Differential Operator Norm

**mia25** · November 28th 2011, 03:11 PM

Any Idea that help in solving this problem is Highly appreciated

**girdav** · November 29th 2011, 06:13 AM

Consider a function $\phi$ smooth, such that \int_{\mathbb R}\phi^2(t)dt=1, and put $\phi_n(t)=\sqrt n\phi(nt)$ . Then $\lVert \phi_n\rVert_{L^2}=1$ , but $\frac d{dt}\phi_n(t)=n^{\frac 32}\phi(nt)$ has a norm $L^2$ which cannot be bounded by a constant which doesn't depend on $n$ .

For the second problem, note that $||f||^2\geq ||\frac d{dt}f||^2_{L^2}$ .

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