A note on the existence and boundedness of singular integrals
Let \(K\in C({\mathbb R}^n\setminus\{0\})\) satisfy \(\int_{\varepsilon<|x|<N}K(x)dx=0\), for all \(0<\varepsilon<N<\infty\); there exists a constant \(C_1>0\) such that \[ |k(x)|\leq C_1|x|^{-n} \] for all \(x\neq 0\); there exists two positive constants \(C_2\) and \(\alpha\in (0,1]\) such that \[ |K(x-y)-K(x)|\leq C_2|y|^\alpha |x|^{-n-\alpha},\quad \forall |x|>2|y|. \] Define the operator \(T\) as follows: \[ T(f)(x)=\lim_{\varepsilon\rightarrow 0}\int_{|x-y|>\varepsilon}K(x-y)f(y)dy \] for \(f\in L^p({\mathbb R}^n)\), \(1\leq p<\infty\). In this paper, the authors prove that the operator \(T\) defines a bounded operator from \(L^\infty\) to the space BMO.
scientific article; zbMATH DE number 1866570