Let
(
T
t
)
t
>
0
{({T_t})_{t > 0}}
be a symmetric contraction semigroup on the spaces
L
p
(
M
)
(
1
≤
p
≤
∞
)
{L^p}(M)\;(1 \leq p \leq \infty )
, and let the functions
ϕ
\phi
and
ψ
\psi
be "regularly related". We show that
(
T
t
)
t
>
0
{({T_t})_{t > 0}}
is
ϕ
\phi
-ultracontractive, i.e., that
(
T
t
)
t
>
0
{({T_t})_{t > 0}}
satisfies the condition
‖
T
t
f
‖
∞
≤
C
ϕ
(
t
)
−
1
‖
f
‖
1
{\left \| {{T_t}f} \right \|_\infty } \leq C\phi {(t)^{ - 1}}{\left \| f \right \|_1}
for all
f
f
in
L
1
(
M
)
{L^1}(M)
and all
t
t
in
R
+
{{\mathbf {R}}^ + }
, if and only if the infinitesimal generator
G
\mathcal {G}
has Sobolev embedding properties, namely,
‖
ψ
(
G
)
−
α
f
‖
q
≤
C
‖
f
‖
p
{\left \| {\psi {{(\mathcal {G})}^{ - \alpha }}f} \right \|_q} \leq C{\left \| f \right \|_p}
for all
f
f
in
L
p
(
M
)
{L^p}(M)
, whenever
1
>
p
>
q
>
∞
1 > p > q > \infty
and
α
=
1
/
p
−
1
/
q
\alpha = 1/p - 1/q
. We establish some new spectral multiplier theorems and maximal function estimates. In particular, we give sufficient conditions on
m
m
for
m
(
G
)
m(\mathcal {G})
to map
L
p
(
M
)
{L^p}(M)
to
L
q
(
M
)
{L^q}(M)
, and for the example where there exists
μ
\mu
in
R
+
{{\mathbf {R}}^ + }
such that
ϕ
(
t
)
=
t
μ
\phi (t) = {t^\mu }
for all
t
t
in
R
+
{{\mathbf {R}}^ + }
, we give conditions which ensure that the maximal function
sup
t
>
0
|
t
α
T
t
f
(
∙
)
|
{\sup _{t > 0}}|{t^\alpha }{T_t}f( \bullet )|
is bounded.