We consider a metric measure space
(
M
,
d
,
μ
)
(M,d,\mu )
and a heat kernel
p
t
(
x
,
y
)
p_{t}(x,y)
on
M
M
satisfying certain upper and lower estimates, which depend on two parameters
α
\alpha
and
β
\beta
. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space
(
M
,
d
,
μ
)
(M,d,\mu )
. Namely,
α
\alpha
is the Hausdorff dimension of this space, whereas
β
\beta
, called the walk dimension, is determined via the properties of the family of Besov spaces
W
σ
,
2
W^{\sigma ,2}
on
M
M
. Moreover, the parameters
α
\alpha
and
β
\beta
are related by the inequalities
2
≤
β
≤
α
+
1
2\leq \beta \leq \alpha +1
.
We prove also the embedding theorems for the space
W
β
/
2
,
2
W^{\beta /2,2}
, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on
M
M
of the form
−
L
u
+
f
(
x
,
u
)
=
g
(
x
)
,
\begin{equation*} -\mathcal {L}u+f(x,u)=g(x), \end{equation*}
where
L
\mathcal {L}
is the generator of the semigroup associated with
p
t
p_{t}
.
The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpiński carpet in
R
n
{\mathbb {R}^{n}}
.