Mieczysław Mastyło and Julian Musielak

vol. 55, no. 2 (2015), 45–78

Henryk Hudzik – vita et opera

Mieczysław Mastyło and Julian Musielak

Vita

It is the end of winter in 1945. World War Two is drawing to a close in Europe, and Hitler’s Grossdeutschland is dying amid the roar of cannons and the howl of bombs of the allied American, British, and Soviet armies. There are three weeks left till the unconditional Ger- man surrender. In these historic days, on March 16, 1945, a baby son is born to Marcin and Jadwiga Hudzik, slave farm laborers in exile near Potsdam, and they give him the name of Henryk. Taking advantage of the first opportunity, they return to Poland and start working on a farm. After completing primary school, young Henryk, whose mother has just died, for two years helps his father on the farm and in raising his two younger sisters. In 1960 he starts learning to become a carpenter, and in 1967 he completes his se- condary education. The same year, feeling a proclivity towards mathematics, he becomes a student at the Department of Mathematics, Physics, and Chemistry of the Adam Mickie- wicz University in Poznań. Barely in his first year, he was attending my lectures, and later, my seminars. He wrote his master’s thesis Modular and Countably Modular Spaces under my supervision. This work was highly evaluated, and Henryk Hudzik received his degree with honors in 1972.

During our meetings at the seminar, I could not but observe his growing scientific ac- tivity, and I decided to ask the director of the Faculty of Mathematics, professor Andrzej Alexiewicz, to employ Henryk Hudzik at the Faculty of Mathematics, more specifically, in my Chair of the Theory of Real-Valued Functions. Soon he was affected to the Gra- duate School at the Faculty of Mathematics of the Adam Mickiewicz University, which he completed under my supervision. In 1977, he received his PhD for the dissertation

Mieczysław Mastyło, Faculty of Mathematics and Computer Sciences, Adam Mickiewicz University in Poznań, ul. Umultowska 87, 61-614 Poznań, Poland (e-mail: mastylo@amu.edu.pl)

Julian Musielak, Faculty of Mathematics and Computer Sciences, Adam Mickiewicz University in Poznań, ul. Umultowska 87, 61-614 Poznań, Poland (e-mail: musielak@amu.edu.pl)

DOI 10.14708/cm.v55i2.1245 © 2015 Polish Mathematical Society

During the school time, Henryk Hudzik first from the right

On the Generalized Orlicz–Sobolev Spaces, which won him the first prize in his scientific career, the Award of the Minister for Science and Higher Education of Third Degree.

He becomes an assistant in the Chair of the Theory of Real-Valued Functions and continues to investigate the Orlicz–Sobolev spaces, to which he devoted some more papers at the time. In those years, on the initiative of assistants in the Chair of the Theory of Real-Valued Functions, namely, doctors Henryk Hudzik, Anna Kamińska, and Ryszard Urbański, a new, “private” seminar was launched, which was not paid by the university, and which was a source of a number of papers. I will mention, in particular, the joint papers of Hudzik and Urbański concerning the Riesz–Thorin theorem and the interpolation of operators in generalized Orlicz spaces.

For some time an idea had been brewing to make Poznań a venue for systematic in-

ternational science conferences. It called for considerable involvement, in particular, from

on the part of the members of the Chair of the Theory of Real-Valued Functions; nothing

could have been achieved without the activity of doctors Henryk Hudzik, Anna Kamiń-

ska, and Ryszard Urbański, as well as many others, including mathematicians from the

Poznań University of Technology. The first conference, under the name Function Spaces I,

was organized in Poznań on August 24–30, 1986. Conferences under this name are held

every three years, partly also in other cities (Kraków, Wrocław, Zielona Góra). A deep

involvement of Henryk Hudzik in the organizing activities is a key element here. Let me

From the left Henryk Hudzik, Władysław Orlicz, Julian Musielak

say a few words about Function Spaces II, which was held in late August and early Septem-

ber of 1989. Without the assistance and dedication of professor by then Henryk Hudzik

and many other colleagues this conference could not possibly have taken place. It was held

at a time of complete economic chaos accompanying the political transformation: Poland

was just leaving the “socialist bloc” in Europe. The communist government was still in pla-

ce, but it did its best to destabilize the country before transferring power. In spite of this,

however, actually the situation was favorable for us. We could do what we pleased, neither

the communist party nor the authorities were in the least interested in us. But we also had

to take care of the money to finance the conference by ourselves. The hyperinflation came

to the rescue: the participation fees of a few guests from “the West”, which were ridiculously

low for them, went a long way towards putting in order the conference’s finances.

Meanwhile the scientific development of Henryk Hudzik shows no signs of slowing down. He makes numerous scientific contacts, both domestic and foreign, resulting in common research papers, in which he plays a leading role, and the leitmotif is usually the geometry of Banach spaces, and in particular of Orlicz spaces and their various ge- neralizations. This is also the problematic of his dissertation Geometric and Topological Properties of Orlicz and Musielak–Orlicz Spaces, for which he was awarded professorship in 1986. Already at that point he is the author of 35 scientific publications.

It was clear to me by that time that Henryk Hudzik needed a broader scope for action also in the organizational sense, i.e. he needed to have his own chair within the Faculty of Mathematics at the Adam Mickiewicz University. I was not a partisan of oversize chairs, so I decided to have an informal talk with the new professors (and there were quite a few of them by that time). To make them feel at ease and under no pressure I invited the new professors of the Chair of the Theory of Real-Valued Functions to a café in Głogowska street in Poznań. Over coffee and ice-cream I first proposed to start addressing each other by first names rather than family names. Next I proposed a fundamental organizational transformation consisting in creating a few independent chairs out of the Chair of the Theory of Real-Valued Functions. As it happens, professors Henryk Hudzik and Ryszard Urbański, who already had disciples in their fields, accepted my proposal to organize new chairs out of the staff of the Chair of the Theory of Real-Valued Functions. Some further deliberations brought up the idea to create the Chair of Function Spaces and the Chair of Optimization and Control Theory with prof. Hudzik and prof. Urbański as their respective heads. Among other staff, prof. Magdalena Jaroszewska and prof. Anna Kamińska, as well as dr. Stanisław Stoiński, who was on the point of attaining professorship, and dr. Tomasz Kubiak and Leszek Skrzypczak, decided to stay in Chair of the Theory of Real-Valued Functions. Dr. Marian Nowak became member of the Chair of Function Spaces, and dr Marek Wisła and dr Tomasz Kubiak of the Chair of Optimization and Control Theory.

There were a certain administrative hurdles to overcome in order to carry through all the changes, which required the approval of the Board of the Faculty, the Board of the Depart- ment, and the Senate of the Adam Mickiewicz University, but we got what we wanted.

The creation of the new chairs together with their seminars proved very fruitful. The new heads of chairs, especially prof. Hudzik, soon started to promote doctors in their special fields. Our chairs collaborated closely, both scientifically and organizationally, and I felt that I was not being an obstacle to the development of young talents.

