, called also Nakano spaces, independenly for p(⋅) ∈ (1, ∞) and p(⋅) ∈ (0, 1). The case when p(⋅) ∈ [1, ∞] is also included.

vol. 55, no. 2 (2015), 119–126

Landau-type theorem for variable Lebesgue spaces

Lech Maligranda and Witold Wnuk

Summary. We describe, using elementary methods, the Köthe dual of variable Lebesgue spaces L

, called also Nakano spaces, independenly for p(⋅) ∈ (1, ∞) and p(⋅) ∈ (0, 1). The case when p(⋅) ∈ [1, ∞] is also included.

Keywords Köthe dual spaces;

associate spaces;

Nakano spaces;

variable Lebesgue spaces

MSC 2010

46E30; 46B20; 46B42 Received: 2016-02-24, Accepted: 2016-03-11

Dedicated to Professor Henryk Hudzik on the occasion of his 70th birthday.

1. Introduction

In 1907, E. Landau [15] proved the reverse Hölder inequality: if (x n y n ) ^∞ _{n =1} ∈ ℓ ¹ for all sequences (x n ) ∈ ℓ ^p (1 < p < ∞), then ( y n ) ∈ ℓ ^q , where q is the conjugate exponent of p, that is 1/ p + 1/q = 1. Landau’s original proof is based on the classical Dini’s theorem that if a n ⩾ 0 and ∑ ^∞ 1 a n = ∞, then ∑ ^∞ 1 (a n /s n ) = ∞ and ∑ ^∞ 1 (a n /s n 1+ε

) < ∞ for any ε > 0, where s n is the n-th partial sum s n = ∑

n

k =1 a k (details of his proof can be found, for example, in [3, I, pp. 529–531; 8, pp. 120–121; 12, pp. 1–2]). New proofs and generalizations were given in [5; 9; 10; 14; 16; 18; 19; 21; 23, pp. 237–238; 24–26]. Today we have more tools with which to check Landau’s theorem – the uniform boundedness principle (cf. [1, p. 179;

Lech Maligranda, Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden (e-mail: lech.maligranda@ltu.se)

Witold Wnuk, Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland (e-mail: wnukwit@amu.edu.pl)

DOI 10.14708/cm.v55i2.1119 © 2015 Polish Mathematical Society

4, p. 2524; 20, pp. 137–138]), or the general form of continuous linear functionals on ℓ ^p . In fact, Landau described the form of the Köthe dual space of ℓ ^p .

Let us recall that if E is an ideal in the space L ⁰ (µ) = L ⁰ (S , Σ, µ) of measurable func- tions with respect to the σ -finite measure space (S , Σ, µ) (i.e. E is a linear subspace, and

∣ f (s)∣ ⩽ ∣g(s)∣ µ-almost everywhere on S and f ∈ L ⁰ (µ), g ∈ E imply f ∈ E; additionally, E contains a function that is strictly positive µ-a.e.), then the Köthe dual (or associated space) E ^′ of E is defined as follows

E ^′ = {g ∈ L ⁰ (µ) ∶ f g ∈ L ¹ (µ) for all f ∈ E}.

The space E ^′ coincides with the space of order continuous functionals F on E, i.e. f α ↓ 0 ⇒ F ( f α ) → 0 (see [28, Theorem 86.3]).

Our notation is standard. Let us only explain that χ A denotes the characteristic func- tion of the set A and f ∣ A denotes the restriction of the function f to A.

2. Results

We will devote attention to a special type of spaces E called variable (exponent) Lebesgue spaces, or Nakano spaces (see [6; 7, Chapter 3; 17, Chapter 3a; 22, pp. 281–284; 23, Chap- ter XI § 89]). For a measurable function p(⋅)∶ S → (0, ∞) we define the variable Lebesgue space as follows

L ^{p (⋅)} (µ) = { f ∈ L ⁰ (µ) ∶ ∃ λ >0 I p (⋅) (λ f ) = ∫

S ∣λ f (t)∣ ^{p (t)} d µ < ∞}.

The space L ^{p (⋅)} (µ) is an F -space (i.e. a complete metrizable topological vector space) with respect to an F -norm given by the formula

∥ f ∥ = inf {λ > 0 ∶ I p (⋅) ( f /λ) ⩽ λ}.

If p(t) ∈ [1, ∞) (µ-a.e.), then the topology generated by ∥ ⋅ ∥ is equivalent to the topology associated with the Luxemburg–Nakano norm

∥ f ∥ p (⋅) = inf {λ > 0 ∶ I p (⋅) ( f /λ) ⩽ 1}. (1) Applying Orlicz’s idea from [24] with slight modifications, we can easily obtain the following result (compare [6, Proposition 2.79] and [7, Theorem 3.2.13]).

