Calderón-Zygmund operator on the weighted Hardy-block spaces
The author defines weighted Hardy block spaces in the obvious way. A function \(b\) is a \(H^{1,q}_ w\) atom if \(b\) satisfies three conditions: \[ supp \;b \subseteq Q, \quad\| b\|_{q,w} \leq w(Q)^{(1/q) - 1} \quad\text{and}\quad \int b = 0. \] Here \(Q\) is any cube and \[ \| b\|_{q,w} = \left( \int | b(x)|^ q w(x) dx \right)^{1/q}. \] The weighted Hardy block space \(B^{\circ}_{q,w}\) is the set of all sums of such atoms, \[ f(x) = \sum_ k m_ k b_ k(x), \] equipped with the ``norm'' defined by \[ N(\left\{m_ k\right\}) = \sum_ k | m_ k| \left\{1 + \log \left(\sum_ l | m_ l|/| m_ k| \right) \right\} , \] which is \[ N^{\circ}_{q,w} (f) = \inf N(\left\{ m_ k \right \}). \] The infimum is taken over all such representations of \(f\). A \(\delta\)-Calderón-Zygmund operator is an operator which is bounded from \(L^ 2\) into \(L^ 2\) and which is given by a kernel on smooth functions. The kernel has a singularity of order \(n\) on the diagonal and satisfies a Lipschitz condition of order \(\delta\) with a singularity of order \(n+\delta\) outside a neighborhood of the origin. The author shows that if \(w \in A_ 1\), the Muckenhoupt class, and if \(T\) is a \(\delta\)-Calderón-Zygmund operator such that \(T^*(1) = 0\), then \(T\) extends to a bounded map of \(B^{\circ}_{q,w}\) into itself.
scientific article; zbMATH DE number 24289