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I think the following should work: Let $M$ be a compact manifold (just to be safe) and $\pi :E \to M$ a vector bundle. Since $E$ carries an action of $\mathbb{R}^{\times}$ there's an invariant notion …

Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE: $$u_t = grad[V(u)]$$ For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-dimens …

I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new c …

Motivating examples: Let $V$ be a real vector space with Haar measure $dv$. The fourier transform induces the following topological isomorphism: $$H^s(V,dv) \cong L^2(V^*,(1+|v^*|^2)^sdv^*)$$ The L …

Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the …