Skip to main content
SpringerLink
Log in

Weighted Fourier Inequalities and Boundedness of Variation

  • Research Articles
  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We study the trigonometric series \(\sum_{n=1}^\infty \lambda_n \cos nx\) and \(\sum_{n=1}^\infty \lambda_n \sin nx\) with \(\{\lambda_n\}\) being a sequence of bounded variation. Let \(\psi\) denote the sum of such a series. We obtain necessary and sufficient conditions for the validity of the weighted Fourier inequality \(\left(\intop_0^\pi\mathopen|\psi(x)|^q \omega(x)\,dx\right)^{1/q} \le C\!\left(\sum_{n=1}^\infty u_n\left(\sum_{k=n}^\infty\mathopen|\lambda_{k}-\lambda_{k+1}|\right)^p \right)^{1/p}\), \(0<p\le q<\infty\), in terms of the weight \(\omega\) and the weighted sequence \(\{u_n\}\). Applications to the series with general monotone coefficients are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K. F. Andersen, “On the transformation of Fourier coefficients of certain classes of functions. II,” Pac. J. Math. 105 (1), 1–10 (1983).

    Article  MathSciNet  Google Scholar 

  2. K. F. Andersen and H. P. Heinig, “Weighted norm inequalities for certain integral operators,” SIAM J. Math. Anal. 14 (4), 834–844 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  3. K. F. Andersen and B. Muckenhoupt, “Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions,” Stud. Math. 72, 9–26 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Ariño and B. Muckenhoupt, “Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions,” Trans. Am. Math. Soc. 320 (2), 727–735 (1990).

    MathSciNet  MATH  Google Scholar 

  5. R. Askey, “Norm inequalities for some orthogonal series,” Bull. Am. Math. Soc. 72, 808–823 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Askey, “A transplantation theorem for Jacobi coefficients,” Pac. J. Math. 21, 393–404 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Askey and S. Wainger, “Integrability theorems for Fourier series,” Duke Math. J. 33, 223–228 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Askey and S. Wainger, “A transplantation theorem for ultraspherical coefficients,” Pac. J. Math. 16, 393–405 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  9. N. K. Bari, Trigonometric Series (Fizmatgiz, Moscow, 1961). Engl. transl.: N. K. Bary, A Treatise on Trigonometric Series (Pergamon Press, Oxford, 1964), Vols. I, II.

    Google Scholar 

  10. N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two conjugate functions,” Tr. Mosk. Mat. Obshch. 5, 483–522 (1956).

    MathSciNet  Google Scholar 

  11. J. J. Benedetto and H. P. Heinig, “Weighted Fourier inequalities: New proofs and generalizations,” J. Fourier Anal. Appl. 9 (1), 1–37 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  12. G. Bennett and K.-G. Grosse-Erdmann, “Weighted Hardy inequalities for decreasing sequences and functions,” Math. Ann. 334 (3), 489–531 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  13. R. P. Boas Jr., Integrability Theorems for Trigonometric Transforms (Springer, Berlin, 1967).

    Book  MATH  Google Scholar 

  14. C. Carton-Lebrun and H. P. Heinig, “Weighted Fourier transform inequalities for radially decreasing functions,” SIAM J. Math. Anal. 23 (3), 785–798 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  15. L. De Carli, D. Gorbachev, and S. Tikhonov, “Pitt inequalities and restriction theorems for the Fourier transform,” Rev. Mat. Iberoam. 33 (3), 789–808 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Debernardi, “Weighted norm inequalities for generalized Fourier-type transforms and applications,” Publ. Mat. 64 (1), 3–42 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Debernardi, “The Boas problem on Hankel transforms,” J. Fourier Anal. Appl 25 (6), 3310–3341 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  18. Ó. Domínguez, D. D. Haroske, and S. Tikhonov, “Embeddings and characterizations of Lipschitz spaces,” J. Math. Pures Appl. 144, 69–105 (2020); arXiv: 1911.08369 [math.FA].

