Some estimates for the commutators of multilinear maximal function on Morrey-type space
From the abstract: ``We study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.'' The two main results of the paper are Theorem 1.1 and 1.3. They read as follows: Theorem 1.1 Let \(b_{i}\) be a real-valued, locally integrable function in \(\mathbb{R}^{n}\) with \(i=1, \ldots, m\). Then, the following three assertions are equivalent. (i) The commutator \([\vec{b}, \mathcal{M}](\vec{f})(x)\) is bounded from \(L^{p_{1}, \lambda}\left(\mathbb{R}^{n}\right) \times \cdots \times L^{p_{m}, \lambda}\left(\mathbb{R}^{n}\right)\) to \(L^{p, \lambda}\left(\mathbb{R}^{n}\right)\) for \(1<p_{i}<\infty\) and \(1 / p=\sum_{i=1}^{m} 1 / p_{i}\) with \(p>1\) and \(0 \leq \lambda<n\). (ii) \(b_{i}\) is in \(B M O\) and \(b_{i}^{-}\)belongs to \(L^{\infty}\). (iii) For \(p \in(1, \infty)\), there is \[ \sup _{Q} \frac{1}{|Q|} \int_{Q}\left|b_{i}(x)-M_{Q}\left(b_{i}\right)(x)\right|^{p} \mathrm{~d} x<\infty \] where \(M_{Q}\left(b_{i}\right)(x)=\sup _{Q_{0} \ni x, Q_{0} \subset Q} \frac{1}{\left|Q_{0}\right|} \int_{Q_{0}}\left|b_{i}(t)\right| \mathrm{d} t\). and Theorem 1.3 Let \(b_{i} \in \operatorname{BMO}\left(\mathbb{R}^{n}\right)\) and \(\omega\) satisfy \( \omega(a, 2 t) \leq C_{0} \omega(a, t) \) with \(1<C_{0} \leq 2^{y}\) for any \(0<\gamma<n\). Then, for any \(\beta>0\) and cube \(B=B(x, t)\) with center \(x \in \mathbb{R}^{n}\) and radius \(t>0\), there is \[ \frac{1}{\omega(B)^{m}}\left|\left\{z \in B(x, t): \mathcal{M}_{\vec{b}}(\vec{f})(z)>\beta^{m}\right\}\right|^{m} \leq C\|\vec{b}\|_{\mathrm{BMO}} \prod_{i=1}^{m}\left\|\Phi\left(\frac{\left|f_{i}\right|}{\beta}\right)\right\|_{L_{L \log L}^{1, \omega}\left(\mathbb{R}^{n}\right)}, \] where \(\Phi(x)=t\left(1+\log ^{+} t\right),\|\vec{b}\|_{\mathrm{BMO}}=\sup _{1 \leq i \leq m}\left\|b_{i}\right\|_{\mathrm{BMO}}\) and \(C\) is a positive constant depending on the dimension \(n\). The definition of the operators used in the previous theorems are: Multilinear maximal operator \[ \mathcal{M}(\vec{f})(x)=\sup _{Q \ni x} \prod_{i=1}^{m} \frac{1}{|Q|} \int_{Q}\left|f_{i}\left(y_{i}\right)\right| d y_{i}. \] Commutator generated by the BMO function and the multilinear maximal functions \[ [\vec{b}, \mathcal{M}](\vec{f})(x)=\sum_{j=1}^{m}[\vec{b}, M]_{j}(\vec{f})(x), \quad \mathcal{M}_{\vec{b}}(\vec{f})(x)=\sum_{j=1}^{m} \mathcal{M}^{b_{j}}(\vec{f})(x) . \] Here \[ [\vec{b}, M]_{j}(\vec{f})(x)=b_{j}(x) \mathcal{M}(\vec{f})(x)-\mathcal{M}\left(f_{1}, \ldots, f_{j-1}, b_{j} f_{j}, f_{j+1}, \ldots, f_{m}\right)(x) \] and \[ \mathcal{M}^{b_{j}}(\vec{f})(x)=\sup _{Q \ni x} \frac{1}{|Q|^{m}} \int_{Q^{m}}\left|b_{j}(x)-b_{j}\left(y_{j}\right)\right| \prod_{i=1}^{m}\left|f_{i}\left(y_{i}\right)\right| d \vec{y} \] where \(\vec{y}=\left(y_{1}, \ldots, y_{m}\right)\).
scientific article; zbMATH DE number 7435663
scientific article