Regularity properties of higher order maximal commutators with Lipschitz symbols
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In the present paper we establish the boundedness and continuity of the higher order maximal commutators with Lipschitz symbols on the Sobolev spaces, Triebel--Lizorkin spaces and Besov spaces. More precisely, let $0\leq\alpha<d$ and $\mathfrak{M}_{b,\alpha}^{k}$ $(k\geq1)$ be the $k$-th order fractional maximal commutator. When $\alpha=0$, we denote $\mathfrak{M}_{b,\alpha}^{k}=\mathfrak{M}_{b}^{k}$. We prove that $\mathfrak{M}_{b,\alpha}^{k}$ maps the first order Sobolev spaces $W^{1,p}(\mathbb{R}^d)$ boundedly and continuously to $W^{1,q}(\mathbb{R}^d)$ for $1<p<q<\infty$ and $1/q=1/p-\alpha/d$ if $b$ belongs to the inhomogeneous Lipschitz space ${\rm Lip}(\mathbb{R}^d)$. We also show that if $0<\gamma\leq1$, $0<s<\gamma$, $1<p,q<\infty$ and $b\in{\rm Lip}_\gamma(\mathbb{R}^d)$, then $\mathfrak{M}_{b}^{k}$ is bounded and continuous from the fractional Sobolev spaces $W^{s,p}(\mathbb{R}^d)$ to itself, from the inhomogeneous Triebel--Lizorkin spaces $F^{p,q}_{s}(\mathbb{R}^d)$ to itself and from the inhomogeneous Besov spaces $B^{p,q}_{s}(\mathbb{R}^d)$ to itself.
Abstract
In the present paper we establish the boundedness and continuity of the higher order maximal commutators with Lipschitz symbols on the Sobolev spaces, Triebel--Lizorkin spaces and Besov spaces. More precisely, let $0\leq\alpha<d$ and $\mathfrak{M}_{b,\alpha}^{k}$ $(k\geq1)$ be the $k$-th order fractional maximal commutator. When $\alpha=0$, we denote $\mathfrak{M}_{b,\alpha}^{k}=\mathfrak{M}_{b}^{k}$. We prove that $\mathfrak{M}_{b,\alpha}^{k}$ maps the first order Sobolev spaces $W^{1,p}(\mathbb{R}^d)$ boundedly and continuously to $W^{1,q}(\mathbb{R}^d)$ for $1<p<q<\infty$ and $1/q=1/p-\alpha/d$ if $b$ belongs to the inhomogeneous Lipschitz space ${\rm Lip}(\mathbb{R}^d)$. We also show that if $0<\gamma\leq1$, $0<s<\gamma$, $1<p,q<\infty$ and $b\in{\rm Lip}_\gamma(\mathbb{R}^d)$, then $\mathfrak{M}_{b}^{k}$ is bounded and continuous from the fractional Sobolev spaces $W^{s,p}(\mathbb{R}^d)$ to itself, from the inhomogeneous Triebel--Lizorkin spaces $F^{p,q}_{s}(\mathbb{R}^d)$ to itself and from the inhomogeneous Besov spaces $B^{p,q}_{s}(\mathbb{R}^d)$ to itself.
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