Let
\[ \square_s=\frac{\partial^2}{\partial x_1^2}+ \cdots + \frac{\partial^2}{\partial x_s^2}= \frac{\partial^2}{\partial x_{s+1}^2}-\frac{\partial^2}{\partial x_n^2}, \;\;\;s=1,2,\dots,n-1, \]
and \(D_s=\square_s+i\frac{\partial}{\partial t}\).
The authors study the fractional powers \(D_s^{-\frac{\alpha}{2}}\) of the operator \(D_s\) introduced on nice functions \(\varphi\) via multiplication of Fourier transforms \(\hat{\varphi}(\xi,\tau)\) by \(\left(\tau-r_s^2(\xi)+i0\right)^{-\frac{\alpha}{2}}\), where \(r_s(\xi)\) is the Lorentz distance.
In the case \(\operatorname{Re}\alpha>0\), these fractional powers are realized as the convolution operator \[ D_s^{-\frac{\alpha}{2}} \varphi (x,t)= \int\limits_{\mathbb{R}^{n+1}}h^\alpha_s(y,\eta)\varphi(x-y,t-\eta)dyd\eta \] with the following oscillating kernel
\[ h^\alpha_s(y,\eta)=c\eta_+^{\alpha-\frac{n}{2}-1}e^{i(r_s^2(y))^\prime4\eta}, \quad \text{where }c=\frac{e^\frac{\pi i}{4(\alpha+n-2s)}}{(4\pi)^\frac{n}{2}\Gamma\left(\frac{\alpha}{2}\right)}. \]
Within the framework of \(L_p\)-functions \(\varphi\), this operator is interpreted as acting from \(L_p\) to a certain space of distributions.
The main goal of the present paper is to construct the powers \(D_s^{\frac{\alpha}{2}}\) with \(\alpha >0\). The explicit expression for them is obtained on the basis of the method of approximative inverse operators (inverse to \(D_s^{-\frac{\alpha}{2}}\)). The characterization of their domain is also given.