- #1
Only a Mirage
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Suppose, for a suitable class of real-valued test functions T(\mathbb{R}^n), that \{G_x\} is a one-parameter family of distributions. That is, \forall x \in \mathbb{R}^n, G_x: T(\mathbb{R}^n) \to \mathbb{R}.
Now, suppose L is a linear differential operator. That is, \forall g \in T(\mathbb{R}^n) makes sense in terms of the normal definitions of derivates (assuming, of course, that g is sufficiently smooth). L also has meaning when acting on distributions by interpreting all derivatives as distributional derivatives. For example, the derivative of the distribution \frac{\partial}{\partial x_i} G_{x_0} is defined by: \forall g \in T(\mathbb{R}^n), \frac{\partial}{\partial x_i} G_{x_0}(g) = G_{x_0}(- \frac{\partial}{\partial x_i}g).
Note that smooth functions can multiply distributions to form a new distribution in the following way. Suppose f:\mathbb{R}^n \to \mathbb{R} is a smooth function. Then f G_x is defined by: \forall g \in T(\mathbb{R}^n), (f G_x) (g) = G_x(f g)
These facts give L G_x meaning.
Also, for fixed g \in T(\mathbb{R}^n), (x \mapsto G_x (g)) is a possibly (non-smooth?) function. Define the function \psi_g : \mathbb{R}^n \to \mathbb{R} by \psi_g (x) = G_x(g)
Now, here is my question: When is the following equality true?
L (\psi_g) (x_0) = (L G_{x_0}) (g), \forall x_0 \in \mathbb{R}^n
Now, suppose L is a linear differential operator. That is, \forall g \in T(\mathbb{R}^n) makes sense in terms of the normal definitions of derivates (assuming, of course, that g is sufficiently smooth). L also has meaning when acting on distributions by interpreting all derivatives as distributional derivatives. For example, the derivative of the distribution \frac{\partial}{\partial x_i} G_{x_0} is defined by: \forall g \in T(\mathbb{R}^n), \frac{\partial}{\partial x_i} G_{x_0}(g) = G_{x_0}(- \frac{\partial}{\partial x_i}g).
Note that smooth functions can multiply distributions to form a new distribution in the following way. Suppose f:\mathbb{R}^n \to \mathbb{R} is a smooth function. Then f G_x is defined by: \forall g \in T(\mathbb{R}^n), (f G_x) (g) = G_x(f g)
These facts give L G_x meaning.
Also, for fixed g \in T(\mathbb{R}^n), (x \mapsto G_x (g)) is a possibly (non-smooth?) function. Define the function \psi_g : \mathbb{R}^n \to \mathbb{R} by \psi_g (x) = G_x(g)
Now, here is my question: When is the following equality true?
L (\psi_g) (x_0) = (L G_{x_0}) (g), \forall x_0 \in \mathbb{R}^n