I am confused with the following example taken from page 6 of Sobolev Met Poincaré, by Hajłasz and Koskela (MR1683160, Zbl 0954.46022).

Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset X$ be an open subset. Assume that $u\in L_{\text{loc}}^1(\Omega)$ and a measurable function $g\geqslant 0$ satisfy the inequality $$\label{1}\tag{$*$} \dfrac{1}{\mu(B)}\int_B|u-u_B|\,\mathrm{d}\mu\leqslant C_P r\Bigl(\dfrac{1}{\mu(B)}\int_{\sigma B} g^p\,\mathrm{d}\mu\Bigr)^\frac{1}{p}, $$ on each ball $B$ with $\sigma B\subset \Omega$, where $r$ is the radius of $B$ and $p\geqslant 1$, $\sigma\geqslant 1$, $C_P>0$ are fixed constants.

Now take two three-dimensional planes in $\mathbb{R}^5$ whose intersection is a line $L$, and let $X$ be the union of these two planes. The metrics and measures induced from the planes have natural extensions to a metric and a measure on $X$. If $u$ is a smooth function on $X$ then we define $g(x)$ to be $|\nabla u(x)|$ whenever $x$ does not belong to $L$, where $\nabla u$ is the nsual gradient of $u$ in the appropriate plane, and define $g(x)$ to be the su of the lengths of the two gradients corresponding to the different planes when $x\in L$. One can then check that \eqref{1} holds for $p>2$ but fails for $p\leqslant 2$.

My question is, why \eqref{1} holds for $p>2$ but fails for $p\leqslant 2$?

I am confused with the following example taken from page 6 of Sobolev Met Poincaré, by Hajłasz and Koskela (MR1683160, Zbl 0954.46022).

Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset X$ be an open subset. Assume that $u\in L_{\text{loc}}^1(\Omega)$ and a measurable function $g\geqslant 0$ satisfy the inequality $$\label{1}\tag{$*$} \dfrac{1}{\mu(B)}\int_B|u-u_B|\,\mathrm{d}\mu\leqslant C_P r\Bigl(\dfrac{1}{\mu(\sigma B)}\int_{\sigma B} g^p\,\mathrm{d}\mu\Bigr)^\frac{1}{p}, $$ on each ball $B$ with $\sigma B\subset \Omega$, where $r$ is the radius of $B$ and $p\geqslant 1$, $\sigma\geqslant 1$, $C_P>0$ are fixed constants.

Now take two three-dimensional planes in $\mathbb{R}^5$ whose intersection is a line $L$, and let $X$ be the union of these two planes. The metrics and measures induced from the planes have natural extensions to a metric and a measure on $X$. If $u$ is a smooth function on $X$ then we define $g(x)$ to be $|\nabla u(x)|$ whenever $x$ does not belong to $L$, where $\nabla u$ is the nsual gradient of $u$ in the appropriate plane, and define $g(x)$ to be the su of the lengths of the two gradients corresponding to the different planes when $x\in L$. One can then check that \eqref{1} holds for $p>2$ but fails for $p\leqslant 2$.

My question is, why \eqref{1} holds for $p>2$ but fails for $p\leqslant 2$?

I am confused with the following example taken from page 6 of Sobolev Met Poincaré, by Hajłasz and Koskela (MR1683160, Zbl 0954.46022).

Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset X$ be an open subset. Assume that $u\in L_{\mathrm{loc}}^1(\Omega)$ and a measurable function $g\geqslant 0$ satisfy the inequality $$\label{1}\tag{$*$} \dfrac{1}{\mu(B)}\int_B|u-u_B|\,\mathrm{d}\mu\leqslant C_P r\Bigl(\dfrac{1}{\mu(B)}\int_{\sigma B} g^p\,\mathrm{d}\mu\Bigr)^\frac{1}{p}, $$ on each ball $B$ with $\sigma B\subset \Omega$, where $r$ is the radius of $B$ and $p\geqslant 1$, $\sigma\geqslant 1$, $C_P>0$ are fixed constants.

Now take two three-dimensional planes in $\mathbb{R}^5$ whose intersection is a line $L$, and let $X$ be the union of these two planes. The metrics and measures induced from the planes have natural extensions to a metric and a measure on $X$. If $u$ is a smooth function on $X$ then we define $g(x)$ to be $|\nabla u(x)|$ whenever $x$ does not belong to $L$, where $\nabla u$ is the nsual gradient of $u$ in the appropriate plane, and define $g(x)$ to be the su of the lengths of the two gradients corresponding to the different planes when $x\in L$. One can then check that \eqref{1} holds for $p>2$ but fails for $p\leqslant 2$.

My question is, why \eqref{1} holds for $p>2$ but fails for $p\leqslant 2$?

