Skip to main content

Advertisement

SpringerLink
Log in

Logarithmic L p Bounds for Maximal Directional Singular Integrals in the Plane

  • Published:
Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

Let K be a Calderón–Zygmund convolution kernel on ℝ. We discuss the L p-boundedness of the maximal directional singular integral

$$T_{\mathbf{V}} f (x)= \sup_{v \in \mathbf{V}} \bigg| \int_{\mathbb{R}} f(x+t v) K(t) \, \mathrm{d} {t} \bigg| $$

where V is a finite set of N directions. Logarithmic bounds (for 2≤p<∞) are established for a set V of arbitrary structure. Sharp bounds are proved for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set.

We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques. As a further application of the latter, we derive an L p almost orthogonality principle for Fourier restrictions to cones.

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

Notes

  1. By CJ we mean the interval with the same center as J and C times longer. If R=I×J is a rectangle, CR=CI×CJ.

References

  1. Bateman, M.: Kakeya sets and directional maximal operators in the plane. Duke Math. J. 147(1), 55–77 (2009). MR 2494456 (2009m:42029)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bateman, M., Thiele, C.: L p Estimates for the Hilbert Transforms Along a One-variable Vector Field (2011). http://arxiv.org/abs/1109.6396

  3. Chang, S.-Y.A., Wilson, J.M., Wolff, T.H.: Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv. 60(2), 217–246 (1985). MR 800004 (87d:42027)

    Article  MATH  MathSciNet  Google Scholar 

  4. Christ, M., Duoandikoetxea, J., Rubio de Francia, J.L.: Maximal operators related to the Radon transform and the Calderón–Zygmund method of rotations. Duke Math. J. 53(1), 189–209 (1986). MR 835805 (88d:42032)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cordoba, A., Fefferman, R.: On the equivalence between the boundedness of certain classes of maximal and multiplier operators in Fourier analysis. Proc. Natl. Acad. Sci. USA 74(2), 423–425 (1977). MR 0433117 (55 #6096)

    Article  MATH  MathSciNet  Google Scholar 

  6. Demeter, C.: Singular integrals along N directions in ℝ2. Proc. Am. Math. Soc. 138(12), 4433–4442 (2010). MR 2680067 (2011i:42023)

    Article  MATH  MathSciNet  Google Scholar 

  7. Demeter, C.: L 2 Bounds for a Kakeya Type Maximal Operator in ℝ3 (2011). http://arxiv.org/abs/1105.1115v1

  8. Demeter, C., Lacey, M.T., Tao, T., Thiele, C.: Breaking the duality in the return times theorem. Duke Math. J. 143(2), 281–355 (2008). MR 2420509 (2009f:42013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Duoandikoetxea, J., Vargas, A.: Directional operators and radial functions on the plane. Ark. Mat. 33(2), 281–291 (1995). MR 1373025 (97c:42031)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fefferman, R., Pipher, J.: Multiparameter operators and sharp weighted inequalities. Am. J. Math. 119(2), 337–369 (1997). MR 1439553 (98b:42027)

    Article  MATH  MathSciNet  Google Scholar 

  11. García-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. In: Notas de Matemática [Mathematical Notes], 104. North-Holland Mathematics Studies, vol. 116. North-Holland, Amsterdam (1985). MR 807149 (87d:42023)

    Google Scholar 

  12. Grafakos, L., Honzík, P., Seeger, A.: On maximal functions for Mikhlin-Hörmander multipliers. Adv. Math. 204(2), 363–378 (2006). MR 2249617 (2007d:42015)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hunt, R.A.: An estimate of the conjugate function. Stud. Math. 44, 371–377 (1972). Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, IV. MR 0338667 (49 #3431)

    Google Scholar 

  14. Hytonen, T., Lacey, M., Martikainen, H., Orponen, T., Reguera, M., Sawyer, E.: Weak and strong type estimates for maximal truncations of Calderón–Zygmund operators on A p weighted spaces (2011). http://arxiv.org/abs/1103.5229v1