As to professor Henryk Hudzik, his becoming independent opened a period of fur- ther spectacular scientific development – after 10 years the number of his publications went up to 86, after 10 more years to 165, and after another 10 years to a staggering 206. He has become an undisputed authority on an international scale in his field.

Julian Musielak

Opera

Henryk Hudzik has been a very active researcher. He has published more than 200 artic- les. His results are cited in 22 monographs. His papers deal mainly with the geometry of Banach spaces, analysis, the approximation theory, and the fixed point theory. He has also published results showing relationships between functional analysis and other branches of mathematical analysis such as PDEs and integral equations. He has supervised 15 doctoral dissertations, 3 of whose authors are from China.

Below we describe some of the contributions of Henryk Hudzik to the areas men- tioned above. It is not possible in this short note to present even a small fraction of his results. We provide only basic information on Henryk Hudzik’s numerous works, giving only some references to his other papers. The interested reader may easily localize all the papers in the enclosed list of all Henryk Hudzik’s publications.

Orlicz spaces and generalized Orlicz spaces called Musielak–Orlicz spaces are of spe- cial importance in the theory of Banach function spaces. Henryk Hudzik contributed si- gnificantly to the geometry of these spaces. In order to present some of his results, we recall fundamental notions from Musielak–Orlicz spaces.

Let (Ω, Σ, µ) be a measure space. The space of all real-valued measurable and µ-almost everywhere finite functions is denoted by L ⁰ (Ω, Σ, µ), or L ⁰ (µ) for short (as usual, func- tions equal µ-almost everywhere are identified).

A function Φ ∶ [0, ∞) × Ω → [0, ∞] is said to be a Musielak–Orlicz function if it satisfies the following conditions:

(i) For every t ∈ Ω the function Φ(⋅, t)∶ [0, ∞) → [0, ∞] is convex, left continuous, continuous at zero, and Φ (u, t) = 0 if and only if u = 0.

(ii) For every u ∈ [0, ∞) the function Φ(u, ⋅)∶ Ω → [0, ∞] is Σ-measurable.

Every Musielak–Orlicz function Φ generates a functional I Φ ∶ L ⁰ (µ) → [0, ∞] defi- ned by the formula

I Φ ( f ) = ∫ _Ω Φ (∣ f (t)∣, t) dµ, f ∈ L ⁰ (µ).

The subspace L _Φ (Ω, Σ, µ) (L Φ (Ω), L Φ (µ) or L Φ for short) given by L _Φ (µ) = { f ∈ L ⁰ (Ω, Σ, µ) ∶ I Φ (λ∣ f ∣) < ∞ for some λ > 0}

is called the Musielak–Orlicz space. A Musielak–Orlicz space L _Φ (µ) is a Banach lattice on (Ω, Σ, µ) considered with the Luxemburg norm

∥ f ∥ ^Φ = inf{λ > 0 ∶ I ^Φ ( f /λ) ⩽ 1}

and with the Orlicz norm

∥ f ∥ ⁰ Φ = sup{∣∫ _Ω f g d µ ∣ ∶ I ^Φ

(g) ⩽ 1}, f ∈ L ^Φ (µ),

where Φ ^∗ is the Musielak–Orlicz function complementary to Φ in the sense of Young, that is,

Φ ^∗ (u, t) = sup{uv − Φ(v, t) ∶ v ⩾ 0}, (u, t) ∈ [0, ∞) × Ω.

It is well known that for any Musielak–Orlicz function Φ we have the following Amemyiya formula for the Orlicz norm

∥ f ∥ ⁰ Φ = inf

k >0

1

k (1 + I ^Φ (k f )), f ∈ L ^Φ (µ).

If µ is purely atomic and takes the value one on every atom, then the Musielak–Orlicz space associated with such a measure is denoted by ℓ Φ (Γ), where Γ is a set having the same cardinality as the set of all atoms of µ. In the case Γ = N we write ℓ ^Φ for short.

If Φ (u, t) = ϕ(u) for all (u, t) ∈ [0, ∞) × Ω, then Φ is called an Orlicz function and the space L ϕ (µ) is called the Orlicz space and is denoted by L ^ϕ . For every Musielak–

–Orlicz function Φ we define E Φ (Ω, Σ, µ) (E ^Φ (µ) or E ^Φ for short) to be the space of all f ∈ L ^Φ (µ) such that I ^Φ (λ f ) < ∞ for every λ > 0. Clearly, E ^Φ (µ) is a closed ideal in L Φ (µ) and the norm ∥ ⋅ ∥ ^L

restricted to L Φ (µ) is order continuous. If Φ takes only finite values, then E _Φ (µ) = (L ^Φ ) ^a (µ) where (L ^Φ ) ^a (µ) = { f ∈ L ^Φ (µ) ∶ ∣ f ∣ ⩾ f ⁿ ↓ 0, implies ∥ f ⁿ ∥ ^L

→ 0}. If µ is purely atomic and takes the value one on every atom, then we write h _Φ instead of E _Φ (µ).

We say that an Orlicz function Φ satisfies the ∆ ₂ -condition (∆ ⁰ ₂ -condition) [∆ ^∞ ₂ -con- dition] if there is K > 0 such that the inequality Φ(2u) ⩽ KΦ(u) holds for all u > 0 (for all u > 0 satisfying u ⩾ u 0 with some u ₀ > 0 such that Φ(u 0 ) < ∞) [for all 0 < u ⩽ u 0 with some u 0 > 0 such that Φ(u ⁰ ) > 0].

We note that if Φ satisfies the ∆ ⁰ ₂ -condition (resp., ∆ ^∞ ₂ -condition and µ (Ω) < ∞), then h Φ = ℓ ^Φ (resp., E Φ (µ) = L ^Φ (µ)). If (Ω, Σ, µ) is an arbitrary measure space and Φ satisfies the ∆ 2 -condition, then E Φ (µ) = L ^Φ (µ).

1. Sobolev spaces

One of the first fields that attracted Henryk Hudzik’s interest was the theory of Sobolev spaces. Sobolev spaces have been generalized in many different ways, including Orlicz–

–Sobolev or Musielak–Orlicz–Sobolev spaces. Henryk Hudzik’s early papers deal with Orlicz–Sobolev and Musielak–Orlicz–Sobolev spaces (see [H1, H2, H3, H4, H5]).

We recall that given k ∈ N, a Musielak–Orlicz function Φ, and a domain Ω ⊂ R ⁿ , the Orlicz–Sobolev space W ^k

Φ (Ω) is defined by

W _Φ ^k (Ω) = { f ∈ L ⁰ (Ω, Σ, m ⁿ ) ∶ ∀ ∣α∣ ⩽ k, D ^α f ∈ L ^Φ (m ⁿ )},

where (Ω, Σ, m ⁿ ) is the Lebesgue measure space and for a multi-index α = (α ¹ , . . . , α n ) ∈

N ⁿ 0 we put ∣α∣ = α ¹ + . . . + α ⁿ . Here the derivatives of f ∈ L ^Φ are understood in the

distribution sense, that is, D ^α f is such a Lebesgue measurable function on Ω that for any infinitely differentiable in the usual sense function φ ∶ Ω → R with compact support in Ω the following equality is satisfied

∫ Ω

D ^α f (t)φ(t) dt = (−1) ^{∣α ∣} ∫

Ω

f (t)D ^α φ (t) dt, where

D ^α φ = ∂ ^α φ

∂ ^α

t 1 . . . ∂ ^α

t n

is the usual mixed derivative of φ.

