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The boundedness of \(k\)-step Stein functions. (Chinese. English summary) Zbl 1146.42300

Let \(f\in L^p(\mathbb{R}^n)\) and \(u(x,y)=\int_{\mathbb{R}^n}P_y(s)f(x-s)ds\) be the Poisson integral of \(f\), \[ | D^ku| ^2=\left(\frac{\partial^ku}{\partial y^k}\right)^2+\displaystyle\sum_{j=1}^n\left(\frac{\partial^ku}{\partial x_j^k}\right)^2. \] Define the \(k\)-th step Stein function by \[ g_{\lambda,k}^*f(x)=\left(\displaystyle\int_0\infty \displaystyle\int_{\mathbb{R}^n}\left(\frac{y}{| s| +y}\right)^{\lambda n}| D^ku(x-s,y)| ^2y^{2k-1-n}dsdy\right)^{1/2}. \] The author shows that \(g_{\lambda,k}^*\) is essential a singular integral operator in the form of vector and then obtains the \(L^p\)-boundedness (\(1<p<\infty \)), the weak type \((1,1)\) estimate and the boundedness from \(L^\infty \) to BMO for the operator \(g_{\lambda, k}^*\). In the previous paper of the author [J. Univ. Sci. Technol. Suzhou, Nat. Sci. 24, 19, No. 1, (2003)], the corresponding results for \(2\leq p<\infty \) were obtained by the Hardy-Littlewood maximal function.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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