Let $X$ be a Polish space and let $m$ be a locally finite Borel measure on $X$.

Let $\epsilon$ be a strongly local, regular Dirichlet form on $L^2(X,m)$ with Domain $V :=\{f\in L^2(X,m):\epsilon(f)<\infty\}$ dense in $L^2(X,m)$. Any such form can be written as $$ \epsilon(u,v)=\int_X d\Gamma(u,v) $$ Where $\Gamma$ is a positive semidefinite, symmetric bilinear form on $V$ with values in the signed Radon measures on $X$.

The nature notion of a (pseudo-) distance on $X$ associated to $\epsilon$ is the intrinsic $\rho$ defined by $$ \rho(x,y)~:=~\sup\{|f(x)-f(y)|: f \in V_{loc}\cap C(X), \Gamma(f) \le m\}. $$

For the sequel, assume that $\rho$ is a metric on X inducing the given topology.

Fix an arbitrary subset $Y\subset X$. Then Sturm's paper "Analysis on local Dirichlet spaces III" says: If the

Completeness property (For all balls $B_{2r}(x)$ the closed balls $\bar{B}_r(x) $ are compact).

Doubling property

Weak Poincare inequality i.e. There exists a constant $C_p=C_p(Y)$ such that for all balls $B_{2r}(x) \subset Y$ and $u\in V$ $$ \int_{B_r(x)}\left|u(y)-\frac{1}{m(B_r(x))} \int_{B_r(x)}u\, d m\right|\, d m(y) \le C r^2 \int_{B_2r(x)} \Gamma(u)\, d m $$holds on $Y$.

Then there exists a constant $C_S$ such that for all balls $B_{2r}(x) \subset Y$,

$$
\left(\int_{B_r(x)}|u|^{\frac{2N}{N-2}}\right)^{\frac{N-2}{N}} \le C_s \frac{r^2}{{m(B_r(x))}^{2/N}}\int_{B_r(x)}(\Gamma(u)+r^{-2}u^2)\, d m
$$
for all $u \in V\cap C_0(B_r(x))$

Question:

I wonder whether the condition for $u$ is too strict and whether the inequality above can be better.

That is to say, if $u \in V$ with compact support in $Y$. Is there a constant C_1 such that $$ \left(\int_Y |u|^{\frac{2N}{N-2}}\right)^{\frac{N-2}{N}} \le C_1 \frac{(\text{diam}Y)^2}{{m(Y)}^{2/N}}\int_Y\Gamma(u)\,d m $$ And consider the special case when $X$ is compact and $u \in V$, for any $k \in R$, the inequality above holds for $(u-k)_+$ and $X$ instead of $u$ and $Y$?