Knowlpedia

Theorem

Derivative bound between Fourier Sobolev spaces

Each distributional derivative lowers the Sobolev order by its degree.

Statement

For real ss and a multi-index α\alpha, distributional differentiation is a bounded map

∂α:Hs(Rn)⟶Hs−∣α∣(Rn),∥∂αu∥Hs−∣α∣≤∥u∥Hs.\partial^\alpha:H^s(\mathbb R^n)\longrightarrow H^{s-|\alpha|}(\mathbb R^n),\qquad \|\partial^\alpha u\|_{H^{s-|\alpha|}}\le\|u\|_{H^s}.

Here HsH^s uses the (1+4π2∣ξ∣2)s(1+4\pi^2|\xi|^2)^s.

Fourier proof

The derivative multiplies u^\widehat u by (2πiξ)α(2\pi i\xi)^\alpha, whose absolute value is at most (1+4π2∣ξ∣2)∣α∣/2(1+4\pi^2|\xi|^2)^{|\alpha|/2}. Insert this inequality in the norm. The same definition shows Hs1⊂Hs2H^{s_1}\subset H^{s_2} continuously when s1≥s2s_1\ge s_2.