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(Usually one-parameter) semigroups of linear operators and their applications to partial differential equations, stochastic processes such as Markov processes and other branches of mathematics.
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What is dispersive estimate?
Consider free NLS: $i\partial_tu+\Delta u=0, \quad u(0, x)=u_0$
The solution of this IVP, can be written as
$$u(x,t)=e^{it\Delta}u_0(x)$$
It is clear to me that how to prove following estimate:
$$ \|e …