### Definition 2.1

Let \(x = (x_{1}, x_{2}, \ldots, x_{n})\) be a point of *n*-dimensional Euclidean space \(\mathbb{R}^{n}\) and *m* be a positive integer. The hypersurface \(G = G(m,x)\) is defined by

$$ G = G(m,x) = \Biggl(\sum_{i=1}^{p} x_{i}^{2} \Biggr)^{m}- \Biggl(\sum _{j=p+1}^{p+q} x_{j}^{2} \Biggr)^{m}, $$

(2.1)

where \(p+q=n\) is the dimension of \(\mathbb{R}^{n}\). The hypersurface *G* is due to Berndtsson and Passare [11]. We observe that putting \(m =1\) in (2.1), we obtain

$$ G= G(1,x) =\sum_{i=1}^{p} x_{i}^{2}-\sum_{j=p+1}^{p+q} x_{j}^{2}= P(x) = P, $$

(2.2)

where the quadratic form *P* is due to Gel’fand and Shilov [4] and is given by (1.1). The hypersurface \(G = 0\) is a generalization of a hypercone \(P = 0\) with a singular point (the vertex) at the origin.

### Definition 2.2

Let grad \(G \neq0\), which means there is no singular point on \(G = 0\). Then we define

$$ \bigl\langle \delta^{(k)}(G),\phi \bigr\rangle = \int\delta ^{(k)}(G)\phi(x) \,dx, $$

(2.3)

where \(\delta^{(k)}\) is the Dirac delta function with *k*-derivatives, *ϕ* is any testing function in the Schwartz space *S*, \(x = (x_{1},x_{2}, \ldots,x_{n})\in\mathbb{R}^{n}\) and \(dx = dx_{1}\,dx_{2}\,dx_{n}\). In a sufficiently small neighborhood *U* of any point \((x_{1},x_{2},\ldots ,x_{n})\) of the hypersurface \(G = 0\), we can introduce a new coordinate system such that \(G = 0\) becomes one of the coordinate hypersurface. For this purpose, we write \(G = u_{1}\) and choose the remaining \(u_{i}\) coordinates (with \(i = 2,3,\ldots,n\)) for which the Jacobian

where

$$D\binom{x}{u}=\frac{\partial(x_{1},x_{2},\ldots, x_{n})}{\partial (G,u_{2},\ldots u_{n})}. $$

Thus (2.3) can be written as

$$ \bigl\langle \delta^{(k)}(G),\phi \bigr\rangle = (-1)^{k} \int \biggl[\frac {\partial^{k}}{\partial G^{k}} \biggl\{ \phi D\binom{x}{u} \biggr\} \biggr]_{G=0}\,du_{2}\,du_{3} \cdots \,du_{n}. $$

(2.4)

The proof of the following lemma is given in [17].

### Lemma 2.1

*Given the hypersurface*

$$G = \Biggl(\sum_{i=1}^{p} x_{i}^{2} \Biggr)^{m}- \Biggl(\sum _{j=p+1}^{p+q} x_{j}^{2} \Biggr)^{m}, $$

*where*
\(p + q = n\)
*is the dimension of*
\(\mathbb{R}^{n}\), *and*
*m*
*is a positive integer*. *If we transform to bipolar coordinates defined by*

$$x_{1} = r\omega_{1},\qquad \ldots,\qquad x_{p} = r \omega_{p},\qquad x_{p+1} = s\omega_{p+1},\qquad \ldots ,\qquad x_{p+q} = s\omega_{p+q}, $$

*where*

$$\sum_{i=1}^{p} \omega_{i}^{2}=1 $$

*and*

$$\sum_{j=p+1}^{p+q} \omega_{j}^{2}=1. $$

*Then the hypersurface*
*G*
*can be written by*

*and we obtain*

$$ \bigl\langle \delta^{(k)}(G),\phi \bigr\rangle = \int_{0}^{\infty}\biggl[ \biggl( \frac{1}{2ms^{2m-1}} \frac{\partial}{\partial s} \biggr)^{k} \biggl\{ s^{q-2m} \frac{\psi(r,s)}{2m} \biggr\} \biggr]_{s=r}r^{p-1}\,dr $$

(2.5)

*or*

$$ \bigl\langle \delta^{(k)}(G),\phi \bigr\rangle =(-1)^{k} \int_{0}^{\infty}\biggl[ \biggl( \frac{1}{2mr^{2m-1}} \frac{\partial}{\partial r} \biggr)^{k} \biggl\{ r^{p-2m} \frac{\psi(r,s)}{2m} \biggr\} \biggr]_{r=s}s^{q-1}\,ds, $$

(2.6)

*where*

$$\psi(r,s) = \int\phi \, d\Omega^{(p)}\, d\Omega^{(q)}, $$

*and*
\(d\Omega^{(p)}\)
*and*
\(d\Omega^{(q)}\)
*are the elements of surface area on the unit sphere in*
\(\mathbb{R}^{p}\)
*and*
\(\mathbb{R}^{q}\), *respectively*.

Now, we assume that *ϕ* vanishes in the neighborhood of the origin, so that these integrals will converge for any *k*. Now for

or

$$k< \frac{1}{2m}(p+q-2m), $$

the integrals in (2.5) converge for any \(\phi(x)\in S\). Similarly, for

or

$$k< \frac{1}{2m}(p+q-2m), $$

the integrals in (2.6) also converge for any \(\phi(x)\in S\). Thus we take (2.5) and (2.6) to be the defining equation for \(\delta^{(k)}(G)\). On the other hand, if

$$k\ge\frac{1}{2m}(p+q-2m), $$

then we shall define \(\langle\delta_{1}^{*}(G),\phi \rangle\) and\(\langle\delta_{2}^{*}(G),\phi \rangle\) as the regularization of (2.5) and (2.6), respectively. For \(p>1\) and \(q>1\), the generalized function \(\delta_{1}^{*(k)}(G)\) and \(\delta_{2}^{*(k)}(G)\) are defined by

$$\bigl\langle \delta_{1}^{*(k)}(G),\phi \bigr\rangle = \int_{0}^{\infty}\biggl[ \biggl( \frac{1}{2ms^{2m-1}} \frac{\partial}{\partial s} \biggr)^{k} \biggl\{ s^{q-2m} \frac{\psi(r,s)}{2m} \biggr\} \biggr]_{s=r}r^{p-1}\,dr $$

for all

$$k\ge\frac{1}{2m}(p+q-2m), $$

we have

$$ \bigl\langle \delta_{2}^{*(k)}(G),\phi \bigr\rangle =(-1)^{k} \int_{0}^{\infty}\biggl[ \biggl( \frac{1}{2mr^{2m-1}} \frac{\partial}{\partial r} \biggr)^{k} \biggl\{ r^{p-2m} \frac{\psi(r,s)}{2m} \biggr\} \biggr]_{r=s}s^{q-1}\,ds $$

(2.7)

for

$$k\ge\frac{1}{2m}(p+q-2m). $$

In particular, for \(m = 1\), \(\delta_{1}^{*(k)}(G)\) is reduced to \(\delta _{1}^{(k)}(G)\), and \(\delta_{2}^{*(k)}(G)\) is reduced to \(\delta _{2}^{(k)}(G)\) (see [4, p.250]).