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Using a weighted $H^s$-contraction mapping argument based on the macro-microdecomposition of Liu and Yu, we give an elementary proof of existence, withsharp rates of decay and distance from the Chapman--Enskog approximation, ofsmall-amplitude shock profiles of the Boltzmann equation with hard-spherepotential, recovering and slightly sharpening results obtained by Caflisch andNicolaenko using different techniques. A key technical point in both analysesis that the linearized collision operator $L$ is negative definite on itsrange, not only in the standard square-root Maxwellian weighted norm for whichit is self-adjoint, but also in norms with nearby weights. Exploring this issuefurther, we show that $L$ is negative definite on its range in a much widerclass of norms including norms with weights asymptotic nearly to a fullMaxwellian rather than its square root. This yields sharp localization invelocity at near-Maxwellian rate, rather than the square-root rate obtained inprevious analyses

Existence and sharp localization in velocity of small-amplitude Boltzmann shocks