  15. Karagulyan, G.A.: On unboundedness of maximal operators for directional Hilbert transforms. Proc. Am. Math. Soc. 135(10), 3133–3141 (2007) (electronic). MR 2322743 (2008e:42044)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kashin, B.S., Saakyan, A.A.: Orthogonal Series. Translations of Mathematical Monographs, vol. 75. American Mathematical Society, Providence (1989). Translated from the Russian by Ralph P. Boas, Translation edited by Ben Silver. MR 1007141 (90g:42001)

    MATH  Google Scholar 

  17. Hawk Katz, N.: Maximal operators over arbitrary sets of directions. Duke Math. J. 97(1), 67–79 (1999). MR 1681088 (2000a:42036)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hawk Katz, N.: Remarks on maximal operators over arbitrary sets of directions. Bull. Lond. Math. Soc. 31(6), 700–710 (1999). MR 1711029 (2001g:42041)

    Article  MATH  Google Scholar 

  19. Kim, J.: Sharp L 2 bound of maximal Hilbert transforms over arbitrary sets of directions. J. Math. Anal. Appl. 335(1), 56–63 (2007). MR 2340304 (2009a:42012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lacey, M., Li, X.: On a Conjecture of E.M. Stein on the Hilbert Transform on Vector Fields. Mem. Amer. Math. Soc., vol. 205, (965) (2010). viii+72 pp. MR 2654385 (2011c:42019)

    Google Scholar 

  21. Lacey, M., Metcalfe, J.: Paraproducts in one and several parameters. Forum Math. 19(2), 325–351 (2007). MR 2313844 (2008b:42026)

    MATH  MathSciNet  Google Scholar 

  22. Lacey, M., Thiele, C.: A proof of boundedness of the Carleson operator. Math. Res. Lett. 7(4), 361–370 (2000). MR 1783613 (2001m:42009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lacey, M.T., Li, X.: Maximal theorems for the directional Hilbert transform on the plane. Trans. Am. Math. Soc. 358(9), 4099–4117 (2006) (electronic). MR 2219012 (2006k:42018)

    Article  MATH  MathSciNet  Google Scholar 

  24. Muscalu, C., Pipher, J., Tao, T., Thiele, C.: Multi-parameter paraproducts. Rev. Mat. Iberoam. 22(3), 963–976 (2006). MR 2320408 (2008b:42037)

    Article  MATH  MathSciNet  Google Scholar 

  25. Muscalu, C., Tao, T., Thiele, C.: L p estimates for the biest. II. The Fourier case. Math. Ann. 329(3), 427–461 (2004). MR 2127985 (2005k:42054)

    MATH  MathSciNet  Google Scholar 

  26. Petermichl, S.: The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p characteristic. Am. J. Math. 129(5), 1355–1375 (2007). MR 2354322 (2008k:42066)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pipher, J.: Bounded double square functions. Ann. Inst. Fourier (Grenoble) 36(2), 69–82 (1986). MR 850744 (88h:42021)

    Article  MATH  MathSciNet  Google Scholar 

  28. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970). MR 0290095 (44 #7280)

    MATH  Google Scholar 

  29. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993). With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. MR 1232192 (95c:42002)

    MATH  Google Scholar 

  30. Stein, E.M., Street, B.: Multi-parameter singular Radon transforms. Math. Res. Lett. 18(2), 257–277 (2011). MR 2784671 (2012b:44007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Thiele, C.: Wave Packet Analysis. CBMS Regional Conference Series in Mathematics, vol. 105 (2006). Published for the Conference Board of the Mathematical Sciences, Washington, DC. MR 2199086 (2006m:42073)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Ciprian Demeter & Francesco Di Plinio

  2. Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Francesco Di Plinio

Authors
  1. Ciprian Demeter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Di Plinio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Di Plinio.

Additional information

Communicated by Michael Lacey.

The first author is partially supported by a Sloan Research Fellowship and by NSF Grant DMS-0901208. The second author was partially supported by the National Science Foundation under the grant NSF-DMS-0906440, and by the Research Fund of Indiana University.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Demeter, C., Di Plinio, F. Logarithmic L p Bounds for Maximal Directional Singular Integrals in the Plane. J Geom Anal 24, 375–416 (2014). https://doi.org/10.1007/s12220-012-9340-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-012-9340-2

Keywords

Mathematics Subject Classification

Search

Navigation