The continuity of commutators on Triebel-Lizorkin spaces
In the paper, the author gives the following definition: Let \(K(x,t)\) be defined on \(\mathbb{R}^n\times \mathbb{R}^+\), and let \(K_t f(x)= \int_{\mathbb{R}^n} K(x-y,t) f(y)\). Define the space \[ H= \Biggl\{h: \| h\|= \Biggl(\int_{\mathbb{R}^+} |h(t)|^2 t^{-1} dt\Biggr)^{1/2}< \infty\Biggr\}. \] For each \(x\in \mathbb{R}^n\), view \(K_t f(x)\) as a function in \(H\). Let \(b\) be a locally integrable function. The commutator related to \(K_t\) is defined by \[ T_bf(x)= \| K_t(bf)(x)- b(x) K_t f(x)\|, \] and denote \(Tf(x)=\| K_t f(x)\|\). The above definition is informal. However, by choosing suitable kernels \(k(x,t)\) and functions \(b\), \(T_bf(x)\) includes commutators related to the Littlewood-Paley operator, the Marcinkiewicz integral and the Bochner-Riesz operator. Assuming that \(b\) is in the Lipschitz space \(\dot\Lambda_\beta\), \(0< \beta< 1\), that \(Tf\) is bounded on \(L^q(\mathbb{R}^n)\), \(1< q< \infty\), and satisfies a certain size condition, the author discusses the continuity of \(T_b\) in some Triebel-Lizorkin spaces.
scientific article; zbMATH DE number 2111522