Pointwise Gradient Estimates and Sobolev Boundedness for Commutators of Fractional New Maximal Operators

In this article, we consider the Sobolev boundedness for the commutators of fractional new maximal operators both in the global and local settings. More precisely, in the global case, we prove that \(M_{b,\varphi ,\beta }\) $M_{b,\phi ,\beta }$ and \([b,{M}_{\varphi ,\beta }]\) $[b,M_{\phi ,\beta }]$ are bounded on Sobolev space \(W^{1,q}(\mathbb {R}^{n})\) $W^{1,q}({\mathbb{R}}^n)$ ; in the local case, the boundedness for \(M_{b,\varphi ,\beta ,\Omega }\) $M_{b,\phi ,\beta ,\mathrm{\Omega }}$ and \([b,{M}_{\varphi ,\beta ,\Omega }]\) $[b,M_{\phi ,\beta ,\mathrm{\Omega }}]$ on Sobolev space \(W^{1,q}(\Omega )\) $W^{1,q}(\mathrm{\Omega })$ and Sobolev space with zero boundary values \(W_{0}^{1,q}(\Omega )\) $W_0^{1,q}(\mathrm{\Omega })$ is given. It is worth noting that the analysis and calculation for the commutators of fractional new maximal operators in the local case are more complex than those in the global case. Moreover, the pointwise gradient inequalities of the above commutators will also be showed.