    Article  MathSciNet  MATH  Google Scholar 

  19. Ó. Domínguez and S. Tikhonov, “Function spaces of logarithmic smoothness: Embeddings and characterizations,” Mem. Am. Math. Soc. (in press); arXiv: 1811.06399 [math.FA].

    Google Scholar 

  20. Ó. Domínguez and M. Veraar, “Extensions of the vector-valued Hausdorff–Young inequalities,” arXiv: 1904.07930 [math.FA].

  21. M. Dyachenko, A. Mukanov, and S. Tikhonov, “Hardy–Littlewood theorems for trigonometric series with general monotone coefficients,” Stud. Math. 250 (3), 217–234 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Dyachenko, E. Nursultanov, and A. Kankenova, “On summability of Fourier coefficients of functions from Lebesgue space,” J. Math. Anal. Appl. 419 (2), 959–971 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Dyachenko and S. Tikhonov, “Convergence of trigonometric series with general monotone coefficients,” C. R., Math., Acad. Sci. Paris 345 (3), 123–126 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Dyachenko and S. Tikhonov, “Integrability and continuity of functions represented by trigonometric series: Coefficients criteria,” Stud. Math. 193 (3), 285–306 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Gogatishvili and V. D. Stepanov, “Reduction theorems for weighted integral inequalities on the cone of monotone functions,” Russ. Math. Surv. 68 (4), 597–664 (2013) [transl. from Usp. Mat. Nauk 68 (4), 3–68 (2013)].

    Article  MathSciNet  MATH  Google Scholar 

  26. M. L. Gol’dman, “Estimates of norms of Hardy-type integral and discrete operators on cones of quasimonotone functions,” Dokl. Math. 63 (2), 250–255 (2001) [transl. from Dokl. Akad. Nauk 377 (6), 733–738 (2001)].

    MATH  Google Scholar 

  27. M. L. Goldman, “Sharp estimates for the norms of Hardy-type operators on the cones of quasimonotone functions,” Proc. Steklov Inst. Math. 232, 109–137 (2001) [transl. from Tr. Mat. Inst. Steklova 232, 115–143 (2001)].

    MathSciNet  MATH  Google Scholar 

  28. B. I. Golubov, A. V. Efimov, and V. A. Skvortsov, Walsh Series and Transforms: Theory and Applications (Nauka, Moscow, 1987; Kluwer, Dordrecht, 1991).

    MATH  Google Scholar 

  29. D. Gorbachev, E. Liflyand, and S. Tikhonov, “Weighted norm inequalities for integral transforms,” Indiana Univ. Math. J. 67 (5), 1949–2003 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  30. D. Gorbachev and S. Tikhonov, “Moduli of smoothness and growth properties of Fourier transforms: Two-sided estimates,” J. Approx. Theory 164 (9), 1283–1312 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  31. G. H. Hardy and J. E. Littlewood, “Notes on the theory of series. XIII: Some new properties of Fourier constants,” J. London Math. Soc. 6, 3–9 (1931).

    Article  MathSciNet  MATH  Google Scholar 

  32. H. Heinig, “Weighted norm inequalities for classes of operators,” Indiana Univ. Math. J. 33 (4), 573–582 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  33. W. Jurkat and G. Sampson, “On rearrangement and weight inequalities for the Fourier transform,” Indiana Univ. Math. J. 33 (2), 257–270 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  34. C. N. Kellogg, “An extension of the Hausdorff–Young theorem,” Mich. Math. J. 18, 121–127 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  35. A. A. Konyushkov, “Best approximations by trigonometric polynomials and Fourier coefficients,” Mat. Sb. 44 (1), 53–84 (1958).

    MathSciNet  Google Scholar 

  36. A. Kufner and L.-E. Persson, Weighted Inequalities of Hardy Type (World Scientific, Singapore, 2003).

    Book  MATH  Google Scholar 

  37. E. Liflyand and S. Tikhonov, “A concept of general monotonicity and applications,” Math. Nachr. 284 (8–9), 1083–1098 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  38. E. Liflyand and S. Tikhonov, “Two-sided weighted Fourier inequalities,” Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. 5, 11 (2), 341–362 (2012).

