The following estimate arises in the proof of Tomas-Stein restriction theorem. $$ \sigma_{\mathbb{S}^{d-1}} (B(x,r)\cap \mathbb{S}^{d-1}) \leq C r^{d-1} $$ The estimate is very intuitive, and I have a vague idea to prove it (essentially estimating the intersected area with the area of the circle of radius $r$ in $\mathbb{R}^{d-1}$ up to global constants, thanks to the compactness of $\mathbb{S}^{d-1}$).

However, when trying to fill in the details, I got stuck, trying to produce a nice and clean proof. In particular, I would love to have a simple and clean proof of this statement possibly extending it to the case $S$ is a compact d-1 dimensional manifold, so any contribution or suggestion is deeply appreciated.

The following estimate arises in the proof of Tomas-Stein restriction theorem. $$ \sigma_{\mathbb{S}^{d-1}} (B(x,r)\cap \mathbb{S}^{d-1}) \leq C r^{d-1} $$ The estimate is very intuitive, and I have a vague idea to prove it (essentially estimating the intersected area with the area of the circle of radius $r$ in $\mathbb{R}^{d-1}$ up to global constants, thanks to the compactness of $\mathbb{S}^{d-1}$).

However, when trying to fill in the details, I got stuck, trying to produce a nice and clean proof. In particular, I would love to have a simple and clean proof of this statement possibly extending it to the case where $S$ is a compact $d-1$ dimensional manifold, so any contribution or suggestion is deeply appreciated.

The following estimate arises in the proof of Tomas-Stein restriction theorem. $$ \sigma_{\mathbb{S}^{d-1}} (B(x,r)\cap \mathbb{S}^{d-1}) \leq C r^{d-1} $$ The estimate is very intuitive, and I have a vague idea to prove it (essentially estimating the intersected area with the area of the circle of radius $r$ in $\mathbb{R}^{d-1}$ up to global constants, thanks to the compactness of $\mathbb{S}^{d-1}$).

However, when trying to fill in the details, I got stuck, trying to produce a nice and clean proof. I would love to have a simple and clean proof of this statement, so any contribution or suggestion is deeply appreciated.

The following estimate arises in the proof of Tomas-Stein restriction theorem. $$ \sigma_{\mathbb{S}^{d-1}} (B(x,r)\cap \mathbb{S}^{d-1}) \leq C r^{d-1} $$ The estimate is very intuitive, and I have a vague idea to prove it (essentially estimating the intersected area with the area of the circle of radius $r$ in $\mathbb{R}^{d-1}$ up to global constants, thanks to the compactness of $\mathbb{S}^{d-1}$).

However, when trying to fill in the details, I got stuck, trying to produce a nice and clean proof. In particular, I would love to have a simple and clean proof of this statement possibly extending it to the case $S$ is a compact d-1 dimensional manifold, so any contribution or suggestion is deeply appreciated.