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Weighted estimates of singular integrals and their applications

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Abstract

We give a survey of research on the problem of single-weighted and double-weighted estimates of strong and weak types for the Hardy-Littlewood maximal function, Riesz potentials, singular integral operators, and harmonic functions. Necessary and sufficient conditions on the weight are given under which weighted estimates are valid (Muckenhoupt's Ap-condition, Sawyer's condition, etc.). Special attention is given to papers which appeared after 1980 and the latest results, published as reports and preprints.

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Literature cited

  1. L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand Reinhold (1966).

  2. K. I. Babenko, “Conjugate functions,” Dokl. Akad. Nauk SSSR,62, 157–160 (1948).

    Google Scholar 

  3. V. M. Badkov, “Convergence in the mean and almost everywhere of Fourier series with respect to polynomials, orthogonal on a segment,” Mat. Sb.,95, No. 2, 229–262 (1974).

    Google Scholar 

  4. N. K. Bari, Trigonometric Series [in Russian], Fizmatgiz, Moscow (1961).

    Google Scholar 

  5. G. M. Verzhbinskii and V. G. Maz'ya, “Asymptotic behavior of solutions of second-order elliptic equations near the boundary. II,” Sib. Mat. Zh.,13, No. 6, 1239–1271 (1972).

    Google Scholar 

  6. V. F. Gaposhkin, “Generalization of a theorem of M. Riesz on conjugate functions,” Mat. Sb.,46, No. 3, 359–372 (1958).

    Google Scholar 

  7. K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall (1962).

  8. S. M. Grudskii, Compactness of an Integral Operator [in Russian], Rostov State Univ. (1980).

  9. R. I. Gurielashvili, “Hilbert transform,” Soobshch. Akad. Nauk Gruz. SSR,103, No. 2, 273–276 (1981).

    Google Scholar 

  10. M. Guzman, Differentiation of Integrals inR n [Russian translation], Mir, Moscow (1978).

    Google Scholar 

  11. I. I. Danilyuk, Nonregular Boundary Problems on the Plane [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  12. N. Dunford and J. T. Schwartz, Linear Operators. Pt. I, Wiley (1958).

  13. N. Dunford and J. T. Schwartz, Linear Operators. Pt. II, Wiley.

  14. O. P. Dzhanelidze, “Boundedness of a multidimensional singular operator in spaces with weight,” Soobshch. Akad. Nauk Gruz. SSR,87, No. 2, 301–303 (1977).

    Google Scholar 

  15. A. É. Dzhrbashyan, “Weighted estimate for the function g *λ ,” Izv. Akad. Nauk Arm. SSR,14, No. 5, 338–347 (1979).

    Google Scholar 

  16. A. É. Dzhrbashyan, “Weighted inequalities for a Littlewood-Paley function in domains with conical points,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,92, 279–282 (1979).

    Google Scholar 

  17. A. É. Dzhrbashyan, “Weighted inequalities for operators of Littlewood-Paley type and. multiplicators of Fourier transforms,” Izv. Akad. Nauk Arm. SSR,16, No. 3, 202–212 (1981).

    Google Scholar 

  18. A. É. Dzhrbashyan, “Littlewood-Paley theory in some spaces of harmonic functions and multiplicators of Fourier transforms,” Candidate's Dissertation, Leningrad State Univ. (1981).

  19. E. M. Dyn'kin, “Estimates of analytic functions in Jordan domains,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,73, 70–90 (1977).

    Google Scholar 

  20. E. M. Dyn'kin, “Sets of free interpolation for Hölder classes,” Mat. Sb.,109, No. 1, 107–128 (1979).

    Google Scholar 

  21. E. M. Dyn'kin, “Free interpolation by functions with derivative from H1,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR (in press).

  22. L. V. Zhizhiashvili, Conjugate Functions and Trigonometric Series [in Russian], Tbilisi State Univ. (1969).

  23. A. Zygmund, Trigonometric Series, Vol. 1, Cambridge Univ. Press.

  24. A. Zygmund, Trigonometric Series, Vol. 2, Cambridge Univ. Press.

  25. K. S. Kazaryan, “Bases and unconditional bases in weighted spaces Lp, 1⩽p<∞,” Dokl. Akad. Nauk Arm. SSR,65, No. 5, 271–275 (1977).

    Google Scholar 

  26. M. K. Karapetyants and B. S. Rubin, “Radial Riesz potentials on a disk and fractional integration operators,” Dokl. Akad. Nauk SSSR,263, No. 6, 1299–1302 (1982).

    Google Scholar 

  27. V. É. Katsnel'son, “When is L2 the direct sum of H 2− and H 2+ ?” in: Theory of Functions, Functional Analysis, and Their Applications [in Russian], No. 35, Kharkov (1981), pp. 45–48.

    Google Scholar 

  28. V. M. Kokilashvili, “Boundedness of singular integral operators in the space Lp with weight,” in: Memoirs of the Symposium on the Mechanics of Continuous Media and Related Problems of Analysis [in Russian], Vol. 1, Metsnieraba, Tbilisi (1973), pp. 125–141.

    Google Scholar 

  29. V. M. Kokilashvili, “Multiplicators of Fourier transforms,” Dokl. Akad. Nauk SSSR,220, No. 1, 19–22 (1975).

    Google Scholar 

  30. V. M. Kokilashvili, “Boundedness of a singular operator with Cauchy kernel in the space Lp with weight,” Soobshch. Akad. Nauk Gruz. SSR,77, No. 3, 529–532 (1975).

    Google Scholar 

  31. V. M. Kokilashvili, “Singular integrals and maximal operators with Cauchy kernel,” Dokl. Akad. Nauk SSSR,223, No. 3, 555–558 (1975).

    Google Scholar 

  32. V. M. Kokilashvili, “Multiplicators of Fourier transforms and imbedding theorems in certain function spaces,” Mat. Zametki,20, No. 4, 605–610 (1976).

    Google Scholar 

  33. V. M. Kokilashvili, “Singular integrals and multiplicators of Fourier transforms,” Tr. Tbilis. Mat. Inst. Akad. Nauk Gruz. SSR,53, 38–61 (1976).

    Google Scholar 

  34. V. M. Kokilashvili, “Maximal singular integral operators with Cauchy kernel,” Tr. Tbilis. Mat. Inst. Akad. Nauk Gruz. SSR,55, 38–58 (1977).

    Google Scholar 

  35. V. M. Kokilashvili, “Weighted inequalities for singular integrals with Cauchy kernel on smooth contours,” Soobshch. Akad. Nauk Gruz. SSR,90, No. 3, 537–540 (1978).

    Google Scholar 

  36. V. M. Kokilashvili, “Maximal inequalities and multiplicators in weighted Lizorkin-Tribel' spaces,” Dokl. Akad. Nauk SSSR,239, No. 1, 42–45 (1978).

    Google Scholar 

  37. V. M. Kokilashvili, “Anisotropic Bessel potentials. Imbedding theorems with limit exponent,” Tr. Tbilis. Mat. Inst. Akad. Nauk Gruz. SSR,58, 134–149 (1978).

    Google Scholar 

  38. V. M. Kokilashvili, “Integral operators in weighted spaces,” Preprint of the School on Operator Theory in Function Spaces, Aug. 24–Sept. 1, 1979, Inst. Mat. Sib. Otd. Akad. Nauk SSSR, Novosibirsk (1979).

    Google Scholar 

  39. V. M. Kokilashvili, “Singular integrals of the type of the Calderon integrals,” Tr. Tbilis. Mat. Inst. Akad. Nauk Gruz. SSR,61, 5–14 (1979).

    Google Scholar 

  40. V. M. Kokilashvili, “Maximal functions in weighted spaces,” Tr. Tbilis. Mat. Inst. Akad. Nauk Gruz. SSR,65, 110–121 (1980).