Sobolev spaces play an important role in various areas of modern analysis, in parti- cular the theory of nonlinear partial differential equations (see [1]).

For a fixed 1 < p < ∞ we define on W Φ ^k (Ω) the norm

∥ f ∥ ^{k , Φ} = ( ∑

∣α ∣⩽k

∥D ^α f ∥ Φ ^p ) ^{1/ p} , f ∈ W Φ ^k (Ω).

Then (W Φ ^k (Ω), ∥ ⋅ ∥ k , Φ ) is a Banach space called the Musielak–Orlicz–Sobolev space.

In [H170], monotonicity properties of (W Φ ^k (Ω), ⩽) with a partial order given by f ⩽ g ⇔ ∀ ∣α∣ ⩽ k, D ^α f ⩽ D ^α g , f , g ∈ W Φ ^k (Ω)

are studied. There are given criteria for strict monotonicity, upper (lower) locally uniform monotonicity, and uniform monotonicity of Orlicz–Sobolev spaces with the Luxemburg norm. Some applications to the problem of a best approximation are also presented.

Critical imbeddings of Sobolev spaces have attracted a lot of attention in recent years.

We mention here Trudinger’s theorem on the limiting imbedding [35], which states that the Sobolev space W _p ^m (Ω), where Ω ⊂ R ^N is a bounded domain with sufficiently smooth boundary, 1 < p < ∞, and mp = N, is imbedded into the Orlicz space L Φ (Ω) with Φ (t) = exp t ^{N /(N −m)} − 1.

We wish to point to the paper by Edmunds, Hudzik, and Krbec [H187], which studies the relationships between weighted and non-weighted exponential spaces, and show their role as a target spaces of the critical imbedding of Sobolev spaces. Namely, the authors consider the weighted Orlicz spaces L exp t

(Ω, ρ) with α > 0 on a bounded domain Ω in R ^N , generated by the Orlicz functions Φ α (t) = exp t ^α − 1 for all t ⩾ 0, and prove that L exp t

(Ω) = L ^{exp t}

(Ω, ρ) if and only if ρ ∈ L ^q (Ω) for some q > 1. This result is applied to prove that if ρ is a radial, non-increasing, and integrable weight function with ρ ⩾ 1 a.e. on a bounded domain in R ^N with a sufficiently smooth boundary, then the continuous inclusion W _N ¹ (Ω) ↪ L exp t

(Ω, ρ) implies that L exp t

(Ω) = L exp t

(Ω, ρ), where N ^′ = N/(N − 1). This result shows that there exists no effective improvement on the standard target space L

exp t

(Ω).

2. Geometry of Banach spaces

A next subject of Hudzik’s interest was the geometry of Banach spaces. First we need to recall some definitions and notions.

Let X be a Banach space with dim X ⩾ 2. Geometric properties of X are determined by the unit sphere S X and the closed unit ball B X . X is said to be uniformly non-square in the sense of James if there exists δ > 0 such that

min {∥x + y∥, ∥x − y∥} ⩽ 2(1 − δ), x, y ∈ S ^X . The modulus of convexity δ X of X is defined by

δ X (ε) = inf{1 − ∥ ^{x + y} ₂ ∥ ∶ x, y ∈ S ^X , ∥x − y∥ ⩾ ε}, ε ∈ [0, 2].

It is well known that the function δ X is continuous on [0, 2) and nondecreasing on the inte- rval [0, 2], but it need not be convex (see [ 25]). δ X is increasing on the interval [ε ⁰ (X), 2], where ε 0 (X) is the characteristic of convexity of X defined by

ε 0 (X) = inf{ε ∈ (0, 2] ∶ δ ^X (ε) > 0}.

A Banach space X is called uniformly convex if δ X (ε) > 0 for all 0 < ε ⩽ 2, i.e. ε 0 (X) = 0.

We stress that these are significant concepts in the theory of Banach spaces. James [17]

proved that uniformly-non-square Banach spaces are reflexive. Later James [18] proved that a uniformly-non-square Banach space has an equivalent uniformly non-square norm.

These interesting facts are connected with the concept of superreflexivity important to the local theory of Banach spaces. We recall that a Banach space X is said to be super- reflexive if any Banach space Y which is finitely representable in X (i.e. for any ε > 0 every finite-dimensional subspace of Y is (1 + ε)-isomorphic to a subspace of X) is itself reflexive.

We note that James [18] also proved that a Banach space is superreflexive if and only if it has an equivalent uniformly non-square norm. This, combined with Enflo’s result [10], yields the well-known result that for every Banach space X the following conditions are equivalent:

(i) X is superreflexive.

(ii) X has an equivalent uniformly non-square norm.

(iii) X has an equivalent uniformly convex norm.

It should be added that it is difficult to provide a formula for the modulus of convexity or even its asymptotic behavior. The characteristic of convexity has important applications in the geometric fixed point theory. In [13] it was proved that any uniformly non-square Banach space has the fixed point property.

We recall that a Banach space X is said to have the fixed (resp., the weak fixed) po-

int property if for any nonempty closed, convex, and bounded (resp., nonempty closed,

convex, and weakly compact) subset A of X and any nonexpansive mapping P ∶ A → A, i.e.

satisfying

∥Px − Py∥ ⩽ ∥x − y∥, x, y ∈ A,

there exists a fixed point x ₀ ∈ A, that is, a point x 0 ∈ A such that Px 0 = x 0 .

Undoubtedly, these deep results motivated Hudzik in the study of the geometric pro- perties of Banach spaces which resulted in valuable contributions described below.

3. Selected topics in the geometry of Köthe–Bochner spaces

Let E be a Banach function lattice on a measure space (Ω, Σ, µ) and let E(X) be the Köthe–

–Bochner space of all strongly measurable functions x ∶ Ω → X with [x](⋅) ∶= ∥x(⋅)∥ ^X ∈ E and the norm ∥x∥ E ( X) = ∥[x]∥ ^E .

We note that Day [7] showed that L _p (X)is uniformly convex if and only if 1 < p < ∞ and X is uniformly convex. Then, Smith and Turett [33] proved that L _p (X) is uniformly non-square if and only if 1 < p < ∞ and X is uniformly non-square. Further, Dowling and Turett [8] generalized these results to the formula ε ₀ (L p (X)) = max{ε 0 (L p ), ε 0 (X)}. Fi- nally, Hudzik [H40] proved for the Orlicz–Bochner space L _Φ (X) over an infinite non-ato- mic measure space that ε 0 (L ^Φ (X)) = max{ε ⁰ (L ^Φ ), ε ⁰ (X)} provided that Φ is a uniform- ly convex function. A remarkable result due to Halperin [15] states that the Köthe–Bochner space E (X) is uniformly convex if both spaces E and X are uniformly convex.

Hudzik and Landes [H49] proved the following result.

