A modified Riesz summation multiplier \((1-r(x))_+^{\lambda}\) is considered. r is the norm in \({\mathbb{R}}^ n\) defined by \(r(x)=\max \{| y|,| z| \}\) for \(x=(y,z)\in {\mathbb{R}}^ j\times {\mathbb{R}}^ k={\mathbb{R}}^ n\) with some j,k\(\geq 1\), \(j+k=n\). It is shown that for \(n=2\) and \(n=3\) the conditions \(\lambda >0\) or \(\lambda >\), respectively, imply \((1-r)_+^{\lambda}\in [L^ 1({\mathbb{R}}^ n)]{\hat{\;}}\). Hence the critical index is smaller than that of the radial Fourier multiplier \((1- | x|)_+^{\lambda}\). In contrast to this, \((1- r)_+^{\lambda}\) is not a Fourier multiplier in \(L^ p({\mathbb{R}}^ n)\), \(n\geq 4\), if \(| 1/p-| \geq 3/2(n-1),\) independently of \(\lambda\). In particular, \(e^{-r}\) is not in \([L^ 1({\mathbb{R}}^ n)]{\hat{\;}}\), \(n\geq 4\).