We consider similar questions for the maximal fractional operator, too.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we investigate the weighted product Hardy spaces $H_w^{p}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$.
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Key Findings & Data:Under some conditions on the weight, we prove that the Riesz potential operator $I_\alpha$ is bounded from $L_w^{p}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ to $L_{w^{q/p}}^{q}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ when $\alpha=(\alpha_1, \alpha_2)$ and $\frac{1}{p}- \frac{1}{q} = \frac{\alpha_1}{d_1} =\frac{\alpha_2}{d_2}$.
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Conclusion & Impact:We consider similar questions for the maximal fractional operator, too.
In this paper, we investigate the weighted product Hardy spaces $H_w^{p}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$. Under some conditions on the weight, we prove that the Riesz potential operator $I_\alpha$ is bounded from $L_w^{p}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ to $L_{w^{q/p}}^{q}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ when $\alpha=(\alpha_1, \alpha_2)$ and $\frac{1}{p}- \frac{1}{q} = \frac{\alpha_1}{d_1} =\frac{\alpha_2}{d_2}$. We also verify the boundedness of $I_\alpha$ from $H_w^{p}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ to $H_{w^{q/p}}^{q}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ and from $H_w^{p}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$ to $L_{w^{q/p}}^{q}(\mathbb{R}^{d_1}\times \mathbb{R}^{d_2})$. We consider similar questions for the maximal fractional operator, too.
M. Bownik, B. Li, D. Yang and Y. Zhou: Weighted anisotropic product Hardy spaces and boundedness of sublinear operators, Math. Nachr., 283 (2010), 392–442.
J. Cao, D.-C. Chang, D. Yang and S. Yang: Boundedness of fractional integrals on weighted Orlicz-Hardy spaces, Math. Methods Appl. Sci., 36 (15) (2013), 2069–2085.
S. Y. A. Chang, R. Fefferman: Some recent developments in Fourier analysis and $H^{p}$-theory on product domains, Bull. Amer. Math. Soc., 12 (1985), 1–43.
X. Fan, J. He, B. Li and D. Yang: Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces, Sci. China, Math., 60 (11) (2017), 2093–2154.
R. Fefferman: Calderon-Zygmund theory for product domains: $H^{p}$ spaces, Proc. Nat. Acad. Sci. USA, 83 (1986), 840–843.
R. Fefferman: $A^{p}$ weights and singular integrals, Am. J. Math., 110 (5) (1988), 975–987.
J. Garcia-Cuerva, J. L. Rubio de Francia: Weighted Norm Inequalities and Related Topics, volume 116 of Mathematics Studies, North-Holland, Amsterdam (1985).
L. Grafakos: Classical and Modern Fourier Analysis, Pearson Education, New Jersey (2004).
G. H. Hardy, J. E. Littlewood: Some properties of fractional integrals. I, Math. Z., 27 (1928), 565–606.
K.-P. Ho: Strong maximal operator and singular integral operators in weighted Morrey spaces on product domains, Math. Nachr., 290 (16) (2017), 2629–2640.
D. Q. Huy, L. D. Ky: Boundedness of fractional integral operators on Musielak-Orlicz Hardy spaces, Math. Nachr., 294 (12) (2021), 2340–2354.
M. Izuki, E. Nakai and Y. Sawano: Atomic and wavelet characterization of Musielak-Orlicz Hardy spaces for generalized Orlicz functions, Integral Equations Oper. Theory, 94 (1) (2022), 33. Id/No 3.
H. Jia, F. Weisz, D. Yang, W. Yuan and Y. Zhang: Atomic Characterization of Musielak–Orlicz–Lorentz Hardy Spaces and Its Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators, J. Geom. Anal., 33 (6) (2023), Article ID: 188.
H. Jia, F. Weisz, D. Yang, W. Yuan and Y. Zhang: Musielak–Orlicz–Lorentz Hardy Spaces: Maximal Function, Finite Atomic, and Littlewood–Paley Characterizations with Applications to Dual Spaces and Summability of Fourier Transforms, Acta Math. Sinica, English Series., 41 (2025), 1–77.
J. L. Journé: Calderon-Zygmund operators on product spaces, Rev. Math. Iberoamericana, 1 (1985), 55–91.
L. D. Ky: New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Oper. Theory, 78 (1) (2014), 115–150.
B. Muckenhoupt, R. L. Wheeden: Weighted norm inequalities for fractional integrals, Trans. Am. Math. Soc., 192 (1974), 261–274.
F. Nursultanov, D. Suragan, M. A. Zaighum: On the boundedness of Riesz potential operators: insights from net spaces, Constr. Math. Anal., 8 (3) (2025), 165–175.
E. M. Stein: Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, N.J. (1993).
J.-O. Strömberg, A. Torchinsky: Weighted Hardy Spaces, volume 1381 of Lecture Notes in Math., Springer, Berlin (1989).
J.-O. Strömberg, R. L. Wheeden: Fractional integrals on weighted $H^{p}$ and $L^{p}$ spaces, Trans. Am. Math. Soc., 287 (1985), 293–321.
M. H. Taibleson, G. Weiss: The molecular characterization of certain Hardy spaces, In Representation theorems for Hardy spaces, pages 67–151, Paris: Société Mathématique de France (SMF) (1980).
F. Weisz: Product Musielak–Orlicz Hardy spaces and applications in harmonic analysis, (preprint).
F. Weisz: Summability of Multi-dimensional Fourier Series and Hardy Spaces, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, Boston, London (2002).
F. Weisz: Convergence and Summability of Fourier Transforms and Hardy Spaces, Applied and Numerical Harmonic Analysis, Springer, Birkhäuser, Basel (2017).
F. Weisz: Multi-parameter product martingale Musielak-Orlicz Hardy spaces and inequalities, J. Func. Anal., 290 (2026), Article ID: 111229.