3.1. Theorem. The characteristic of convexity of any Köthe–Bochner space E(X) satisfies the following estimates

max {ε 0 (E), ε ⁰ (X)} ⩽ ε ⁰ (E(X)) ⩽ ε ⁰ (E) + ε ⁰ (X) − 2 ¹ ε ₀ (E)ε ⁰ (X). (1) This result implies the following fact which covers all of the above-mentioned results.

3.2. Corollary.

(i) E (X) is uniformly convex if and only if both E and X are uniformly convex.

(ii) E (X) is uniformly non-square if both E and X are uniformly non-square.

Below we use the standard symbols ∨ and ∧ to denote the lattice operations. Follo- wing [H49], we define the binary operator ∗ on [0, 2] by a ⋆ b = a + b − ¹ ₂ ab for all a, b ∈ [0, 2].

In [H49], it was shown that the estimates in the above theorem are optimal. More precisely, it is proved that

(i) For all α, η ∈ (0, 2) and ε ∈ (η ∨ α, η ⋆ α), there is a Banach function lattice E with

ε ₀ (E) = η such that ε ⁰ (E(X)) = ε whenever ε ⁰ (X) = α.

(ii) For every η ∈ (0, 2), there is a Banach function space E with ε ⁰ (E) = η such that ε ₀ (E(X)) = η whenever ε 0 (X) ⩽ ₂ ¹ η ² .

(iii) If ε ₀ (E) ⩾ 1 or ε 0 (X) ⩾ 1, then the upper bound in ( 1) can be attained, namely:

For every η ⩾ 1, there is a function lattice E with ε 0 (E) = η such that ε 0 (E(X)) = ε ₀ (E) ⋆ ε 0 (X) for every X.

One of the most important geometric notions is the (β)-property introduced by Rolewicz in [32]. Recall that for any subset C of a Banach space X , the Kuratowski me- asure of C is the infimum α (C) of those ε > 0 for which there is a covering of C by a finite number of sets of diameter less than ε.

A Banach space X is said to have the (β)-property if for every ε > 0 there exists δ > 0 such that

α (conv({x} ∪ B ^X ) ∖ B ^X )) < ε whenever 1 < ∥x∥ < 1 + δ.

We note that for applications the following equivalent form of the (β)-property pro- ved by Kutzarova [24] is useful: a Banach space has the (β)-property if and only if for every ε > 0 there exists δ > 0 such that for every x ∈ B ^X and every sequence in B _X with sep ({x ⁿ }) ∶= inf{∥x ⁿ − x ^m ∥ ∶ n ≠ m} ⩾ ε, there exists k ∈ N such that ∥x + x ^k ∥ ⩽ 2(1− δ).

Rolewicz [32] showed that the (β)-property follows from uniform convexity and that the (β)-property implies nearly uniform convexity. Recall, following Huff [ 16], that a Ba- nach space X is said to be nearly uniformly convex if for every ε > 0 there exists δ ∈ (0, 1) such that for every sequence {x ⁿ } in B ^X with sep ({x ⁿ }) > ε, we have

conv ({x ⁿ }) ∩ (1 − δ)B ^X ≠ ∅.

The importance of the (β)-property resides in the fact that if a Banach space X has the (β)-property, then both X and X ^∗ have the fixed point property.

In their joint papers, Hudzik and Kolwicz studied the (β)-property in Köthe–Bochner spaces. To present the results, we will need the notion of orthogonal uniform convexity in- troduced in [22]: A Banach function lattice E on a measure space is said to be orthogonally uniform convex if for every ε > 0 there exists δ > 0 such that for any x, y ∈ B ^X the inequality

∥x χ ^A

∥ ^E ∨∥yχ ^A

∥ ^E ⩾ ε implies that ∥x + y∥ ^E ⩽ 2(1−δ), where A ^{x y} = supp(x)÷supp(y) is the symmetric difference of supp (x) and supp(y).

The main results from [H140] can be stated as follows.

3.3. Theorem. Let E be a Banach sequence lattice and X an infinite-dimensional Banach space. Then E (X) has the (β)-property if and only if X has the (β)-property and E is or- thogonally uniformly convex.

3.4. Theorem. Let E be a symmetric Banach sequence space and X a Banach space. Then

the following statements are equivalent:

(i) E (X) has the (β)-property.

(ii) E and X have the (β)-property.

(iii) E is orthogonally uniformly convex and X has the (β)-property.

4. Selected topics from the geometry of Orlicz spaces

In modern Banach space theory, spaces with special properties are considered to be of powerful interest in applications. We will discuss a class of Banach spaces important in probability theory, namely the B-convex spaces. This class of spaces was studied by Hudzik as well. To state some of his results in this area we will need some definition and notions.

A Banach space X is said to be uniformly non-ℓ ⁽¹⁾ _n (n ∈ N, n ⩾ 2) if there exists ε > 0 such that for any elements x ₁ , . . . , x _n ∈ B X and some choice of signs, we have

∥x ¹ ± . . . ± x ⁿ ∥ ⩽ n(1 − ε).

A remarkable result due to Beck [4] provides a characterization of Banach spaces X for which the strong law of large numbers for random variables which take values in X holds true: A Banach space X is uniformly non-ℓ ^{(1) n} for some integer n ⩾ 2 if and only if for every sequence {X ⁱ } of independent random X -variables with E(X ⁱ ) = 0 for each i ∈ N and sup

i ⩾1 Var (X i ) < ∞, we have 1 n

n

∑

i =1

X i → 0 strongly almost surely.

This result motivates the study of uniformly non-ℓ ⁽¹⁾ _n Banach spaces for some integer n ⩾ 2, which in literature are called the B-convex spaces.

In what follows we consider only σ -finite measure spaces. In the paper [H26], Hudzik gives some criteria for Orlicz spaces with the Luxemburg norm to be uniformly non-ℓ ⁽¹⁾ n

which are simpler than Sundaresan’s [34].

4.1. Theorem. Let Φ be an Orlicz function. The Orlicz space L ^Φ (µ) on a measure space (Ω, Σ, µ) is uniformly non-ℓ ⁽¹⁾ n if and only if the following conditions are satisfied:

(i) Φ satisfies the ∆ ₂ -condition and the following inequality for some σ ∈ (0, 1) Φ ( u

n ) ⩽ σ Φ (u) n

(2) for every u ⩾ 0 if µ is a non-atomic and infinite measure.

(ii) Φ is finite, satisfies the ∆ ^∞ ₂ -condition and inequality (2) for all u ⩾ Φ ⁻¹ (n/µ(Ω)) =

sup {v ⩾ 0 ∶ Φ(v) ⩽ n/µ(Ω)}, if µ is a non-atomic and finite measure.

(iii) Φ satisfies the ∆ ⁰ ₂ -condition and inequality (2) in some interval [0, u ⁰ ], where u ⁰ > 0, if µ is purely atomic with a countably infinite number of atoms of measure 1 and Φ (c) = 1 for c > 0.

