\(L^ p\) bounds on singular integrals in homogenization. (English) Zbl 0761.42008

The authors study the \(L^ p\)-boundedness of operators \[ (\partial/\partial x^ \alpha)(\partial/\partial x^ \beta)(-L)^{- 1},\;(\partial/\partial x^ \alpha)(-L)^{-1/2},\;(-L)^{- 1/2}(\partial/\partial x^ \alpha),\;(\partial/\partial x^ \alpha)(- L)^{-1}(\partial/\partial x^ \beta), \] \(1\leq\alpha,\beta\leq n\), where \(L\) belongs to a class of the second-order elliptic operators arising in the theory of homogenization.


42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
35J99 Elliptic equations and elliptic systems
Full Text: DOI


[1] Applications of homogenization theory in harmonic analysis II: Parabolic Harnack inequalities and Riesz transforms on solvable Lie groups of polynomial growth, preprint, and La conjecture de Kato pour des opérateurs différentiels à coefficients periocliques, C. R. Acad. Sci. Paris 312, Série I, 1991, pp. 263–266.
[2] Agmon, Comm. Pure Appl. Math. 12 pp 623– (1959)
[3] Comm. Pure Appl. Math. 17 pp 35– (1963)
[4] Avellaneda, Comm. Pure Appl. Math. 40 pp 803– (1987)
[5] Avellaneda, Comm. Pure Appl. Math. 42 pp 139– (1989)
[6] Avellaneda, C. R. Acad. Sci. Paris 309 pp 245– (1989)
[7] Avellaneda, Appl. Math. and Optimization 15 pp 109– (1987)
[8] , and , Asymptotic Analysis of Periodic Structures, North-Holland, Amsterdam 1978.
[9] Fabes, Ann. of Math. 119 pp 121– (1984)
[10] Kozlov, Math. USSR Sbornik 41 pp 249– (1982)
[11] Kozlov, Math. USSR Sbornik 35 pp 418– (1979)
[12] Kozlov, Russian Math. Surveys 34 pp 69– (1979)
[13] Lin, Comm. P. D. E. 10 pp 1767– (1985)
[14] Sevastjanova, Math. USSR Sbornik 43 pp 181– (1982)
[15] Topics in Harmonic Analysis, Princeton Univ. Press, 1970.
[16] Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970. · Zbl 0207.13501
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.