Hardy's inequalities for Hermite and Laguerre expansions

Hardy's inequalities for Hermite and Laguerre expansions

著者 Kanjin Yuichi

journal or

publication title

Bulletin of the London Mathematical Society

volume 29

number 3

page range 331‑337

year 1997‑05‑01

URL http://hdl.handle.net/2297/3509

AND LAGUERRE EXPANSIONS

Dedicated to Professor Satoru Igari on his 60th birthday

YUICHI KANJIN

A

The well-known inequality of Hardy for Fourier coefficients of functionsf(t)C3^¢_n=−¢b_ne^intin the real Hardy space is3^¢_n=−¢rb_nr}(rnr­1)!¢. We shall establish analogues of this inequality for the Hermite function expansions and also for the Laguerre function expansions.

1. Introduction

Hardy’s inequality says that iff(t)C3^¢_n=−¢b_ne^int is in ReH", then 3^¢

n=−^¢

rb_nr rnr­1!¢,

where ReH" is the real Hardy space consisting of the boundary values of the real parts of functions in the Hardy spaceH"on the unit disk in the plane. The aim of this paper is to establish analogues of this inequality for the Hermite function expansions and also for the Laguerre function expansions.

Let(_n(x) be the Hermite function defined by

(_n(x)¯ ²π"^/#2ⁿΓ(n­1)´⁻"^/#H_n(x)e^−x#/#, whereH_n(x) is the Hermite polynomial of degreengiven by H_n(x)¯(®1)ⁿexp (x#) (d}dx)ⁿexp (®x#).

Then the system²(_n´^¢_n=!is complete orthonormal on the real line2with respect to the ordinary Lebesgue measuredx(compare [6, 5.7]). This system leads to the formal expansion of a function f(x) on2:

f(x)C3^¢

n=!

c_n(f)(_n(x),

wherec_n(f)¯!₋^¢_¢f(x)(_n(x)dxis thenth Hermite–Fourier coefficient off(x).

Received 8 September 1994 ; revised 29 May 1996.

1991Mathematics Subject Classification42C10, 33C45.

Partly supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan.

Bull.London Math.Soc. 29 (1997) 331–337

332  

Let,^α_n(x),α"®1, be the Laguerre function defined by

,^α_n(x)¯

(

*

where L^α_n(x)¯(n!)⁻"x^−αe^x(d}dx)ⁿ²x^n+αe^−x´ is the Laguerre polynomial of degree n and of order α. Then ²,^α_n´^¢_n=! is a complete orthonormal system on the interval (0,¢) with respect todx(compare [6, 5.7]). We have the formal expansion

g(x)C3^¢

n=!

c^α_n(g),^α_n(x) of a function g(x) on (0,¢), where c^α_n(g)¯!^¢

! g(x),^α_n(x)dx is the nth Laguerre–

Fourier coefficient.

LetH"(2) be the real Hardy space consisting of the boundary values of the real

parts of functions in the Hardy spaceH"(2#₊) on the upper half plane2#₊. InH"(2), we consider the norm induced byH"(2#₊). For a functiong(x) on (0,¢), we define the extensiongh(x) ofg(x) to2bygh(x)¯g(x) if 0!x, andgh(x)¯0 ifx%0. We define

H"(0,¢)¯ ²g(x) on (0,¢) ;gh(x)`H"(2)´,

sgs_H"(!^,¢)¯sghs_H"(2).

Our main results of this paper are as follows.

T. (i) There exists a constant C such that 3^¢

n=!

rc_n(f)r

(n­1)#*^/$'%Csfs_H"(2) (1)

for f(x)C3^¢_n=!c_n(f)(_n(x)in H"(2).

(ii) Letα&0.Then there exists a constant C such that 3^¢

n=! rc^α_n(g)r

n­1 %Csgs_H"(!^,¢) (2)

for g(x)C3^¢_n=!c^α_n(g),^α_n(x)in H"(0,¢).

The proof of the Theorem will be given in the next section. The atomic decomposition characterization of Hardy spaces will play an essential role in the proof. Also, the transplantation theorem (see (9) below) of R. Askey [1, p. 401, line 14] for Laguerre coefficients will be of use in the proof of part (ii) of the Theorem. The deduction of Paley’s theorem from the Theorem will be discussed in Section 3 with a remark.

