The multiplier for the ball and radial functions
Consider the multiplier \(^{\wedge}(\xi)=\chi_ B(\xi)\hat f(\xi)\), where \^{\ } denotes the Fourier transform and \(B=\{\xi:| \xi | \leq 1\}\). We show that for \(\chi_ E(x)\) radial, \(\lambda>0\), \(| \{x\in {\mathbb{R}}^ n:| T\chi_ E(x)|>\lambda \}| \leq c\lambda^{-2n/(n+1)}| E|\), where c does not depend on \(\lambda\) or the set E. Using the interpolation theorem of E. Stein and G. Weiss, and interpolating the estimate above with \(\| Tf\|_ 2\leq c\| f\|_ 2\), we obtain the following theorem of C. Herz. Let f(x) be radial, then \(\| Tf\|_ p\leq c_ p\| f\|_ p,\quad 2n/(n+1)<p<2n/(n-1)\). The operator T is thus restricted weak type \((2n/n+1,2n/n+1)\) on radial functions. T is not weak type on radial functions as was proved by C. E. Kenig and P. A. Tomas. The result above has analogues for orthogonal series, where T is now replaced by the partial sum operator. These results are contained in [Trans. Am. Math. Soc. 268, 367-376 (1981)].
scientific article; zbMATH DE number 3842455