s. v. astashkin- lieb–thirring inequality for lp norms

8/3/2019 S. V. Astashkin- LiebThirring Inequality for Lp Norms

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ISSN 0001-4346, Mathematical Notes, 2008, Vol. 83, No. 2, pp. 145151. c Pleiades Publishing, Ltd., 2008.Original Russian Text c S. V. Astashkin, 2008, published in Matematicheskie Zametki, 2008, Vol. 83, No. 2, pp. 163169.

LiebThirring Inequality for Lp Norms

S. V. Astashkin*

Samara State University

Received May 31, 2007; in final form, August 28, 2007

AbstractIn this paper, we obtain the LiebThirring inequality for Lp-norms. The proof uses onlythe standard apparatus of the theory of orthogonal series.

DOI: 10.1134/S0001434608010173

Key words: LiebThirring inequality, Lp-norm, orthonormal system, orthogonal series,Marcinkiewicz theorem, Fourier multiplier, Rademacher function.

In 1976, Lieb and Thirring obtained the following result [1].

Theorem A. If = {j}Nj=1 L

2(Rd) is an arbitrary orthonormal system,

max

1,

d

2

< p 1 +

d

2,

and

:=N

j=1

2j , (1)

then

Rd

p/(p1) dx2(p1)/d Cp,dN

j=1

j2L2(Rd), (2)

where, as is customary, = (/x1, . . . ,/ xd).

More recently, inequalities of type (2) for finite orthonormal systems were established by differentauthors (they were of interest mainly because of their applications to the theory of partial differentialequations). The well-known methods of proof based on nontrivial results from spectral theory used bythese authors were quite complicated (for details, see [2] and [3]).

Recently, Kashin [4] proposed a new approach to the proof of inequalities of LiebThirring type, inwhich he used only the standard apparatus of the theory of orthogonal series: Lp-inequalities for vector-valued Rademacher series and the classical LittlewoodPaley theorem. Let us recall some definitions

and notation from [4]. Suppose that

= {j}Nj=1 L

2(S1), where j1, 1 j N. (3)

Let us define the operator P : lN2 L

2(S1) as follows:

P({cj}Nj=1) =

Nj=1

cjj.

*E-mail: [email protected] .

145


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146 ASTASHKIN

Besides, suppose that m : L2(S1) T2m is the orthogonal projection onto the space T2m , m =

0, 1, 2, . . . , of trigonometric polynomials t(z) of the form

t(z) =

2m|k| 0 such that, for an arbitrary system of theform (3), the following inequality holds:

S12 d C

m=0

m()

Nj=1

2m|i| 0 and afunction

f(z) =j=0

f(j)zj L2(S1),the derivative of order is defined by the relation

f()(z) :=j=0

|j|f(j)zj .In this paper, we show that, following the approach used in [4], one can generalize inequalities (6)

and (7) to the case of Lp-norms. In the proof, we shall use the Marcinkiewicz theorem on Fouriermultipliers.

We begin by stating the results and making a few remarks.

MATHEMATICAL NOTES Vol. 83 No. 2 2008


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LIEBTHIRRING INEQUALITY FOR Lp NORMS 147

Theorem 1. For any k = 2, 3, . . . , there exists a constant C = C(k) such that, for an arbitrarysystem of the form (3), the following inequality holds:

S1k d C

m=0

m()N

j=1

2m|i|


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148 ASTASHKIN

where the rj(t) are Rademacher functions on [0, 1]. Then, in view of the vector-valued version ofKhinchines inequality [7, p. 52] and the LittlewoodPaley inequality [8, p. 63, inequalities (9)], we find

Q :=

S1

k d =

S1

Nj=1

2j (z)

kd

C1 1

0 N

j=1 rj(t)j2k

L2kdt C2

1

0 S1

m=0 2

m(t, z)

k

d dt,

where the constants depend only on k. Hence

Q

C2

m1=0

mk=0

10

S1

2m1(t, z)2m2(t, z)

2mk

(t, z) ddt. (11)

For each i = 1, 2, . . . , k, we have

2mi(t, z) =

Nj=1

rj(t)mi(j)(z)

2=

Nj,q=1

rj(t)rq(t)mi(j)(z)mi(q)(z)

and, therefore, for all t [0, 1],S1

2m1(t, z)2m2(t, z)

2mk

(t, z) d =N

j1=1

Nq1=1

N

jk=1

Nqk=1

rj1(t)rq1(t)rj2(t)rq2(t) rjk(t)rqk(t)

S1

m1(j1)m1(q1) mk(jk)mk(qk) d.

Note that the integral 10

rj1(t)rq1(t)rj2(t)rq2(t) rjk(t)rqk(t) dt

is nonzero (is equal to 1) if and only if, in the collection of the natural numbers (j1, q1, . . . , jk, qk), eachnumber occurs an even number of times. Therefore, the desired quantity

Im1,m2,...,mk :=10

S1

2m1(t, z) 2mk

(t, z) ddt =S1

10

2m1(t, z) . . . 2mk

(t, z) dtd

is the sum of integrals of the formS1

m1(j1)m(1)(j1)m2(j2)m(2)(j2) mk(jk)m(k)(jk) d, (12)

where is a permutation of the set {1, 2, . . . , k}. Conversely, expressing any such permutation asthe product of cycles, we can readily verify that the corresponding integral (12) is one of the summandsmaking up Im1,m2,...,mk . Thus,

Im1,m2,...,mk =

N

j1=1N

j2=1 N

jk=1 S1 m1(j1)m(1)(j1)m2(j2) m(2)(j2) mk (jk)m(k)(jk) d

=

S1

Nj1=1

m1(j1)m(1)(j1)

N

j2=1

m2(j2)m(2)(j2) N

jk=1

mk(jk)m(k)(jk) d.



