Two-sided weighted Fourier inequalities
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 2, pp. 341-362.

Fourier transform estimates for f ^ L q,w ˜ via f L p,w from above and from below are studied. For p=q, equivalence results, i.e.,

C 1 f L p,w f ^ L p,w ˜ C 2 f L p,w ,w ˜(x)=w(1/x)x p-2 ,1p<,

are shown to be valid for functions from certain classes under the Muckenhoupt conditions: wA p or wA 2p . Sharpness of these conditions is proved.

Publié le :
Classification : 42A38, 26D15, 46E30
Liflyand, Elijah 1 ; Tikhonov, Sergey 2

1 Department of Mathematics Bar-Ilan University 52900 Ramat-Gan, Israel
2 ICREA and Centre de Recerca Matemàtica Apartat 50 08193 Bellaterra Barcelona, Spain
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Liflyand, Elijah; Tikhonov, Sergey. Two-sided weighted Fourier inequalities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 2, pp. 341-362. http://archive.numdam.org/item/ASNSP_2012_5_11_2_341_0/

[1] M. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990), 727–735. | MR | Zbl

[2] R. Askey and S. Wainger, Integrability theorems for Fourier series, Duke Math. J. 33 (1966), 223–228. | MR | Zbl

[3] W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), 1897–1905. | MR | Zbl

[4] J. J. Benedetto and H. P. Heinig, Weighted Hardy spaces and the Laplace transform, In: “Harmonic Analysis”, Proc. Lecture Notes Math., Springer, Vol. 992, 1983, 240–277. | MR | Zbl

[5] J. J. Benedetto and H. P. Heinig, Fourier transform inequalities with measure weights, Adv. Math. 96 (1992), 194– 225. | MR | Zbl

[6] J. J. Benedetto, H. P. Heinig and R. Johnson, Fourier inequalities with A p -weights, In: “General Inequalities”, 5 (Oberwolfach, 1986), Vol. 80, Birkhäuser, Basel, 1987, 217–232. | MR | Zbl

[7] J. J. Benedetto and H. P. Heinig, Weighted Fourier inequalities: new proofs and generalizations, J. Fourier Anal. Appl. 9 (2003), 1–37. | MR | Zbl

[8] J. J. Benedetto and J. D. Lakey, The definition of the Fourier transform for weighted inequalities, J. Funct. Anal. 120 (1994), 403–439. | MR | Zbl

[9] S. Bloom and G. Sampson, Weighted spherical restriction theorems for the Fourier transform, Illinois J. Math. 36 (1992), 73–101. | MR | Zbl

[10] R. P. Boas, The integrability class of the sine transform of a monotonic function, Studia Math. 44 (1972), 365–369. | EuDML | MR | Zbl

[11] R. P. Boas, “Integrability Theorems for Trigonometric Transforms”, Springer-Verlag, New York, 1967. | MR | Zbl

[12] S. Bochner, “Lectures on Fourier Integrals”, Princeton University Press, Princeton, N.J., 1959. | MR | Zbl

[13] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405–408. | MR | Zbl

[14] J. Cerdà and J. Martín, Weighted Hardy inequalities and Hardy transforms of weights, Studia Math. 139 (2000), 189–196. | EuDML | MR | Zbl

[15] D. Cruz-Uribe, Piecewise monotonic doubling measures, Rocky Mountain J. Math. 26 (1996), 545–583. | MR | Zbl

[16] M. I. D’yachenko and E. D. Nursultanov, The Hardy-Littlewood theorem for trigonometric series with α-monotone coefficients, Mat. Sb. 200 (2009), 45–60. | MR | Zbl

[17] M. L. Gol’dman, Estimates for multiple Fourier transforms of radially symmetric monotone functions, Sib. Math. J. 18 (1978), 391–406. | Zbl

[18] H. P. Heinig, Weighted norm inequalities for classes of operators, Indiana Univ. Math. J. 33 (1984), 573–582. | MR | Zbl

[19] R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. | MR | Zbl

[20] R. Johnson, Recent Results on Weighted Inequalities for the Fourier Transform, In: “Seminar Analysis of the Karl-Weierstrass-Institute of Mathematics”, 1986/87 (Berlin, 1986/87), Teubner-Texte Math., Vol. 106, Leipzig, 1988, 287–296. | MR | Zbl

[21] W. B. Jurkat and G. Sampson, On maximal rearrangement inequalities for the Fourier transform, Trans. Amer. Math. Soc. 282 (1984), 625–643. | MR | Zbl

[22] W. B. Jurkat and G. Sampson, On rearrangement and weight inequalities for the Fourier transform, Indiana Univ. Math. J. 33 (1984), 257–270. | MR | Zbl

[23] E. Liflyand and S. Tikhonov, Extended solution of Boas’ conjecture on Fourier transforms, C.R. Acad. Sci. Paris, Ser.1 346 (2008), 1137–1142. | MR | Zbl

[24] E. Liflyand and S. Tikhonov, The Fourier Transforms of General Monotone Functions, In: “Analysis and Mathematical Physics”, Trends in Mathematics, Birkhäuser, 2009, 373–391. | MR | Zbl

[25] E. Liflyand and S. Tikhonov, A concept of general monotonicity and applications, Math. Nachr. 284 (2011), 1083–1098. | MR | Zbl

[26] B. Muckenhoupt, Weighted norm inequalities for the Fourier transform, Trans. Amer. Math. Soc. 276 (1983), 729–742. | MR | Zbl

[27] H. R. Pitt, Theorems on Fourier series and power series, Duke Math. J. 3 (1937), 747–755. | MR | Zbl

[28] C. Sadosky and R. L. Wheeden, Some weighted norm inequalities for the Fourier transform of functions with vanishing moments, Trans. Amer. Math. Soc. 300 (1987), 521–533. | MR | Zbl

[29] Y. Sagher, Integrability conditions for the Fourier transform, J. Math. Anal. Appl. 54 (1976), 151–156. | MR | Zbl

[30] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integrals and Derivatives: Theory and Applications”, New York, NY: Gordon and Breach, 1993. | MR | Zbl

[31] J.-O. Strömberg and R. L. Wheeden, Weighted norm estimates for the Fourier transform with a pair of weights, Trans. Amer. Math. Soc. 318 (1990), 355–372. | MR | Zbl

[32] B. Sz.-Nagy, Séries et intégrales de Fourier des fonctions monotones non bornées, Acta Sci. Math. (Szeged) 13 (1949), 118–135. | MR | Zbl

[33] E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492. | MR | Zbl

[34] E. M. Stein, “Harmonic Analysis, Real Variable Methods, Orthogonality, and Oscillatory Integrals”, Princeton Univ. Press, Princeton, NJ, 1993. | MR | Zbl

[35] S. Tikhonov, Trigonometric series with general monotone coefficients, J. Math. Anal. Appl. 326 (2007), 721–735. | MR | Zbl

[36] E. C. Titchmarsh, “Introduction to the Theory of Fourier Integrals”, Oxford, 1937. | JFM | MR | Zbl