Find $a,b>0$ for which $||\langle x\rangle^{-b}|\partial_x|^{1/2}f||_{L^2}\lesssim||\partial_x f||_{L^2}+||\langle x\rangle^{-a}f||_{L^2}$

Question

Consider the following problem.

Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\varphi\in \mathscr D(\mathbb R)$: $$ ||\langle x\rangle^{-\beta}\;|\partial_x|^{1/2}\varphi||_{L^2 (\mathbb R)}\lesssim ||\partial_x \varphi||_{L^2 (\mathbb R)}+||\langle x\rangle^{-\alpha}\;\varphi||_{L^2 (\mathbb R)}.\qquad(1) $$

Notation. The “half derivative” is defined as a Fourier multiplier $$ \mathscr F(|\partial_x|^{1/2}\varphi)(\xi)=|2\pi\xi|^{1/2}\widehat\varphi(\xi) $$ and I define the japanese bracket $\langle x\rangle:=(1+x^2)^{1/2}$.

A few preliminary observations.

• Taking $\alpha$ larger and larger eventually does not change the right hand side up to equivalent norms. In fact, a function such that $\partial_x\varphi\in L^2$ satisfies $$ |\varphi(x)-\varphi(y)|\lesssim |x-y|^{1/2}, $$ and in particular the quantity $||\langle x\rangle^{-\alpha}\varphi||_{L^2}$ is automatically finite if $\alpha>1$. With this (being a bit careful since $\partial_x\varphi$ determines $\varphi$ only up to a constant) it is feasible to prove that, if we take $\alpha>1$, the right hand side of $(1)$ is equivalent to $$ ||\partial_x \varphi||_{L^2}+|\varphi(0)|. $$ So the problem is split in two cases: the case $0<\alpha\leq 1$ and the case $\alpha>1$, the latter consisting in fact of one single estimate.

• It is possible to show through the Fourier transform that $|\partial_x|^{1/2}\varphi(x)=[\operatorname{sgn}(\cdot)|\cdot|^{-1/2}*\partial_x\varphi](x)$ and since $\partial_x\varphi\in L^2$, then $|\partial_x|^{1/2}\varphi\in BMO(\mathbb R)$. This implies that $|\partial_x|^{1/2}\varphi$ is, for instance, locally in $L^2$, so that (unless my eyes are cheated by some spell) the following estimate should hold $$ ||\mathbb \chi_{[-1,1]}(\cdot) |\partial_x|^{1/2}\varphi ||_{L^2} \lesssim ||\partial_x \varphi||_{L^2}+ |\varphi(0)| $$ and by a summation trick (i.e., taking infinitely many shifted copies of the above estimate to cover all the real line), using the Hölder-continuity of $\varphi$ which I stated above, one should obtain that estimate $(1)$ holds whenever $\alpha>1$ and $\beta>1$.

My questions:

1. Does what I have said above in the second observation make sense to you? I put some dust under the carpet, but I think that what I wrote works.
2. Do you have any clue on how one would prove the estimate for $\alpha<1$, besides doing it from scratch? Simply put, does this kind of estimates fall into some more general framework for which there are strong techniques to prove the estimates whenever I need to use one in a paper without reinventing the wheel? Like, would abstract interpolation theory help in this context? Any references you think could be useful in understanding how to work out these estimates, or research papers dealing with this kind of estimates?