We introduce a polynomial extremal function $$\Phi (E,{\mathbb {F}},z)$$
Φ
(
E
,
F
,
z
)
which is one of possible generalizations of the classical Siciak extremal function, restricted to subspaces $${\mathbb {F}}$$
F
of the linear space of all polynomials of N variables that are invariant under differentiation. We show that the so-called HCP condition in this situation: $$\log \Phi (E,{\mathbb {F}},z)\le A\text { dist}(z,E)^s,\ z\in {\mathbb {C}}^N$$
log
Φ
(
E
,
F
,
z
)
≤
A
dist
(
z
,
E
)
s
,
z
∈
C
N
is equivalent to a generalization of the classical V. Markov’s inequality: $$||D^\alpha P||_E\le A_1^{|\alpha |}\frac{(\deg P)^{m|\alpha |}}{(|\alpha |!)^{m-1}}||P||_E,\ P\in {\mathbb {F}}$$
|
|
D
α
P
|
|
E
≤
A
1
|
α
|
(
deg
P
)
m
|
α
|
(
|
α
|
!
)
m
-
1
|
|
P
|
|
E
,
P
∈
F
with dependence $$m=1/s$$
m
=
1
/
s
. The situation is similar to the basic case (cf. [8]) $${\mathbb {F}}={\mathbb {K}}[z_1,\dots ,z_N],$$
F
=
K
[
z
1
,
⋯
,
z
N
]
,
where a V. Markov’s type inequality was introduced and the above-mentioned equivalence was proved. As a byproduct, we prove new results related to V. Markov’s inequality for an important class of subsets of $${\mathbb {R}}^N$$
R
N
, which are then applied to obtain the first versions of this inequality for some thin sets, such as spheres in $${\mathbb {R}}^{N+1}$$
R
N
+
1
and Euclidean spheres in particular. Furthermore, we prove an interesting fact on the polynomial convex hull of the circle $$S^1$$
S
1
, as a subset of $${\mathbb {R}}^2$$
R
2
.
