×

Hörmander type multiplier theorems on bi-parameter anisotropic Hardy spaces. (English) Zbl 1439.42012

Summary: The main purpose of this paper is to establish, using the bi-parameter Littlewood-Paley-Stein theory (in particular, the bi-parameter Littlewood-Paley-Stein square functions), a Calderón-Torchinsky type theorem for the following Fourier multipliers on anisotropic product Hardy spaces \(H^p(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2};\vec{A})\) \((0<p\leq 1)\):
\[T_mf(x,y)=\int_{\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}}m(\xi,\eta)\hat{f}(\xi,\eta)e^{2\pi i(x\cdot\xi+y\cdot\eta)}\mathop{}\!d\xi\mathop{}\!d\eta. \] Our main theorem is the following: Assume that \(m(\xi,\eta)\) is a function on \(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}\) satisfying \[ \sup_{j,k\in\mathbb{Z}}\lVert m_{j,k}\rVert_{W^{(s_1,s_2)}(\vec{A})}<\infty \] with \(s_1>\zeta_{1,-}^{-1}(\frac{1}{p}-\frac{1}{2}), s_2>\zeta_{2,-}^{-1}(\frac{1}{p}-\frac{1}{2})\), where \(\zeta_{1,-}\) and \(\zeta_{2,-}\) depend only on the eigenvalues and are defined in the first section. Then \(T_m\) is bounded from \(H^p(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2};\vec{A})\) to \(H^p(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2};\vec{A})\) for all \(0<p\leq 1\) and
\[ \lVert T_m\rVert_{H^p(\vec{A})\to H^p(\vec{A})}\leq C_{\vec{A},s_1,s_2,p}\sup_{j,k\in\mathbb{Z}}\lVert m_{j,k}\rVert_{W^{(s_1,s_2)}(\vec{A})}, \] where \(W^{(s_1,s_2)}(\vec{A})\) is a bi-parameter anisotropic Sobolev space on \(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}\) with \(C_{\vec{A},s_1,s_2,p}\) is a positive constant that depends on \(\vec{A},s_1,s_2,p\). Here we use the notations \(m_{j,k}(\xi,\eta)=m(A_1^{\ast j}\xi,A_2^{\ast k}\eta)\varphi^{(1)}(\xi) \varphi^{(2)}(\eta)\), where \(\varphi^{(1)}(\xi)\) is a suitable cut-off function on \(\mathbb{R}^{n_1}\) and \(\varphi^{(2)}(\eta)\) is a suitable cut-off function on \(\mathbb{R}^{n_2} \), respectively.

MSC:

42B15 Multipliers for harmonic analysis in several variables
42B25 Maximal functions, Littlewood-Paley theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc. 781 (2003), 1-122. · Zbl 1036.42020
[2] M. Bownik, B. Li, D. Yang and Y. Zhou, Weighted anisotropic product Hardy spaces and boundedness of sublinear operators, Math. Nachr. 283 (2010), no. 3, 392-442. · Zbl 1205.42021
[3] A.-P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution. II, Adv. Math. 24 (1977), no. 2, 101-171. · Zbl 0355.46021
[4] A. Carbery and A. Seeger, H^p- and L^p-variants of multiparameter Calderón-Zygmund theory, Trans. Amer. Math. Soc. 334 (1992), no. 2, 719-747. · Zbl 0770.42010
[5] S.-Y. A. Chang and R. Fefferman, A continuous version of duality of H^1 with BMO on the bidisc, Ann. of Math. (2) 112 (1980), no. 1, 179-201. · Zbl 0451.42014
[6] S.-Y. A. Chang and R. Fefferman, Some recent developments in Fourier analysis and H^p-theory on product domains, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 1-43. · Zbl 0557.42007
[7] J. Chen and G. Lu, Hörmander type theorem on bi-parameter Hardy spaces for bi-parameter Fourier multipliers with optimal smoothness, Rev. Mat. Iberoam. 34 (2018), no. 4, 1541-1561. · Zbl 1422.42023
[8] L.-K. Chen, The multiplier operators on the product spaces, Illinois J. Math. 38 (1994), no. 3, 420-433. · Zbl 0801.42007
[9] M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601-628. · Zbl 0758.42009
[10] W. Ding and G. Lu, Duality of multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular integral operators, Trans. Amer. Math. Soc. 368 (2016), no. 10, 7119-7152. · Zbl 1338.42025
[11] W. Ding, G. Lu and Y. Zhu, Multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular integrals and their atomic decomposition, Forum Math. 28 (2016), no. 1, 25-42. · Zbl 1335.42021
[12] W. Ding, G. Lu and Y. Zhu, Discrete Littlewood-Paley-Stein characterization of multi-parameter local Hardy spaces, Forum Math. 31 (2019), no. 6, 1467-1488. · Zbl 1440.42104
[13] W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 (2019), 352-380. · Zbl 1418.42038
[14] Y. Ding, G. Z. Lu and B. L. Ma, Multi-parameter Triebel-Lizorkin and Besov spaces associated with flag singular integrals, Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 4, 603-620. · Zbl 1192.42014
[15] R. Fefferman, Harmonic analysis on product spaces, Ann. of Math. (2) 126 (1987), no. 1, 109-130. · Zbl 0644.42017
[16] R. Fefferman and E. M. Stein, Singular integrals on product spaces, Adv. Math. 45 (1982), no. 2, 117-143. · Zbl 0517.42024
[17] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Reg. Conf. Ser. Math. 79, American Mathematical Society, Providence, 1991. · Zbl 0757.42006
[18] R. F. Gundy and E. M. Stein, H^p theory for the poly-disc, Proc. Natl. Acad. Sci. USA 76 (1979), no. 3, 1026-1029. · Zbl 0405.32002
[19] Y. Han, G. Lu and E. Sawyer, Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group, Anal. PDE 7 (2014), no. 7, 1465-1534. · Zbl 1318.42026
[20] L. Hörmander, Estimates for translation invariant operators in L^p spaces, Acta Math. 104 (1960), 93-140. · Zbl 0093.11402
[21] J.-L. Journé, Calderón-Zygmund operators on product spaces, Rev. Mat. Iberoam. 1 (1985), no. 3, 55-91. · Zbl 0634.42015
[22] J.-L. Journé, Two problems of Calderón-Zygmund theory on product-spaces, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 1, 111-132. · Zbl 0638.47026
[23] H. V. Le, Multiplier operators on product spaces, Studia Math. 151 (2002), no. 3, 265-275. · Zbl 1050.42011
[24] G. Lu and Z. Ruan, Duality theory of weighted Hardy spaces with arbitrary number of parameters, Forum Math. 26 (2014), no. 5, 1429-1457. · Zbl 1298.42025
[25] S. G. Mihlin, On the multipliers of Fourier integrals (in Russian), Dokl. Akad. Nauk SSSR (N. S.) 109 (1956), 701-703. · Zbl 0073.08402
[26] A. Miyachi, On some Fourier multipliers for H^p({\mathbf{R}}^n), J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 1, 157-179. · Zbl 0433.42019
[27] J. Pipher, Journé’s covering lemma and its extension to higher dimensions, Duke Math. J. 53 (1986), no. 3, 683-690. · Zbl 0645.42018
[28] Z. Ruan, Multi-parameter Hardy spaces via discrete Littlewood-Paley theory, Anal. Theory Appl. 26 (2010), no. 2, 122-139. · Zbl 1224.42039
[29] S. Sato, Lusin functions on product spaces, Tohoku Math. J. (2) 39 (1987), no. 1, 41-59. · Zbl 0652.42007
[30] S. Sato, An atomic decomposition for parabolic H^p spaces on product domains, Proc. Amer. Math. Soc. 104 (1988), no. 1, 185-192. · Zbl 0674.42004
[31] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813-874. · Zbl 0783.42011
[32] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I, World Scientific, Hackensack, 2011. · Zbl 1254.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.