Extending definition of weighted $L^2$ norm

Question

Is there a simple characterization of the domain of the semi-norm $\| \nabla (g \ast x) \|_{L^2 ( \mathbb{R}^3)}$, where $g$ is a gaussian convolution kernel? It is finite on $L^2$, but probably on a much larger space. I am mainly interested in functions $x$ from (some subset of) $L^\infty + L^{3/2}$.

Answer 1

I don't think that you get a much larger space on the Lebesgue scale. The Gaussian will smoothen $x$, but it will not eliminate its heavy tail, should $x$ have one. Indeed, let $x=\sum c_k \chi_{B_k}$ where $B_k$ is a sequence of balls of unit radius spread far apart. Then $$|\nabla (g*x)|=|(\nabla g)*x|\approx \sum |c_k|\,|(\nabla g)*\chi_{B_k}| \tag{1}$$ because there is very little cancellation between the contributions of different $\chi_{B_k}$. Consequently, $$\int |\nabla (g*x)|^2 \approx \sum |c_k|^2 \tag{2}$$ So your function space must not contain $x=\sum c_k \chi_{B_k}$ with $\sum |c_k|^2=\infty$. In particular, $L^p$ with $p>2$ would not work.