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Simplyorange
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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.

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.

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.

Minor Math Jaxing (norm sign and bracket scaling)
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Daniele Tampieri
Daniele Tampieri
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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 $\epsilon>0$

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

My question is: what is the meaning of $(1+\sqrt{-\Delta})^\epsilon f$$\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.

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 $\epsilon>0$

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

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

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.

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Simplyorange
Simplyorange
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Notation for right hand side of local smoothing conjecture

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 $\epsilon>0$

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

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