    MathSciNet  MATH  Google Scholar 

  39. B. Muckenhoupt, “Hardy’s inequality with weights,” Stud. Math. 44, 31–38 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  40. E. D. Nursultanov, “On the coefficients of multiple Fourier series in \(L_p\)-spaces,” Izv. Math. 64 (1), 93–120 (2000) [transl. from Izv. Ross. Akad. Nauk, Ser. Mat. 64 (1), 95–122 (2000)].

    Article  MathSciNet  MATH  Google Scholar 

  41. E. Nursultanov and S. Tikhonov, “Weighted Fourier inequalities in Lebesgue and Lorentz spaces,” J. Fourier Anal. Appl. 26 (4), 57 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  42. H. R. Pitt, “Theorems on Fourier series and power series,” Duke Math. J. 3 (4), 747–755 (1937).

    Article  MathSciNet  MATH  Google Scholar 

  43. J. Rastegari and G. Sinnamon, “Weighted Fourier inequalities via rearrangements,” J. Fourier Anal. Appl. 24 (5), 1225–1248 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  44. Y. Sagher, “An application of interpolation theory to Fourier series,” Stud. Math. 41, 169–181 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  45. E. M. Stein, “Interpolation of linear operators,” Trans. Am. Math. Soc. 83, 482–492 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  46. S. Yu. Tikhonov, “On the integrability of trigonometric series,” Math. Notes 78 (3–4), 437–442 (2005) [transl. from Mat. Zametki 78 (3), 476–480 (2005)].

    Article  MathSciNet  MATH  Google Scholar 

  47. S. Tikhonov, “Trigonometric series with general monotone coefficients,” J. Math. Anal. Appl. 326 (1), 721–735 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  48. S. Tikhonov, “Best approximation and moduli of smoothness: Computation and equivalence theorems,” J. Approx. Theory 153 (1), 19–39 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  49. P. L. Ul’yanov, “Application of \(A\)-integration to a class of trigonometric series,” Mat. Sb. 35 (3), 469–490 (1954).

    MathSciNet  Google Scholar 

  50. W. H. Young, “On the Fourier series of bounded functions,” Proc. London Math. Soc. 12, 41–70 (1913).

    Article  MathSciNet  MATH  Google Scholar 

  51. D. Yu, P. Zhou, and S. Zhou, “On \(L^p\) integrability and convergence of trigonometric series,” Stud. Math. 182 (3), 215–226 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 2002).

    MATH  Google Scholar 

Download references

Acknowledgments

I would like to thank A. Gogatishvili, M. L. Goldman, and V. D. Stepanov for their comments on Hardy’s inequality. I am also grateful to Amiran Gogatishvili for his remarks regarding Proposition 2.2.

Funding

This research was partially supported by Ministerio de Ciencia, Innovación y Universidades (grant MTM 2017-87409-P) and Generalitat de Catalunya (grant 2017 SGR 358).

Author information

Authors and Affiliations

  1. Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193, Bellaterra (Barcelona), Spain

    Sergey Yu. Tikhonov

  2. ICREA, Pg. Lluís Companys 23, 08010, Barcelona, Spain

    Sergey Yu. Tikhonov

  3. Universitat Autònoma de Barcelona, Plaça Cívica, 08193, Bellaterra (Cerdanyola del Vallès), Spain

    Sergey Yu. Tikhonov

Authors
  1. Sergey Yu. Tikhonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey Yu. Tikhonov.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 312, pp. 294–312 https://doi.org/10.4213/tm4130.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tikhonov, S.Y. Weighted Fourier Inequalities and Boundedness of Variation. Proc. Steklov Inst. Math. 312, 282–300 (2021). https://doi.org/10.1134/S0081543821010193

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543821010193

Keywords

Search

Navigation