An oscillating operator related to wave equations in the block spaces (Q546273)

The authors study certain oscillating multipliers related to the Cauchy problem for the wave equations on the Euclidean space and on the torus. Precisely, let \(T_{\alpha,m}(f)=K_{\alpha,m}*f\), where \(K_{\alpha,m}\) is a distribution kernel whose Fourier transform is \[ \widehat{K_{\alpha,m}}(y)=e^{i|y|}|y|^{-\alpha}\Omega(y')\Psi(|y|)(\log |y|)^m,\quad m\geq 0, \] and denote the corresponding operator on the \(n\)-torus by \(\widetilde{T_{\alpha,m}}\). For \(m=0\), \textit{A. Miyachi} [J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 331--354 (1980; Zbl 0437.35042)] and \textit{J. C. Peral} [J. Funct. Anal. 36, 114--145 (1980; Zbl 0442.35017)] proved that \(T_{\alpha,0}\) is bounded on \(L^p\) if and only if \(|1/p-1/2|\leq \alpha/(n-1)\). The corresponding result for \(\widetilde{T_{\alpha,0}}\) is also true. Denote \(\alpha_p=|1/p-1/2|(n-1)\). It is shown that when \(m>0\) the \(L^p\) boundedness fails for the operator \(T_{\alpha_p,m}\) and \(\widetilde{T_{\alpha_p,m}}\), and a good substitute for \(L^p\) at the end point case is found. The authors prove that for \(m>0\), \(T_{\alpha, m}\) or \(\widetilde{T_{\alpha,m}}\) is bounded on \(L^p\) if and only if \(|1/p-1/2|<\alpha/(n-1)\), and for the end point case \(\alpha_p=(1/p-1/2)(n-1)\) with \(1\leq p\leq 2\), \(T_{\alpha_p, m}\) or \(\widetilde{T_{\alpha_pm}}\) is bounded from \(L^p\) to certain block spaces.