Some multiplicative products of \(n\)-dimensional distributions. Part I (Q2913868)

Let \(\phi_s\) be a distribution of one variable \(s\), and let \(u(x)\in C^\infty(\mathbb{R}^n)\) be such that \(\{x\in\mathbb{R}^n: u(x_1,x_2,\dots, x_n)= 0\}\) has no critical point. \(\phi_u(x)\) denotes the distribution by the Leray formula: NEWLINE\[NEWLINE\int_{\mathbb{R}^n} f(x)\phi_u(x) \,dx_1\dots dx_n= \int_{\mathbb{R}} \phi_s \,ds\int_{u(x)= s} f(x)\omega_u(x, dx).NEWLINE\]NEWLINE Definition. \(P= P(x)= x_1^2+\dots+ x^2_p- x_{p+1}^2-\dots- x^2_{p+q}\), \(n= p+q\), \((m^2+ P\pm i0)^\lambda= \lim_{\varepsilon\to 0} (m^2+ P\pm i\varepsilon|x|^2)^\lambda\), where \(\varepsilon> 0\), \(\lambda\in \mathbb C\). Let \((m^2+P)^\lambda_+= (m^2+ P)^\lambda\) if \((m^2+P)\geq 0\), \(=0\) if \((m^2+P)< 0\); \((m^2+P)^\lambda_-= (m^2+P)^\lambda\) if \((m^2+P)\leq 0\), \(=0\) if \((m^2+P)> 0\). Then the entire distributional function of \(\lambda\), \((m^2+P\pm i0)^\lambda= (m^2+P)^\lambda_++ e^{\pm i\pi\lambda} (m^2+P)^\lambda_0\) holds, NEWLINE\[NEWLINE\ln(m^2+P= i0)= \ln|m^2+P|+ i\pi H(-(m^2+ P)).NEWLINE\]NEWLINE The author proves ten lemmas that are generalization of results due to B. Fischer on the products of distributions.NEWLINENEWLINE For example, we have in Lemma 1: For \(r,s= 1,2,3,\dots\), NEWLINE\[NEWLINE\begin{multlined} (-1)^s\pi(m^2+P)^{- r+ 1/2}_+(m^2+P)^{- s}_-- ((-1)^{r+s} \pi^2/2(s- 1)!)(m^2+ P)^{-r+1/2}, \delta^{(s-1)}+\\ (-1)^r(m^2+ P)^{-r+1/2}_-\times [(m^2+ P)^{-s}|\ln(m^2+ P)|]= (-1)^{r+ s}(m^2+ P)^{-r-s-1/2}_-\ln(m^2+ P)_-.\end{multlined}NEWLINE\]