J. of Inequal. & Appl., 2000, Vol. 5, pp. 1-9
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A Remark on Polynomial Norms and
Their Coefficients
SUNG GUEN KIM*
Topology and Geometry Research Center, Kyungpook National
University, Taegu, Korea (702-701)
(Received 20 December 1998; Revised 29 January 1999)
This paper presents new lower bounds for the norms of 2-homogeneous real-valued
polynomials on lp spaces for 0 < p < which are sharper than those recently given by the
author.
Keywords: Coefficients; lp spaces; Polynomial norms
1991 Mathematics Subject
Classification: 46B20; 46E15
This note is concerned with the general problem of the relation between
the norm of a polynomial and its coefficients. This type of problem
has been studied in many contexts [1-6,8-11] over the years, because of
both its relevance to non-trivial problems in mathematics and because
of its our inherent interest. In this note we focus our attention on lower
bounds for the norms of 2-homogeneous real-valued polynomials on lp
spaces. Recently the author [9] gave lower bounds for the norms of 2We
homogeneous real-valued polynomials on lp spaces for 0 < p <
.
here improve them.
Let E be a real Banach space with the unit sphere SE and rn > 2, a
natural number. 79(mE) denotes the Banach space of m-homogeneous
real-valued polynomials on E, endowed with the polynomial norm
* E-mail: sgkim@gauss.kyungpook.ac.kr.
S.G. KIM
2
IIPII supllxll IP(x)]. See Dineen [7] for more details about the theory
of polynomials on Banach spaces.
LEMMA
LetO7ScS’ cN andx, aijERfori, jES’ with i<j. Then
max
ek=d:l, kS’
Ix+ Z
aijeiej
i,jS’, i<j
Proof Let m > 2 be a positive integer. It is enough to show that
max
ek=-t-1, <k<_m
Ix+
Z
ek=’+-l, l<k<m+l
l<i<j<m
a#’eij
l<i<j<m+l
Let
max
=:1:1, <k<m
[x
nt-
Z
aijeif-j
--x+
<_i<j<_m
<i<j<m
for some sign choices f-(,..., %. Let
f-m+
aim+
if x + 21<i<j<_m aijf-[f-j > 0 and
aim+ i
otherwise. Then we have
max
et=+l, l<_k<_m+l
Ix
>x+
+
x+
<i<j<m
<i<j<m+
<_i<j<_m+
Z aim+lf-
l<i<m
A REMARK ON POLYNOMIAL NORMS
3
LEMMA 2 Let rn >_ 2 be a positive integer. Let x, aij (1 < <j < m) E R.
Then
max
ek=4-1, l<_k<_m
x+ 2_
aijeiej
Ixl+
l<i<j<_m
max
<i<j<m
lail.
The equality holds if and only if the following conditions are satisfied.
Without loss ofgenerality, assume that max1 _< i<j<_m [a01- la21.
(a)
(b)
(C)
aij O for 3 < < j < m.
xal2alia2i < 0 and [ale[-
la2glfor each 3 < < m.
E3 _<i_< m [ali[ _< min{lx[, la121}.
Proof Use induction on m. If m 2, then the lemma is true because
max{Ix + al, Ix a12[} Ixl + lal. Suppose that the lemma is true
for 2,3,...,m-1. Without loss of generality, we may assume that
max <_i<j<_m [a/[- [a34[. Put
M=
max
e=-t-
l<k<m
x+
Z
aijeiej
l<i<j<m
Substituting el- 4-1, we get
max
e=+l, 2<_k<_m
3<i<j<m
<M
aijeiej)
(1)
and
max
ek=-t-1, 2<k<_ m
3<i<j<m
(2)
S.G. KIM
4
By adding (1) and (2) and the triangle inequality, we get
aijeiej
ek=4-1, 2<k<m
3<j<m
_<.
(3)
3<_i<j<m
Again, by substituting 2--+1 into (3) and adding each other and the
triangle inequality, we get
max
ek=+l 3<k<m
Ix
-’t-
ao’e.iej
_< M.
