Notes on singular integrals on some inhomogeneous Herz spaces
A central \((p,q)\) block is a function \(a\) supported in \(\{|x|<R\}\) such that \(\|a\|_{L^q}\leq |\{|x|<R\}|^{1/q-1/p}\). A central \((p,q)\)-atom is a central \((p,q)\)-block having integral zero. The block space \(K^p_q\) consists of those distributions that can be expressed as superpositions \(\sum_k \lambda_k a_k\) of central \((p,q)\)-blocks such that \(\sum_{k} |\lambda_k|^p<\infty\) and is \(p\)-normed by \(\|f\|_{K^p_q}^p=\inf \sum_k |\lambda_k|^p\), with infimum taken over block representations of \(f\). The space \(HK^p_q\) is defined in exactly the same way with blocks replaced by atoms. The \(p\)-norm then makes sense when \(p>n/(n+1)\). The author introduces a notion intermediate to that of a block and an atom, namely, a \((p,q,\varepsilon)\)-block is a \((p,q)\)-block \(a\) supported in \(\{|x|<R\}\) for some \(R\geq 1\) such that \(|\int a|\leq |\{|x|<R\}|^{\varepsilon -1/p}\). This notion allows the author to define a new block space \(K_p^{1,\varepsilon}\) just as above, but in terms of \((p,q,\varepsilon)\) blocks, and to extend boundedness of certain singular integrals to these spaces. Specifically, a \((q,\theta)^t\)-central singular integral is a linear operator \(T:\mathcal{D}\to \mathcal{D}'\) that is bounded on \(L^q(\mathbb{R}^n)\) and has integral kernel \(K\) satisfying \[ \sup_{R\geq 1} \sup_{|y|<R} R^{n(q-1)} \int_{2^jR<|x|<2^{j+1}R}\, |K(x,y)-K(x,0)|^q\, dx < e_j \text{ such that }\sum_{j=1}^\infty 2^{j\theta} e_j<\infty . \] The author's main result says the following: Suppose that \(n/(n+1)<p\leq 1<q<\infty\), \(q/(q-1)\leq s\), \(\lambda\leq \varepsilon -1\), and \(T\) is a \((q,\theta)^t\)-central singular integral with \(\theta> n(1/p-1/q)\). If \(T^t(1)\in {\mathrm CMO}^{s,\lambda}(\mathbb{R}^n)\) then \(T\) is bounded from \(HK_p^q (\mathbb{R}^n)\) to \(K_p^{q,\varepsilon}(\mathbb{R}^n)\). Here, the finite central oscillation space \({\mathrm CMO}^{s,\lambda}\) consists of those \(f\) such that \[ \sup_{R\geq 1} \left(\frac{1}{R^{n(1+\lambda q)}} \int_{|x|<R} |f(x)-{\mathrm ave}(f,\{|x|<R\})|^q\, dx\right)^{1/q} \] is finite. The result extends previous work of \textit{J. Alvarez, J. Lakey} and \textit{M. Guzmán-Partida} [Collect. Math. 51, No. 1, 1--47 (2000; Zbl 0948.42013)] concerning boundedness of operators from block spaces into Herz-Hardy spaces. The author also corrects a minor error in that work.
scientific article; zbMATH DE number 2148915