In [H34] Hudzik, Kamińska, and Musielak initiated a novel approach to the study of the characteristic of convexity of Orlicz spaces. In this, for a given real-valued Orlicz function Φ vanishing only at zero the following parameter p (Φ) plays an essential role

p (Φ) = sup {a ∈ (0, 1) ∶ sup

u>0

f _a (u) < 1},

where for any a ∈ (0, 1)

f _a (u) = 2Φ ((u + au)/2)

Φ (u) + Φ(au) , u > 0.

Let us state here the interesting result from [H34].

4.2. Theorem. Assume that Φ is a real-valued Orlicz function vanishing only at zero and let L _Φ be an Orlicz space on a non-atomic and infinite measure space, equipped with the Luxemburg norm. Then

(i) ε 0 (L ^Φ ) = 2 if Φ does not satisfy the ∆ ² -condition.

(ii) ε 0 (L ^Φ ) = ^{2(1− p(Φ))} _{1+ p(Φ)} if Φ does satisfy the ∆ 2 -condition.

Interesting immediate consequences stem from the above result.

(i) L Φ (µ) is uniformly convex if and only if Φ satisfies the ∆ ² -condition and p (Φ) = 1, that is, there is a ∈ (0, 1) such that sup a>0 f a (u) < 1 (see [ 20]).

(ii) L Φ (µ) is uniformly non-square if and only if Φ satisfies the ∆ ² -condition and p (Φ) >

0, this is, there is a ∈ (0, 1) such that sup a>0 f _a (u) < 1. This means that the space L ^Φ is reflexive (see [H26]).

These results can be applied to the study of the normal structure of Orlicz spaces.

We recall that a Banach space has the (weak) normal structure if for every non-singleton (weakly compact) nonempty bounded, closed, and convex subset C of X , there exists x ∈ C such that

r _c (x) ∶= sup{∥x − y∥ ∶ y ∈ C} < diam(C) ∶= sup{∥u − v∥ ∶ u, v ∈ C}.

Moreover, if there exists h ∈ (0, 1) such that for every non-singleton nonempty bounded,

closed, and convex subset C of X , there exists x ∈ C such that r ^c (x) ⩽ (1 − h)diam(C),

then X is said to have the uniform normal structure.

It is well known that these concepts play an important role in the fixed point theory via the fact that if a Banach space has the weak normal structure, then it has the weak fixed point property (see [14,21]).

Since Banach spaces with ε ₀ (X) < 1 have the uniform normal structure (see [ 14]), the above theorem implies that the Orlicz space L _Φ on any non-atomic measure space has the uniform normal structure whenever Φ satisfies the ∆ 2 -condition and p (Φ) > 1/3.

We note that in [H155] the following formula for the parameter p (Φ) was proved:

p (Φ) = β(Φ) ∶= sup {a ∈ (0, 1) ∶ sup

u>0

ϕ (au) ϕ (u) < 1}, where ϕ (u) denotes the right-hand derivative of Φ at u > 0.

It is interesting to observe (see [H155]) that from this formula it follows that for every ε ∈ (0, 1) there is an Orlicz function Φ such that ε ⁰ (L ^Φ ) = ε. To see this take a = ^2−ε _2+ε and put ϕ (t) ∶= a ^{− i} for every t ∈ [a ^{− i +1} , a ^{− i} ] and each i ∈ Z. Then for Φ(u) ∶= ∫ ^{0 u} ϕ (t) dt for all u ⩾ 0, we have that Φ satisfies the ∆ 2 -condition and β (Φ) = a, hence ε 0 (L Φ ) = ε.

For further results on the computation of the characteristic of convexity of Orlicz spaces over a finite measure, we refer to [H47], and for Musielak–Orlicz function spaces, to [H59]. We also refer to the paper [H40] where the convexity properties of Orlicz spaces of vector-valued functions are studied.

Packing constants in Orlicz spaces. This topic concerns some important constants related to the theory of Banach spaces. The Kottman constant of a Banach space X is defined as follows:

D (X) = sup{inf{∥x ⁿ − x ^m ∥ ∶ n ≠ m} ∶ {x ⁿ } ^∞ n=1 ⊂ S ^X }.

By a result of Kottman [22], we have Λ (X) = D(X)/(2+D(X)) if X is infinite-dimensional, where Λ (X) is a packing constant of X given by

Λ (X) = sup{r > 0 ∶ ∃{x ⁿ } ⊂ X, ∥x ⁿ ∥ ⩽ 1 − r, ∥x ⁿ − x ^m ∥ ⩾ 2r, m ≠ n}.

From the Riesz lemma it follows that for any infinite-dimensional Banach space D (X) ⩾ 1.

We also note that Elton and Odell proved in [9] that if X is infinite-dimensional, then there exists ε > 0 such that D(X) ⩾ 1 + ε.

In [H64], Hudzik proved that D (X) = 1/2 and so Λ(X) = 2 for every non-reflexive Banach space X . In [H68], a formula for the packing constant for sequence Musielak–

–Orlicz spaces with the Luxemburg norm is given.

In [H77], lower and upper bounds for the Kottman constant of reflexive Orlicz sequ- ence spaces ℓ _Φ generated by a normalized Orlicz function Φ (i.e., Φ (1) = 1) is proved:

2 ^1/q

0

⩽ 1

g ˜ ⁻¹

Φ (1/2) ⩽ D(ℓ ^Φ ) ⩽ 1

( f Φ ⁰ ) ⁻¹ (1/2) ⩽ 2 ^{1/ p}

,

where

p ⁰ ⩽ inf

0<t⩽1

t ϕ (t)

Φ (t) , q ⁰ ⩽ sup

0<t⩽1

t ϕ (t) Φ (t) (ϕ denotes the right-hand derivative of Φ) and

f _Φ ⁰ (u) = sup

0<v ⩽1

Φ (uv)

Φ (v) , g ˜ Φ (u) = lim inf

v →0+

Φ (uv) Φ (v) .

Geometric properties related to the fixed point theory. The subject covers the geometric properties of Banach spaces that have applications in the fixed point theory.

An important geometric property playing a role in the fixed point theory is the Opial property. We recall that a Banach space X is said to have the Opial property if for every weakly-null sequence {x ⁿ } and every x ≠ 0 in X, we have

lim inf

n→∞ ∥x ⁿ ∥ < lim inf

n→∞ ∥x ⁿ + x∥.

In line with [30], a Banach space X is said to have the uniform Opial property if for every ε > 0 there exists τ such that for any weakly-null sequence {x n } and x ∈ X with ∥x∥ ⩾ ε, we have

lim inf

n→∞ ∥x n + x∥ ⩾ 1 + τ.

We note that Opial property was originally defined in [27].

Hudzik has worked on Opial properties in Orlicz sequence spaces. In [H131] authors investigated, in the case of Orlicz sequence spaces, the Opial modulus δ (ε, X) defined for any Banach space X and 0 < ε ⩽ 1 by

δ (ε, X) = inf{lim inf

n→∞ ∥x ⁿ + x∥ ∶ ∥x ⁿ ∥ = 1, x ⁿ Ð→ 0, ∥x∥ = ε}. ^w

Formulas for the Opial modulus for Orlicz sequence spaces ℓ _Φ with the Luxemburg norm and the Orlicz norm were established. They proved the following results.

