On multivariate 𝐿^{𝑝} Bernstein-Markov type inequalities
Authors: Kroó András 
Published: June 2024

It was established in a recent paper by Kroó [J. Approx. Theory 281/282 (2022), p. 5] that the following sharp

L
2

L^2

Bernstein-Markov type inequality on the unit ball

B
d

B^d

holds for any polynomial

p
p

of degree at most

n
n

in

d
d

variables,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar left-parenthesis 1 minus StartAbsoluteValue x EndAbsoluteValue squared right-parenthesis Superscript StartFraction mu plus 1 Over 2 EndFraction Baseline upper D p double-vertical-bar Subscript upper L 2 left-parenthesis upper B Sub Superscript d Subscript right-parenthesis Baseline less-than-or-equal-to upper M Subscript n Baseline left-parenthesis d right-parenthesis double-vertical-bar left-parenthesis 1 minus StartAbsoluteValue x EndAbsoluteValue squared right-parenthesis Superscript StartFraction mu Over 2 EndFraction Baseline p double-vertical-bar Subscript upper L 2 left-parenthesis upper B Sub Superscript d Subscript right-parenthesis Baseline comma mu greater-than negative 1 comma">

‖
(
1
−

|

x

|

2

)

μ
+
1

2

D
p

‖

L
2

(

B
d

)

≤

M
n

(
d
)
‖
(
1
−

|

x

|

2

)

μ
2

p

‖

L
2

(

B
d

)

,

μ
>
−
1
,

\begin{equation*} \|(1-|x|^2)^{\frac {\mu +1}{2}}D p\|_{L_2(B^d)}\leq M_n(d)\|(1-|x|^2)^{\frac {\mu }{2}}p\|_{L_2(B^d)}, \;\;\mu >-1, \end{equation*}

with

M
n

(
d
)
=

n
(
n
+
d
+
2
μ
)

M_n(d)=\sqrt {n(n+d+2\mu )}

or

n
(
n
+
d
+
2
μ
)
−
d
+
1

\sqrt {n(n+d+2\mu )-d+1}

when

n
n

is even or odd, respectively, where

D
p

Dp

denotes the

l
2

l^2

norm of the gradient of

p
p

. And all of the estimates listed above were sharp with equalities being attained for certain polynomials. In this paper the uniqueness of the corresponding extremal polynomials is verified.

For homogeneous polynomials

h
n

∈

H
n
d

h_n\in H^d_n

of degree

n
n

in

d
d

variables we will prove sharp

L
2

L_2

Markov type inequalities
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi Subscript n Baseline double-vertical-bar h Subscript n Baseline double-vertical-bar Subscript upper L 2 left-parenthesis upper S Sub Superscript d minus 1 Subscript right-parenthesis Baseline less-than-or-equal-to double-vertical-bar upper D h Subscript n Baseline double-vertical-bar Subscript upper L 2 left-parenthesis upper S Sub Superscript d minus 1 Subscript right-parenthesis Baseline less-than-or-equal-to StartRoot n left-parenthesis 2 n plus d minus 2 right-parenthesis EndRoot double-vertical-bar h Subscript n Baseline double-vertical-bar Subscript upper L 2 left-parenthesis upper S Sub Superscript d minus 1 Subscript right-parenthesis">

ξ
n

‖

h
n

‖

L
2

(

S

d
−
1

)

≤
‖
D

h
n

‖

L
2

(

S

d
−
1

)

≤

n
(
2
n
+
d
−
2
)

‖

h
n

‖

L
2

(

S

d
−
1

)

\begin{equation*} \xi _n\|h_n\|_{L_2(S^{d-1})}\leq \|D h_n\|_{L_2(S^{d-1})}\leq \sqrt {n(2n+d-2)}\|h_n\|_{L_2(S^{d-1})} \end{equation*}

with

ξ
n

=
n

\xi _n=n

if

n
n

is even and

ξ
n

=

n
2

+
d
−
1

\xi _n=\sqrt {n^2+d-1}

if

n
n

is odd. The upper bound is attained if and only if

h
n

h_n

is a spherical harmonic while the lower bound is attained if and only if

h
n

(
x
)
=
c

|

x

|

n

h_n(x)=c|x|^n

, or

h
n

(
x
)
=

|

x

|

n
−
1

q
(
x
)
,
q
∈

H
1
d

h_n(x)=|x|^{n-1}q(x), q\in H^d_1

when

n
n

is even or odd, respectively.

In addition, we will study possible extensions of these results to the

L
p

L^p

case. In particular it will be established that the

L
p

L^p

Markov factor for homogeneous polynomials of degree

n
n

on

C
2

C^2

star like domains with non degenerate outer normals is of asymptotically optimal order

n
n

.