Boundedness of partial difference transforms for heat semigroups generated by discrete Laplacian. (English) Zbl 1477.42017
Summary: We analyze the convergence of the following type of series
\[ T_N f(n)=\sum_{j=N_1}^{N_2} v_j\left(e^{a_{j+1}{\varDelta }_d} f(n)-e^{a_j{\varDelta }_d} f(n)\right),\quad n\in{\mathbb{Z}}, \]
where \(\{e^{t{\varDelta }_d} \}_{t>0}\) is the heat semigroup of the discrete Laplacian \({\varDelta }_d\), \(N=(N_1, N_2)\in{\mathbb{Z}}^2\) with \(N_1<N_2\), \(\{v_j\}_{j\in{\mathbb{Z}}}\) is a bounded real sequence and \(\{a_j\}_{j\in{\mathbb{Z}}}\) is an increasing real sequence. Our analysis will consist in the boundedness on \(\ell^p({\mathbb{Z}}, \omega )\) of the operators \(T_N\) and its maximal operator \( T^\ast f(n)= \sup_N \left| T_N f(n)\right| \), where \(1\le p<\infty\) and \(\omega\) is a discrete Muckenhoupt weight. Moreover, we also get the alternative behavior of the maximal operator \(T^\ast \) on \(\ell^\infty ({\mathbb{Z}})\).
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B30 | \(H^p\)-spaces |
| 42B15 | Multipliers for harmonic analysis in several variables |
References:
| [1] | Almeida, V., Betancor, J., Rodríguez-Mesa, L.: Discrete Hardy spaces and heat semigroup associated with the discrete Laplacian. Mediterr. J. Math. 16(4), Article No. 91 (2019) · Zbl 1416.42027 |
| [2] | Bernardis, AL; Lorente, M.; Martín-Reyes, FJ; Martínez, MT; de la Torre, A.; Torrea, JL, Differential transforms in weighted spaces, J. Fourier Anal. Appl., 12, 83-103 (2006) · Zbl 1115.42003 · doi:10.1007/s00041-005-5064-z |
| [3] | Chao, Z.; Ma, T.; Torrea, JL, Boundedness of differential transforms for one-sided fractional Poisson type operator sequence, J. Geom. Anal., 31, 67-99 (2021) · Zbl 1460.42013 · doi:10.1007/s12220-019-00251-x |
| [4] | Chao, Z.; Torrea, JL, Boundedness of differential transforms for heat semigroups generated by Schrödinger operators, Can. J. Math., 73, 622-655 (2021) · Zbl 1467.42020 · doi:10.4153/S0008414X20000097 |
| [5] | Ciaurri, O.; Gillespie, T.; Roncal, L.; Torrea, JL; Varona, JL, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132, 109-131 (2017) · Zbl 1476.39003 · doi:10.1007/s11854-017-0015-6 |
| [6] | Grünbaum, FA; Bustoz, J.; Ismail, ME; Suslov, S., The bispectral problem: an overview, Special Functions 2000: Current Perspective and Future Directions, 129-140 (2001), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 0999.47018 · doi:10.1007/978-94-010-0818-1_6 |
| [7] | Grünbaum, FA; Iliev, P., Heat kernel expansions on the integers, Math. Phys. Anal. Geom., 5, 183-200 (2002) · Zbl 0996.35077 · doi:10.1023/A:1016258207606 |
| [8] | Gunawan, H., Schwanke, C.: The Hardy-Littlewood maximal operator on discrete Morrey spaces. Mediterr. J. Math. 16(1), Article No. 24 (2019) · Zbl 1414.42028 |
| [9] | Hunt, R.; Muckenhoupt, B.; Wheeden, R., Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Am. Math. Soc., 176, 227-251 (1973) · Zbl 0262.44004 · doi:10.1090/S0002-9947-1973-0312139-8 |
| [10] | Jones, RL; Rosenblatt, J., Differential and ergodic transforms, Math. Ann., 323, 525-546 (2002) · Zbl 1005.28012 · doi:10.1007/s002080200313 |
| [11] | Lebedev, NN, Special Functions and Their Applications (1972), New York: Dover, New York · Zbl 0271.33001 |
| [12] | Pierce, L.B.: Discrete Analogues in Harmonic Analysis. Ph.D. thesis, Princeton University, Princeton (2009) |
| [13] | Prudnikov, AP; Brychkov, AY; Marichev, OI, Integrals and Series. Volume 1. Elementary Functions (1986), New York: Gordon and Breach Science Publishers, New York · Zbl 0733.00004 |
| [14] | Rubio de Francia, JL; Ruiz, FJ; Torrea, JL, Calderón-Zygmund theory for operator-valued kernels, Adv. Math., 62, 7-48 (1986) · Zbl 0627.42008 · doi:10.1016/0001-8708(86)90086-1 |
| [15] | Stein, EM; Wainger, S., Discrete analogues in harmonic analysis, I: \( \ell^2\) estimates for singular Radon transforms, Am. J. Math., 121, 1291-1336 (1999) · Zbl 0945.42009 · doi:10.1353/ajm.1999.0046 |
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