Spaces of anisotropic potentials. Applications (Q1095344)

Let \(r=(r_ 1,...,r_ n)\), where \(r_ i\) are positive, \((1/r^*)=n^{-1}\sum^{n}_{j=1}1/r_ j\), \(\lambda_ j=(r^*/r_ j)\), \(\lambda =(\lambda_ 1,...,\lambda_ n)\) and let \(\rho\) be a function such that \(\rho (x)>0\) for \(x\neq 0\) and \(\sum^{n}_{i=1}x^ 2_ i\rho (x)^{-2\lambda_ i}=1\). The space \(I^ r(L_ p)\) of anisotropic potentials \[ f(x)=\int_{R^ n}(\phi (y)dy)/(\rho^{n- r^*}(x-y)), \] 0\(<r^*<n\), where \(1\leq p<(n/r^*)\) and \(\phi \in L_ p\), is described in terms introduced by \textit{P. I. Lizorkin} [cf. Mat. Sb., N. S. 81(123), 79-91 (1969; Zbl 0198.188)].