    Google Scholar 

  41. V. M. Kokilashvili, “Weighted inequalities for maximal functions with respect to Vitali families,” Soobshch. Akad. Nauk Gruz. SSR,98, No. 3, 545–547 (1980).

    Google Scholar 

  42. V. M. Kokilashvili, “Multiple Hilbert transforms and multiplicators in weighted spaces,” Soobshch. Akad. Nauk Gruz. SSR,98, No. 2, 285–288 (1980).

    Google Scholar 

  43. V. M. Kokilashvili, “Bisingular integral operators in weighted spaces,” Soobshch. Akad. Nauk Gruz. SSR,101, No. 2, 289–292 (1981).

    Google Scholar 

  44. V. M. Kokilashvili, “Singular integrals and imbedding theorems in weighted function spaces. Boundary properties of functions,” Doctoral Dissertation, Tbilis Univ. (1981).

  45. A. S. Krantsberg, “Basis property of a Haar system in weighted spaces,” Tr. Mosk. Inst. Elektron. Mashinostr., No. 24, 14–26 (1972).

    Google Scholar 

  46. N. Ya. Krupnik, “Consequences of a theorem of Hunt, Muckenhoupt, and Wheeden,” Mat. Issled. Kishinev, No. 47, 64–70 (1978).

    Google Scholar 

  47. M. A. Lavrent'ev, “Boundary problems in the theory of schlicht functions,” Mat. Sb.,1, 815–844 (1936).

    Google Scholar 

  48. N. S. Landkof, Foundations of Modern Potential Theory [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  49. V. G. Maz'ya and S. P. Preobrazhenskii, “Estimates of (ℓ, p)-capacity and traces of potentials,” Technische Hochschule Karl-Marx-Stadt, Sektion Mathematik, Wissenschaftliche Informationen,28, Karl-Marx-Stadt (1981).

  50. V. G. Maz'ya, “The negative spectrum of the Schrödinger operator,” Dokl. Akad. Nauk SSSR,144, No. 4, 721–722 (1962).

    Google Scholar 

  51. V. G. Maz'ya and Yu. E. Khaikin, “Continuity of singular integral operators in normed spaces,” Vestn. Leningr. Univ., No. 1, 28–34 (1975).

    Google Scholar 

  52. A. V. Maslov, “Estimates of Hardy-Littlewood-Paley type for Fourier coefficients with respect to general orthonormal systems,” Moscow State Univ. (1982). (Manuscript deposited in VINITI, March 21, 1980, No. 1370-82 Dep.)

  53. A. D. Nakhman, “Elliptic means of binary series and Fourier integrals,” Editors of Sib. Mat. Zh. Sib. Otd. Akad. Nauk SSSR, Novosibirsk (1981). (Manuscript deposited in VINITI December 3, 1981, No. 5505-81 Dep.)

  54. A. D. Nakhman and B. P. Osilenker, “Weighted estimates for linear means of Fourier series,” Mosk. Inzh.-Stroit. Inst., Moscow (1979). (Manuscript deposited in VINITI, May 25, 1979, No. 1861-79 Dep.)

  55. A. D. Nakhman and B. P. Osilenker, “Estimates of weighted norms of certain operators generated by multiple trigonometric Fourier series,” Izv. Vyssh. Uchebn. Zaved., Mat. No. 4, 39–50 (1982).

    Google Scholar 

  56. B. P. Osilenker, “Weighted polynomial approximation by linear means of functions of several variables,” Dokl. Akad. Nauk SSSR,247, No. 6, 1320–1324 (1979).

    Google Scholar 

  57. B. P. Osilenker, “Weighted estimates of the majorant of linear means of Fourier series with respect to orthogonal polynomials,” Usp. Mat. Nauk,35, No. 5, 239–240 (1980).

    Google Scholar 

  58. B. P. Osilenker, “Weighted estimates of the majorant of Cesaro means of Fourier series of ℓ-valued vector-functions,” Funkts. Anal. Prilozhen.,15, No. 1, 80–81 (1981).

    Google Scholar 

  59. V. A. Paatashvili, “Singular Cauchy integral on a countable set of closed contours,” Tr. Tbilis. Mat. Inst. Akad. Nauk Gruz. SSR,65, 122–130 (1980).

    Google Scholar 

  60. B. S. Pavlov, “Basis property of a system of exponentials and Muckenhoupt's condition,” Dokl. Akad. Nauk SSSR,247, No. 1, 37–40 (1979).

    Google Scholar 

  61. A. L. Rozin, “Singular integrals and maximal functions in the space L1,” Soobshch. Akad. Nauk Gruz. SSR,87, No. 1, 28–32 (1977).

    Google Scholar 

  62. S. G. Samko, “Proof of a theorem of Babenko-Stein,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 47–51 (1975).

    Google Scholar 

  63. G. Szego, Orthogonal Polynomials, Amer. Math. Soc. (1975).

  64. A. M. Sedletskii, “Biorthogonal expansions of functions in series of exponentials on intervals of the real axis,” Usp. Mat. Nauk,37, No. 5, 51–95 (1982).

    Google Scholar 

  65. I. B. Simonenko, “Closedness of the set {(ρ, γ):|ρ|1+γWρ(Γ)} and some other properties of weight functions for a Cauchy singular integral in the case of contours of type ℜ,” Rostov State Univ. (1980). (Manuscript deposited in VINITI February 27, 1981, No. 965-81 Dep.)

  66. I. B. Simonenko, “More on the Muckenhoupt conditions,” Rostov State Univ. (1981) (Manuscript deposited in VINITI May 20, 1982, No. 2557-82 Dep.)

  67. I. B. Simonenko, “Weight properties of exponentials of a multidimensional singular integral with continuous and bounded density,” Rostov State Univ. (1981) (Manuscript deposited in VINITI May 20, 1982, No. 2559-82 Dep.)

  68. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press (1971).

  69. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press (1971).

  70. T. A. Timan (T. A. Makanina), “Conditions for existence and estimation of transformations of Calderon-Zygmund type in weighted Lp-spaces,” Tr. Mat. Inst. Akad. Nauk SSSR,105, 213–229 (1969).

    Google Scholar 

  71. T. A. Timan, “Invariance of weighted Lp-classes with respect to singular integral transformations,” Tr. Mosk. Inzh.-Stroit. Inst.,153, 58–63 (1977).

    Google Scholar 

  72. Kh. Tribel', Theory of Interpolation, Function Spaces, Differential Operators [Russian translation], Mir, Moscow (1981).

    Google Scholar 

  73. G. Ts. Tumarkin, “Problems arising in considering classes of domains defined by properties of integrals of Cauchy type,” J. Sov. Math.,36, No. 5 (1984).

  74. C. Fefferman, “Multiplicator problem for the ball,” in: Mathematics [Russian translation], Vol. 18, No. 1 (1974), pp. 138–145.

    Google Scholar 

  75. G. H. Hardy, D. Littlewood, and G. Polya, Inequalities, Cambridge Univ. Press (1952).

  76. B. V. Khvedelidze, “Method of Cauchy-type integrals in discontinuous problems of the theory of holomorphic functions of one complex variable,” in: Contemporary Problems of Mathematics [in Russian], Vol. 7, Itogi Nauki i Tekh. VINITI Akad. Nauk SSSR, Moscow (1975), pp. 5–162.

    Google Scholar 

  77. S. V. Khrushchev, “Perturbation theorem for bases of exponentials and Muckenhoupt's condition,” Dokl. Akad. Nauk SSSR,247, No. 1, 44–48 (1979).