2.1. Theorem. Let p(t) ∈ (1, ∞) and let q(⋅)∶ S → (1, ∞) be such that _{p (t)} ¹ + _{q (t)} ¹ = 1 for all t ∈ S. Then the following statements are equivalent:

(i) f g ∈ L ¹ (µ) for every f ∈ L ^{p (⋅)} (µ).

(ii) g ∈ L ^{q (⋅)} (µ).

Proof. (i)⇒(ii). Since the measure space (S , Σ, µ) is σ -finite, it follows that there exists a sequence (S k ) of sets with finite positive measures and increasing to S. The function ∣g∣

determines a linear functional φ on L ^{p (⋅)} (µ) by φ( f ) = ∫ S f ∣g∣ d µ. Since φ is positive (i.e.

φ transforms nonnegative functions onto nonnegative numbers), then it is continuous (see [2, Theorem 5.19]). For fixed arbitrary η > 0, let

f (t) = (

∣g(t)∣

∥φ∥ p (⋅) + η )

q(t) p(t)

. Clearly,

q(t) p(t)

+ 1 = 1 p(t) − 1

+ 1 = p(t) p(t) − 1

= q(t), and so

1

∥φ∥ p (⋅) + η

f (t)∣g(t)∣ = (

∣g(t)∣

∥φ∥ p (⋅) + η )

q (t)

. We claim that

∫ S f (t) ^{p (t)} d µ ⩽ 1.

Indeed, otherwise we can find natural numbers N , k such that for A _{N ,k} = {t∶ f (t) ^{p (t)} ⩽ N } ∩ S k

we obtain

1 < ∫

A

f (t) ^{p (t)} d µ = ρ < ∞.

Hence, by the convexity of (⋅) ^{p (t)} ,

∫ _A

N , k

( f (t)

ρ )

p (t)

d µ ⩽ 1 ρ ∫ _A

N , k

f (t) ^{p (t)} d µ = 1, and so ∥ f χ A

∥ p (⋅) ⩽ ρ. The equality (2) implies

ρ = ∫ _A

N , k

f (t) ^{p (t)} d µ = ∫ _A

N , k

(

∣g(t)∣

∥φ∥ p (⋅) + η )

q (t)

d µ = ∫ _A

N , k

1

∥φ∥ p (⋅) + η f ∣g∣ d µ

= 1

∥φ∥ p (⋅) + η φ( f χ A

N , k

) ⩽ 1

∥φ∥ p (⋅) + η

∥φ∥ p (⋅) ∥ f χ A

N , k

∥ p (⋅) ⩽

∥φ∥ p (⋅)

∥φ∥ p (⋅) + η ρ, a contradiction because ρ > 0. Since (2) holds, then f ∈ L ^{p (⋅)} (µ), which implies

∫ S (

∣g(t)∣

∥φ∥ p (⋅) + η )

q (t)

d µ = ∫

S

1

∥φ∥ p (⋅) + η

f ∣g∣ d µ = 1

∥φ∥ p (⋅) + η

φ( f ) < ∞,

i.e. g ∈ L ^{q (⋅)} (µ). This means that (L ^{p (⋅)} ) ^′ ⊂ L ^{q (⋅)} . Moreover, ∥g∥ q (⋅) ⩽ ∥φ∥ p (⋅) + η, and so

∥g∥ q (⋅) ⩽ sup

∥ f ∥

⩽1 ∫

S ∣ f (t)g(t)∣d µ = ∥g∥ ^′ _{p (⋅)} .

(ii)⇒(i). Fix f ∈ L ^{p (⋅)} (µ) and let λ f , λ g > 0 be such that ∫ S (λ f ∣ f (t)∣) ^{p (t)} d µ < ∞ and

∫ S (λ g ∣g(t)∣) ^{q (t)} d µ < ∞. Remembering that p(t), q(t) > 1, by Young’s inequality for fixed t ∈ S, we obtain

uv ⩽ u ^{p (t)}

p(t) +

v ^{q (t)} q(t)

⩽ u ^{p (t)} + v ^{q (t)} for all u, v ⩾ 0.

