RESEARC 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 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 arclength, convolution operators 1 Introduction Let I ⊂ ℝ be an open interval and ψ : I ® ℝ be a C 3 function. Let g : I ® ℝ 3 be the curve given by g(t)=(t, t 2 /2,ψ(t)), t Î I. Associated to g is the affine arclength measure ds g on ℝ 3 determined by R 3 fdσ γ = I f (γ (t))λ(t) dt, f ∈ C ∞ 0 (R 3 ) with λ(t)= ψ (3) (t ) 1 6 , t ∈ I . The L p - L q mapping propertie s 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 [1-8]. The use of the affine arclength measure was suggested by Drury [2] to mitigate the effect of degen eracy and has been helpful to obtain uniform estimates. We denote by Δ the closed convex hull of {(0, 0), (1, 1), (p 0 -1 , q 0 -1 )(p 1 -1 , q 1 -1 )} in the plane, where p 0 =3/2,q 0 =2,p 1 =2andq 1 = 3. The line segment joining (p 0 -1 , q 0 -1 ) and (p 1 -1 , q 1 -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 (ℝ 3 )toL q (ℝ 3 ) 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 on T σ γ for a wider class of curves g. Before we state our main result, we introduce certain conditions on functions defined o n intervals. For an interval J 1 in ℝ, a locally integrable function F : J 1 ® ℝ + , Choi Journal of Inequalities and Applications 2011, 2011:91 http://www.journalofinequalitiesandapplications.com/content/2011/1/91 © 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 p ermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. and a positive real number A, we let G(, A):= ω : J 1 → R + | ω(s 1 )ω( s 2 ) ≤ A s 2 − s 1 s 2 s 1 (s)ds whenever s 1 < s 2 and [s, s 2 ] ⊂ J 1 and E 1 ( A ) := { : J → R + | ∈ G ( , A ) } . An interesting subclass of E 1 ( 2A ) is the collection E 2 ( A ) ,introducedin[9],offunc- tions F : J ® ℝ + such that 1. F is monotone; and 2. whenever s 1 <s 2 and [s 1 , s 2 ] ⊂ J, (s 1 )(s 2 ) ≤ A((s 1 + s 2 )/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 C 3 func- tion such that 1. ψ (3) (t) ≥ 0, whenever t Î I ; 2. the re exists A Î (0, ∞ ) such that, for each u Î (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 L p (ℝ 3 ) to L q (ℝ 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 Î (0, b - a). The theorem generalize s 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 i nteresting in itself will be useful: Theorem 2.1. Let J be an open interval in ℝ, and j : J ® ℝ be a C 2 function such that j″ ≥ 0. Let ω : J ® ℝ 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 2 s 1 φ (v)d v holds whenever s 1 <s 2 and [s 1 , s 2 ] ⊂ J. Let S be the operator given by Sg ( x 2 , x 3 ) = J g(x 2 − s, x 3 − φ(s))ω 1/3 (s)d s Choi Journal of Inequalities and Applications 2011, 2011:91 http://www.journalofinequalitiesandapplications.com/content/2011/1/91 Page 2 of 6 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 | |Sg|| 3 3 = R R J J J 3 j=1 g x 2 − s j , x 3 − φ s j ω 1/3 s j ds 1 ds 2 ds 3 dx 2 dx 3 = R R R G g ( z 1 , · ) , g ( z 2 , · ) , g ( z 3 , · ) ( z 1 , z 2 , z 3 ) dz 1 dz 2 dz 3 , where for z 1 , z 2 , z 3 Î ℝ and suitable functions h 1 , h 2 , h 3 defined on ℝ, [G(h 1 , h 2 , h 3 )(z 1 , z 2 , z 3 ):= R J(z 1 ,z 2 ,z 3 ) 3 j=1 [h j (x 3 − φ(x 2 − z j ))ω 1/3 (x 2 − z j )] dx 2 dx 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 h 1 , h 2 , h 3 , z 1 , z 2 , and z 3 . 