    Google Scholar 

  78. N. A. Shirokov, “Generalizations of a theorem of Littlewood-Paley,” J. Sov. Math.,8, No. 1 (1977).

  79. N. A. Shirokov, “Some imbedding theorems for spaces of harmonic functions,” J. Sov. Math.,14, No. 2 (1980).

  80. N. A. Shirokov, “Estimation in Lp(C) of some singular integral operators,” Izv. Akad. Nauk Arm. SSR,15, No. 1, 63–76 (1980).

    Google Scholar 

  81. D. R. Adams, Lectures on Lp-Potential Theory, Dep't. Math. Univ. Umea, No. 2 (1981).

  82. E. Adams, “Weighted norm inequalities for the Hilbert transform of functions with moments zero,” Trans. AMS,272, No. 2, 487–500 (1982).

    Google Scholar 

  83. N. E. Aguilera and E. O. Harboure, “Some inequalities for maximal operators,” Indiana Univ. Math. J.,29, No. 4, 559–576 (1980).

    Article  Google Scholar 

  84. N. E. Aguilera and C. Segovia, “Weighted norm inequalities relating the g *λ and the area functions,” Stud. Math. (PRL),61, No. 3, 293–303 (1977).

    Google Scholar 

  85. K. F. Andersen, “Weighted norm inequalities for Hilbert transforms and conjugate functions of even and odd functions,” Proc. AMS,56, 99–107 (1976).

    Google Scholar 

  86. K. F. Andersen, “Discrete Hilbert transforms and rearrangement invariant sequence spaces,” Appl. Anal.,5, 193–200 (1976).

    Google Scholar 

  87. K. F. Andersen, “On inequality for the Hilbert transform,” J. London Math. Soc.,16, No. 2, 290–296 (1977).

    Google Scholar 

  88. K. F. Andersen, “Inequalities with weights for discrete Hilbert transforms,” Can. Math. Bull.,20, No. 1, 9–16 (1977).

    Google Scholar 

  89. K. F. Andersen, “Weighted inequalities for the Stieltjes transformation and Hilbert's double series,” Proc. R. Soc. Edinburgh,A86, Nos. 1–2, 75–84 (1980).

    Google Scholar 

  90. K. F. Andersen, “Inequalities for scalar-valued linear operators that extend to their vector-valued analogues,” J. Math. Anal. Appl.,77, No. 1, 264–269 (1980).

    Article  Google Scholar 

  91. K. F. Andersen, “Weighted inequalities for the disc multiplier,” Proc. Am. Math. Soc.,83, No. 2, 269–275 (1981).

    Google Scholar 

  92. K. F. Andersen and R. T. John, “Weighted inequalities for vector-valued maximal functions and singular integrals,” Stud. Math. (PRL),69, No. 1, 19–31 (1980).

    Google Scholar 

  93. K. F. Andersen and R. A. Kerman, “Weighted norm inequalities for generalized Hankel conjugate transformation,” Stud. Math. (PRL),71, No. 1, 15–26 (1981).

    Google Scholar 

  94. K. F. Andersen and B. Muckenhoupt, “Weighted weak-type Hardy inequalities with applications to Hilbert transforms and maximal functions,” Stud. Math. (PRL),72, No. 1, 9–26 (1982).

    Google Scholar 

  95. R. Askey, “Norm inequalities for some orthogonal series,” Bull. Am. Math. Soc.,72, No. 5, 808–823 (1966).

    Google Scholar 

  96. R. Askey and I. I. Hirschman, “Mean summability for ultraspherical polynomials,” Math. Scand.,12, 167–177 (1963).

    Google Scholar 

  97. A. Baernstein, “Univalence and bounded mean oscillation,” Mich. Math. J.,23, No. 3, 217–223 (1976).

    Article  Google Scholar 

  98. P. R. Beesack and H. P. Heinig, “Hardy's inequalities with indices less than 1,” Proc. Am. Math. Soc.,83, No. 3, 532–536 (1981).

    Google Scholar 

  99. D. Bekolle, “Inegalites a poids pour le projecteur de Bergman dans la boule unite de Cn,” Stud. Math. (PRL),71, No. 3, 305–323 (1982).

    Google Scholar 

  100. D. Bekolle and A. Bonami, “Inegalites a poids pour le noyau de Bergman,” C. R. Acad. Sci.,AB 286, No. 18, 775–778 (1978).

    Google Scholar 

  101. A. Bonami and J. L. Clerk, “Sommes de Cesaro et multiplicateurs des developpements en harmoniques spheriques,” Trans. Am. Math. Soc.,183, 223–263 (1973).

    Google Scholar 

  102. J. S. Bradley, “Hardy inequalities with mixed norms,” Can. Math. Bull.,21, No. 4, 405–408 (1978).

    Google Scholar 

  103. J. Bruna, “Muckenhoupt's weights in some boundary problems of a complex variable,” Lect. Notes Math.,908, 74–85 (1982).

    Google Scholar 

  104. J. Burbea, “The Bergman projection on weighted norm spaces,” Can. J. Math.,32, No. 4, 979–986 (1980).

    Google Scholar 

  105. N. Burger, “Espace des fonctions a variation moyenne bornee sur un espace de nature homogene,” C. R. Acad. Sci.,A286, No. 3, 139–142 (1978).

    Google Scholar 

  106. D. L. Burkholder and R. F. Gundy, “Extrapolation and interpolation of quasilinear operators on martingales,” Acta Math.,124, Nos. 3–4, 249–304 (1970).

    Google Scholar 

  107. D. L. Burkholder and R. F. Gundy, “Distribution function inequalities for the area integral,” Stud. Math. (PRL),44, No. 6, 527–544 (1972).

    Google Scholar 

  108. A. P. Calderon, “Commutators of singular integral operators,” Proc. Nat. Acad. Sci. USA,53, 1092–1099 (1965).

    Google Scholar 

  109. A. P. Calderon, “Inequalities for the maximal function relative to a metric,” Stud. Math. (PRL),57, No. 3, 297–306 (1976).

    Google Scholar 

  110. A. P. Calderon, “On an integral of Marcinkiewicz,” Stud. Math. (PRL),57, No. 3, 279–284 (1976).

    Google Scholar 

  111. A. P. Calderon, “Cauchy integrals on Lipschitz curves and related operators,” Proc. Nat. Acad. Sci. USA,74, No. 4, 1324–1327 (1977).

    Google Scholar 

  112. A. P. Calderon, “Commutators, singular integrals on Lipschitz curves and applications,” in: Proc. Int. Congress Math., Helsinki, 15–23 Aug., 1978, Vol. 1, Helsinki (1980), pp. 85–86.

    Google Scholar 

  113. A. P. Calderon, C. P. Calderon, E. B. Fabes, M. Jodeit, and N. Riviere, “Applications of the Cauchy integral along Lipschitz curves,” Bull. Am. Math. Soc.,84, No. 2, 287–290 (1978).

    Google Scholar 

  114. A. P. Calderon and A. Torchinsky, “Parabolic maximal functions associated with a distribution,” Adv. Math.,16, No. 1, 1–64 (1975).

    Article  Google Scholar 

  115. A. P. Calderon and A. Torchinsky, “Parabolic maximal functions associated with a distribution. I,” Adv. Math.,24, No. 2, 101–171 (1977).

    Article  Google Scholar 

  116. A. P. Calderon and R. Vaillancourt, “On the boundedness of pseudodifferential operators,” J. Math. Soc. Jpn.,23, No. 2, 374–378 (1971).

    Google Scholar 

  117. A. P. Calderon and R. Vaillancourt, “A class of bounded pseudodifferential operators,” Proc. Nat. Acad. Sci. USA,69, No. 5, 1185–1187 (1972).

