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The answer to your question is negative. Take the smooth function $\chi$ defined by $$ \chi(t)=H(t) e^{-t^{-1}-t^2}, \quad H=\mathbf 1_{(0,+\infty)}. $$ This function is in $L^1(\mathbb R)$, $C^\infty …

You have the explicit Hadamard formula $$ \frac{1}{R(x)}=\limsup_n\vert a_n(x)\vert^{1/n}=\inf_n\bigl(\sup_{k\ge n}\vert a_k(x)\vert^{1/k}\bigr), $$ triggering semi-continuity properties for $1/R$: se …

Let $g,f$ be real-valued functions defined on the real line. Let $s$ be a real number. Assuming that $g,f$ are both in the Gevrey class $G^{s}$, it is true that $g\circ f$ belongs to $G^{s}$ if $s\ge …

A strong argument is given above on the heat equation; let me be more specific. The heat equation, one of the most basic in PDE and mathematical physics, already known to Fourier, is $$ L=\frac{\parti …

Take the Lewy operator $\mathcal L$ in $\mathbb R^3_{x,y,t}$ $$ \mathcal L=\partial_x+i\partial_y+i(x+iy)\partial_t. $$ There exists a set $S$ of second category in $C^\infty(\mathbb R^3)$ such that f …