Boundedness of singular integral operators of Calderón type. V
Let BMO denote the totality of real functions f such that \(\| f\|_{BMO}=\sup (1/| I|)\int_{I}| f(x)-m_ If| dx<\infty,\) where \(m_ If=(1/| I|)\int_{I}f(x)dx\) and the supremum is taken over all finite intervals I. For a real locally integrable function \(\phi\) define a kernel \(C[\phi](x,y)=1/\{(x-y)+i(\Phi (x)-\Phi (y))\},\) where \(\Phi (x)=\int^{x}_{0}\phi (s)ds.\) C[\(\phi\) ] denotes the corresponding singular integral operator from \(L^ 2\) to \(L^ 2\). Using severals lemmas, the author proves the following theorem. For any \(\phi \in BMO_ r\), \(\| C[\phi]\| \leq Const(1+\| \phi \|^ 2_{BMO}).\) [For part IV see Hiroshima Math. J. 14(1985), 511-525 (1984; Zbl 0562.42008).]
scientific article; zbMATH DE number 3983820