    Google Scholar 

  118. A. P. Calderon and A. Zygmund, “On singular integrals with variable kernels,” Appl. Anal.,7, No. 3, 221–238 (1978).

    Google Scholar 

  119. L. Carleson, “On interpolation problem for bounded analytic functions,” Am. J. Math.,80, No. 4, 921–930 (1958).

    Google Scholar 

  120. L. Carleson, “On convergence and growth of partial sums of Fourier series,” Acta Math.,116, Nos. 1–2, 135–157 (1966).

    Google Scholar 

  121. L. Carleson, “Two remarks on H1 and BMO,” Adv. Math.,22, No. 3, 269–277 (1976).

    Article  Google Scholar 

  122. L. Carleson, “BMO — 10 years' development,” in: 18 Scand. Congr. Math., Aarhus, 18–22 Aug. 1980. Proc., Boston (1981), pp. 3–21.

  123. L. Carleson and P. Jones, “Weighted norm inequalities and a theorem of Koosis,” Mittag-Leffler Inst., Rep., No. 2 (1981).

  124. Y. Chen, “Theorems of asymptotic approximation,” Math. Ann.,140, 360–407 (1960).

    Article  Google Scholar 

  125. Y. Chen, “On conjugate functions,” Can. J. Math.,15, No. 3, 486–494 (1963).

    Google Scholar 

  126. H.-M. Chung, R. A. Hunt, and D. S. Kurtz, “The Hardy-Littlewood maximal functions on L(p, q) spaces with weights,” Indiana Univ. Math. J.,31, No. 1, 109–120 (1982).

    Article  Google Scholar 

  127. R. R. Coifman, “Distribution function inequalities for singular integrals,” Proc. Nat. Acad. Sci. USA,69, No. 10, 2838–2839 (1972).

    Google Scholar 

  128. R. R. Coifman, “A real variable characterization of HP,” Stud. Math. (PRL),51, No. 3, 269–274 (1974).

    Google Scholar 

  129. R. R. Coifman, G. David, and Y. Meyer, “La solution des conjectures de Calderon,” Prepublications Univ. Paris-Sud., Dept. Math., 82T04 (1982).

  130. R. R. Coifman and C. Fefferman, “Weighted norm inequalities for maximal functions and singular integrals,” Stud. Math. (PRL),15, No. 3, 241–250 (1974).

    Google Scholar 

  131. R.R. Coifman, P. W. Jones, and J. L. Rubio de Francia, “On a constructive decomposition of BMO functions and factorization of Ap weights,” Preprint (1982).

  132. R. R. Coifman, A. McIntosh, and Y. Meyer, “L'integrale de Cauchy definit un operateur borne sur L2 pour les courbes Lipschitziennes,” Prepublications Univ. de Paris-Sud, Dept. Math., 81T16 (1981).

  133. R. R. Coifman and Y. Meyer, “Au dela des operateurs pseudodifferentiels,” Asterisque, No. 57 (1978).

  134. R. R. Coifman and R. Rochberg, “Another characterization of BMO,” Proc. Am. Math. Soc.,79, No. 2, 249–254 (1980).

    Google Scholar 

  135. R. R. Coifman and R. Rochberg, “Projections in weighted spaces, skew projections and inversion of Toeplitz operators,” Integr. Equat. Oper. Theory,5, No. 2, 145–159 (1982).

    Article  Google Scholar 

  136. R. R. Coifman, R. Rochberg, and G. Weiss, “Factorization theorems for Hardy spaces in several variables,” Ann. Math.,103, No. 3, 611–635 (1976).

    Google Scholar 

  137. R. R. Coifman and G. Weiss, “Maximal functions and HP-spaces defined by ergodic transformation,” Proc. Nat. Acad. Sci. USA,70, No. 6, 1761–1763 (1973).

    Google Scholar 

  138. R. R. Coifman and G. Weiss, “Analyse harmonique noncommutative sur certains espaces homogenes,” Lect. Notes Math.,242 (1971).

  139. R. R. Coifman and G. Weiss, “Extensions of Hardy spaces and their use in analysis,” Bull. Am. Math. Soc.,83, No. 4, 569–645 (1977).

    Google Scholar 

  140. W. G. Connett and A. L. Schwartz, “A multiplier theorem for ultraspherical series,” Stud. Math. (PRL),51, No. 1, 51–70 (1974).

    Google Scholar 

  141. W. G. Connett and A. L. Schwartz, “A multiplier theorem for Jacobi expansions,” Stud. Math. (PRL),52, No. 3, 243–261 (1975).

    Google Scholar 

  142. W. G. Connett and A. L. Schwartz, “The Littlewood-Paley theory for Jacobi expansions,” Trans. Am. Math. Soc.,251, 219–234 (1979).

    Google Scholar 

  143. A. Cordoba and C. Fefferman, “A weighted norm inequality for singular integrals,” Stud. Math. (PRL),57, No. 1, 97–101 (1976).

    Google Scholar 

  144. A. Cordoba and B. Lopez-Melero, “Spherical summation: A problem of E. M. Stein,” Ann. Inst. Fourier,31, No. 3, 147–152 (1981).

    Google Scholar 

  145. M. Cotlar, “A unified theory of Hilbert transforms and ergodic theory,” Rev. Math. Guyana,1, 105–167 (1955).

    Google Scholar 

  146. M. Cotlar and C. A. Sadosky, “A moment theory approach to the Riesz theorem on the conjugate function with general weights,” Stud. Math. (PRL),53, No. 1, 75–101 (1975).

    Google Scholar 

  147. M. Cotlar and C. A. Sadosky, “Transformee de Hilbert, theoreme de Bochner et le probleme de moments. I,” C. R. Acad. Sci.,A285, No. 6, 433–436 (1977).

    Google Scholar 

  148. M. Cotlar and C. A. Sadosky, “Transformee de Hilbert, theoreme de Bochner et le probleme des moments. II,” C. R. Acad. Sci.,A285, No. 10, 661–664 (1977).

    Google Scholar 

  149. M. Cotlar and C. A. Sadosky, “On the Helson-Szegö theorem and a related class of modified Toeplitz kernels,” in: Harm. Anal. Euclidean Spaces, Proc Symp. Pure Math., Am. Math. Soc., Williamstown, Mass., Part 1, 1978, Providence, R. I.(1979), pp. 383–407.

    Google Scholar 

  150. M. Cotlar and C. A. Sadosky, “Majorized Toeplitz forms and weighted inequalities with general norms,” Lect. Notes Math.,908, 139–168 (1982).

    Google Scholar 

  151. B. E. J. Dahlberg, “Estimates of harmonic measures,” Arch. Rat. Mech. Anal.,65, 272–288 (1977).

    Google Scholar 

  152. B. E. J. Dahlberg, “On the Poisson integral for Lipschitz and C1-domains,” Stud. Math. (PRL),66, No. 1, 13–24 (1979).

    Google Scholar 

  153. B. E. J. Dahlberg, “Harmonic functions in Lipschitz domains,” in: Harmonic Analysis in Euclidean Spaces, Proc. Symposium in Pure Math., Am. Math. Soc., Williamstown, Mass., 1978, Part 1, Providence, R.I. (1979), pp. 313–322.

    Google Scholar 

  154. B. E. J. Dahlberg, “Weighted norm inequalities for the Lusin area integrals and the non-tangential maximal functions for functions harmonic in a Lipschitz domain,” Stud. Math., (PRL),67, No. 3, 297–314 (1980).

    Google Scholar 

  155. B. E. J. Dahlberg, “Approximation of harmonic functions,” Ann. Inst. Fourier,30, No. 2, 97–107 (1980).

    Google Scholar 

  156. B. E. J. Dahlberg, “A note on H1 and BMO,” Preprint (1980).

  157. G. David, “L'integrale de Cauchy sur les courbes rectifiables,” Prepublications Univ. Paris-Sud, Dept. Math., 82T05 (1982).

