On Marcinkiewicz integral with homogeneous kernels
The authors improve the boundedness theorems of the Marcinkiewicz integral \(\mu_\Omega\) on \(\roman{BMO}(\mathbb R^n)\) and the Campanato spaces \(\mathcal E^{\alpha, p}(\mathbb R^n)\). Recall the Campanato spaces. A locally integrable function \(f(x)\) is said to belong to \(\mathcal E^{\alpha, p}(\mathbb R^n)\) if \(\|f\|_{\alpha,p}=\sup_Q |Q|^{-\alpha/n}\bigl(|Q|^{-1}\int_Q|f(x)-f_Q|^p dx \bigr)^{1/p}<\infty\), where the supremum is taken over all cubes in \(\mathbb R^n\) with sides parallel to the coordinate axes, and \(f_Q\) is the integral average of \(f\) over \(Q\). \(\mathcal E^{\alpha, p}(\mathbb R^n)={\roman {Lip}}_\alpha(\mathbb R^n)\) \((0<\alpha<1)\), \(=\roman{BMO}(\mathbb R^n)\) \((\alpha=0)\), and \(=L_{}^{p,n+p\alpha}\): the Morrey space \((0>\alpha\geq -n/p)\). We also recall the Marcinkiewicz integral with homogeneous symbols, defined first by E.~M.~Stein. Let \(\Omega\in L^1(S^{n-1}) \) be homogeneous of degree zero and have the cancellation property \(\int_{S^{n-1}}\Omega(x') d\sigma(x')=0\). Then the Marcinkiewicz integral of higher dimension is defined by \(\mu_\Omega (f)(x)=\bigl(\int_{0}^{\infty}|F_{\Omega, t}(x)|^2t^{-3} dt\bigr) ^{1/2}\), where \(F_{\Omega, t}(x)=\int_{|x-y|\leq t}\Omega(x-y)|x-y|^{1-n}f(y) dy\). To state the authors' results, recall the \(L^q\) modulus of continuity, \(\omega_q(\delta)\), defined by \(\omega_q(\delta) =\sup_{\|\rho-I\|\leq \delta}\bigl(\int_{S^{n-1}}|\Omega(\rho x')-\Omega(x')|^q d\sigma(x') \bigr)^{1/q}\), where \(\rho\) is a rotation on \(S_{n-1}\) and \(I\) is the identity operator. The authors' results are as follows: Let \(\Omega \in L^q(S^{n-1})\) satisfy the cancellation property and \(\int_{0}^{1}\omega_q(\delta)\delta^{-1}(1+|\log \delta|)^\varepsilon d\delta <\infty\) for some \(\varepsilon >2\). Then, for \(f\in \roman{BMO}(\mathbb R^n)\) satisfying \(\mu_\Omega(f)(x)<\infty \) for some measurable set \(E\) with positive measure, it holds \(\mu_\Omega(f)(x)<\infty \) \({\roman {a.e.}}\) on \(\mathbb R^n\) and \(\|\mu_\Omega(f)\|_{\roman{BMO}}\leq C\|f\|_{\roman{BMO}}\), where \(C\) is independent of \(f\). For \(1<p<\infty\) and \(1\leq q'\leq p\), similar results hold, if the continuity condition on \(\Omega\) is replaced by \(\int_{0}^{1}\omega_q(\delta)\delta^{-1-\varepsilon} d\delta<\infty\) for some \(\varepsilon >0\) in the case \(0<\alpha<1\), and by \(\int_{0}^{1}\omega_q(\delta)\delta^{-1} d\delta<\infty\) in the case \(\alpha<0\). These results improve the known ones by \textit{Y. Han} [Acta Sci. Nat. Univ. Pekinensis 5, 799-839 (1987)]and by \textit{S. Qiu} [J. Math. Res. Exp. 12, No. 1, 41-50 (1992; Zbl 0773.42014)].
Campanato spaces \(\mathcal E^{\alpha,p}\)
scientific article; zbMATH DE number 1471625
scientific article