4.3. Theorem. Let Φ be a real-valued Orlicz function such that lim t→0 Φ (t)/t = 0 and let X be one of the spaces ℓ _Φ , h _Φ equipped with the Luxemburg norm. Then the following statements are equivalent.

(i) X has the uniform Opial property.

(ii) X has the Opial property.

(iii) Φ satisfies the ∆ ⁰ ₂ -condition.

In the case of Orlicz sequence spaces with the Orlicz norm we have the following variant.

4.4. Theorem. Let Φ be an arbitrary real-valued Orlicz function and let X be one of the spaces ℓ Φ , h Φ equipped with the Orlicz norm. Then X has the uniform Opial property if and only if Φ satisfies the ∆ 2 -condition at zero.

We refer to [H99], where the Opial properties of Musielak–Orlicz spaces and general modular spaces are further studied.

We mention another geometric notion that found application in the fixed point the- ory. In [12], Garcia–Falset introduced the following coefficient of any Banach space X

R (X) = sup { lim inf

n→∞ ∥x ⁿ − x∥ ∶ x ⁿ Ð→ 0, x ∈ B ^{w X} },

and in [13] it was proved that if R (X) < 2, then X has the weak fixed point property.

In [H124], the Galset–Falset coefficient were calculated for the Orlicz space h _Φ of finite elements.

4.5. Theorem. For any Orlicz function Φ, the equality holds R (h ^Φ ) = sup{c ^x > 0 ∶ x =

m

∑

i =1

x i e i ∈ S ^ℓ

for some m ∈ N, I ^Φ ( c ^x

) = 2 ¹ }.

Yet another geometric property relevant to the fixed point theory studied by Hudzik concerns noncreasy and uniformly noncreasy Banach spaces. Let us recall the definition of these notions introduced by Prus in [30]. Let X be a Banach space and X ^∗ its dual space.

For any x ^∗ ∈ S ^X

and any δ ∈ [0, 1], we set

S (x ^∗ , δ ) = {x ∈ B ^X ∶ x ^∗ (x) ⩾ 1 − δ}.

For any functionals x ^∗ , y ^∗ ∈ S X

and any scalar δ ∈ [0, 1], we put S (x ^∗ , y ^∗ , δ ) = S(x ^∗ , δ ) ∩ S(y ^∗ , δ ).

If x ^∗ , y ^∗ ∈ S ^X

and x ^∗ ≠ y ^∗ , then S (x ^∗ , Y ^∗ , 0 ) is a crease lying on the unit sphere. A Banach space X is said to be noncreasy if X has no crease of positive diameter on its unit sphere, that is, diam S (x ^∗ , y ^∗ , 0 ) = 0 for any two different functionals x ^∗ , y ^∗ ∈ S ^X

. X is said to be uniformly noncreasy if for any ε > 0 there exists δ > 0 such that if x ^∗ , y ^∗ ∈ S ^X

with

∥x ^∗ − y ^∗ ∥ ⩾ ε, we have diam S(x ^∗ , y ^∗ , δ ) ⩽ ε.

It is easy to see that both strictly convex and smooth Banach spaces are noncreasy

and that both uniformly convex and uniformly smooth Banach spaces are uniformly non-

creasy.

In [H127], a complete characterization of noncreasy Orlicz spaces equipped with the Orlicz norm is presented.

4.6. Theorem. An Orlicz space L ^Φ (µ) equipped with the Orlicz norm generated by a real- -valued Orlicz function Φ with lim _u→0 Φ (u)/u = 0 and lim u→∞ Φ (u)/u = ∞ is noncreasy

if and only if Φ is strictly convex on R ⁺ or Φ satisfies a suitable ∆ 2 -condition (that is, the

∆ 2 -condition if the measure µ is infinite, and the ∆ 2 -condition at infinity id µ is finite) and the right-hand side derivative ϕ of Φ is continuous which means that L Φ is strictly convex or smooth.

In [H127], we also find a description of uniformly noncreasy Orlicz spaces equipped with the Luxemburg norm.

4.7. Theorem. If the Orlicz function Φ satisfy the conditions form the above theorem, then the Orlicz space L Φ (µ) is uniformly noncreasy if and only if

(i) Φ satisfies a suitable ∆ 2 -condition.

(ii) Φ is strictly convex or smooth on the whole of R ⁺ .

Smoothness of Orlicz spaces. Smooth Banach spaces form an important class in the ca- tegory of Banach spaces. Let X be a Banach space. Recall that a functional x ^∗ ∈ X ^∗ is said to be a support functional at x ∈ X if x ^∗ ∈ S ^X

and x ^∗ (x) = ∥x∥. The set of all support functionals at x is denoted by Grad (x). A point x ∈ X is said to be smooth if card (Grad(x)) = 1.

In a series of papers, Henryk Hudzik studied the criteria of smoothness in Orlicz spaces (see [H52,H67,H81,H83,H100,H127]) and Musielak–Orlicz spaces (see [H42,H62, H93,H120,H121,H138]). Here we only mention that in [H67] the support functionals at a

point of the unit sphere of L _Φ are described both for the Luxemburg norm and the Orlicz norm. These results are next applied to obtain a complete description of the smooth points of the unit sphere of L Φ and E Φ . Combining the results the authors of the paper conclude that in the case of a non-atomic, σ -finite measure space the Orlicz space L Φ equipped with the Orlicz norm is smooth if and only if the Orlicz funtion is smooth and satisfies a suitable ∆ 2 -condition.

Some geometric properties of Orlicz–Lorentz spaces. Let (Ω, Σ, µ) be a non-atomic σ -finite measure space. Given x ∈ L ⁰ , its distribution function is defined by d x (λ) = µ{t ∈ Ω ∶ ∣x∣(t) > λ} for λ ⩾ 0, and its decreasing rearrangement by

x ^∗ (s) = inf{λ > 0 ∶ d ^x (λ) ⩽ s}, s > 0.

A Banach lattice X on (Ω, Σ, µ) is called a rearrangement invariant space if y ∈ X and

∥x∥ ^X = ∥y∥ ^X whenever x ∈ X, y ∈ L ⁰ (µ) and d ^x (λ) = d ^y (λ) for every λ > 0.

Given an Orlicz function Φ and a non-increasing, locally integrable function ω on the interval [0, γ) with γ = µ(Ω), we define

ρ Φ, ω (x) = ∫ ₀ ^γ Φ (x ^∗ (t))ω(t) dt.

The Orlicz–Lorentz space Λ Φ, ω is defined to be the space of all x ∈ L ⁰ (µ) such that ρ _{Φ, ω} (λx) < ∞ for some λ > 0. The space Λ ^{Φ, ω} is a rearrangement invariant space equip- ped with the norm

∥x∥ = inf {λ > 0 ∶ ρ Φ, ω (x/λ) ⩽ 1}.

