A generalization of distributional product of Dirac's delta in a hypercone (Q2765963)

Let \(m^2+ P\) be a quadratic form in \(n\) variables defined by NEWLINE\[NEWLINEm^2+ P= m^2+ x^2_1+ x^2_2+\cdots+ x^2_p- x^2_{p+1}- x^2_{p+2}-\cdots- x^2_{p+q},NEWLINE\]NEWLINE where \(m^2\) is a real number and \(p+q=n\) is the dimension of the space. Using a known definition for the distribution \(\delta^j(m^2+p)\) and the distributional product NEWLINE\[NEWLINE\delta^{k=1}(P)\cdot \delta^{l-1}(P),NEWLINE\]NEWLINE where \(P\) is a quadratic form in \(n\)-variables, the author generalizes the above product and obtains a new definition for the product of \(\delta^{k-1}(m^2+ P)\) and \(\delta^{l-1}(m^2+P)\). It is also proved that this product is the zero distribution if \(n\), the dimension of the space, is odd.