Thus,

λ f λ g ∣ f (t)∣ ∣g(t)∣ ⩽ (λ f ∣ f (t)∣) ^{p (t)} + (λ g ∣g(t)∣) ^{q (t)}

for all t ∈ S, i.e. f g ∈ L ¹ (µ). Moreover, ∥ f g∥ 1 ⩽ 2∥ f ∥ p (⋅) ∥g∥ q (⋅) and so L ^{q (⋅)} ⊂ (L ^{p (⋅)} ) ^′ with

∥g∥ ^′ _{p (⋅)} ⩽ 2 ∥g∥ q (⋅) .

We can extend the definition of the space L ^{p (⋅)} for functions p taking the values in [1, ∞] or in (0, ∞]. If we consider a measurable function p(⋅)∶ S → [1, ∞], then

I p (⋅) ( f ) = ∫

S ∖P

∣ f (t)∣ ^{p (t)} d µ + ess sup

t ∈P

∣ f (t)∣,

where P _∞ = {t ∈ S ∶ p(t) = ∞}, is a convex modular (cf. [6, p. 17] and [7, p. 74]; see also [16,22,23]), and with the norm (1) the space L ^{p (⋅)} is a Banach ideal space. The first ma- thematician who allowed this possibility was probably Sharapudinov [26] (Nakano in [23]

allowed it also but for a little different modular).

It is necessary to consider the sets P ₁ = {t ∈ S ∶ p(t) = 1} and P 0 = S ∖ (P 1 ∪ P _∞ ).

If for t ∈ S we define the conjugate measurable function q(⋅)∶ S → [1, ∞] by

q(t) =

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

p (t)

p (t)−1 if t ∈ P 0 , 1 if t ∈ P _∞ ,

∞ if t ∈ P 1 ,

then Theorem 1 is still true. In fact, if L ^{p (⋅)} (µ) is generated by a function p taking values in [1, ∞], then

L ^{p (⋅)} (µ) = L ^p

^(⋅) (µ 0 ) + L ¹ (µ 1 ) + L ^∞ (µ _∞ ),

where p 0 means the restriction of p to P 0 and µ i , i ∈ {0, 1, ∞} are the restrictions of µ

to σ -algebras Σ ∩ P i = {A ∩ P i ∶ A ∈ Σ}. Hence, Theorem 2.1 and the classical result

about the form of the dual space L ¹ (µ) ^∗ , as well as the representation theorem for order continuous functionals over L ^∞ (µ _∞ ), imply that f g ∈ L ¹ (µ) for every f ∈ L ^{p (⋅)} (µ) if and only if g ∈ L ^q

^(⋅) (µ 0 ) + L ^∞ (µ 1 ) + L ¹ (µ _∞ ) = L ^{q (⋅)} (µ) (clearly q 0 denotes the restriction of q to P ₀ ).

Now we investigate the spaces L ^{p (⋅)} (µ) generated by functions p(t) ∈ (0, 1). We have to consider two cases: the atomless and the purely atomic measures µ.

2.2. Theorem. Let p(t) ∈ (0, 1) and let µ be atomless. Then the following statements are

equivalent:

(i) f g ∈ L ¹ (µ) for every f ∈ L ^{p (⋅)} (µ).

(ii) g = 0 µ-almost everywhere.

Proof. (i)⇒(ii). Let the sequence (S k ) have the same meaning as in the proof of The- orem 2.1. Put I _{n ,k} = {t ∈ S k ∶ p(t) < p n }, where p n = 1 − ¹

2 n . For every k there exists n k such that µ(I _{n ,k} ) > 0 for n ⩾ n k . If n ⩾ n k and 0 ⩽ f ∈ L ^p

(µ∣ _Σ∩I

), then

∫ I

f (t) ^{p (t)} d µ = ∫

I

∩{t∶ f (t)⩽1} f (t) ^{p (t)} d µ + ∫

I

∩{t∶ f (t)>1} f (t) ^{p (t)} d µ

⩽ µ(S k ) + ∫

I

f (t) ^p

d µ < ∞.

We have just shown that L ^p

(µ∣ _Σ∩I

) ⊂ L ^{p (⋅)} (µ). Since µ∣ _Σ∩I

is atomless and p n ∈ (0, 1) then, by the classical Day’s result, g∣ I

= 0 µ-a.e. for n ⩾ n k (see [2, Theorem 5.24]

or [16, p. 41]). Finally, g = 0 µ-a.e. because S = ⋃ ^∞ _{n ,k=1} I _{n ,k} .

The next result concerns variable Lebesgue spaces over purely atomic measures, and our proof of this result is inspired by [27, Lemma 4 and Theorem 7]. Let us recall that a measurable function is (almost everywhere) constant on an atom.

