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\(H^ p\) continuity properties of Calderón-Zygmund-type operators. (English) Zbl 0596.42006

In a recent paper, G. David and J. L. Journé obtained a characterization of the Caldéron-Zygmund operators, in the sense of R. Coifman and Y. Meyer. These operators are an interesting generalization of the classical singular integral operators of Calderón-Zygmund. In particular, they include some classes of pseudodifferential operators, Cauchy integrals on Lipschitz curves, commutators of order \(n\), etc. The \(L^ p\) theory of these operators is now well understood.
In this paper we consider the \(H^ p\) theory (in the sense of R. Coifman and G. Weiss) and obtain the following result: Let \(T\) be a \(\delta\) C-Z operator such that \(T^*(1)=0\). Then, \(T\) extends to a continuous operator from \(H^ p\) into itself for \(1\geq p>n/n+\delta\). The proof of this theorem consists in showing that the image under \(T\) of a \((p,\infty)\)-atom is what we call a \((p,q,\alpha)\)-molecule.
In the second part of the paper we introduce the class of strongly singular Calderón-Zygmund operators. Our study of these operators is motivated by the multiplier operators whose symbols are given by \(\exp (i| \xi |^ a)/| \xi |^{\beta}\) away from the origin. These multiplier operators have been studied by several authors. We show that our strongly singular C-Z operators map \(L^{\infty}\) into BMO, extending a well-known result of C. Fefferman and E. M. Stein. Moreover, we show that a strongly singular C-Z operator \(T\) satisfying \(T^*(1)=0\) acts continuous on \(H^ p\) spaces for a certain range of values of \(p\), \(1\geq p>p_ 0\). For convolution operators on \({\mathbb{R}}\), R. Coifman has obtained the continuity for the critical index \(p_ 0\). We are not able to proof \(H^{p_ 0}\) continuity in the general case. However, we single out a class of convolution operators of strongly singular type, on \({\mathbb{R}}^ n\), for which a general version of Coifman’s result holds.
In the last part of the paper we consider a certain class of pseudodifferential operators. In fact, following a suggestion of E. M. Stein, we show that our class of strongly singular Calderón-Zygmund operators includes the pseudodifferential operators with symbols in \(S^{-b}_{a,\delta}\) where \(0<\delta \leq a<1\), \((1- a)n/2\leq b<n/2\). The case \(\delta <a\) was considered by C. Fefferman.
Reviewer: J. Alvarez

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
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[1] Alvarez, J., An algebra of \(L^p\) bounded pseudo-differential operators, J. Math. Anal. Appl., 94, 268-282 (1983) · Zbl 0519.35084
[2] Alvarez, J.; Calderón, A. P., Functional calculi for pseudo-differential operators I, (de Guzmán, M.; Peral, I., Proceedings Seminar on Fourier Analysis (1979)), 1-61, El Escorial
[3] Bordin, B., \(H^p\) estimates for weakly strongly singular integral operators on spaces of homogeneous type, Studia Math., 75, 217-234 (1983) · Zbl 0464.47029
[4] Ching, Chin-Hung, Pseudodifferential operators with non-regular symbols, J. Differential Equations, 11, 436-447 (1972) · Zbl 0248.35106
[5] Coifman, R., A real variable characterization of \(H^p\), Studia Math., 51, 269-274 (1974) · Zbl 0289.46037
[6] Coifman, R.; Meyer, Y., Au delà des opérateurs pseudo-différentiels, Astérisque, 57, 1-184 (1979)
[7] Coifman, R.; Weiss, G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83, 569-645 (1977) · Zbl 0358.30023
[8] David, G.; Journé, J. L., A boundness-criterion for generalized Calderón-Zygmund operators, Ann. of Math., 120, 371-397 (1984) · Zbl 0567.47025
[9] Fefferman, C., Inequalities for strongly singular convolution operators, Acta Math., 124, 9-36 (1970) · Zbl 0188.42601
[10] Fefferman, C., \(L^p\) bounds for pseudo-differential operators, Israel J. Math., 14, 413-417 (1973) · Zbl 0259.47045
[11] Fefferman, C.; Stein, E. M., \(H^p\) spaces of several variables, Acta Math., 129, 137-193 (1972) · Zbl 0257.46078
[12] Journé, J. L., Calderón-Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón, (Lecture Notes in Math., Vol. 994 (1983), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0508.42021
[13] Hirschman, I. I., On multiplier transformations, Duke Math. J., 26, 221-242 (1959) · Zbl 0085.09201
[14] Sjölin, P., An \(H^p\) inequality for strongly singular integrals, Math. Z., 165, 231-238 (1979) · Zbl 0378.47028
[15] Stein, E. M., Singular Integrals Harmonic functions and differentiability properties of functions of several variables, (Princeton Math. Series, Vol. 34 (1981), Princeton Univ. Press: Princeton Univ. Press Princeton, N. J) · Zbl 0177.39101
[16] Taylor, M. E., Pseudo-differential operators, (Princeton Math. Series, Vol. 34 (1981), Princeton Univ. Press: Princeton Univ. Press Princeton N. J) · Zbl 0207.45402
[17] Wainger, S., Special trigonometric series in \(k\) dimensions, Mem. Amer. Math. Soc., 59 (1965) · Zbl 0136.36601
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