Open Access

Convolution estimates related to space curves

Journal of Inequalities and Applications20112011:91

https://doi.org/10.1186/1029-242X-2011-91

Received: 27 April 2011

Accepted: 25 October 2011

Published: 25 October 2011

Abstract

Based on a uniform estimate of convolution operators with measures on a family of plane curves, we obtain optimal L p -L q 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.

Keywords

affine arclengthconvolution operators

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
3 f d σ γ = I f ( γ ( t ) ) λ ( t ) d t , f C 0 ( 3 )
with
λ ( t ) = ψ ( 3 ) ( t ) 1 6 , t I .
The L p - L q 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 L p (3) to L q (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
( Φ , A ) : = ω : J 1 + ω ( s 1 ) ω ( s 2 ) A s 2 - s 1 s 1 s 2 Φ ( s ) d s whenever s 1 < s 2 and s , s 2 J 1
and
E 1 ( A ) : = { Φ : J + Φ ( 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 ) + 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 L p (3) to L q (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 ( 2 ) . Then, there exists a constant C that depends only on A such that
S g L 3 ( 2 ) C g L 3 2 ( 2 )

holds uniformly in g C 0 ( 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 = 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 = 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 ) : = 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 h 2 L 3 2 h 3 L 3 2 ( 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 ) : = 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 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 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 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 ( 2 ) . Then, there exists a constant C that depends only on A such that
S g L 3 ( 2 ) C g L 3 2 ( 2 )

holds uniformly in g C 0 ( 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 ( 3 ) C f L 3 2 ( 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 ( 3 ) C 1 f L 3 2 ( 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 ( 3 ) C 1 f L 3 2 ( 3 ) .
Next we make a change of variables, u = t - s and write for u (0, b - a)
I u = { s : 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 ( 2 ) L 3 ( 2 ) is uniformly bounded. Hence, we obtain
T ( 1 ) f 3 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 0 b - a 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 ) 0 b - a u 1 3 f u x 1 - u , , L 3 2 ( 2 ) d u u 2 3 3 d x 1 1 3 C ( A ) 0 b - a f ( x 1 - u , , ) L 3 2 ( 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.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Ajou University

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Copyright

© Choi; licensee Springer. 2011

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