The parameterized local fractional singular integral operator $ \mathit{ T}^{(\tau)} $ is defined on the space $ L_p^{\tau\mu}(\mathbb{R}_+^\tau) $ as $ \mathit{T}^{(\tau)}: L_p^{\tau \mu}(\mathbb{R}_+^\tau)\rightarrow L_p^{\tau\nu(1-p)}(\mathbb{R}_+^\tau) $, $ \mathit{T} ^{(\tau)}(f_\tau)(y) = \, _0\mathscr{Y}_{+\infty}^{(\tau)}\Big[\frac{ |x-y|^{\tau\alpha}}{(x+y)^{\tau\beta}}f_\tau(x)\Big], y\in\mathbb{R}_+ $. By employing the weight function method and analysis techniques on the fractal real line number set $ \mathbb{R}_+^\tau $, a general Hilbert-type local fractional integral inequality has been established, thereby demonstrating the boundedness of the defined integral operator. Through optimization of parameters, it was determined that the necessary and sufficient condition for the constant factor in this general Hilbert-type local fractional inequality to be the best possible is that the power parameters $ \sigma $ and $ \sigma_1 $ satisfy $ \sigma+\sigma_1 = \beta-\alpha $. Consequently, the formula for calculating the operator norm has been derived.
Citation: Ling Peng, Qiong Liu. The construction conditions of a Hilbert-type local fractional integral operator and the norm of the operator[J]. AIMS Mathematics, 2025, 10(1): 1779-1791. doi: 10.3934/math.2025081
The parameterized local fractional singular integral operator $ \mathit{ T}^{(\tau)} $ is defined on the space $ L_p^{\tau\mu}(\mathbb{R}_+^\tau) $ as $ \mathit{T}^{(\tau)}: L_p^{\tau \mu}(\mathbb{R}_+^\tau)\rightarrow L_p^{\tau\nu(1-p)}(\mathbb{R}_+^\tau) $, $ \mathit{T} ^{(\tau)}(f_\tau)(y) = \, _0\mathscr{Y}_{+\infty}^{(\tau)}\Big[\frac{ |x-y|^{\tau\alpha}}{(x+y)^{\tau\beta}}f_\tau(x)\Big], y\in\mathbb{R}_+ $. By employing the weight function method and analysis techniques on the fractal real line number set $ \mathbb{R}_+^\tau $, a general Hilbert-type local fractional integral inequality has been established, thereby demonstrating the boundedness of the defined integral operator. Through optimization of parameters, it was determined that the necessary and sufficient condition for the constant factor in this general Hilbert-type local fractional inequality to be the best possible is that the power parameters $ \sigma $ and $ \sigma_1 $ satisfy $ \sigma+\sigma_1 = \beta-\alpha $. Consequently, the formula for calculating the operator norm has been derived.
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Ling Peng, Qiong Liu. The construction conditions of a Hilbert-type local fractional integral operator and the norm of the operator[J]. AIMS Mathematics, 2025, 10(1): 1779-1791. doi: 10.3934/math.2025081
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