2.3. Theorem. Let p(t) ∈ (0, 1) and let µ be purely atomic. Assume that (S k ) ^∞ _{k =1} is a sequence of atoms. If µ(S k ) = u k and p(S k ) = p k (k = 1, 2, . . .), then the following statements are equivalent:

(i) f g ∈ L ¹ (µ) for every f ∈ L ^{p (⋅)} (µ).

(ii) sup _{k ∈N} ∣g(S k )∣ ^p

u k p

k

−1

< ∞.

Proof. (i)⇒(ii). Put b k = ∣g(S k )∣ and suppose, to the contrary of (ii), that b

p

n

u n

p

k

−1

> k ² for some subsequence (n k ).

Let

f (S n

) = 1 b n

u n

and f (S j ) = 0 for j ∉ {n 1 , n ₂ , . . . }.

Then

∫ S

f (t)

p (t)

d µ =

∞

∑

k =1

f (S n

) ^p

u n

=

∞

∑

k =1

1 b n

p

k

u n

p

k

−1 <

∞

∑

k =1

1 k ²

< ∞,

i.e. f ∈ L ^{p (⋅)} (µ), but ∣ f (S n

)g(S n

)∣µ(S n

) = f (S n

)b n

u n

= 1 for all k, and hence f g ∉ L ¹ (µ), a contradiction.

(ii)⇒(i). Let sup _{k ∈N} ∣g(S k )∣ ^p

u k p

−1

< M < ∞ and fix f ∈ L ^{p (⋅)} (µ). Put x k = ∣ f (S k )∣. We can choose a natural number j such that

∫ S ( 1 j

f (t))

p (t)

d µ = ∑

k

( 1 j

x k )

p

u k <

1 M

. Therefore,

( 1

j x k )

p

∣g(S k )∣ ^p

u k p

= ( 1 j

x k )

p

u k ∣g(S k )∣ ^p

u k p

−1

< 1.

Hence, ¹ _j x k ∣g(S k )∣u k < 1, and we obtain 1

j

x k ∣g(S k )∣u k ⩽ ( 1

j x k )

p

∣g(S k )∣ ^p

u k p

= ( 1

j x k )

p

u k ∣g(S k )∣ ^p

u k p

−1

⩽ M(

1 j

x k )

p

u k . Thus

∫ S ∣ f (t)g(t)∣ d µ = j ∑

k

1 j

x k ∣g(S k )∣u k ⩽ jM ∑

k

( 1

j x k )

p

u k < j, i.e. f g ∈ L ¹ (µ).

If (S , Σ, µ) is a σ -finite measure space, then there exists a sequence (A k ) of µ-atoms such that for S ₀ = ⋃ k A k the measure µ∣ _{Σ ∩(S∖S}

0

) is atomless, or else µ(Σ ∩ (S ∖ S ₀ )) = {0}.

Considering a Σ-measurable function p(⋅)∶ S → (0, ∞), we can separate three sets:

P _<1 = {t ∈ S ∶ p(t) < 1}, P ₁ = {t ∈ S ∶ p(t) = 1}, P _>1 = {t ∈ S ∶ p(t) > 1}.

Putting together the theorems presented above we obtain the following result.

2.4. Theorem. Let p(⋅)∶ S → (0, ∞) be a Σ-measurable function. For a σ -finite measure µ

the following statements are equivalent:

(i) f g ∈ L ¹ (µ) for every f ∈ L ^{p (⋅)} (µ).

(ii) g χ P

∈ L ^{q (⋅)} (µ∣ _Σ∩P

), where q(⋅)∶ P _>1 → (1, ∞) satisfies _{p (t)} ¹ + _{q (t)} ¹ = 1 for t ∈ P _>1 ; mo- reover, g χ P

∈ L ^∞ (µ∣ _Σ∩P

) and sup _k ∣g(A k ∩ P _<1 )∣ ^{p (A}

^∩P

⁾ µ(A k ∩ P _<1 )

p (A

∩P

)−1

<

∞ and g∣ _(P

_)∖S

= 0 µ-a.e.

Finally, we should mention that quite recently in a Kamińska–Kubiak paper [13, Ap- pendix A] appeared the proof of the Köthe duality of the Musielak–Orlicz spaces L ^M (Mu- sielak–Orlicz spaces include Orlicz spaces and variable Lebesgue spaces) in the case when the function M∶ S × [0, ∞) → [0, ∞] is convex in the second variable, which means that the space L ^M is a Banach space. The proof in [13] uses a Hudzik–Maligranda [11] result on Orlicz functions characterizing equality in Young’s inequality.