  158. E. M. Dyn'kin, “The rate of polynomial approximation in the complex domain,” Lect. Notes Math.,864, 90–142 (1981).

    Google Scholar 

  159. A. Erdélyi, “The Stieltjes transformation of weighted LP spaces,” Appl. Anal.,7, No. 3, 213–219 (1978).

    Google Scholar 

  160. E. Fabes, “Various notions of the Hardy space H1 of a C1-domain,” Rend. Circ. Mat. Palermo, No. 1, Suppl., 129–137 (1981).

    Google Scholar 

  161. E. Fabes, C. E. Kenig, and U. Neri, “Carleson measures, H1-duality and weighted BMO in nonsmooth domains,” Indiana Univ. Math. J.,30, No. 4, 547–581 (1981).

    Article  Google Scholar 

  162. E. Fabes and U. Neri, “Dirichlet problem in Lipschitz domains with BMO data,” Proc Am. Math. Soc.,78, No. 1, 33–39 (1980).

    Google Scholar 

  163. C. Fefferman, “The multiplier problem for the ball,” Ann. Math.,94, No. 2, 330–336 (1971).

    Google Scholar 

  164. C. Fefferman and B. Muckenhoupt, “Two nonequivalent conditions for weight funtions,” Proc. Am. Math. Soc.,45, No. 1, 99–104 (1974).

    Google Scholar 

  165. C. Fefferman and E. M. Stein, “Some maximal inequalities,” Am. J. Math.,93, No. 1, 107–115 (1971).

    Google Scholar 

  166. C. Fefferman and E. M. Stein, “HP-spaces of several variables,” Acta Math.,129, Nos. 3–4, 137–193 (1972).

    Google Scholar 

  167. R. Fefferman, “A note on singular integrals,” Proc. Am. Math. Soc.,74, No. 2, 266–270 (1979).

    Google Scholar 

  168. R. Fefferman, “Strong differentiation with respect to measures,” Am. J. Math.,103, No. 1, 33–40 (1981).

    Google Scholar 

  169. R. Fefferman, “Singular integrals on product domains,” Bull. Am. Math. Soc.,4, No. 2, 195–201 (1981).

    Google Scholar 

  170. R. Fefferman and E. M. Stein, “Singular integrals on product spaces,” Adv. Math.,45, 117–143 (1982).

    Article  Google Scholar 

  171. T. M. Flett, “Some theorems on odd and even functions,” Proc. London Math. Soc.,8, No. 29, 135–148 (1958).

    Google Scholar 

  172. T. M. Flett, “On a theorem of Pitt,” J. London Math. Soc.,7, No. 2, 376–384 (1973).

    Google Scholar 

  173. F. Forelli, “The Marcel Riesz theorem of conjugate theorems,” Trans. Am. Math. Soc.,106, No. 3, 369–390 (1963).

    Google Scholar 

  174. G. Freud, Orthogonale Polynome, Birkhäuser, Basel-Stuttgart (1969).

    Google Scholar 

  175. N. Fujii, “Weighted bounded mean oscillation and singular integrals,” Math. Jpn.,22, No. 5, 529–534 (1978).

    Google Scholar 

  176. J. B. Garnett, Bounded Analytic Functions, Academic Press, New York-London (1981).

    Google Scholar 

  177. J. B. Garnett and P. W. Jones, “The distance in BMO to L∞,” Ann. Math.,108, No. 2, 373–393 (1978).

    Google Scholar 

  178. J. B. Garnett and P. W. Jones, “BMO from dyadic BMO,” Pac. J. Math.,99, No. 2, 351–371 (1982).

    Google Scholar 

  179. J. Garsia-Cuerva, “Weighted Hardy spaces,” in: Harmonic Analysis in Euclidean Spaces, Proc. Symp. Pure Math., Am. Math. Soc., Williamstown, Mass., 1978, Part 1, Providence, R.I. (1979), pp. 253–261.

    Google Scholar 

  180. J. Garsia-Cuerva, “Weighted HP spaces,” Rozp. Mat., No. 162 (1979).

  181. G. Gasper and W. Trebels, “Multiplier criteria of Marcinkiewicz type for Jacobi expansions,” Trans. Am. Math. Soc.,231, No. 1, 117–132 (1977).

    Google Scholar 

  182. G. Gasper and W. Trebels, “Multiplier criteria of Hörmander type for Jacobi expansions,” Stud. Math. (PRL),68, No. 2, 187–197 (1980).

    Google Scholar 

  183. F. W. Gehring, “The LP-integrability of the partial derivatives for quasiconformal mapping,” Acta Math.,130, Nos. 3–4, 265–278 (1973).

    Google Scholar 

  184. J. A. Gosselin, “Weighted norm inequalities for Vilenkin-Fourier series,” Proc. Am. Math. Soc.,49, No. 2, 349–353 (1975).

    Google Scholar 

  185. R. P. Gundy and R. L. Wheeden, “Weighted integral inequalities for the nontangential maximal functions, Lusin area integral and Walsh-Paley series,” Stud. Math. (PRL),49, No. 2, 107–124 (1974).

    Google Scholar 

  186. E. O. Harboure and N. E. Aguilera, “Das designaldades en normas con pesos para series de Walsh,” Rev. Union Mat. Argent.,26, No. 3, 143–159 (1972).

    Google Scholar 

  187. G. H. Hardy and J. E. Littlewood, “A maximal theorem with function theoretic application,” Acta Math.,54, 81–116 (1930).

    Google Scholar 

  188. G. H. Hardy and J. E. Littlewood, “Some theorems of Fourier series and Fourier power series,” Duke Math. J.,2, 354–381 (1936).

    Article  Google Scholar 

  189. Harmonic Analysis in Euclidean Spaces, Proc. Symp. Pure Math., Am. Math. Soc., Williamstown, Mass., 1978, Part I, Providence, R.I. (1979).

  190. Harmonic Analysis in Euclidean Spaces, Proc. Symp, Pure Math., Am. Math. Soc., Williamstown, Mass., 1978, Part II, Providence, R.I. (1979).

  191. Harmonic Analysis, Proc. Conf. University of Minnesota, Min., April, 1981. Lect. Notes Math.,908 (1982).

  192. L. I. Hedberg, “On certain convolution inequalities,” Proc. Am. Math. Soc.,36, No. 2, 505–510 (1972).

    Google Scholar 

  193. L. I. Hedberg, “Spectral synthesis in Sobolev spaces and uniqueness of solutions of the Dirichlet problem,” Acta Math.,147, 237–264 (1981).

    Google Scholar 

  194. H. P. Heinig, “Some extensions of Hardy's inequality,” SIAM J. Math. Anal.,6, No. 4, 698–713 (1975).

    Article  Google Scholar 

  195. H. P. Heinig, “Weighted maximal inequalities forl r-valued functions,” Can. Math. Bull.,19, No. 4, 445–453 (1976).

    Google Scholar 

  196. H. P. Heinig, “Variations of Hardy's inequality,” Real Anal. Exch.,5, No. 1, 61–81 (1979–1980).

    Google Scholar 

  197. H. Helson and D. Szegö, “A problem in prediction theory,” Ann. Math. Pura Appl.,51, 107–138 (1960).

    Google Scholar 

  198. I. I. Hirschman, “The decomposition of Walsh and Fourier series,” Mem. Am. Math. Soc., No. 15 (1955).

  199. I. I. Hirschman, “A note on orthogonal systems,” Pac. J. Math.,6, No. 1, 47–56 (1956).

    Google Scholar 

  200. I. I. Hirschman, “Multiplier transformations. II,” Duke Math. J.,28, No. 1, 45–56 (1961).

    Article  Google Scholar 

  201. L. Hörmander, “LP-estimates for (pluri) subharmonic functions,” Math. Scand.,20, No. 1, 65–78 (1967).

    Google Scholar 

  202. S. V. Hrusčev, N. K. Nikol'skii, and B. S. Pavlov, “Unconditional bases of exponentials and of reproducing kernels,” Lect. Notes Math.,864, 214–335 (1981).

