Classes \(L_{p,r}^\alpha\) of Lizorkin-Samko type, related to complex powers of the telegraph operator (Q1397524)

The author considers the fractional powers \(\left(\square+\frac{\partial}{\partial x_1}\right)^{-\alpha}\) with \(\Re\alpha >0\), where \(\square =\frac{\partial^2}{\partial x_1^2} -\frac{\partial^2}{\partial x_2^2}-\cdots - \frac{\partial^2}{\partial x_n^2}\) is the wave operator. \ (The equation \(\left(\square+\frac{\partial}{\partial x_1}\right) u =0\) is known as the telegraph equation). He shows that these fractional powers may be realized in the form of the potential operator \[ H^\alpha\varphi (x)= c_n(\alpha)\int_{K^+_+}\frac{e^{-\frac{y_1}{2}} I_{\frac{\alpha -n}{2}}\left( \frac{r(y)}{2}\right)}{r^\frac{n-\alpha}{2}(y)} \varphi(x-y) dy \] where \(K^+_+\) is the forward light cone and \(r(y)\) is the Lorentz distance. The mapping properties of the operator \(H^\alpha\) within the spaces \(L_p\) are given, including the estimate \(\|H^\alpha\varphi\|_q\leq C \|\varphi\|_p\) with \(\frac{1}{q}=\frac{1}{p}=\frac{\Re\alpha+n-1}{2n}\). The main attention is paid to the characterization of the range \(H^\alpha(L_p)\) of this potential operator and its intersection \(L_r\cap H^\alpha(L_p)\) with the Lebesgue space with an arbitrary \(r>1.\) Their characterization is given in terms of some limiting process (the method of approximative inverse operators).