Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
Littlewood-Paley function; non-isotropic dilation
Summary:
We consider Littlewood-Paley functions associated with a non-isotropic dilation group on $\Bbb R^n$. We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted $L^p$ spaces, $1<p<\infty $, with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).\looseness -1
References:
[1] Benedek, A., Calderón, A. P., Panzone, R.: Convolution operators on Banach space valued functions. Proc. Natl. Acad. Sci. USA 48 (1962), 356-365. DOI 10.1073/pnas.48.3.356 | MR 0133653 | Zbl 0103.33402
[2] Calderón, A. P.: Inequalities for the maximal function relative to a metric. Stud. Math. 57 (1976), 297-306. DOI 10.4064/sm-57-3-297-306 | MR 0442579 | Zbl 0341.44007
[3] Calderón, A. P., Torchinsky, A.: Parabolic maximal functions associated with a distribution. Adv. Math. 16 (1975), 1-64. DOI 10.1016/0001-8708(75)90099-7 | MR 0417687 | Zbl 0315.46037
[4] Capri, O. N.: On an inequality in the theory of parabolic $H^p$ spaces. Rev. Unión Mat. Argent. 32 (1985), 17-28. MR 0873913 | Zbl 0643.42013
[5] Cheng, L. C.: On Littlewood-Paley functions. Proc. Am. Math. Soc. 135 (2007), 3241-3247. DOI 10.1090/S0002-9939-07-08917-4 | MR 2322755 | Zbl 1124.42013
[6] Ding, Y., Sato, S.: Littlewood-Paley functions on homogeneous groups. Forum Math. 28 (2016), 43-55. DOI 10.1515/forum-2014-0058 | MR 3441105 | Zbl 1332.42007
[7] Duoandikoetxea, J.: Weighted norm inequalities for homogeneous singular integrals. Trans. Am. Math. Soc. 336 (1993), 869-880. DOI 10.2307/2154381 | MR 1089418 | Zbl 0770.42011
[8] Duoandikoetxea, J.: Sharp $L^p$ boundedness for a class of square functions. Rev. Mat. Complut. 26 (2013), 535-548. DOI 10.1007/s13163-012-0106-y | MR 3068610 | Zbl 1334.42040
[9] Duoandikoetxea, J., Francia, J. L. Rubio de: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84 (1986), 541-561. DOI 10.1007/BF01388746 | MR 0837527 | Zbl 0568.42012
[10] Duoandikoetxea, J., Seijo, E.: Weighted inequalities for rough square functions through extrapolation. Stud. Math. 149 (2002), 239-252. DOI 10.4064/sm149-3-2 | MR 1890732 | Zbl 1015.42016
[11] Fan, D., Sato, S.: Remarks on Littlewood-Paley functions and singular integrals. J. Math. Soc. Japan 54 (2002), 565-585. DOI 10.2969/jmsj/1191593909 | MR 1900957 | Zbl 1029.42010
[12] Garcia-Cuerva, J., Francia, J. L. Rubio de: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies 116, Notas de Matemática 104, North-Holland, Amsterdam (1985). DOI 10.1016/s0304-0208(08)x7154-3 | MR 0807149 | Zbl 0578.46046
[13] Hörmander, L.: Estimates for translation invariant operators in $L^p$ spaces. Acta Math. 104 (1960), 93-140. DOI 10.1007/BF02547187 | MR 0121655 | Zbl 0093.11402
[14] Rivière, N.: Singular integrals and multiplier operators. Ark. Mat. 9 (1971), 243-278. DOI 10.1007/BF02383650 | MR 0440268 | Zbl 0244.42024
[15] Francia, J. L. Rubio de: Factorization theory and $A_p$ weights. Am. J. Math. 106 (1984), 533-547. DOI 10.2307/2374284 | MR 0745140 | Zbl 0558.42012
[16] Sato, S.: Remarks on square functions in the Littlewood-Paley theory. Bull. Aust. Math. Soc. 58 (1998), 199-211. DOI 10.1017/S0004972700032172 | MR 1642027 | Zbl 0914.42012
[17] Sato, S.: Estimates for Littlewood-Paley functions and extrapolation. Integral Equations Oper. Theory 62 (2008), 429-440. DOI 10.1007/s00020-008-1631-4 | MR 2461129 | Zbl 1166.42009
[18] Sato, S.: Estimates for singular integrals along surfaces of revolution. J. Aust. Math. Soc. 86 (2009), 413-430. DOI 10.1017/S1446788708000773 | MR 2529333 | Zbl 1182.42019
[19] Sato, S.: Littlewood-Paley equivalence and homogeneous Fourier multipliers. Integral Equations Oper. Theory 87 (2017), 15-44. DOI 10.1007/s00020-016-2333-y | MR 3609237 | Zbl 1364.42025
[20] Stein, E. M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30, Princeton University Press, Princeton (1970). DOI 10.1515/9781400883882 | MR 0290095 | Zbl 0207.13501
[21] Stein, E. M., Wainger, S.: Problems in harmonic analysis related to curvature. Bull. Am. Math. Soc. 84 (1978), 1239-1295. DOI 10.1090/S0002-9904-1978-14554-6 | MR 0508453 | Zbl 0393.42010
Partner of
EuDML logo