Bilinear Θ-type Calderón–Zygmund operators and their commutators on product generalized fractional mixed Morrey spaces,Mathematische Nachrichten - X-MOL

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Bilinear Θ-type Calderón–Zygmund operators and their commutators on product generalized fractional mixed Morrey spaces
Mathematische Nachrichten ( IF 1 ) Pub Date : 2024-02-12 , DOI: 10.1002/mana.202200481
Guanghui Lu ₁ , Shuangping Tao ₁ , Miaomiao Wang ₁

Affiliation

The aim of this paper is to investigate the boundedness of the bilinear θ-type Calderón–Zygmund operator and its commutator on the product of generalized fractional mixed Morrey spaces. Under assumption that the positive and increasing functions

φ (\cdot) $\varphi (\cdot)$

defined on [0, ∞) satisfy doubling conditions, we prove that the bilinear θ-type Calderón–Zygmund operator

{\tilde{T}}_{θ} $\widetilde{T}_{\theta }$

is bounded from the product of generalized fractional mixed Morrey spaces

L^{{\vec{p}}_{1}, η_{1}, φ} (R^{n}) \times L^{{\vec{p}}_{2}, η_{2}, φ} (R^{n}) $L^{\vec{p}_{1},\eta _{1},\varphi }({\bf R}^{n})\times L^{\vec{p}_{2},\eta _{2},\varphi }({\bf R}^{n})$

into spaces

L^{\vec{p}, η, φ} (R^{n}) $L^{\vec{p},\eta,\varphi }({\bf R}^{n})$

, where

{\vec{p}}_{1} = (p_{11}, …, p_{1 n}) $\vec{p}_{1}=(p_{11},\ldots ,p_{1n})$

,

{\vec{p}}_{2} = (p_{21}, …, p_{2 n}) $\vec{p}_{2}=(p_{21},\ldots ,p_{2n})$

,

\vec{p} = (p_{1}, …, p_{n}) $\vec{p}=(p_{1},\ldots ,p_{n})$

,

\frac{1}{{\vec{p}}_{1}} + \frac{1}{{\vec{p}}_{2}} = \frac{1}{\vec{p}} $\frac{1}{\vec{p}_{1}}+\frac{1}{\vec{p}_{2}}=\frac{1}{\vec{p}}$

for

1 < {\vec{p}}_{1}, {\vec{p}}_{2} < \infty $1<\vec{p}_{1}, \vec{p}_{2}<\infty$

, and

η = η_{1} + η_{2} $\eta =\eta _{1}+\eta _{2}$

for

0 \leq η_{1}, η_{2}, η < n $0\le \eta _{1},\eta _{2}, \eta <n$

. Furthermore, the boundedness of the commutator

{\tilde{T}}_{θ, b_{1}, b_{2}} $\widetilde{T}_{\theta,b_{1},b_{2}}$

formed by

b_{1}, b_{2} \in BMO (R^{n}) $b_{1}, b_{2}\in \mathrm{BMO}({\bf R}^{n})$

and

{\tilde{T}}_{θ} $\widetilde{T}_{\theta }$

on spaces

L^{\vec{p}, η, φ} (R^{n}) $L^{\vec{p},\eta,\varphi }({\bf R}^{n})$

is also obtained.

更新日期：2024-02-12

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