For a historical survey on Hardy’s inequality and Paley’s theorem, we may refer to [2, p. 398, Comments]. The exposition of the role of the atomic decomposition characterization in the classial harmonic analysis is in R. R. Coifman and G. Weiss [3]. We shall also consult J. Garcia-Cuerva and J. L. Rubio de Francia [4] for the Hardy space theory.

2. Proof of the Theorem

AnH"atom is a functiona(x),x`2, supported in an interval (b,b­h) satisfying

ra(x)r%h⁻" and !a(x)dx¯0. The space H"(2) is characterized in terms of atoms

(compare [4, Chapter 3]) : f(x) belongs toH"(2) if and only if f(x)¯3^¢_j=!λ_ja_j(x),

where every a_j(x) is an H" atom and 3^¢_j=!rλ_jr!¢. The H"(2) norm of f(x) is equivalent to inf3^¢_j=!rλ_jr, the infimum being taken over all decompositions. The space H"(0,¢) is also characterized as follows [4, Lemma 7.40] : g(x) belongs to H"(0,¢) if and only ifg(x)¯3^¢_j=!λ_ja_j(x), where everya_j(x) is anH"atom satisfying suppa(x)Z(0,¢) and3^¢_j=!rλ_jr!¢.

We shall first give a lemma. Let²φ_n´^¢_n=!be an orthonormal system on an interval (c,d). For a function f(x) on (c,d), we consider the formal expansion

f(x)C3^¢

n=!

b_n(f)φ_n(x), whereb_n(f)¯!^b

cf(x)φ_n(x)dxis thenth Fourier coefficient with respect to the system

²φ_n´.

L. Suppose that max

c^!x^!d

)

)

with a constant K independent of n.Then there exist constants C

"and C

#satisfying the following.

(i) For an H"atom a(x)withsuppa(x)Z(c,d),

rb_n(a)r%C"n^δsas⁻## (3)

and

3^¢

n=!

rb_n(a)r

(n­1)^(δ+#^)/$%C#. (4) (ii) Let f(x)be a function of the form f(x)¯3^¢_j=!λ_ja_j(x),where eŠery a_j(x)is an H"

atom withsuppa(x)Z(c,d)and 3^¢_j=!rλ_jr!¢.Then 3^¢

n=!

rb_n(f)r (n­1)^(δ+#^)/$%C

#3^¢

j=!rλ_jr. (5)

Proof. Leta(x) be anH"atom with support contained in (b,b­h)Z(c,d) such

thatra(x)r%h⁻". Since

b_n(a)¯

&

a(x)φ_n(x)dx®φ_n(b)

&

a(x)dx¯

&

a(x)²φ_n(x)®φ_n(b)´dx, it follows from Schwarz’s inequality and the mean value theorem that

rb_n(a)r%sas#

(&

(x®b)#dx

*

)

)

%C"n^δsas#h$^/#,

whereC"¯K}3"^/#. By the facth%sas⁻##, we have the inequality (3).

For simplicity, we putγ¯sas'#^/(#^δ+"⁾and σ¯(δ­2)}3. It follows from (3) that 3^¢

n=! rb_n(a)r

(n­1)^σ¯

(

n^"γ

*

%C

"sas⁻##3

n^%γ

n^δ

(n­1)^σ­sas#

(

1

(n­1)#^σ

*

%C!"(sas⁻##γ⁽#^δ+"^)/$­sas#γ⁻⁽#^δ+"^)/')%C

#, whereC!"andC#are positive constants independent ofa(x).

334  

Sincesas"%1, the series3^¢_j=!λ_ja_j(x) converges inL"(c,d). We see that (4) implies (5).

We now come to the proof of the Theorem. By virtue of the Lemma and the atomic decomposition characterization, it is enough to show, for part (i) of the Theorem, that

)

)

forx`2. Here and below, Cdenotes a positive constant which may differ at each different occurrence. It follows from the equations

H_n(x)¯2xH_n−

"(x)®2(n®1)H_n−

#(x) and (d}dx)H_n(x)¯2nH_n−

"(x) (compare [6, (5.5.8), (5.5.10)]) that

(d}dx)(_n(x)¯(n}2)"^/#(_n−"(x)­((n­1)}2)"^/#(_n+"(x).