In the case of counting measure on 2 ^N and the non-increasing weight sequence ω = {ω ⁿ } ^∞ n=1 , the Orlicz–Lorentz sequence space λ _{Φ, ω} is defined to be a Banach space of all x = {x ⁿ } ∈ ℓ ∞ , equipped with the norm

∥x∥ = inf {λ > 0 ∶ ∑ ^∞

n=1

Φ (x ^∗ (n))ω ⁿ ⩽ 1}.

If 1 ⩽ p < ∞ and Φ(t) = t ^p for all t ⩾ 0, we write Λ ^{p , ω} (resp., λ p , ω ) instead of Λ Φ, ω

(resp., λ Φ, ω ).

Henryk Hudzik contributed significantly to the geometry of Orlicz–Lorentz spaces.

Before presenting some of his results recall some definitions. A positive weight function ω on [0, γ) is said to be regular if there exists K > 1 such that S(2t) ⩾ KS(t) for every t ∈ [0, γ/2), where

S (t) = ∫ ^t

0

ω (s) ds, t ∈ [0, γ).

A positive non-increasing weight sequence ω = {ω ⁿ } is said to be regular if there is a con- stant K > 1 such that S(2n) ⩾ KS(n), where S(n) = ∑ ⁿ k =1 ω k for each n ∈ N.

The study of the geometry of Lorentz spaces was initiated by Halperin and Altschuler.

In [15] (resp., in [3]), they proved that the Lorentz space Λ p , q (resp., λ p , ω ) is uniformly convex if and only if 1 < p < ∞ and the weight function (resp., sequence) ω is regular.

In [H97] the following result was proved.

4.8. Theorem. The Orlicz sequence space λ ^{Φ, ω} is uniformly convex if and only if the we- ight sequence ω is regular, Φ satisfies the ∆ ⁰

2 -condition, Φ (b(Φ)ω(1)) ⩾ 1, where b(Φ) ∶=

sup {u ⩾ 0 ∶ Φ(u) < ∞}, and Φ is uniformly convex on the interval [0, Φ ⁻¹ (1/(ω ¹ + ω ² )].

In order to state a result from [H82] we recall that a normed space X is said to be locally uniformly convex if for any x ∈ X and any sequence {x ⁿ } in X such that ∥x ⁿ ∥ → ∥x∥

and ∥x ⁿ + x∥ → 2∥x∥, we have ∥x ⁿ − x∥ → 0. A normed space X is said to be midpoint

locally uniformly convex if every point x ∈ S ^X is a strong extreme point of B _X , i.e. for any

sequences {x ⁿ } and {y ⁿ } in B ^X such that ∥x ⁿ + y ⁿ − 2x∥ → 0, we have ∥x ⁿ − x∥ → 0 and

∥y n − x∥ → 0.

4.9. Theorem. For the Orlicz–Lorentz space Λ ^{Φ, ω} on a non-atomic measure space (Ω, Σ, µ) the following statements are equivalent.

(i) Φ is strictly convex on R ⁺ , ω is strictly positive on [0, γ), ∫ ^{0 γ} ω (s) ds = ∞ if γ = µ(Ω) =

∞, and Φ satisfies the ∆ ^∞ 2 -condition if γ < ∞, Φ satisfies the ∆ ² -condition if γ = ∞.

(ii) Λ _{Φ, ω} is locally uniformly convex.

(iii) Λ _{Φ, ω} is midpoint locally uniformly convex.

(iv) Λ _{Φ, ω} is convex.

We refer to [H194], where the criteria for non-squareness properties (non-squareness, local uniform non-squareness and uniform non-squareness) of Orlicz–Lorentz sequence spaces as well as their order continuous parts are given.

Geometry of a class of Calderón–Lozanovski˘ı spaces. Hudzik has also been active in the investigation of the properties of a special class of Calderón–Lozanovski˘ ı spaces which are a generalization of the Calderón spaces defined in [6]. Calderón–Lozanovski˘ ı spaces play an important role in interpolation theory and have many deep applications to various areas of modern analysis (see [19, 28,29]).

To discuss the topic, we need to introduce some notation. Let U denote the set of all functions ψ ∶ R ⁺ ×R ⁺ → R ⁺ that are positively homogeneous (i.e., ψ (λs, λt) = λψ(s, t) for every s, t , λ ⩾ 0) and concave. Recall that ψ is concave whenever ψ(αs ¹ + βs ² , α t 1 + βt ² ) ⩾ αψ (s ¹ , t 1 ) + βψ(s ² , t 2 ) for all α, β ∈ [0, 1] with α + β = 1 and s ⁱ , t i ⩾ 0 for i = 1, 2.

Given ψ ∈ U and a couple of Banach lattices (X ⁰ , X 1 ) on (Ω, Σ, µ), the Calderón–

–Lozanovski˘ ı space is defined as follows (see [26, 31])

ψ (X ⁰ , X ₁ ) = {x ∈ L ⁰ ∶ ∣x∣ = ψ(∣x ⁰ ∣, ∣x ¹ ∣) for some x ⁱ ∈ X ⁱ , i = 0, 1}.

For any 1 ⩽ p < ∞, define a norm on the space ψ(X 0 , X ₁ ) as

∥x∥ ψ

( X

, X

) = inf {(∥x 0 ∥ ^p X

+ ∥x 1 ∥ ^p X

) ^{1/ p} ∶ ∣x∣ = ψ(∣x ⁰ ∣, ∣x 1 ∣), x i ∈ X i , i = 0, 1}, and for p = ∞ as

∥x∥ ψ

( X

, X

) = inf{max(∥x ⁰ ∥ ^X

, ∥x ¹ ∥ ^X

) ∶ ∣x∣ = ψ(∣x ⁰ ∣, ∣x ¹ ∣), x ⁱ ∈ X ⁱ , i = 0, 1}.

Under each norm ∥ ⋅ ∥ ψ

( X

, X

) , the space ψ (X ⁰ , X 1 ) is a Banach lattice [ 32]. Notice that all the norms ∥ ⋅ ∥ ψ

( X

, X

) , 1 ⩽ p ⩽ ∞, are in fact equivalent on ψ(X ⁰ , X 1 ). Denote by ψ p (X ⁰ , X 1 ) the space ψ(X ⁰ , X 1 ) equipped with the norm ∥ ⋅ ∥ ψ

( X

, X

) .

In the case when E is a Banach function lattice on (Ω, Σ, µ), Φ is a finite Orlicz function and ψ (s, t) = tΦ ⁻¹ (s/t), for all s ⩾ 0, t > 0, and ψ(0, 0) = 0

ψ (E, L ^∞ ) = E ^Φ ∶= {x ∈ L ⁰ (µ) ∶ Φ(∣x∣/λ) ∈ E for some λ > 0}

and

∥x∥ ψ

(E , L

) = ∥x∥ ^E

∶= inf {λ > 0 ∶ ∥Φ(∣x∣/λ)∥ ^E ⩽ 1},

∥x∥ ^ψ

(E , L

) = inf

k >0

1 + ∥Φ(k∣x∣)∥ ^E k

.

In particular, ψ _∞ (L ¹ , L _∞ ) (resp., ψ ¹ (L ¹ , L _∞ )) is the classical Orlicz space L ^Φ equipped with the Luxemburg (resp., the Orlicz norm).