    Google Scholar 

  203. R. A. Hunt, “On the convergence of Fourier series,” in: Proc. Conf. on Orthog. Expansions and Their Continuous Analogues, Edwardsville, Ill., 1967, Southern Ill. Press, Carbondale, Ill. (1968), pp. 235–255.

    Google Scholar 

  204. R. A. Hunt, D. Kurtz, and C. Neugebauer, “A note on the equivalence of Ap and Sawyer's condition for equal weights,” Preprint (1981).

  205. R. A. Hunt, B. Muckenhoupt, and R. L. Wheeden, “Weighted norm inequalities for the conjugate function and Hilbert transform,” Trans. Am. Math. Soc.,176, 227–251 (1973).

    Google Scholar 

  206. R. A. Hunt and R. L. Wheeden, “On the boundary values of harmonic functions,” Trans. Am. Math. Soc.,132, No. 2, 307–322 (1968).

    Google Scholar 

  207. R. A. Hunt and R. L. Wheeden, “Positive harmonic functions in Lipschitz domains,” Trans. Am. Math. Soc.,147, No. 2, 507–527 (1970).

    Google Scholar 

  208. R. A. Hunt and W.-S. Young, “A weighted norm inequality for Fourier series,” Bull. Am. Math. Soc.,80, No. 2, 274–277 (1974).

    Google Scholar 

  209. D. Jerison and C. E. Kenig, “An identity with applications to harmonic measure,” Bull. Am. Math. Soc.,2, No. 3, 447–451 (1980).

    Google Scholar 

  210. D. Jerison and C. E. Kenig, “Boundary behavior of harmonic functions in nontangentially accessible domains,” Preprint (1980).

  211. D. Jerison and C. E. Kenig, “Hardy spaces, A∞, and singular integrals on chord-arc domains,” Math. Scand.,50, No. 2, 221–247 (1982).

    Google Scholar 

  212. F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Commun. Pure. Appl. Math.,14, No. 3, 415–426 (1961).

    Google Scholar 

  213. R. Johnson, “Weighted estimates for fractional powers of partial differential operators,” Trans. Am. Math. Soc.,265, No. 2, 511–525 (1981).

    Google Scholar 

  214. P. W. Jones, “On construction for BMO(R) and Ap(R n),” in: Harmonic Anal. in Euclidean Spaces. Proc. Symp. Pure Math. Am. Math. Soc., Williamstown, Mass., 1978, Part 1, Providence, R.I. (1979), pp. 417–419.

  215. P. W. Jones, “Factorization of Ap-weights,” Ann. Math.,111, No. 3, 511–530 (1980).

    Google Scholar 

  216. J. Junggeburth, “Multipliers for (C, k)-bounded Fourier expansions in weighted locally convex spaces and approximation,” Rev. Union Math. Argent.,27, No. 3, 127–146 (1975–1976).

    Google Scholar 

  217. M. Kanenko and S. Jano, “Weighted norm inequalities for singular integrals,” J. Math. Soc. Jpn.,27, No. 4, 570–588 (1975).

    Google Scholar 

  218. K. S. Kazarian, “On bases and unconditional bases in the spaces LP(dμ)(1⩽ p<∞),” Stud. Math. (PRL),71, No. 3, 227–249 (1982).

    Google Scholar 

  219. C. E. Kenig, “Weighted Hardy spaces on Lipschitz domains,” in: Harmonic Anal. in Euclidean Spaces. Proc. Symp. Pure Math. Am. Math. Soc., Williamstown, Mass., 1978, Part 1, Providence, R.I. (1979), pp. 263–274.

  220. C. E. Kenig, “Weighted HP spaces on Lipschitz domains,” Am. J. Math.,102, No. 1, 129–163 (1980).

    Google Scholar 

  221. R. Kerman and A. Torchinsky, “Integral inequalities with weights for the Hardy maximal function,” Stud. Math. (PRL),71, No. 3, 277–284 (1982).

    Google Scholar 

  222. N. Kerzman, “Singular integrals in complex analysis,” in: Harmonic Anal. in Euclid. Spaces. Proc. Symp. Pure Math. Am. Math. Soc., Williamstown, Mass., 1978, Part 2, Providence, R.I. (1979), pp. 3–41.

  223. B. D. Khanh, “Integrales singulieres, commutateurs et la fonction f#,” Bull. Sci. Math.,103, No. 3, 241–253 (1979).

    Google Scholar 

  224. B. D. Khanh, “Integrales de Marcinkiewicz du type des commutateurs de A. P. Calderon,” C. R. Acad. Sci., Ser. A,292, No. 17, 781–784 (1981).

    Google Scholar 

  225. A. Knapp and E. M. Stein, “Singular integrals and principal series,” Proc. Nat. Acad. Sci. USA,63, No. 2, 281–284 (1969).

    Google Scholar 

  226. A. Knapp and E. M. Stein, “Intertwinning operators for semisimple groups,” Ann. Math.,93, 489–578 (1971).

    Google Scholar 

  227. P. Koosis, “Moyennes quadratiques ponderees de fonctions periodiques et de leurs conjugees harmoniques,” C. R. Acad. Sci.,A291, No. 4, 255–257 (1980).

    Google Scholar 

  228. A. Koranyi and S. Vagi, “Integrales singulieres sur certains espaces homogenes,” C. R. Acad. Sci.,A268, No. 14, 765–768 (1969).

    Google Scholar 

  229. A. Koranyi and S. Vagi, “Singular integrals on homogeneous spaces and some problems of classical analysis,” Ann. Scuola Norm. Sup., Pisa. Sci. Fis. Mat.,25, No. 4, 575–648 (1971).

    Google Scholar 

  230. P. Kree, “Sur les multiplicateurs dans FLP avec poids,” Ann. Inst. Fourier,2, 91–121 (1966).

    Google Scholar 

  231. D. S. Kurtz, “Weighted norm inequalities for the Hardy-Littlewood maximal functions for one-parameter rectangles,” Stud. Math. (PRL),53, No. 1, 39–54 (1975).

    Google Scholar 

  232. D. S. Kurtz, “Littlewood-Paley and multiplier theorem on weighted Lp-spaces,” Trans. Am. Math. Soc.,259, No. 1, 235–254 (1980).

    Google Scholar 

  233. D. S. Kurtz and R. L. Wheeden, “Results on weighted norm inequalities for multipliers,” Trans. Am. Math. Soc.,255, 343–362 (1979).

    Google Scholar 

  234. D. S. Kurtz and R. L. Wheeden, “A note on singular integrals with weights,” Proc. Am. Math. Soc.,81, No. 3, 391–397 (1981).

    Google Scholar 

  235. R. A. Macias and C. Segovia, “Weighted norm inequalities for parabolic fractional integrals,” Stud. Math. (PRL),61, No. 3, 279–291 (1977).

    Google Scholar 

  236. R. A. Macias and C. Segovia, “A decomposition into atoms of distributions on spaces of homogeneous type,” Adv. Math.,33, No. 3, 271–309 (1979).

    Article  Google Scholar 

  237. R. A. Macias and C. Segovia, “Lipschitz functions on spaces of homogeneous type,” Adv. Math.,33, No. 3, 257–270 (1979).

    Article  Google Scholar 

  238. R. A. Macias and C. Segovia, “Singular integrals on generalized Lipschitz and Hardy spaces,” Stud. Math. (PRL),65, No. 1, 55–75 (1979).

