JJ II

(1)

volume 4, issue 3, article 51, 2003.

Received 12 March, 2003;

accepted 31 March, 2003.

Communicated by:S.S. Dragomir

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

THE HARDY-LANDAU-LITTLEWOOD INEQUALITIES WITH LESS SMOOTHNESS

CONSTANTIN P. NICULESCU AND CONSTANTIN BU ¸SE

University of Craiova, Department of Mathematics, Craiova 200585,

Romania.

E-Mail:cniculescu@central.ucv.ro URL:http://www.inf.ucv.ro/~niculescu West University of Timisoara, Timisoara 300223,

Romania.

E-Mail:buse@hilbert.math.uvt.ro URL:http://rgmia.vu.edu.au/BuseCVhtml/

c

2000Victoria University ISSN (electronic): 1443-5756 031-03

(2)

The Hardy-Landau-Littlewood Inequalities with Less

Smoothness

Constantin P. Niculescu and Constantin Bu¸se

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of19

J. Ineq. Pure and Appl. Math. 4(3) Art. 51, 2003

http://jipam.vu.edu.au

Abstract

One proves Hardy-Landau-Littlewood type inequalities for functions in the Lip- schitz space attached to aC₀-semigroup (or to aC₀^?-semigroup).

2000 Mathematics Subject Classification:Primary 26D10, 47D06; Secondary 26A51 Key words: Landau’s inequalities,C0-semigroup, Lipschitz function.

The first author was partially supported by CNCSIS Grant A3/2002.

We are indebted to Sever S. Dragomir (Melbourne), Adrian Duma (Craiova) and Petru Jebelean (Timi¸soara) for many useful conversations which allowed us to im- prove our work in many respects.

1. Introduction

If a function and its second derivative are small, then the first derivative is small too. More precisely, for each p ∈ [1,∞] and each of the intervals I = R+

or I = R, there is a constant C_p(I) > 0 such that if f : I → R is a twice differentiable function withf, D²f ∈L^p(I),thenDf ∈L^p(I)and

(1.1) kDfk_LP ≤C_p(I)kfk^1/2_Lp

D²f

1/2 L^p .

We make the convention to denote byC_p(I)the best constant for which the inequality (1.1) holds.

The investigation of such inequalities was initiated by E. Landau [17] in 1914. He considered the casep=∞and proved that

C∞(R+) = 2 and C∞(R) =√ 2.

In 1932, G.H. Hardy and J.E. Littlewood [12] proved (1.1) forp = 2, with best constants

C₂(R⁺) = √

2 and C₂(R) = 1.

In 1935, G.H. Hardy, E. Landau and J.E. Littlewood [13] showed that C_p(R+)≤2 forp∈[1,∞)

which yieldsC_p(R) ≤ 2forp ∈[1,∞).Actually, C_p(R) ≤√

2.See Theorem 1.1below.

In 1939, A.N. Kolmogorov [16] showed that

(1.2)

D^kf

L^∞ ≤C∞(n, k,R) kfk^1−k/n_L∞ kDⁿfk^k/n_L∞

(4)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of19

for functions f on Rand1 ≤ k < n(D^k denotes thekth derivative of f).As above,C∞(n, k,R)denote the best constant in (1.2). Their explicit formula was indicated also by A.N. Kolmogorov [16]. An excellent account on inequalities (1.1) (and their relatives) are to be found in the monograph of D. S. Mitrinovi´c, J. E. Peˇcari´c, and A. M. Fink [19].

All these results were extended to C0-semigroups (subject to different re- strictions) by R.R. Kallman and G.-C. Rota [15], E. Hille [14] and Z. Ditzian [5]. We shall consider here the case of stableC₀-semigroups on a Banach space E, i.e. of semigroups(T(t))t≥0 such that

sup

t≥0

kT(t)k=M <∞.

Theorem 1.1. Let(T(t))_t≥0be a stableC₀-semigroup onE, and letA:Dom(A)

⊂ E → E be its infinitesimal generator. Then for eachn = 2,3, . . . and each integer numberk ∈(0, n)there exists a constantK(n, k)>0such that

(1.3) ||A^kf|| ≤K(n, k)||Aⁿf||^k/n||f||^1−k/n for allf ∈Dom(Aⁿ).

