In the setting of a Heisenberg group, we first studied the sharp weak estimate for the $ n $-dimensional fractional Hardy operator from $ L^p $ to $ L^{q, \infty} $. Next, we studied the sharp bounds for the $ m $-linear $ n $-dimensional integral operator with a kernel on weighted Lebesgue spaces. As an application, the sharp bounds for Hardy, Hardy-Littlewood-Pólya, and Hilbert operators on weighted Lebesgue spaces were obtained. Finally, according to the previous steps, we also found the estimate for the Hausdorff operator on weighted $ L^p $ spaces.
Citation: Tianyang He, Zhiwen Liu, Ting Yu. The Weighted $ \boldsymbol{L}^{\boldsymbol{p}} $ estimates for the fractional Hardy operator and a class of integral operators on the Heisenberg group[J]. AIMS Mathematics, 2025, 10(1): 858-883. doi: 10.3934/math.2025041
In the setting of a Heisenberg group, we first studied the sharp weak estimate for the $ n $-dimensional fractional Hardy operator from $ L^p $ to $ L^{q, \infty} $. Next, we studied the sharp bounds for the $ m $-linear $ n $-dimensional integral operator with a kernel on weighted Lebesgue spaces. As an application, the sharp bounds for Hardy, Hardy-Littlewood-Pólya, and Hilbert operators on weighted Lebesgue spaces were obtained. Finally, according to the previous steps, we also found the estimate for the Hausdorff operator on weighted $ L^p $ spaces.
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Tianyang He, Zhiwen Liu, Ting Yu. The Weighted $ \boldsymbol{L}^{\boldsymbol{p}} $ estimates for the fractional Hardy operator and a class of integral operators on the Heisenberg group[J]. AIMS Mathematics, 2025, 10(1): 858-883. doi: 10.3934/math.2025041
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