Necessary and sufficient conditions for the bounds of the commutator of a Littlewood-Paley operator with fractional differentiation
Tóm tắt
For $$b\in L_{\mathrm{loc}}({\mathbb {R}}^n)$$ and $$0<\alpha <1$$, we use fractional differentiation to define a new type of commutator of the Littlewood-Paley g-function operator, namely $$\begin{aligned} g_{\Omega ,\alpha ;b}(f )(x) =\bigg (\int _0^\infty \bigg |\frac{1}{t} \int _{|x-y|\le t}\frac{\Omega (x-y)}{|x-y|^{n+\alpha -1}}(b(x)-b(y))f(y)\,dy\bigg |^2\frac{dt}{t}\bigg )^{1/2}. \end{aligned}$$Here, we obtain the necessary and sufficient conditions for the function b to guarantee that $$g_{\Omega ,\alpha ;b}$$ is a bounded operator on $$L^2({\mathbb {R}}^n)$$. More precisely, if $$\Omega \in L(\log ^+ L)^{1/2}{(S^{n-1})}$$ and $$b\in I_{\alpha }(BMO)$$, then $$g_{\Omega ,\alpha ;b}$$ is bounded on $$L^2({\mathbb {R}}^n)$$. Conversely, if $$g_{\Omega ,\alpha ;b}$$ is bounded on $$L^2({\mathbb {R}}^n)$$, then $$b \in Lip_\alpha ({\mathbb {R}}^n)$$ for $$0<\alpha < 1$$.
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