (3.3), we write

$T σ γ * T σ γ f ( x ) = ∫ I ∫ I f ( x - γ ( t ) + γ ( s ) ) λ ( t ) λ ( s ) d t d s e q u i v T ( 1 ) f ( x ) + T ( 2 ) f ( x ) ,$

where

$T ( 1 ) f ( x ) = ∬ t , s ∈ I t > s f ( x - γ ( t ) + γ ( s ) ) λ ( t ) λ ( s ) d t d s , T ( 2 ) f ( x ) = ∬ t , s ∈ I t < s f ( x - γ ( t ) + γ ( s ) ) λ ( t ) λ ( s ) d t d s .$

By symmetry, it suffices to prove

$∥ T ( 1 ) f ∥ L 3 ( ℝ 3 ) ≤ C 1 ∥ f ∥ L 3 ∕ 2 ( ℝ 3 ) .$

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

$I u = { s ∈ ℝ : a < s < b - u } , Ψ u ( s ) = ψ ( s + u ) - ψ ( s ) .$

Then, we obtain:

$T ( 1 ) f ( x ) = ∫ I ∫ 0 b - s f ( x 1 - u , x 2 - u ( s + u ∕ 2 ) , x 3 - Ψ u ( s ) ) λ ( s + u ) λ ( s ) d u d s = ∫ 0 b - a ∫ I u f ( x 1 - u , x 2 - u ( s + u ∕ 2 ) , x 3 - Ψ u ( s ) ) λ ( s + u ) λ ( s ) d s d u ,$

and so

$T ( 1 ) f ( x 1 , x 2 , x 3 ) = ∫ 0 b - a T u [ f u ( x 1 - u , ⋅ , ⋅ ⋅ ) ] ( ( x 2 - u 2 ∕ 2 ) ∕ u , x 3 ) d u u 2 ∕ 3 ,$

where

$f u ( x 1 , x 2 , x 3 ) : = u 1 ∕ 3 f ( x 1 , u x 2 , x 3 ) T u g ( x 2 , x 3 ) : = ∫ I u g ( x 2 - s , x 3 - Ψ u ( s ) ) Λ u 1 ∕ 3 ( s ) d s Λ u ( s ) : = u λ 3 ( s + u ) λ 3 ( s ) = u ψ ( 3 ) ( s + u ) ψ ( 3 ) ( s )$

for x1, x2, x3 , u (0, b - a), s I u .

Notice that for u (0, b - a) and [s1, s2] I u , we have

$Λ u 1 ∕ 2 ( s 1 ) Λ u 1 ∕ 2 ( s 2 ) ≤ A u s 2 - s 1 ∫ s 1 s 2 ψ ( 3 ) ( s + u ) ψ ( 3 ) ( s ) d s ≤ A 2 u s 2 - s 1 ∫ s 1 s 2 1 u ∫ s s + u ψ ( 3 ) ( v ) d v d s = A 2 s 2 - s 1 ∫ s 1 s 2 ( ψ ″ ( s + u ) - ψ ″ ( s ) ) d s = A 2 s 2 - s 1 ∫ s 1 s 2 Ψ ″ u ( s ) d s$

by (1.2) and (3.1). By Theorem 2.1, $∥ T u ∥ L 3 ∕ 2 ( ℝ 2 ) → L 3 ( ℝ 2 )$ is uniformly bounded. Hence, we obtain

$∥ T ( 1 ) f ∥ 3 ≤ ∫ ℝ ∬ ℝ 2 ∫ 0 b - a T u f u ( x 1 - u , ⋅ , ⋅ ⋅ ) x 2 - u 2 ∕ 2 u , x 3 d u u 2 ∕ 3 3 d x 2 d x 3 1 3 ⋅ 3 d x 1 1 3 ≤ ∫ ℝ ∫ 0 b - a ∬ ℝ 2 T u f u ( x 1 - u , ⋅ , ⋅ ⋅ ) x 2 - u 2 ∕ 2 u , x 3 3 d x 2 d x 3 1 3 d u u 2 ∕ 3 3 d x 1 1 3 ≤ C ( A ) ∫ ℝ ∫ 0 b - a u 1 3 ∥ f u x 1 - u , ⋅ , ⋅ ⋅ ∥ L 3 ∕ 2 ( ℝ 2 ) d u u 2 ∕ 3 3 d x 1 1 3 ≤ C ( A ) ∫ ℝ ∫ 0 b - a ∥ f ( x 1 - u , ⋅ , ⋅ ⋅ ) ∥ L 3 ∕ 2 ( ℝ 2 ) d u u 2 ∕ 3 3 d x 1 1 3 .$

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

$∥ T ( 1 ) f ∥ 3 ≤ C 1 ( A ) ∥ f ∥ 3 ∕ 2$

This finishes the proof of Theorem 1.1.

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Correspondence to Youngwoo Choi.

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Choi, Y. Convolution estimates related to space curves. J Inequal Appl 2011, 91 (2011). https://doi.org/10.1186/1029-242X-2011-91

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### Keywords

• affine arclength
• convolution operators