×

Convolution singular integrals on Lipschitz surfaces. (English) Zbl 0763.42009

The authors analyse some classes of convolution singular integral operators on \(\mathbb{R}^{n+1}\), embedded in the Clifford algebra \(\mathbb{R}_{(n)}\) with identity \(e_ 0\). Let \(\Sigma=\{g(x)e_ 0+{\mathbf x}\in\mathbb{R}^{n+1}\): \({\mathbf x}\in\mathbb{R}^ n\}\), where \(g\) is a Lipschitz function which satisfies \(|\nabla g|_ \infty\leq\tan\omega\), \(0\leq\omega<\pi/2\). Consider a right-monogenic (right-Clifford-regular) function \(\varphi\) which satisfies \(|\varphi(x)|\leq C| x|^{-n}\) on a sector \(S_ \mu=\{x=x_ 0+{\mathbf x}\in\mathbb{R}^{n+1}\): \(| x_ 0|<|{\mathbf x}|\tan\mu\}\) where \(\omega<\mu<\pi/2\). Suppose there exists a bounded function \(\Phi\) satisfying \(\Phi(R)- \Phi(r)=\int_{x\in\mathbb{R}^ n, r<| x|<R} \varphi(x)dx\). Then the related convolution singular integral operator \[ (Tu)(x)=\lim_{\varepsilon\to 0+}\bigl\{ \int_{y\in\Sigma, | x- y|\geq\varepsilon} \varphi(x-y)n(y)u(y)dS_ y+\Phi(\varepsilon n(x))u(x)\bigr\} \] is bounded on \(L_ p(\Sigma)\) for \(1<p<\infty\). Here \(dS_ y\) is the surface measure of \(\Sigma\) and \(n(y)\) is the outer unit normal defined for almost all \(y\in\Sigma\). Their result is an extension of the known one-dimensional case, and at the same time gives other proofs of the known results on the singular Cauchy integral operators on \(\Sigma\). They give some comments on bounds on the \(L^ p\)-boundedness of these operators.
Reviewer: K.Yabuta (Nara)

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. · Zbl 0529.30001
[2] A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 4, 1324 – 1327. · Zbl 0373.44003
[3] R. R. Coifman, A. McIntosh, and Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur \?&sup2; pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), no. 2, 361 – 387 (French). · Zbl 0497.42012 · doi:10.2307/2007065
[4] R. R. Coifman and Y. Meyer, Fourier analysis of multilinear convolutions, Calderón’s theorem, and analysis of Lipschitz curves, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 104 – 122. · Zbl 0427.42006
[5] R. R. Coifman, Peter W. Jones, and Stephen Semmes, Two elementary proofs of the \?&sup2; boundedness of Cauchy integrals on Lipschitz curves, J. Amer. Math. Soc. 2 (1989), no. 3, 553 – 564. · Zbl 0713.42010
[6] Björn E. J. Dahlberg, Poisson semigroups and singular integrals, Proc. Amer. Math. Soc. 97 (1986), no. 1, 41 – 48. · Zbl 0595.31009
[7] G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), no. 4, 1 – 56 (French). · Zbl 0604.42014 · doi:10.4171/RMI/17
[8] G. I. Gaudry, R. L. Long, and T. Qian, A martingale proof of \( {L^2}\) -boundedness of Clifford-valued singular integrals, Ann. Mat. Pura Appl. (to appear). · Zbl 0814.42009
[9] John E. Gilbert and Margaret A. M. Murray, \?^{\?}-theory on Euclidean space and the Dirac operator, Rev. Mat. Iberoamericana 4 (1988), no. 2, 253 – 289. · Zbl 0711.35089 · doi:10.4171/RMI/74
[10] V. Iftimie, Fonctions hypercomplexes, Bull. Math. Soc. Sci. Math. R. S. Roumanie 9 (57) (1965), 279 – 332 (1966) (French). · Zbl 0177.36903
[11] Carlos E. Kenig, Weighted \?^{\?} spaces on Lipschitz domains, Amer. J. Math. 102 (1980), no. 1, 129 – 163. · Zbl 0434.42024 · doi:10.2307/2374173
[12] C. Li, A. McIntosh, and T. Qian, Clifford algebras, Fourier transforms, and singular convolution operators on Lipschitz surfaces, submitted for publication. · Zbl 0817.42008
[13] Alan McIntosh, Clifford algebras and the higher-dimensional Cauchy integral, Approximation and function spaces (Warsaw, 1986) Banach Center Publ., vol. 22, PWN, Warsaw, 1989, pp. 253 – 267.
[14] Alan McIntosh and Tao Qian, Fourier theory on Lipschitz curves, Miniconference on harmonic analysis and operator algebras (Canberra, 1987) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 15, Austral. Nat. Univ., Canberra, 1987, pp. 157 – 166. · Zbl 0645.42012
[15] Alan McIntosh and Tao Qian, Convolution singular integral operators on Lipschitz curves, Harmonic analysis (Tianjin, 1988) Lecture Notes in Math., vol. 1494, Springer, Berlin, 1991, pp. 142 – 162. · Zbl 0791.42012 · doi:10.1007/BFb0087766
[16] Yves Meyer, Ondelettes et opérateurs. II, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 (French). Opérateurs de Calderón-Zygmund. [Calderón-Zygmund operators]. · Zbl 0745.42011
[17] Margaret A. M. Murray, The Cauchy integral, Calderón commutators, and conjugations of singular integrals in \?\(^{n}\), Trans. Amer. Math. Soc. 289 (1985), no. 2, 497 – 518. · Zbl 0574.42012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.