I am confused with the following example taken from page 6 of Sobolev Met Poincaré, by Hajłasz and Koskela (MR1683160, Zbl 0954.46022).

Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset X$ be an open subset. Assume that $u\in L_{\text{loc}}^1(\Omega)$ and a measurable function $g\geqslant 0$ satisfy the inequality $$\label{1}\tag{$*$} \dfrac{1}{\mu(B)}\int_B|u-u_B|\,\mathrm{d}\mu\leqslant C_P r\Bigl(\dfrac{1}{\mu(B)}\int_{\sigma B} g^p\,\mathrm{d}\mu\Bigr)^\frac{1}{p}, $$ on each ball $B$ with $\sigma B\subset \Omega$, where $r$ is the radius of $B$ and $p\geqslant 1$, $\sigma\geqslant 1$, $C_P>0$ are fixed constants.

Now take two three-dimensional planes in $\mathbb{R}^5$ whose intersection is a line $L$, and let $X$ be the union of these two planes. The metrics and measures induced from the planes have natural extensions to a metric and a measure on $X$. If $u$ is a smooth function on $X$ then we define $g(x)$ to be $|\nabla u(x)|$ whenever $x$ does not belong to $L$, where $\nabla u$ is the nsual gradient of $u$ in the appropriate plane, and define $g(x)$ to be the su of the lengths of the two gradients corresponding to the different planes when $x\in L$. One can then check that \eqref{1} holds for $p>2$ but fails for $p\leqslant 2$.

My question is, why \eqref{1} holds for $p>2$ but fails for $p\leqslant 2$?

I am confused with the following example taken from page 6 of Sobolev Met Poincar'e, which is written by Haglasz and Koskela.

Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset X$ be an open subset. Assume that $u\in L_{\mathrm{loc}}^1(\Omega)$ and a measurable function $g\geqslant 0$ satisfy the inequality \begin{equation}\tag{*} \dfrac{1}{\mu(B)}\int_B|u-u_B|\,\mathrm{d}\mu\leqslant C_P r\Bigl(\dfrac{1}{\mu(B)}\int_{\sigma B} g^p\,\mathrm{d}\mu\Bigr)^\frac{1}{p}, \end{equation} on each ball $B$ with $\sigma B\subset \Omega$, where $r$ is the radius of $B$ and $p\geqslant 1$, $\sigma\geqslant 1$, $C_P>0$ are fixed constants.

Now take two three-dimensional planes in $\mathbb{R}^5$ whose intersection is a line $L$, and let $X$ be the union of these two planes. The metrics and measures induced from the planes have natural extensions to a metric and a measure on $X$. If $u$ is a smooth function on $X$ then we define $g(x)$ to be $|\nabla u(x)|$ whenever $x$ does not belong to $L$, where $\nabla u$ is the nsual gradient of $u$ in the appropriate plane, and define $g(x)$ to be the su of the lengths of the two gradients corresponding to the different planes when $x\in L$. One can then check that (*) holds for $p>2$ but fails for $p\leqslant 2$.

My question is, why (*) holds for $p>2$ but fails for $p\leqslant 2$?

I am confused with the following example taken from page 6 of Sobolev Met Poincaré, by Hajłasz and Koskela (MR1683160, Zbl 0954.46022).

Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset X$ be an open subset. Assume that $u\in L_{\mathrm{loc}}^1(\Omega)$ and a measurable function $g\geqslant 0$ satisfy the inequality $$\label{1}\tag{$*$} \dfrac{1}{\mu(B)}\int_B|u-u_B|\,\mathrm{d}\mu\leqslant C_P r\Bigl(\dfrac{1}{\mu(B)}\int_{\sigma B} g^p\,\mathrm{d}\mu\Bigr)^\frac{1}{p}, $$ on each ball $B$ with $\sigma B\subset \Omega$, where $r$ is the radius of $B$ and $p\geqslant 1$, $\sigma\geqslant 1$, $C_P>0$ are fixed constants.

Now take two three-dimensional planes in $\mathbb{R}^5$ whose intersection is a line $L$, and let $X$ be the union of these two planes. The metrics and measures induced from the planes have natural extensions to a metric and a measure on $X$. If $u$ is a smooth function on $X$ then we define $g(x)$ to be $|\nabla u(x)|$ whenever $x$ does not belong to $L$, where $\nabla u$ is the nsual gradient of $u$ in the appropriate plane, and define $g(x)$ to be the su of the lengths of the two gradients corresponding to the different planes when $x\in L$. One can then check that \eqref{1} holds for $p>2$ but fails for $p\leqslant 2$.

My question is, why \eqref{1} holds for $p>2$ but fails for $p\leqslant 2$?