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 )]dx 2 dx 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 ) 3 j=2 |h j (x 3 − φ(x 2 − z j ))|ω 1/2 (x 2 − z j ) dx 2 dx 3 . For z 2 , z 3 Î ℝ and x 2 Î J (z 1 , z 2 , z 3 ), we have φ )(x 2 − z 2 ) − φ (x 2 − z 3 ) = x 2 −z 3 x 2 −z 2 φ (s)ds ≥ 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 ) 3 j=2 |h j (x 3 − φ(x 2 − z j ))| |φ ( x 2 − z 2 ) − φ ( x 2 − z 3 ) |dx 2 dx 3 . Choi Journal of Inequalities and Applications 2011, 2011:91 http://www.journalofinequalitiesandapplications.com/content/2011/1/91 Page 3 of 6 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 )|dz 2 dz 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 = j″ provides a unifo rm 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 C 2 function such that j″ ≥ 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 2 s 1 φ (v)d v holds whenever s 1 <s 2 and [s 1 , s 2 ] ⊂ J. Let S be the operator given by Sg(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 Î (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 Î L 3/2 (ℝ 3 ). Recall the following lemma observed by Oberlin [3]: Lemma 3.1. Suppose there exists a constant C 1 such 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 Î L 3/2 (R 3 ). Then, (3.2) holds for each f Î L 3/2 (ℝ 3 ). Choi Journal of Inequalities and Applications 2011, 2011:91 http://www.journalofinequalitiesandapplications.com/content/2011/1/91 Page 4 of 6 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) dtd s 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 | |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 Î (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 (x 1 − u, x 2 − u(s + u/2), x 3 − u (s))λ(s + u)λ(s) duds = b−a 0 I u f (x 1 − u, x 2 − u(s + u/2), x 3 − u (s))λ(s + u)λ(s) dsdu , and so T (1) f (x 1 , x 2 , x 3 )= b −a 0 T u [f u (x 1 − u, ·, ··)]((x 2 − u 2 /2)/u, x 3 ) du u 2/3 , where f u (x 1 , x 2 , x 3 ):=u 1/3 f (x 1 , ux 2 , x 3 ) T u g(x 2 , x 3 ):= I u g(x 2 − s, x 3 − u (s)) 1/3 u (s) ds u (s):=uλ 3 (s + u)λ 3 (s) = u ψ (3) (s + u)ψ (3) (s) for x 1 , x 2 , x 3 Î ℝ , u Î (0, b - a), s Î I u . Notice that for u Î (0, b - a) and [s 1 , s 2 ] ⊂ I u , we have 1/2 u (s 1 ) 1/2 u (s 2 ) ≤ Au s 2 − s 1 s 2 s 1 ψ (3) (s + u)ψ (3) (s)d s ≤ A 2 u s 2 − s 1 s 2 s 1 1 u s+u s ψ (3) (v)dvds = A 2 s 2 − s 1 s 2 s 1 (ψ (s + u) − ψ (s))ds = A 2 s 2 − s 1 s 2 s 1 u (s)ds Choi Journal of Inequalities and Applications 2011, 2011:91 http://www.journalofinequalitiesandapplications.com/content/2011/1/91 Page 5 of 6 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 b−a 0 T u f u (x 1 − u, ·, ··) x 2 − u 2 /2 u , x 3 du u 2/3 3 dx 2 dx 3 ⎤ ⎦ 1 3 ·3 dx 1 ⎞ ⎟ ⎟ ⎠ 1 3 ≤ ⎛ ⎜ ⎝ R ⎡ ⎢ ⎣ b−a 0 R 2 T u f u (x 1 − u, ·, ··) x 2 − u 2 /2 u , x 3 3 dx 2 dx 3 1 3 du u 2/3 ⎤ ⎥ ⎦ 3 dx 1 ⎞ ⎟ ⎠ 1 3 ≤ C(A) ⎛ ⎜ ⎝ R ⎡ ⎣ b−a 0 u 1 3 ||f u ( x 1 − u, ·, ·· ) || L 3/2 (R 2 ) du u 2/3 ⎤ ⎦ 3 dx 1 ⎞ ⎟ ⎠ 1 3 ≤ C(A) ⎛ ⎝ R b−a 0 ||f (x 1 − u, ·, ··)|| L 3/2 (R 2 ) du u 2/3 3 dx 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. Competing interests The author declares that they have no competing interests. Received: 27 April 2011 Accepted: 25 October 2011 Published: 25 October 2011 References 1. Choi, Y: The L p -L q mapping properties of convolution operators with the affine arclength measure on space curves. 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RESEARC H Open Access Convolution estimates related to space curves Youngwoo Choi Correspondence: youngwoo@ ajou. ac.kr Department of Mathematics, Ajou University,. 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. 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