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Hence, by Cauchys inequality, we have

Im1,m2,...,mk

S1

Nj1=1

2m1(j1)

1/2 Nj1=1

2m(1)(j1)

1/2

Njk=1

2mk(jk)

1/2

Njk=1

2m(k)(jk)

1/2d

=

S1

Nj=1

2m1(j) N

j=1

2mk(j) d

= k!

S1

Nj=1

2m1(j) N

j=1

2mk(j) d.

Thus,

Im1,m2,...,mk k! Jm1,m2,...,mk ,

where

Jm1,m2,...,mk := S1

Nj=1

2m1(j)

Nj=1

2mk(j) d.

Since, for any permutation : {1, 2, . . . , k} {1, 2, . . . , k},

Jm1,m2,...,mk = Jm(1),m(2),...,m(k) ,

in view of (11), we can write

Q

C2 k!

m1=0

mk=0

Jm1,m2,...,mk = (k!)2

0m1mk

Jm1,m2,...,mk

(k!)2

0m1mk

N

j=12m1(j)

L(S1)

N

j=12mk1(j)

L(S1)

S1

Nj=1

2mk(j) d.

By a lemma from [4], we have N

j=1

2i (j)

L(S1)

2i+12i () (13)

for each i = 0, 1, 2, . . . . Therefore,

Q

C2 (k!)2

mk=0mk

mk1=0 2mk1+12mk1()

mk1mk2=0

2mk2+12mk2() m2

m1=0

2m1+12m1()

S1

Nj=1

2mk(j) d

= (k!)22k1

m=0

m()N

j=1

2m|i|


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150 ASTASHKIN

Proof of Theorem 2. Preserving the notation from the proof of Theorem 1, we can write

Q

C2 k!

m1=0

mk=0

Jm1,m2,...,mk ,

where, as before,

Jm1,m2,...,mk := S1N

j=1

2m1(j)

N

j=1

2mk(j) d.

Suppose that n = k/l. Then again, in view of (13) and (5) (see also Remark 2), we have

Q

C2 k!(l!)n

0m1ml

l1i=1

N

j=1

2mi(j)

L(S1)

0ml+1m2l

l1i=1

N

j=1

2ml+i(j)

L(S1)

0ml(n1)+1mk

l1i=1

N

j=1

2m(n1)l+i(j)

L(S1)

S1N

j=12ml(j)

N

j=12m2l(j)

N

j=12mk(j) d

k!(l!)n2(l1)(n+k/2)

ml=0

2(l1)ml

m2l=0

2(l1)m2l

mk=0

2(l1)mk

S1

Nj=1

2ml(j)N

j=1

2m2l(j) N

j=1

2mk(j) d

= k!(l!)n2(l1)(n+k/2)S1

Nj=1

m=0

2(l1)m2m(j)

nd.

Hence applying the Minkowski and LittlewoodPaley inequalities, we find

QC2

k!(l!)n2(l1)(n+k/2) Nj=1

S1

m=0

2(l1)m2m(j)n1/n

dn

k!(l!)n2(l1)(n+k/2)C3

Nj=1

j22n

n, (14)

where C3 > 0 is a constant depending only on k and

j(z) =

m=0

2m(l1)/2

2m|i|


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where C4 > 0 is a constant depending only on p. Combining this with inequality (14), we obtain (10).The theorem is proved.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (grant no. 07-01-96603).

REFERENCES

1. E. Lieb and W. Thirring, Inequalities for the moments of the eigenvalues of the Schr odinger Hamiltonianand their relation to Sobolev inequalities, in Studies in Mathematical Physics (Princeton Univ. Press,Princeton, NJ, 1976), pp. 269303.

2. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, in Applied Mathemat-ical Sciences (Springer-Verlag, New York, 1997), Vol. 68.

3. A. A. Ilin, Lieb-Thirring integral inequalities and their applications to attractors of Navier-Stokes equations,Mat. Sb. 196 (1), 3366 (2005) [Russian Acad. Sci. Sb. Math. 196 (1), 2961 (2005)].

4. B. S. Kashin, On a class of inequalities for orthonormal systems, Mat. Zametki 80 (2), 204208 (2006)[Math. Notes 80 (12), 199203 (2006)].

5. J.-M. Ghidaglia, M. Marion, and R. Temam, Generalization of the SobolevLiebThirring inequalities andapplications to the dimension of attractors, Differential Integral Equations 1 (1), 121 (1988).

6. A. Eden and C. Foias, A simple proof of the generalized LiebThirring inequalities in one-space dimension,

J. Math. Anal. Appl. 162 (1), 250

254 (1991).7. B. S. Kashin and A. A. Saakyan, Orthogonal Series (AFTs, Moscow, 1999) [in Russian].8. S. M. Nikolskii, Approximation of Functions of Several Variables and Embedding Theorems (Nauka,

Moscow, 1969) [in Russian].9. I. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press,

Princeton, NJ, 1970; Mir, Moscow, 1973).


s. v. astashkin- lieb–thirring inequality for lp norms

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