By the inequality

r(_n(x)r%Cn⁻"^/"# (7)

(compare [5, (2.23)]), we have (6), which completes the proof of part (i) of the Theorem.

We turn to the proof of part (ii) of the Theorem. Similarly, it suffices to show that

)

)

But we shall be able to prove this inequality only in the case thatα&2 orα¯0. The case 0!α!2 will be treated by the transplantation theorem for Laguerre coefficients betweenαandα­2 [1, p. 401, line 14] :

rc^α_n(f)r%C

0

0

1

1

To prove (8), we use the equation (d}dx)L^α_n(x)¯®L^α_n−⁺""(x) (compare [6, (5.1.14)]).

It follows that d

dx,^α_n(x)¯τ^α_n

0

1

whereτ^α_n¯ ²Γ(n­1)}Γ(n­α­1)´"^/#. By the inequality

r,^α_n(x)r%C, x"0, α&0 (10) (compare [5, (2.9)]), we have

)

)

0

1

%C(A­B­1), say.

We divide the matter into two cases,nx&1 and 0!nx!1. Fornx&1, we have, by (10) and τ^α_n%Cn^−α/#,

A%τ^α_nrL^α_n−⁺""(x)re^−x/#x^(α+"^)/#x⁻"^/#%Cn(nx)⁻"^/#%Cn,

B%α

2τ^α_nrL^α_n(x)re^−x/#x^α/#x⁻"%Cn(nx)⁻"%Cn.

Thus we have (8) for nx&1 when α&0. For 0!nx!1, we use the inequality rL^α_n(x)r%Cn^α, 0!nx!1 [6, (7.6.8)]. It follows that A%Cn^−α/#n^α+"x^α/#%Cn(nx)^α/#

%Cn when α&0, and that B%Cn^−α/#n^αx^α/#x⁻"%Cn(nx)^α/#⁻"%Cn when α&2.

Note that the termBdoes not appear whenα¯0. We conclude that (8) holds when α&2 orα¯0. Therefore we have the desired inequality (2) whenα&2 orα¯0.

We shall interpolate between α¯0 and α¯2 by using (9). Let 0!α!2 and

g(x)`H"(0,¢). By (10), we note thatrc^α!(g)r%Csgs_L"(!^,¢)%Csgs_H"(!^,¢). It follows

from (9) that 3^¢

n=! rc^α_n(g)r

n­1¯rc^α

!(g)r­3^¢

n="

rc^α_n(g)r n­1

%Csgs_H"(!^,¢)­C

0

n="

1 n­13^¢

j=n

rc^α_j⁺#(g)r

0

1

1

We treat the last sum. Since²3^j_n="n^α/#(n­1)⁻"´j^−α/#%Cforα"0, it follows that 3^¢

n="

1 n­13^¢

j=n

rc^α_j⁺#(g)r

0

1

(

n^α/#

n­1

*

%C3^¢

j="j⁻"rc^α_j⁺#(g)r. Therefore we have

3^¢

n=! rc^α_n(g)r

n­1%C

0

1

forα"0, which completes the proof of part (ii) of the Theorem since inequality (2) withα&2 orα¯0 has been proved.

3. Paley type theorem

The following inequalities, (11) and (12), with respect to the Laguerre function system follow from Paley’s theorem (compare [7, Vol. II, Theorem (5.1)]) with respect to general systems ²φ_n´ of functions orthonormal and uniformly bounded, rφ_n(x)r%C.

Letα&0. Forg(x)C3^¢_n=!c^α_n(g),^α_n(x)`L^p(0,¢), 1!p%2, 3^¢

n=!rc^α_n(g)r^p(n­1)^p−#%Csgs^p_Lp(!^,¢). (11) For²b_n´^¢_n=!with 3^¢_n=!rb_nr^q(n­1)^q−#, 2%q!¢, there exists a function

g(x)C3^¢_n=!b_n,^α_n(x)`L^q(0,¢) such that

sgs^q_L^q₍!^,¢)%C3^¢

n=!rb_nr^q(n­1)^q−#. (12) Also, by applying Paley’s theorem to the Hermite function system, we have the same inequalities as (11) and (12), where we substitute c^α_n(g), g(x)`L<sup>p</sup>(0,¢), by c<sub>n</sub>(f), f(x)`L^p(2), respectively. But, for the Hermite function system, we can obtain sharper inequalities by interpolating Hardy’s inequality (1) and Parseval’s equation.