Moreover,K(2,1) = 2M in the case of semigroups, andK(2,1) =M√ 2in the case of groups. The other constantsK(n, k)can be estimated by recursion.

The aim of this paper is to prove similar inequalities with less smoothness assumptions, i.e. outside Dom(A²). See Theorem 2.1 below. The idea is to replace twice differentiability by the membership of the first differential to the Lipschitz class. In the simplest case our result is equivalent with the following fact: Letf :R→ Rⁿ be a differentiable bounded function, whose derivative is Lipschitz. ThenDf is bounded and

(1.4) kDfk²_L∞ ≤2kfk_L∞ · kDfk_Lip.

(5)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of19

See Section3for details.

An important question concerning the above inequalities is their signifi- cance. One possible physical interpretation of the inequality studied by Landau is as follows: Suppose that a massmparticle moves along a curver =r(t), t≥ 0, under the presence of a continuous force F, according to Newton’s Law of motion,

m¨r=F.

If the entire evolution takes place in a ballB_R(0),then the kinetic energy of the particle,

E = mk˙rk² 2 , satisfies an estimate of the form

E ≤







2RkFk_L∞, if the temporal interval isR+

RkFk_L∞, if the temporal interval isR,

which relates the level of energy and the size of ambient space where motion took place.

The same inequality of Landau reveals an obstruction concerning the extension properties of smooth functions outside a given compact interval I. Does there exist a constantC > 0such that for each functionf ∈ C²(I) there is a corresponding functionF ∈C²(R)such that

F =f onI

(6)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of19

and

sup

x∈R

|D^kF(x)| ≤Csup

x∈I

|D^kf(x)| fork= 0,1,2 ?

By assuming a positive answer, an immediate consequence would be the relation

sup

x∈I

|f⁰(x)|² ≤2C²

sup

x∈I

|f(x)| sup

x∈I

|f⁰⁰(x)|

.

Or, simple examples (such as that one at the end of section 3 below) show the impossibility of such a universal estimate.

A recent paper by G. Ramm [21] describes still another obstruction derived from (1.1), concerning the stable approximation off⁰.

(7)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of19

2. Taylor’s Formula and the Extension of the Hardy- Landau-Littlewood Inequality

Throughout this section we shall deal withσ(E, X)-continuous semigroups of linear operators on a Banach spaceE,whereX is a (norm) closed subspace of E^? which satisfies the following three technical conditions:

S1) kxk= sup{|x^?(x)|; x^? ∈X, kx^?k= 1}.

S2) Theσ(E, X)-closed convex hull of everyσ(E, X)-compact subset ofEis σ(E, X)-compact as well.

S3) Theσ(X, E)-closed convex hull of everyσ(X, E)-compact subset ofXis σ(X, E)-compact as well.

For example, these conditions are verified when X is the dual space ofE or its predual (if any), so that our approach will include both the case of C0- semigroups and ofC₀^?-semigroups. See [3], Section 3.1.2, for details.

(A, Dom(A))will always denote the generator of such a semigroup T = (T(t))t≥0.

The Lipschitz space of order α ∈ (0,1] attached to A is defined as the set Λ^α(A)of allx∈Esuch that

kxk_Λα = sup

s>0

kT(s)x−xk s^α <∞.

This terminology is (partly) motivated by the case ofA=d/dt ,with domain Dom(A) = {f ∈L^∞(R); f is absolutely continuous andf⁰ ∈L^∞(R)},

(8)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of19

which generates theC₀^?-semigroup of translations onL^∞(R) : T(t)f(s) = f(s+t), for everyf ∈L^∞(R).

In this case, the elements ofΛ^α(A)are the usual Lipschitz mappingsf :R→C of orderα(which are essentially bounded).

