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In Tao's "Recent progress on the restriction conjecture"

On page 53, Tao introduced the local smoothing conjecture: let $u(t,x)$ be the solution to the wave equation $u_{tt}=\Delta u$, $u(0,x)=f(x)$ and $u_t(0,x)=0$ in $n$ spatial dimensions, and the conjecture for $\epsilon>0$

$$\|u\|_{L^p([1,2]\times\mathbb{R}^n)}\leq C_{p,\epsilon} \big\|\big(1+\sqrt{-\Delta}\big)^\epsilon f\big\|_{L^p(\mathbb{R}^n)}$$

My question is: what is the meaning of $\big(1+\sqrt{-\Delta}\big)^\epsilon f$? This seems to be a standard notation but I can't find a place where it is explained.

Edit: I see, this the Bessel potential space, equivalent to the Sobolev spaces.

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    $\begingroup$ The easiest interpretation in this context is as the Fourier multiplier $(1+|\xi|)^{\epsilon}$ (that is, applying the operator to an $f$ is the same as multiplying $\widehat{f}(\xi)$ by the multiplier I gave). Alternatively, you could use functional calculus, at least when $p=2$, since $-\Delta$ is self-adjoint on $L^2(\mathbb R^n)$. $\endgroup$
    – Christian Remling
    Commented Jun 28, 2023 at 17:33

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