Regularity and continuity of higher order maximal commutators

Let \(k\ge 1\), \(0\le \alpha <d\) and \(\mathfrak {M}_{b,\alpha }^k\) be the k-th order fractional maximal commutator. When \(\alpha =0\), we denote \(\mathfrak {M}_{b,\alpha }^k=\mathfrak {M}_{b}^k\). We show that \(\mathfrak {M}_{b,\alpha }^k\) is bounded from the first order Sobolev space \(W^{1,p_1}(\mathbb {R}^d)\) to \(W^{1,p}(\mathbb {R}^d)\) where \(1<p_1,p_2,p<\infty \), \(1/p=1/p_1+k/p_2-\alpha /d\). We also prove that if \(0<s<1\), \(1<p_1,p_2,p,q<\infty \) and \(1/p=1/p_1+k/p_2\), then \(\mathfrak {M}_b^k\) is bounded and continuous from the fractional Sobolev space \(W^{s,p_1}(\mathbb {R}^d)\) to \({W^{s,p}(\mathbb {R}^d)}\) if \(b\in W^{s,p_2}(\mathbb {R}^d)\), from the inhomogeneous Triebel–Lizorkin space \(F_s^{p_1,q}(\mathbb {R}^d)\) to \(F_s^{p,q}(\mathbb {R}^d)\) if \(b\in F_s^{p_2,q} (\mathbb {R}^d)\) and from the inhomogeneous Besov space \(B_s^{p_1,q}(\mathbb {R}^d)\) to \(B_s^{p,q}(\mathbb {R}^d)\) if \(b\in B_s^{p_2,q}(\mathbb {R}^d)\). It should be pointed out that the main ingredients of proving the above results are some refined and complex difference estimates of higher order maximal commutators as well as some characterizations of the Sobolev spaces, Triebel–Lizorkin spaces and Besov spaces.