336  

P. (i) Let1!p%2.Then there exists a constant C such that 3^¢

n=!rc_n(f)r^p(n­1)#*^(p−#^)/$'%Csfs^p_L^p₍2) (13) for f(x)C3^¢_n=!c_n(f)(_n(x)`L^p(2).

(ii) Let2%q!¢. Then there exists a constant C such that if ²b_n´^¢_n=! satisfies 3^¢_n=!rb_nr^q(n­1)#*^(q−#^)/$'!¢, then

sfs^q_Lq(²)%C3^¢

n=!rb_nr^q(n­1)#*^(q−#^)/$', (14) where f(x)C3^¢_n=!b_n(_n(x)`L^q(2).

Proof. Forκ"0, we denote by l^p_κ,p&1, the space of sequences²b_n´^¢_n=! such thats²b_n´s_p¯ ²3^¢_n=!rb_nr^p(n­1)⁻#^κ´"^/p!¢. We define an operatorT_κofL^p(2) tol^p_κ byT_κf¯ ²c_n(f) (n­1)^κ´^¢_n=!. We consider the caseκ¯29}36. Then by (1), we see that T_κis a bounded linear operator ofH"(2) tol"_κ, which implies thatT_κis of weak type

(H"(2),l"_κ). Parseval’s equation says thatT_κis an isometry ofL#(2) tol#_κ; in particular,

T_κis of weak type (L#(2),l#_κ). We follow line by line the proof [4, pp. 308–310] of the interpolation theorem between H" space and L^p space. We conclude that T_κ is a bounded operator ofL^p(2) tol^p_κ, 1!p%2, which means (13).

The inequality (14) is obtained by a standard duality argument (compare [7, Vol. II, proof of Theorem (5.1)]).

Lastly, we give a remark. Instead of the interpolation theorem betweenH"space

andL"space, we may use the interpolation theorem between weak-L^pspaces. But it

will become clear in the sequel that the inequality (15) below, obtained by this method, is weaker than our inequality (13).

We now considerl^p_κ,T_κwithκ¯11}12. Parseval’s equation implies thatT_κis of weak type (L#(2),l#_κ). Moreover, we see thatT_κ is of weak type (L"(2),l"_κ). For, we put/_t¯ ²n;rc_n(f) (n­1)^κr"t´fort"0, wheref(x)`L"(2). It follows from (7) that

t!rc_n(f) (n­1)^κr!Cn⁻"^/"#n^κsfs_L"(²)¯Cn"!^/"#sfs_L"(²). Thus we have

²t}sfs_L"(2)´"#^/"!!n.

This gives3_n`/t(n­1)⁻#^κ%C²t}sfs_L"(2)´"#⁽⁻#^κ+"^)/"!¯Csfs_L"(2)}t, which means that T_κis of weak type (L"(2),l"_κ). By the Marcinkiewicz interpolation theorem, we see that T_κis a bounded operator of L^p(2) tol^p_κ, 1!p!2, that is,

3^¢

n=!rc_n(f)r^p(n­1)""^(p−#^)/"#%Csfs_L^pp₍2). (15)

I should like to thank the referee for informing me of Kwon and Littlejohn’s work (Annals of Numerical Mathematics2 (1995) 289–303) which might lead to a further study.

References

1. R. A, ‘A transplantation theorem for Jacobi coefficients ’,Pacific J.Math. 21 (1967) 393–404.

2. L. S. B et al. (eds), Collected papers of G.H.Hardy, Vol. III (Oxford University Press, London, 1969).

3. R. R. Cand G. W, ‘ Extensions of Hardy spaces and their use in analysis ’,Bull.Amer.Math.

Soc. 83 (1977) 569–645.

4. J. G-Cand J. L. R F,Weighted norm inequalities and related topics(North- Holland, Amsterdam, 1985).

5. C. M, ‘ Nikolskii-type inequalities for Laguerre and Hermite expansions ’, Functions, series, operators, Colloq. Math. Soc. Ja!nos Bolyai 35 (ed. J. Szabados, North-Holland, Amsterdam, 1982) 811–834.

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Department of Mathematics College of Liberal Arts Kanazawa University Kanazawa 920-11 Japan