Extension of a result of Benedek, Calderón and Panzone
A Banach space X is said to be UMD, or to have UMD property, as defined in \textit{D. L. Burkholder}, Ann. Probab., 9, 997-1011 (1981; Zbl 0474.60036) if, for \(1<p<\infty\), the martingale difference sequence \(d=(d_ 1,d_ 2,...,...)\) in \(L^ p_ X(0,1)\) is unconditional, so that for \(\epsilon_ j=+1\) or -1, \(j=1,2,...\), \(\|\epsilon_ 1d_ 1+\epsilon_ 2d_ 2+...\|_ p\leq C(X)\| d_ 1+d_ 2+...\|_ p.\) Suppose that the singular integral \(T_ j\) is defined by \(T_ j(f)(x)=(P.V)\quad (K_ j*f)(x),\) where \((a)\quad\| \hat K_ j\|_{\infty}\leq C_ j;\quad (b)\quad| K_ j(x)|\leq C_ j| x|^{-n};\quad (c)\quad| K_ j(x)-K_ j(x- y)|\leq C_ j| y|| x|^{-n-1}\) for \(| y|\leq (1/2)| x|\); where \(C_ j\) is a fixed constant. If \((e_ j)\) is a normalized unconditional basis for X, then the operator T is defined by \(T(\Sigma f_ je_ j)=\Sigma T_ j(f_ j)e_ j.\) In the main theorem of this paper the author indicates that \(T:L^ p_ X(R^ n)\to L^ p_ X(R^ n)\) is bounded. The special case of this result in which \(X=\ell^ r\) has previously been considered by \textit{A. Benedek, A. P. Calderon} and \textit{R. Panzone}, Proc. Nat. Acad. Sci. USA 48, 356-365 (1962; Zbl 0103.33402).
scientific article; zbMATH DE number 3873987
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