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From microlocal to local

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From microlocal to local

Let f be a smooth compactly supported function on the cotangent bundle (phase space) T*(R^n) which is equal to 1 in a neighborhood of a point (x_0, z_0). Assume that its quantization Op(f), defined as the pseudodifferential operator having f as its symbol, satisfies

|Op(f)u| < e

for some u in L^2(R^n) where || is the L^2-norm. Can we deduce that, for some neighborhood U of x_0,

|u|_L^2(U) < e

as well?

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