Boundedness of commutators on homogeneous Herz spaces
Let \(B_k=\{x\in \mathbb R^n; | x| \leq 2^k\}\) and \(C_k=B_k \setminus B_{k-1}\) for \(k\in \mathbb Z\). Let \(\chi_k\) denote the characteristic function of the set \(C_k\). The homogeneous Herz space \(\dot K_q^{\alpha, p}(\mathbb R^n)\) is defined in terms of \(\| f\| _{\dot K_q^{\alpha, p}(\mathbb R^n)}\) \(=\bigl\{\sum_{k=-\infty}^\infty 2^{k\alpha p}\| f\chi_{k}\| _{L^q(\mathbb R^n)}^p\bigr\}^{1/p}\) by letting \(\dot K_q^{\alpha, p}(\mathbb R^n) =\{f\in L_{\roman{loc}}^q(\mathbb R^n\setminus \{0\}); \| f\| _{\dot K_q^{\alpha, p}(\mathbb R^n)}<\infty\}\). \(\dot K_q^{0, q}(\mathbb R^n)=L^q(\mathbb R^n)\) and \(\dot K_q^{\alpha/q, q}(\mathbb R^n)=L_{| x| ^\alpha}(\mathbb R^n)\). The authors' main theorem is as follows: Let \(b\in \roman{BMO}(\mathbb R^n)\) and \(\Omega\in L^r(S^{n-1})\) for some \(1<r\leq \infty\) be a homogeneous function of degree 0. Assume that \(T\) is a linear operator satisfying \(| Tf(x)| \leq C| x| ^{-n}\int_{\mathbb R^n}| f(y)\Omega(x-y)| dy\) for \(f\in L^1\), \(\roman{supp} f\subset C_k\) and \(| x| \geq 2^{k+1}\) \((k\in \mathbb Z)\), and \(| Tf(x)| \leq C2^{-nk}\int_{\mathbb R^n}| f(y)\Omega(x-y)| dy\) for \(f\in L^1\), \(\roman{supp} f\subset C_k\) and \(| x| \leq 2^{k-2}\) \((k\in \mathbb Z)\). Then, if \(1<p\leq \infty\) and the commutator \([b, T]\) is bounded on \(L^q(\mathbb R^n)\) for some \(1<q<\infty\), it follows that \([b, T]\) is also bounded on \(\dot K_q^{\alpha, p}(\mathbb R^n)\), provided that \(r\), \(q\) and \(\alpha\) satisfy one of the following conditions; (i) \(r'<q\) and \(-n/q<\alpha<n(1/r'-1/q)+1/r\), (ii) \(r>q\) and \(n(1/r-1/q)-1/r<\alpha<n(1-1/q)\). To prove their theorem, they establish boundedness of the corresponding maximal operators on the Herz spaces, such as \(\sup_{r>0}| B(x,r)| ^{-1}\int_{B(x,r)}| \Omega (x-y)(b(x)-b(y))f(y)| dy\). They also mention the fractional versions of their main theorem.
scientific article; zbMATH DE number 1281446