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This paper is dedicated to studying some Hausdorff operators on the Heisenberg group $\mathbb{H}^{n}$. The sharp bounds on the strong-type weighted Lorentz spaces $\Lambda _{u}^{p}(w)$ and the weak-type weighted Lorentz spaces $ \Lambda _{u}^{p,\infty }(w)$ are investigated. Especially, the results cover the classical power weighted space $L_{\alpha}^{p,q}$. The results are also extended to the product spaces $\Lambda _{u_{1}}^{p_{1}}(w_{1})\times \Lambda_{u_{2}}^{p_{2}}(w_{2})$, especially for $L_{\alpha_{1}}^{p_{1},q_{1}}\times L_{\alpha _{2}}^{p_{2},q_{2}}$. Our proofs are quite different from those in previous documents since the duality principle, and some well-known inequalities concerning the weights are adopted. The results recover the existing results as well as we obtain new results in the new and old settings.
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참고문헌 (38)
참고문헌 신청
M. A. Arino / 1990 / Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for non-increasing functions / Trans. Amer. Math. Soc. 320:727~735
A. Benyi / 2006 / Best constants for certain multilinear integral operators / J. Inequal. Appl. :1~12
C. Bennett / 1988 / Interpolation of Operators, Pure and Applied Mathematics, 129 / Academic Press
S. Boza / 2017 / Norm estimates for the Hardy operator in terms of $B_p$ weights / Proc. Amer. Math. Soc 145:2455~2465
J. Chen / 2012 / Hausdorff operators on function spaces / Chin. Ann. Math. Ser. B 33:537~556
A. Benyi / 2006 / Best constants for certain multilinear integral operators / J. Inequal. Appl. :1~12
C. Bennett / 1988 / Interpolation of Operators, Pure and Applied Mathematics, 129 / Academic Press
S. Boza / 2017 / Norm estimates for the Hardy operator in terms of
J. Chen / 2012 / Hausdorff operators on function spaces / Chin. Ann. Math. Ser. B 33:537~556
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