An \(L^ p\)-\(L^ q\) transference theorem
Let \(S^{n - 1}\) be the unit sphere in \(\mathbb{R}^n\), \(\omega \in S^{n - 1}\); the norm \(|\cdot |_{p, \omega}\) will be taken with respect to the Lebesgue measure on \(S^{n - 1}\) and \(|\cdot |_{p - q}\) is the norm of an operator from \(L^p (\mathbb{R})\) to \(L^q (\mathbb{R})\). For \(\alpha \in \mathbb{R}\) the fractional integration operator \(I^\alpha\) on \(\mathbb{R}\) is defined by \(\widehat {(I^\alpha f)} (\xi) = |\xi |^{- \alpha} \widehat f(\xi)\), \(\xi \in \mathbb{R}\). The author obtains a transference theorem: Theorem. Fix \(n \geq 2\), \(p \in [1,2)\), and \(q \in (2, \infty)\). Let \(1/r = 1/p - 1/q\). Suppose that \(\{A_\omega\}_{\omega \in S^{n - 1}}\) is a weakly continuous family of convolution operators on \(\mathbb{R}\) such that \[ \bigl |\;|I^{(n - 1) (1/q - 1/p)} A_\omega |_{p - q} \bigr |_{r, \omega} < \infty. \] Then the convolution operator \(A\) on \(\mathbb{R}^n\) with the symbol \(A(s \omega) = \widehat A_\omega (s)\), \(\omega \in S^{n - 1}\) and \(s > 0\), is bounded from \(L^p (\mathbb{R}^n)\) to \(L^q (\mathbb{R}^n)\).
scientific article; zbMATH DE number 617444
scientific article