Elementary Sobolev space problem

Question

For which $k$ does the following function belong to Sobolev space $H^k(-1,1)$:

$$f(x) = \begin{cases} x e^{- \frac{1}{x} } & x > 0\\0 & x \leq 0 \end{cases}$$

Answer 1

HINT: A function $\,f\,$ is in Sobolev space $\, H^k\,$ if and only if the $\,k$-th derivative $\,f^{(k)}\,$ is in $\,L^2\,$:

$$ f\in H^k\big(\Omega \big) \iff f^{(k)} = \frac{d^kf}{dx^k}\in L^2\big(\Omega \big) $$

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Let us compute a few derivatives of your function:

\begin{align} f\left(x\right) &= \begin{cases} x \,e^{ - \frac{1}{x} } &&&&& x \in \left(0, 1\right) \\ 0, &&&&& x \in \left(-1, 0\right] \end{cases} \\ f'\left(x\right) &= \begin{cases} \dfrac{1+x}{x}\,e^{ -\frac{1}{x} }, &&& x \in \left(0, 1\right) \\ 0 , &&& x \in \left(-1, 0\right] \end{cases} \\ f''\left(x\right) &= \begin{cases} \dfrac{1}{x^3}\,e^{ -\frac{1}{x^3} }, &&&& x \in \left(0, 1\right) \\ 0 , &&&& x \in \left(-1, 0\right] \end{cases} \\ f'''\left(x\right) &= \begin{cases} \dfrac{1-3 x}{x^5}\,e^{-\frac{1}{x}} , &&& x \in \left(0, 1\right) \\ 0 , &&& x \in \left(-1, 0\right] \end{cases} \\ f''''\left(x\right) &= \begin{cases} \dfrac{12 x^2 - 8 x + 1}{x^7}\,e^{-\frac{1}{x}} , & x \in \left(0, 1\right) \\ 0 , & x \in \left(-1, 0\right] \end{cases} \end{align}

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What happens to the order of multiplier of exponent $\displaystyle\,e^{-\frac{1}{x}}\,$ in $\;f^{\left(k\right)}\,$ as $\, k\to \infty$? What can we conclude about integrability of $\;f^{\left(k\right)}$? Hope you can pick it from here.

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PS As a reminder, because of the Poincaré inequality we only need to worry about integrability of a highest derivative of $\,f,\,$ because it provides a bound for norms of all derivatives of lower order.