Coming back to the general case, notice that (2.1) T(t)x=x+tAx+

Z t 0

(T(s)−I)Ax ds, forx∈Dom(A)andt >0 (possibly, in the weak^? sense, if the given semigroup isC₀^?-continuous). In the classical approach, the remainder is estimated via “higher derivatives”, i.e. via A².In the framework of semigroups, we need the inequality

Z t 0

(T(s)−I)Ax ds

≤ t^α+1

α+ 1kAxk_Λα,

which works for everyx∈ Dom(A)withAx ∈Λ^α(A)and everyt > 0.Then, from Taylor’s formula (2.1), we can infer immediately the relation

kAxk ≤ (1 +kT(t)k)kxk

t + t^α

α+ 1kAxk_Λα,

for everyx∈Dom(A)withAx ∈Λ^α(A)and everyt >0.Taking in the right- hand side the infimum over t > 0,we arrive at the following generalization of the Hardy-Landau-Littlewood inequality:

(9)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of19

Theorem 2.1. If(A, Dom(A))is the generator of aC₀-(or of aC₀^?-)semigroup (T(t))t≥0 such that

sup

t≥0

kT(t)k ≤M < ∞, then

kAxk ≤M_sg(A)kxk^α/(1+α)· kAxk^1/(1+α)_Λα , for everyx∈Dom(A)withAx∈Λ^α(A),where

M_sg(A) = (1 +M)^α/(1+α)

"

α 1 +α

1/(1+α)

+ 1

1 +α ·

1 +α α

α/(1+α)# .

In the case of (C₀- orC₀^?-continuous) groups of isometries, again by Taylor’s formula (2.1),

(2.2) T(−t)x=x−tAx+ Z 0

−t

(T(s)−I)Axds, forx∈Dom(A)andt >0 so that subtracting (2.2) from (2.1) we get

kAxk ≤ (kT(t)k+kT(−t)k)kxk

2t + t^α

α+ 1 kAxk_Λα

which leads to a better constant in the Hardy-Landau-Littlewood inequality, more precisely, the boundM_sg should be replaced by

Mg(A) = M^α/(1+α)

"

α 1 +α

1/(1+α)

+ 1

1 +α ·

1 +α α

α/(1+α)# .

(10)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of19

The problem of finding the best constants in the Hardy-Landau-Littlewood inequality is left open. Notice that even the best values of Cp(I), for 1 <

p < ∞, are still unknown; an interesting conjecture concerning this particu- lar case appeared in a paper by J.A. Goldstein and F. Räbiger [9], but only a little progress has been made since then. See [7].

The generalization of Taylor’s formula for higher order of differentiability is straightforward (and it allows us to extend A.N. Kolmogorov’s interpolating inequalities to the case of semigroups).

Theorem2.1outlined the Sobolev-Lipschitz space of order1 +α, WΛ^α(A) = {x∈Dom(A); Ax∈Λ^α(A)}. which can be endowed with the norm

kxk_W_Λα =kxk_W1 +kAxk_Λα. Clearly,

Dom(A²)⊂WΛ¹(A)⊂D(A)

and the following example shows that the above inclusions can be strict.

LetX=C₀(R+)be the Banach space of all continuous functionsf :R+→ R such thatlim_t→∞f(t) = 0 (endowed with the sup-norm). The generator of the translation semigroup onXis

A= d

dt withDom(A) = {f ∈X; f differentiable andf⁰ ∈X}.

See [20]. Then we have

Dom(A²) ={f ∈Dom(A); f⁰⁰ ∈X}

(11)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of19

and

WΛ¹(A) ={f ∈Dom(A); f⁰ is a Lipschitz function}.

(12)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of19

3. The Inequalities of Hadamard

WhenI isR+ orR,the following result (essentially due to J. Hadamard [11]) is a straightforward consequence of Theorem 2 above, applied to the semigroup generated by _dx^d onL^∞_Rn(I) :

Theorem 3.1. Let I be an interval and let f : I → Rⁿ be a differentiable bounded function, whose derivative is Lipschitz, of order1.Thenf⁰is bounded and

kf⁰k_L∞≤











4kfk_L∞

`(I) + `(I)

4 kf⁰k_Lip , if`(I)≤4q

kfk_L∞/kf⁰k_Lip 2q

kfk_L∞ · kf⁰k_Lip, if`(I)≥4q

kfk_L∞/kf⁰k_Lip andI6=R q

2kfk_L∞ · kf⁰k_Lip, ifI =R.

Furthermore, these inequalities are sharp. Here `(I)denotes the length of I.

Proof. Of course, Theorem 3.1 admits a direct argument. Notice first that we can restrict ourselves to the case of real functions.

