Absorbing a subquadratic power

Statement

For $0<\alpha<2$ , $A,b\ge0$ , and $\delta>0$ ,

bA^\alpha\le\delta A^2+C_\alpha\delta^{-\alpha/(2-\alpha)}b^{2/(2-\alpha)}.

This is a subquadratic absorption estimate, a rescaled form of Young's inequality.

Derivation and radius factors

Apply Young with conjugate exponents $2/\alpha$ and $2/(2-\alpha)$ , rescaling the two factors so the first term is $\delta A^2$ . Equivalently, maximize $bA^\alpha-\delta A^2$ over $A\ge0$ . If $b=R^{-1}$ and $\alpha=3/2$ , the remainder is $C_\delta R^{-4}$ . The strict inequality $\alpha<2$ is what permits an arbitrarily small coefficient in front of $A^2$ .