We have a function $f(x(y),y)$ where $x\in \mathbb{R}$ depends on $y\in \mathbb{R}^3$, $x$ is a scalar but $y$ is a vector of $\mathbb{R}^3$ : \begin{align*} f(x,y)&= \frac{\partial}{\partial x} g(x,y) \\ \end{align*} For example $x= x(y)= y\cdot v+ b$

We want to consider cases where $g$ is discontinuous, so $f$ can be regarded as a distribution in the parameter $x$, that is,

$$\text{for } \phi\in S(\mathbb{R}): f(\phi)=\int_{\mathbb{R}} f(x,y)\phi(x)\,dx \text{ , for a fixed vector $y$} $$

We want to sum over the sphere, we know in the case when $x$ does not depend on $y$ we can do this :

We have the family of distributions $\Big(f(x, y)\Big)_{y}$ , to which we apply a notion of distribution-valued integral; the 'Gelfand-Pettis' weak integration, or Bochner integrals.
( under some conditions that $f$ is 'nice' enough as a function of $y$ ) \begin{align*} \left(\int_{S^2} f\,d\sigma(y) \right) (\phi)=& \int_{S^2} f(\phi)\,d\sigma(y) \quad \quad (\text{ Gelfand-Pettis }) \\ =& \int_{S^2} \left(\int_\mathbb{R} f(x,y)\phi(x)\,dx\right) \,d\sigma(y) \\ =& \int_{S^2}\int_\mathbb{R} - g(x,y)\phi'(x)\,dx \,d\sigma(y) \end{align*}

$\textbf{My question is : Can we still do this when } x \textbf{ is a function of }y$ ?

Is there another way to integrate these distributions ? in the sense that we want the integral to be a sort of generalized function of the smooth classical integral for when $g$ is continuous in $x$ .



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