    Google Scholar 

  239. R. A. Macias and C. Segovia, “A well-behaved quasidistance for spaces of homogeneous type,” Trab. Mat. Inst. Argent. Mat., No. 32 (1981).

  240. W. G. Mazja, Einbettungssätze für Sobolewsche Räume. I, II, Teubner, Leipzig (1979, 1980).

  241. W. G. Mazja, Zur Theorie Sobolewsche Räume, Teubner, Leipzig (1981).

    Google Scholar 

  242. L. de Michele and I. R. Inglis, “LP-estimates for strongly singular integrals on spaces of homogeneous type,” J. Funct. Anal.,39, No. 1, 1–15 (1980).

    Article  Google Scholar 

  243. N. Miller, “Weighted Sobolev spaces and pseudodifferential operators with smooth symbols,” Trans. Am. Math. Soc.,269, No. 1, 91–109 (1982).

    Google Scholar 

  244. A. Miyachi, “Some singular integral transformations bounded in the Hardy spaces H1(R n),” J. Fac. Sci. Univ. Tokyo, Sec. 1A,25, No. 1, 93–108 (1978).

    Google Scholar 

  245. A. Miyachi, “On the weakly (strongly) singular integrals,” Jpn. J. Math.,4, No. 1, 221–262 (1978).

    Google Scholar 

  246. B. Muckenhoupt, “Mean convergence of Jacobi series,” Proc. Am. Math. Soc.,23, 306–310 (1969).

    Google Scholar 

  247. B. Muckenhoupt, “Hermite conjugate expansions,” Trans. Am. Math. Soc.,139, 243–260 (1969).

    Google Scholar 

  248. B. Muckenhoupt, “Poisson integrals for Hermite and Laguerre expansions,” Trans. Am. Math. Soc.,139, 231–242 (1969).

    Google Scholar 

  249. B. Muckenhoupt, “Mean convergence of Hermite and Laguerre series. I,” Trans. Am. Math. Soc.,147, No. 2, 419–431 (1970).

    Google Scholar 

  250. B. Muckenhoupt, “Mean convergence of Hermite and Laguerre series. II,” Trans. Am. Math. Soc.,147, No. 2, 433–460 (1970).

    Google Scholar 

  251. B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal functions,” Trans. Am. Math. Soc.,165, 207–226 (1972).

    Google Scholar 

  252. B. Muckenhoupt, “Hardy's inequality with weights,” Stud. Math. (PRL),44, No. 1, 31–38 (1972).

    Google Scholar 

  253. B. Muckenhoupt, “The equivalence of two conditions for weight functions,” Stud. Math. (PRL),49, No. 2, 101–106 (1974).

    Google Scholar 

  254. B. Muckenhoupt, “Weighted norm inequalities for the classical operators,” Int. Ser. Numer. Math. Basel-Stuttgart,25, 265–283 (1974).

    Google Scholar 

  255. B. Muckenhoupt, “Two weight function norm inequalities for the Poisson integrals,” Trans. Am. Math. Soc.,210, 225–231 (1975).

    Google Scholar 

  256. B. Muckenhoupt, “Weighted norm inequalities for classical operators,” Harm. Anal. Euclidean Spaces. Proc. Symp. Pure Math. Am. Math. Soc., Williamstown, Mass., 1978, Part 1, Providence, R.I. (1979), pp. 51–60.

  257. B. Muckenhoupt and E. M. Stein, “Classical expansions and their relation to conjugate harmonic functions,” Trans. Am. Math. Soc.,118, No. 6, 17–92 (1965).

    Google Scholar 

  258. B. Muckenhoupt and R. L. Wheeden, “Weighted norm inequalities for singular and fractional integrals,” Trans. Am. Math. Soc.,161, 249–258 (1971).

    Google Scholar 

  259. B. Muckenhoupt and R. L. Wheeden, “Weighted norm inequalities for fractional integrals,” Trans. Am. Math. Soc.,192, 261–274 (1974).

    Google Scholar 

  260. B. Muckenhoupt and R. L. Wheeden, “Norm inequalities for the Littlewood-Paley function g *λ ,” Trans. Am. Math. Soc.,191, 95–111 (1974).

    Google Scholar 

  261. B. Muckenhoupt and R. L. Wheeden, “Weighted bounded mean oscillation and the Hilbert transform,” Stud. Math. (PRL),54, No. 3, 221–237 (1976).

    Google Scholar 

  262. B. Muckenhoupt and R. L. Wheeden, “Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform,” Stud. Math. (PRL),55, No. 3, 279–294 (1976).

    Google Scholar 

  263. B. Muckenhoupt and R. L. Wheeden, “Some weighted weak-type inequalities for the Hardy-Littlewood maximal function and the Hilbert transform,” Indiana Univ. Math. J.,26, No. 5, 801–816 (1977).

    Article  Google Scholar 

  264. B. Muckenhoupt and R. L. Wheeden, “On the dual of weighted H1 of the half-space,” Stud. Math. (PRL),63, No. 1, 57–79 (1978).

    Google Scholar 

  265. M. K. V. Murthy and G. Stampacchia, “Boundary-value problems for some degenerate-elliptic operators,” Ann. Math. Pure Appl.,80, 1–122 (1968(1969)).

    Google Scholar 

  266. P. G. Nevai, “Orthogonal polynomials,” Mem. Am. Math. Soc.,18, No. 213, 1–185 (1979).

    Google Scholar 

  267. C. B. Pereyra, “An extension of a theorem of E. Stein,” Proc. Am. Math. Soc.,19, No. 6, 1396–1402 (1968).

    Google Scholar 

  268. S. K. Pichorides, “Une propriete de la transformee de Hilbert,” C. R. Acad. Sci.,280, No. 18, A1197-A1199 (1975).

    Google Scholar 

  269. E. L. Poiani, “Mean Cesaro summability of Laguerre and Hermite series,” Trans. Am. Math. Soc.,173, 1–31 (1972).

    Google Scholar 

  270. H. Pollard, “The mean convergence of orthogonal series,” Proc. Nat. Acad. Sci. USA,32, 5–10 (1946).

    Google Scholar 

  271. H. M. Reimann, “Functions of bounded mean oscillation and quasiconformal mapping,” Comment. Math. Helv.,49, No. 2, 260–276 (1974).

    Google Scholar 

  272. H. M. Reimann and T. Rycheher, “Funktionen beschränkter mittlerer Oszillation,” Lect. Notes Math.,487 (1975).

  273. N. Riviere, “Singular integrals and multiplier operators,” Ark. Mat.,9, No. 2, 243–278 (1971).

    Google Scholar 

  274. R. Rochberg, “Toeplitz operators on weighted HP spaces,” Indiana Univ. Math. J.,21, No. 2, 291–298 (1977).

    Article  Google Scholar 

  275. P. G. Rooney, “On the ranges of certain fractional integrals,” Can. J. Math.,24, No. 6, 1198–1216 (1972).

    Google Scholar 

  276. P. G. Rooney, “Multipliers for the Mellin transformation,” Can. Math. Bull.,25, No. 3, 257–262 (1982).

    Google Scholar 

  277. M. Rosenblum, “Summability of Fourier series in LP(dμ),” Trans. Am. Math. Soc.,165, 32–42 (1962).

    Google Scholar 

  278. J. L. Rubio de Francia, “Vector valued inequalities for operators in LP spaces,” Bull. London Math. Soc.,12, No. 3, 211–215 (1980).

    Google Scholar 

  279. J. L. Rubio de Francia, “Vector valued inequalities for Fourier series,” Proc. Am. Math. Soc.,78, No. 4, 525–528 (1980).