(4)
3<i<j<m
By induction hypothesis and (4), we have
max la+l- Ixl + 1a341- Ixl + max [aijl
Ixl + l<i<j<m
3<i<j<m
<
max
Ck--q-1, <_k <_m
Ixq-aije.ie.j
3<i<j<m
Suppose that conditions (a)-(c) are satisfied. From now on, we will
assume that
l<i<j<m
Then by (a),
--al2l--l<_i<_m(ali--a2i)i3
Ix--al21nt-l<i<m(ali--a2i)i3
Without loss of generality, assume that xa12 > O. Then by (b),
Ix + a21 +
(ali + a2i)i
3<i<m
A REMARK ON POLYNOMIAL NORMS
5
for any sign choices 3,’’ m and, by (b) and (c),
Ix a121 +
Ix a21 + 2 3<i<m
Z lai[
<_ IX al2l q- 2 min{Ix i, lal2l}
(ali- a2i)i <_
3<i<m
(6)
--Ixl-t-la12[
for any sign choices 3,’’’, m. Thus M= Ixl + la121. Let us prove the
necessary condition. First we will prove it when m 4. Some computation shows that
Ix+
max
ek=+l,
l<k<4[
aijeiej
<i<j<4
max{Ix + a12 -4- a34[-+-I(al3 + a23) -4- (a14 -+- a24)1,
IX a12 4- a341 + [(a13 a23) 4- (a14 a24)]}.
By some calculation, we get
(a)
(b)
(C)
a34
0.
xa12alia2i < 0
and lal,I--la2/I for i= 3, 4.
E3 _<i_< 4
_< min{lxl, lal2l}.
Let m > 4. Suppose that M Ixl / laa21. Let 3 < i0 <Jo <_ m be fixed.
Let cr be the permutation on { 1,2,..., m} such that
cr(3)=i0, tr(4)=j0, cr(i0)=3, or(j0)=4.
Define bij
ar(i)r(j)
max
ek=+l, l_<k<4
x+
for each
Z
l<i<j<4
<j_< m. By Lemma 1,
x+
b0"i < ek=-+-1,max
l<k<m
bijeij =M
<i< j<rn
and by the first claim of Lemma 2,
max
k--Zt= 1, l_<k_<4
X
+
b0.ij
l<i<j<4
max Ibijl
Ixl + l_<i<j_<4
S.G. KIM
6
SO
max
ek=+l, l<k<4
x-+-
Z
bijeiej
l<i<j<4
max Ib l.
Ixl + l_<i<j<4
By the above argument for m--4 case, we have 0--b34--aiojo and
xal2alioaljo <_ 0 and [ali0[
Ib131 Ib24[--lau0l, showing
(a) and (b). Suppose that (c) is not true. By (a), (b), (5) and the triangle
xb12b13b24
inequality,
M>
x
max
ek=+l, 3<k<m
a12[ +
21
aliei
3<i<m
IX- a121 + 2 3<i<m lail > Ix a=l + 2 min{Ix I, la9.l}
a contradiction. Therefore we complete the proof.
Remark Lemma in [9] can be improved as follows. Let E be a normed
space over a field (C or R) and m > 2, a natural number. Let x, aij
(1 < <j < m) E E. Then
x+
eiejaij
l<i<j<m
max
<i<j<m
{llxll Ila/ll}.
Using Lemma 2, we obtain the main result of this paper.
2i<_j bijxixj 79(2/p), b/ R, 0 < p <_ c. Then
THEOREM 3 Let P(x)
we have
Ilpll
sup
mN, (Wl ,w2
wm,O )ESlp
f
{
Y biiw2i
<i<m
+
max
< < j<_m
Ibijwiwj[).
-
A REMARK ON POLYNOMIAL NORMS
7
Proof It follows from Lemma 2 because
Ilell
Ie(wa,..., f.mwm, O,O,..
l<i<m
biiw"- l<i<j<m
for any (Wl, w2,
Wm, 0,...) E
for any sign choices el,..., em.
COROLLARY 4 (a)LetP(x)
Then we have
(b) Let P(x)
Sll,
X
bijwiwjiJ]
l<i<m biiw
Yi<jbijxixj p(2/p), bij
and aij= bijeiey
R,0 <p <
.
-]i<jbijxixj p(2/), bij R. Then we have
IIPII
mENSUp{ <i<m
bii
max
+ l<i<j<m
Ibil}.
Proof (a) follows by taking wk 1/m lip for k 1, 2,..., m.
(b) follows by taking wk 1 for k 1,2,..., m.
PROPOSITION 5 (a) Let P(x)= Eil inair"inXi
Then we have
IIPII
(b) If
then
lai,...il
Xin
(n/2)
air-i,
K.
S.G. KIM
8
Proof Use induction on n.
Case n 2. For x
(Xk) E St2,
k
(by the H61der inequality)
(by the Hblder inequality)
k
k,j
lajle
Suppose that for n _< k, the proposition is true. For x
aiv..ik+ Xil
(Xk)
Sty,
Xik+
il ik+l
(by the induction hypothesis)
(by the H61der inequality)
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A REMARK ON POLYNOMIAL NORMS
9
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