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Maximal Operators Associated with Vector Polynomials of Lacunary Coefficients

Published online by Cambridge University Press:  20 November 2018

Sunggeum Hong ,
Joonil Kim  and
Chan Woo Yang
Sunggeum Hong
Affiliation:
Department of Mathematics, Chosun University, Gwangju 501-759, Korea, e-mail: skhong@mail.chosun.ac.kr
Joonil Kim
Affiliation:
Department of Mathematics, Yonsei University, Seoul 120-749, Korea, e-mail: jikim7030@yonsei.ac.kr
Chan Woo Yang
Affiliation:
Department of Mathematics, Korea University, Seoul 136-701, Korea, e-mail: cw_yang@korea.ac.kr
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Abstract

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We prove the ${{L}^{P}}({{\mathbb{R}}^{d}})(1<p\le \infty )$ boundedness of the maximal operators associated with a family of vector polynomials given by the form $\left\{ ({{2}^{{{k}_{1}}}}{{\mathfrak{p}}_{1}}(t),...,{{2}^{{{k}_{d}}}}{{\mathfrak{p}}_{d}}(t)):t\in \mathbb{R} \right\}$. Furthermore, by using the lifting argument, we extend this result to a general class of vector polynomials whose coefficients are of the form constant times ${{2}^{k}}$.

Keywords

42B2042B25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 4 , 01 August 2009 , pp. 807 - 827
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Carbery, A., Differentiation in lacunary directions and an extension of the Marinkiewicz multiplier theorem. Ann. Inst. Fourier 38(1988), no. 1, 157–168.Google Scholar
[2] Cordoba, A. and Fefferman, R., On differentiation of integrals. Proc. Nat. Acad. Sci. U. S. A. 74(1977), no. 6, 2211–2213.Google Scholar
[3] Hare, K. and Ricci, F., Maximal function with polynomial densities in lacunary directions. Trans. Amer. Math. Soc. 355(2003), no. 3, 1135–1144.Google Scholar
[4] Nagel, A., Stein, E. M., and Wainger, S., Differentiation in lacunary directions. Proc. Nat. Acad. Sci. U. S. A. 75(1978), no. 3, 1060–1062.Google Scholar
[5] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993.Google Scholar
[6] Stein, E. M., Singular integrals and differentiability of functions. Princeton Mathematical Series 30, Princeton University Press, Princeton, NJ, 1970.Google Scholar
[7] Stromberg, J.-O., Weak estimates on maximal functions with rectangles in certain directions. Ark. Mat. 15(1977), no. 2, 229–240.Google Scholar