    Google Scholar 

  280. J. L. Rubio de Francia, “Boundedness of maximal functions and singular integrals in weighted LP spaces,” Proc. Am. Math. Soc.,83, No. 4, 673–679 (1981).

    Google Scholar 

  281. J. L. Rubio de Francia, “Weighted norm inequalities and vector valued inequalities,” Lect. Notes Math.,908, 86–101 (1982).

    Google Scholar 

  282. J. L. Rubio de Francia, “Factorization and extrapolation of weights,” Bull. Am. Math. Soc.,7, No. 2, 393–396 (1982).

    Google Scholar 

  283. E. T. Sawyer, “Norm inequalities relating singular integrals and the maximal function,” Preprint (1980).

  284. E. T. Sawyer, “Weighted norm inequalities for fractional maximal operators,” Proc. C.M.S.,1, 283–309 (1981).

    Google Scholar 

  285. E. T. Sawyer, “A characterization of a two-weight norm inequality for maximal operators,” Stud. Math. (PRL),75, No. 1, 1–11 (1982).

    Google Scholar 

  286. E. T. Sawyer, “Two weight norm inequalities for certain maximal and integral operators,” Lect. Notes Math.,908, 102–127 (1982).

    Google Scholar 

  287. C. Segovia and R. Wheeden, “On weighted norm inequalities for the Lusin area integral,” Trans. Am. Math. Soc.,176, 103–123 (1973).

    Google Scholar 

  288. J. Serrin, “Local behaviour of quasilinear expansions,” Acta Math.,111, Nos. 3–4, 247–302 (1964).

    Google Scholar 

  289. P. Sjölin, “LP estimates for strongly singular convolution operators inR n,” Ark. Mat.,14, No. 1, 59–64 (1976).

    Google Scholar 

  290. P. Sjölin, “Two inequalities for pseudodifferential operators,” Stockholms Univ. Math. Inst.,10 (1977).

  291. P. Sjölin and P. Sjörgen, “Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets,” Ann. Inst. Fourier,31, No. 1, 157–175 (1981).

    Google Scholar 

  292. K. J. Smith, “A generalization of an inequality of Hardy and Littlewood,” Can. J. Math.,8, No. 2, 157–170 (1956).

    Google Scholar 

  293. R. J. Stanton, “Mean convergence of Fourier series on compact Lie groups,” Trans. Am. Math. Soc.,218, 61–87 (1976).

    Google Scholar 

  294. E. M. Stein, “Note on singular integrals,” Proc. Am. Math. Soc.,8, No. 2, 250–254 (1957).

    Google Scholar 

  295. E. M. Stein, “Some problems in harmonic analysis,” in: Harmonic Analysis in Euclidean Spaces. Proc. Symp. Pure Math. Am. Math. Soc., Williamstown, Mass., 1978, Part 1, Providence, R.I. (1979), pp. 3–20.

  296. E. M. Stein and G. Weiss, “Interpolation of operators with change of measures,” Trans. Am. Math. Soc.,87, No. 1, 159–172 (1958).

    Google Scholar 

  297. R. S. Strichartz, “LP-estimates for integral transforms,” Trans. Am. Math. Soc.,136, 33–50 (1969).

    Google Scholar 

  298. J.-O. Strömberg, “Nonequivalence between two kinds of conditions on weight functions,” in: Harmonic Analysis in Euclidean Spaces. Proc Symp. Pure Math. Am. Math. Soc., Williamstown, Mass., 1978, Part. 1, Providence, R.I. (1979), pp. 141–148.

  299. J.-O. Strömberg and A. Torchinsky, “Weights, sharp maximal functions and Hardy spaces,” Bull. Am. Math. Soc.,3, No. 3, 1053–1056 (1980).

    Google Scholar 

  300. J.-O. Strömberg and R. L. Wheeden, “Relations between H Pu and L Pu with polynomial weights,” Trans. Am. Math. Soc.,270, No. 2, 439–467 (1982).

    Google Scholar 

  301. G. Talenti, “Asservazioni sopra una classe di disuguaglianze,” Rend. Sem. Math. Fis. Milano,39, 171–185 (1969).

    Google Scholar 

  302. G. Tomaselli, “A class of inequalities,” Bull. Unione Mat. Ital.,2, No. 6, 622–631 (1969).

    Google Scholar 

  303. A. Torchinsky, “Weighted norm inequalities for the Littlewood-Paley functions g *λ ,” in: Harmonic Analysis in Euclidean Spaces. Proc. Symp. Pure Math. Am. Math. Soc., Williamstown, Mass., 1978, Part 1, Providence, R.I. (1979), pp. 125–131.

  304. A. Torre, “Weights in ergodic theory,” Lect. Notes Math.,908, 128–138 (1982).

    Google Scholar 

  305. G. Travalgini, “Weyl functions and the Ap-condition on compact Lie groups,” J. Austral. Math. Soc.,A33, No. 2, 185–192 (1982).

    Google Scholar 

  306. A. Uchiyama, “A remark on Carleson's characterization of BMO,” Proc. Am. Math. Soc.,79, 35–41 (1980).

    Google Scholar 

  307. A. Uchiyama, “The factorization of HP on the spaces of homogeneous type,” Pac. J. Math.,92, No. 2, 453–468 (1981).

    Google Scholar 

  308. A. Unterberger, “Extensions du lemme de Cotlar et applications,” C. R. Acad. Sci.,AB288, No. 4, A249-A252 (1979).

    Google Scholar 

  309. N. T. Varopoulos, “BMO functions and the ¯t6 equation,” Pac. J. Math.,71, No. 1, 221–273 (1977).

    Google Scholar 

  310. N. T. Varopoulos, “A remark on bounded mean oscillation and bounded harmonic functions Pac. J. Math.,74, No. 1, 257–259 (1978).

    Google Scholar 

  311. G. V. Welland, “Weighted norm inequalities for fractional integrals,” Proc. Am. Math. Soc.,51, No. 1, 143–148 (1975).

    Google Scholar 

  312. R. L. Wheeden, “On the radial and nontangential maximal functions for the disc,” Proc. Am. Math. Soc.,42, No. 2, 418–422 (1974).

    Google Scholar 

  313. R. L. Wheeden, “A boundary value characterization of weighted H1,” Enseign. Math.,22, Nos. 1–2, 121–134 (1976).

    Google Scholar 

  314. H. Widom, “Singular integral equation in LP,” Trans. Am. Math. Soc.,97, No. 1, 131–160 (1960).

    Google Scholar 

  315. S. Yano, “On Marcinkiewicz integral,” Tohoku Math. J.,27, No. 3, 381–388 (1975).

    Google Scholar 

  316. W.-S. Young, “Weighted norm inequalities for multipliers,” in: Harmonic Anal. in Euclidean Spaces. Symp. Pure Math. Am. Math. Soc., Williamstown, Mass., 1978, Part 1, Providence, R.I. (1979), pp. 133–139.

  317. W.-S. Young, “Weighted norm inequalities for the Hardy-Littlewood maximal functions,” Proc. Am. Math. Soc.,85, No. 1, 24–26 (1982).

    Google Scholar 

  318. M. Zinsmeister, “Courbes de Jordan verifiant une condition corde-arc,” Ann. Inst. Fourier,32, No. 2, 13–21 (1982).

    Google Scholar 

  319. A. Zygmund, “On certain lemmas of Marcinkiewicz and Carleson,” J. Appr. Theory,2, No. 3, 249–257 (1969).

    Article  Google Scholar 

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Authors
  1. E. M. Dyn'kin
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  2. B. P. Osilenker
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Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 21, pp. 42–129, 1983.

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Dyn'kin, E.M., Osilenker, B.P. Weighted estimates of singular integrals and their applications. J Math Sci 30, 2094–2154 (1985). https://doi.org/10.1007/BF02105397

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