According to our hypotheses,f⁰ satisfies onI an estimate of the form

|f⁰(t)−f⁰(s)| ≤ kf⁰k_Lip|t−s|

wherekf⁰k_Lip =kf⁰k_Λ1 is the best constant for which this relation holds. As f(t) = f(a) +f⁰(a)(t−a) +

Z t a

[f⁰(t)−f⁰(a)]dt,

(13)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of19

we have

|f(t)−f(a)−f⁰(a)(t−a)| ≤

Z t a

[f⁰(t)−f⁰(a)]dt

≤ 1

2kf⁰k_Lip|t−a|²

for every t, a ∈ I, t 6= a. The integrability is meant here in the sense of Henstock-Kurzveil [2], [10]. Consequently,

|f⁰(a)| ≤ |f(t)−f(a)|

|t−a| + 1

2kf⁰k_Lip|t−a|

≤ 2kfk_L∞

|t−a| +1

2kf⁰k_Lip|t−a|

for every t, a ∈ I, t 6= a.Now, the problem is how much room is left to t. In the worse case, i.e., when I is bounded and `(I) ≤ 4q

kfk_L∞/kf⁰k_Lip, the infimum overtin the right side hand is 4kfk_L∞

`(I) +`(I)

4 kf⁰k_Lip. IfI is unbounded, then the infimum is at most2q

kfk_L∞kf⁰k_Lip,or even q

2kfk_L∞kf⁰k_Lip, forI =R).

In order to prove that the bounds indicated in Theorem3.1 above are sharp it suffices to exhibit some appropriate examples. The critical case is that of

(14)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of19

bounded intervals, because for half-lines, as well as for R, the sharpness is already covered by Landau’s work.

Restricting to the case ofI = [0,1],we shall consider the following example, borrowed from [4]. Leta∈[0,4].The function

fa(t) = −at² 2 +

2 + a

2

t−1, t∈I = [0,1]

verifies kf_ak_L∞ = 1,kf_a⁰k_L∞ = 2 +a/2 andkf_a⁰k_Lip = a.As `(I) = 1, the relation given by Theorem3.1becomes

2 + a

2 ≤ 2·1 1 +1

2a.

On the other hand, no estimate of the form kf⁰k_L∞ ≤Cq

kfk_L∞kf⁰k_Lip

can work for all functionsf ∈ C²(I),because, taking into account the case of the functionsf_a,we are led to

2 + a

2 2

≤Ca for everya∈[0,4]

a fact which contradicts the finiteness ofC.

(15)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of19

4. The Case of Nonlinear Semigroups

We shall discuss here the case of one of the most popular nonlinear semigroup of contractions, precisely, that generated by thep-Laplacian(p∈(2,∞)),

Au= ∆_pu=div |∇u|^p−2· ∇u

, (p∈(2,∞)) acting onH =L²(Ω)and having as its domain

Dom(A) =

u∈W₀^1,p(Ω); ∆_pu∈H .

Here Ω denotes a bounded open subset of R^N, with sufficiently smooth boundary.

PutV =W₀^1,p(Ω)and denote byj :V →Handj⁰ :H →V⁰the canonical embeddings.

Clearly, A is a dissipative operator. It is also maximal dissipative i.e., the image ofI_H−AequalsH. In fact, letf ∈H.SinceAis dissipative, hemicon- tinuous and coercive as an operator fromV intoV⁰,it follows thatImA=V⁰, so thatIm(j⁰j −A) =V⁰.Therefore the equation

u−Au=f

has a unique solutionu ∈ V.This shows that u ∈ Dom(A)i.e.,A is maximal dissipative and thus it generates a nonlinear semigroup of contractions on H.

See [1].

Suppose there exists a positive constantCsuch that

||Ax||²_H ≤C||A²x||H · ||x||H for everyx∈Dom(A²).

(16)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of19

As||Ax||_V⁰ =||x||^p−1_V ,it would follow that

||x||^2(p−1)_V ≤C1||A²x||H · ||x||H

≤C₂||A²x||_H · ||x||_V

i.e., ||x||^2p−3_V ≤ C₂||A²x||_H for everyx ∈ Dom(A²). Lettingx = εy, where ε >0andy∈Dom(A²), y 6= 0,we are led to

ε^2p−3||y||^2p−3_V ≤C₂ε^(p−1)²||A²y||_H

i.e., to ||y||^2p−3_V ≤ C₂ε^(p−2)²||A²y||_H, which constitutes a contradiction for y fixed andεsmall enough.