Acknowledgements Witold Wnuk acknowledges the support of The National Center of

Science, Poland, Grant no. N N 201 605340.

References

[1] A. Alexiewicz, Functional Analysis, Monografie Mat., vol. 49, PWN, Warszawa 1969; in Polish.

[2] C. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, vol. 105, American Mathematical Sociaty, Providence 2003.

[3] N. K. Bary, A Treatise on Trigonometric Series, Vol. I, II, Pergamon Press, New York 1964.

[4] J. Berger and D. Bridges, A constructive study of Landau’s summability theorem, J. UCS 16 (2010), no. 18, 2523–2534, DOI 10.3217/jucs-016-18-2523.

[5] Z. W. Birnbaum and W. Orlicz, Über die Verallgemeinerung des Begriffes der zueinander konjugierten Po- tenzen, Studia Math. 3 (1931), 1–67; reprinted in Władysław Orlicz. Collected Papers, PWN, Warszawa 1988, 133–199.

[6] D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Birkhäu- ser, Basel 2013.

[7] L. Diening, P. Harjulehto, P. Hästö, and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math., vol. 2017, Springer 2011.

[8] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press 1988.

[9] G. Helmberg, A construction concerning (l

)

⊂ l

, Amer. Math. Monthly 111 (2004), no. 6, 518–520, DOI 10.2307/4145070.

[10] G. Helmberg, A Landau-type construction concerning (L

)

⊂ L

, Indag. Math. (N.S.) 17 (2006), no. 2, 243–249, DOI 10.1016/S0019-3577(06)80019-4.

[11] H. Hudzik and L. Maligranda, Amemiya norm equals Orlicz norm in general, Indag. Math. (N.S.) 11 (2000), no. 4, 573–585, DOI 10.1016/s0019-3577(00)80026-9.

[12] S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, Monogr. Mat., vol. 6, Warsaw 1935.

[13] A. Kamińska and D. Kubiak, The Daugavet property in the Musielak–Orlicz spaces, J. Math. Anal. Appl.

427 (2015), no. 2, 873–898, DOI 10.1016/j.jmaa.2015.02.035.

[14] O. Kováčik and J. Rákosník, On spaces L

and W

, Czechoslovak Math. J. 41 (1991), no. 4, 592–618.

[15] E. Landau, Über einen Konvergensatz, Göttingen Nachrichten 8 (1907), 25–27.

[16] L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Math., vol. 5, Univ. of Campinas, Campinas 1989.

[17] L. Maligranda, Hidegoro Nakano (1909–1974) – on the centenary of his birth, Proc. of the Third Internat.

Symp. on Banach and Function Spaces (ISBFS2009) (Kitakyushu, Japan, 14), Banach and Function Spaces

III (M. Kato, L. Maligranda, and T. Suzuki, eds.), Yokohama Publishers, 2011, 99–171.

[18] L. Maligranda and L. E. Persson, Generalized duality of some Banach function spaces, Indag. Math. 51 (1989), no. 3, 323–338, DOI 10.1016/S1385-7258(89)80007-1.

[19] L. Maligranda and W. Wnuk, Landau type theorem for Orlicz spaces, Math. Z. 208 (1991), 57–64, DOI 10.1016/s1385-7258(89)80007-1.

[20] J. Musielak, An Introduction to Functional Analysis, PWN, Warszawa 1976; in Polish.

[21] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Springer-Verlag, Berlin 1983.

[22] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo 1950.

[23] H. Nakano, Topology of Linear Topological Spaces, Maruzen, Tokyo 1951.

[24] W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200–211; reprinted in Władysław Orlicz, Collected Papers, PWN, Warszawa 1988, 200–211.

[25] S. G. Samko, Differentiation and integration of variable order and the space L

, Operator Theory for Complex and Hypercomplex Analysis (Mexico City, 1994), Contemp. Math., vol. 212, American Mathe- matical Sociaty, Providence, 1998, 203–219, DOI 10.1090/conm/212/02884.

[26] I. I. Sharapudinov, The topology of the space L

([0, 1]), Mat. Zametki 26 (1979), no. 4, 613–632; English translation: Math. Notes 26 (1979), no. 3–4, 796–806 (1980).

[27] S. Simons, The sequence spaces ℓ (p

) and m(p

), Proc. London Math. Soc. (3) 15 (1965), 422–436, DOI 10.1112/plms/s3-15.1.422.

[28] A. Zaanen, Riesz Spaces II, North-Holland, New York Oxford 1983.

© 2015 Polish Mathematical Society