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Revista Integración

Print version ISSN 0120-419X

Integración - UIS vol.33 no.1 Bucaramanga Jan./June 2015

 

Weak-type (1,1) bounds for a class of
operators with discrete kernel

DUVÁN CARDONA*,

Universidad del Valle, Department of Mathematics, A.A. 25360, Cali, Colombia.


Abstract. In this paper we investigate the weak continuity of a certain class of operators with kernel defined on ℤ × ℤ. We prove some results on the weak boundedness of discrete analogues of Calderón-Zygmund operators. The considered operators arise from the study of discrete pseudo-differential operators and discrete analogues of singular integral operators.

Keywords: Lp spaces, discrete operator, pseudo-differential operator, Calderón-Zygmund decomposition.
MSC2010: 47B34, 47G10, 28A25.



Cotas del tipo débil (1,1) para una clase de operadores
con núcleo discreto

Resumen. En este trabajo se investigará el tipo débil (1,1) de una cierta clase de operadores con núcleo definido sobre ℤ × ℤ. Se estudiará la continuidad débil de operadores que son análogos discretos de los ahora conocidos, operadores singulares integrales de Calderón-Zygmund. Los operadores considerados surgen desde el estudio de operadores pseudo diferenciales de tipo discreto y versiones discretas de integrales singulares.

Palabras clave: Espacios Lp, operador discreto, operador pseudo diferencial, descomposición de Calderón-Zygmund.


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References

[1] Bober J., Carneiro E., Hughes K. and Pierce L., "On a discrete version of Tanaka's theorem for maximal functions", Proc. Amer. Math. Soc. 140 (2012), no. 5, 1669-1680.         [ Links ]

[2] Calderón A.P. and Zygmund A., "On the existence of certain singular integrals", Acta Math. 88 (1952), 85-139.         [ Links ]

[3] Cardona D., "Invertibilidad de operadores pseudo diferenciales definidos en ℤn", Lect. Mat. 34 (2013), no. 2, 179-186.         [ Links ]

[4] Carneiro E. and Hughes K., "On the endpoint regularity of discrete maximal operators", Math. Res. Lett. 19 (2012), no. 6, 1245-1262.         [ Links ]

[5] Carro M., "Discretization of linear operators on Lp(ℝn)", Illinois J. Math. 42 (1998), no. 1, 1-18.         [ Links ]

[6] Duoandikoetxea J., Fourier Analysis, American Mathematical Society, Providence, RI, 2001.         [ Links ]

[7] Grafakos L., "An elementary proof of the square summability of the discrete Hilbert transform", Amer. Math. Monthly. 101 (1994), no. 5, 456-458.         [ Links ]

[8] Hughes K.J. Jr., "Arithmetic analogues in harmonic analysis: Results related to Waring's problem", Thesis (Ph.D.), Princeton University, 2012, 112 p.         [ Links ]

[9] Kikuchi N., Nakai E., Tomita N., Yabuta K. and Yoneda T., "Calderón-Zygmund operators on amalgam spaces and in the discrete case", J. Math. Anal. Appl. 335 (2007), no. 1, 198-212.         [ Links ]

[10] Marcinkiewicz J., "Sur l'interpolation d'operations", C. R. Acad. Sci. Paris. 208 (1939), 1272-1273.         [ Links ]

[11] Mirek M., "Weak type (1,1) inequalities for discrete rough maximal functions", arXiv:1305.0575v2 (2014).         [ Links ]

[12] Molahajloo S., "Pseudo-differential operators on ℤ : Pseudo-differential operators: complex analysis and partial differential equations", Oper. Theory. Adv. Appl. 205 (2010), 213-221.         [ Links ]

[13] Pierce L., "Discrete analogues in harmonic analysis", Thesis (Ph.D), Princeton University, 2009, 321 p.         [ Links ]

[14] Riesz M., "Sur les fonctions conjuguées", Math. Z. 27 (1928), no.1, 218-244.         [ Links ]

[15] Rodriguez C.A., "Lp-estimates for pseudo-differential operators on ℤn", J. Pseudo-Differ. Oper. Appl. 1 (2011), 183-205.         [ Links ]

[16] Stein E., Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, 1993.         [ Links ]

[17] Stein E. and Wainger S., "Discrete analogues of singular Radon transforms", Bull. Amer. Math. Soc. (N.S.). 23 (1990), no. 2, 537-544.         [ Links ]

[18] Stein E. and Wainger S., "Discrete analogues in harmonic analysis. I. l2 estimates for singular Radon transforms", Amer. J. Math. 121 (1999), no. 6, 1291-1336.         [ Links ]

[19] Stein E. and Wainger S., "Discrete analogues in harmonic analysis, II. Fractional integration", J. Anal. Math. 80 (2000), 335-355.         [ Links ]

[20] Urban R. and Zienkiewicz J., "Weak type (1,1) estimates for a class of discrete rough maximal functions", Math. Res. Lett. 14 (2007), no. 2, 227-237.         [ Links ]

[21] Wong M.W., Discrete Fourier Analysis. Birkhäuser/Springer Basel AG, Basel, 2011.         [ Links ]

[22] Zygmund A., "On a theorem of Marcinkiewicz concerning interpolation of operations", J. Math. Pures. Appl. 35 (1956), no. 9, 223-248.         [ Links ]


*E-mail: duvanc306@gmail.com.
Received: 09 September 2014, Accepted: 10 March 2015.
To cite this article: D. Cardona, Weak-type (1,1) bounds for a class of operators with discrete kernel, Rev. Integr. Temas Mat. 33 (2015), no. 1, 51-60.