(17)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page17of19

References

[1] V. BARBU, Nonlinear semigroups and differentiable equations in Banach spaces, Ed. Academiei, Bucharest, and Noordhoff International Publish- ing, Leyden, 1976.

[2] R.G. BARTLE, Return to the Riemann Integral, Amer. Math. Monthly, 103 (1996), 625–632.

[3] O. BRATTELIANDD.W. ROBINSON, Operator Algebras and Quantum Statistical Mechanics 1, Springer-Verlag, New York-Heidelberg-Berlin, 1979.

[4] C.K. CHUI ANDP.W. SMITH, A note on Landau’s problem for bounded intervals, Amer. Math. Monthly, 82 (1975), 927–929.

[5] Z. DITZIAN, Some remarks on inequalities of Landau and Kolmogorov, Aequationes Math., 12 (1975), 145–151.

[6] Z. DITZIAN, On Lipschitz classes and derivative inequalities in vari- ous Banach spaces. In vol. Proceedings Conference on Functional Anal- ysis, Holomorphy and Approximation (G. Zapata Ed.), pp. 57–67, North- Holland, Amsterdam, 1984.

[7] Z. DITZIAN, Remarks, questions and conjectures on Landau- Kolmogorov-type inequalities, Math. Inequal. Appl., 3 (2000), 15–24.

[8] K. ENGELANDR. NAGEL, One-parameter semigroups for linear evolu- tion equations, Springer-Verlag, 2000.

(18)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page18of19

[9] J.A. GOLDSTEIN AND F. RÄBIGER, On Hardy-Landau-Littlewood Inequalities, Semesterbericht Funktionanalysis. Workshop on Operator Semigroups and Evolution Equations, Blaubeuren, October 30-November 3, 1989, pp. 61–75, Tübingen, 1990.

[10] R.A. GORDON, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Grad. Studies in Math., 4 (1994), Amer. Math. Soc., Providence.

[11] J. HADAMARD, Sur le module maximum d’une fonction et ses dérivées, Comptes Rendus des séances de la Societé Mathematique de France, 1914, pp. 68–72.

[12] G.H. HARDY ANDJ.E. LITTLEWOOD, Some integral inequalities con- nected with the calculus of variations, Quart. J. Math. Oxford Ser., 3 (1932), 241–252.

[13] G.H. HARDY, E. LANDAUANDJ.E. LITTLEWOOD, Some inequalities satisfied by the integrals or derivatives of real or analytic functions, Math.

Z., 39 (1935), 677–695.

[14] E. HILLE, On the Landau-Kallman-Rota inequality, J. Aprox. Th., 6 (1972), 117–122.

[15] R.R. KALLMAN AND G.-C. ROTA, On the inequality kf⁰k² ≤ 4kfk kf⁰⁰k. In Inequalities, vol. II (O. Shisha, Ed.), pp. 187–192, Aca- demic Press, New York, 1970.

[16] A.N. KOLMOGOROV, On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval,

(19)

Smoothness

Title Page Contents

JJ II

J I

Go Back Close

Quit Page19of19

Ucebn. Zap. Moskov. Gos. Univ. Mat., 30 (1939), 3–13; English translation in Amer. Math. Soc. Transl., 1(4) (1949), 1–19.

[17] E. LANDAU, Einige Ungleichungen für zweimal differentzierbare Funk- tionen, Proc. London Math. Soc., 13 (1913), 43–49.

[18] E. LANDAU, Die Ungleichungen für zweimal differentzierbare Funktio- nen, Danske Vid. Selsk, Math. Fys. Medd., 6(10) (1925).

[19] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Inequalities Involv- ing Functions and Their Integrals and Derivatives, Kluwer, 1991.

[20] R. NAGEL, et al., One parameter semigroups of positive operators, Lec- ture Notes in Math., No. 1184, Springer-Verlag, Berlin, 1986

[21] A.G. RAMM, Inequalities for the derivatives, Math. Inequal. Appl., 3 (2000), 129–132.