The study of the global geometry of Calderón–Lozanovski˘ ı spaces E _Φ was underta- ken by Hudzik in [H82]. Some criteria for order continuity of these spaces were proved, and they play an essential role in the study of many other geometric properties of these spaces. These criteria are used to obtain estimates of the characteristic of convexity of the spaces E Φ under some mild assumptions on Banach lattices over non-atomic measure spaces. Moreover, local uniform convexity and uniform convexity in every direction we- re investigated. The local geometric properties of Claderón–Lozanovski˘ ı spaces E Φ were studied in [H141] and [H159]. Hudzik studied also some geometric and topological pro- perties of generalized Claderón–Lozanovski˘ ı spaces E Φ generated by a Banach lattice E and a Musielak function Φ (for more details, we refer to [H92] and [H110]).

5. Moduli and characteristics of monotonicity in Banach lattices

Monotonicity properties of Banach lattices were introduced and studied in the context of their geometric structure [5]. Let (X, ⩽, ∥⋅∥) be a real Banach lattice. We recall that a norm

∥ ⋅ ∥ is said to be strictly monotone if for any x, y ∈ X with 0 ⩽ x < y ∈ X, then ∥x∥ < ∥y∥.

A space X is said to be uniformly monotone if for any ε ∈ (0, 1) there exists 0<δ(ε) ∈ (0, 1) such that for any 0 ⩽ x ∈ S ^X and 0 ⩽ y ∈ X with ∥y∥ ⩾ ε, we have ∥x − y∥ ⩽ 1 − δ(ε).

We point out that in 1985 Akcoglu and Sucheston [2] showed how the strict and uni- form monotonicity were related to ergodic theory. Next, in 1992, Kurc [23] observed that the role of monotonicity properties of Banach lattices is similar to the role of convexity properties of Banach spaces.

Henryk Hudzik brought a significant contribution to the study of relationships be- tween monotonicity properties and geometric properties of Banach lattices. Relations be- tween convexity properties and monotonicity properties of Banach lattices were studied in [H74].

In the paper [H177], the authors introduce the concepts of moduli and characteristics of monotonicity of general Banach lattices and show their applications to Orlicz spaces.

Following [H177], for a given Banach lattice X , the function δ m , X ∶ [0, 1] → [0, 1] defined by

δ _{m , X} (ε) = inf{1 − ∥x − y∥ ∶ 0 ⩽ y ⩽ x, ∥x∥ = 1, ∥y∥ ⩾ ε}, ε ∈ [0, 1]

is called a modulus of monotonicity of X . Clearly, X is uniformly monotone if and only if δ _{m , X} (ε) > 0 for any ε ∈ (0, 1], and X is strictly monotone if and only if δ m , X (1) = 1. The number

ε _{0 , m} (X) = sup{ε ∈ [0, 1] ∶ δ m , X (ε) = 0}

is called the characteristic of monotonicity of X . The function defined by δ _X (ε) = inf {1 − ∥(x + y)/2∥ ∶ ∣y∣ ⩽ x, ∥x∥ = 1, ∥y∥ ⩽ 1, ∥x − y∥ = ε}

is called a modulus of order convexity of X . The authors proved the following results.

(i) The equality δ _{m , X} (ε/2) = δ X (ε) holds for every ε ∈ [0, 2].

(ii) If Φ is an Orlicz function with Φ (1) = 1 satisfying a suitable ∆ 2 -condition, then 1 − (1 − ε ^q ) ^1/q ⩽ δ ^{m , L}

(ε) ⩽ 1 − (1 − ε ^p ) ^{1/ p} , ε ∈ (0, 1),

where p, q are some indices related to Φ.

For the case of the lattice E (X) generated by a Banach function lattice E and a Banach lattice X , it was proved in [H175] that

max {ε ^{0 , m} (E), ε ^{0 , m} (X)} ⩽ ε ^{0 , m} (E(X)) ⩽ ε ^{0 , m} (E) + ε ^{0 , m} (X) − ε ^{0 , m} (E)ε ^{0 , m} (X) and that these estimates are optimal.

Monotonicity properties of specific lattices such as Lorentz, Orlicz, or in particular Musielak–Orlicz spaces have been extensively studied by many authors. In [H98], Hudzik with coauthors introduced the concept of a locally uniform monotonicity and investigated it for Musielak–Orlicz spaces. We note that monotonicity properties show close relation- ships to complex rotundities and their applications (see [H148]) .

6. Integral and partial differential equations

Henryk Hudzik has done some work in the theory of integral equations. In [H132], the authors proved the existence and uniqueness of local and global solutions to the nonlinear Hammerstein integral equation

x (t) = g(t) + ν∫ ₀ ¹ K (t, s) f (x(s)) ds, t ∈ I, ν ∈ R and the Volterra–Hammerstein integral equation

x (t) = g(t) + ν∫ ₀ ^t K (t, s) f (x(s)) ds, t ∈ I

in the space BV Φ (I) of functions of bounded Φ-variation in the sense of Young [ 36] on

a compact interval I ⊂ R, where Φ is an Orlicz function. For the sake of completeness, we

recall that if Φ is an Orlicz function, then the space BV _Φ ([a, b]) is defined as the space of all functions x ∶ [a, b] → R such that x(a) = 0 and

var _Φ (x) ∶= sup

π n

∑

k =1

Φ (∣x(t ^k ) − x(t k −1 )∣) < ∞,

where the supremum is taken over all partitions π ∶ a = t 0 < t 1 < . . . < t n = b. Since x ∈ BV Φ ([a, b]) implies that x is bounded on [a, b], a standard combination with Helly’s extraction theorem yields that BV _Φ ([a, b]) is a Banach space equipped with the norm

∥x∥ = ∥x∥ ^∞ + inf {λ > 0 ∶ var ^Φ (x/λ) ⩽ 1}.

In the proofs of the existence and uniqueness of solutions to the above equations, the Banach contraction principle and the Leray–Schauder alternative were used. We only state here the following result.

6.1. Theorem. Assume that g ∈ BV ^Φ (I) with I = [0, a], f ∶ R → R is a locally Lipschitz function, and K ∶ I × I → R is a function such that K(t, ⋅) is Lebesgue integrable on I for every t ∈ I, K(0, s) = 0 and there exists a number α > 0 such that var Φ (K(⋅, s)/α) ⩽ M(s) for almost all s ∈ I, where M∶ I → R + is Lebesgue integrable on I. Then there exists a number ρ > 0 such that for every ν with ∣ν∣ ⩽ ρ the nonlinear Hammerstein integral equation has a unique solution x ∈ BV ^Φ [0, a].

Recall that the Leray–Schauder alternative states: Let U be an open subset of a Banach space X with 0 ∈ U and let U denote the closure of U in X. Let F∶ U → X be a function such that ∥F(x) − F(y)∥ ⩽ ϕ(∥x − y∥) for all x, y ∈ U, where ϕ∶ [0, ∞) → [0, ∞) is a continuous non-decreasing function satisfying ϕ (z) < z for all z > 0. If in addition F(U) is bounded, x ≠ λF(x) for x ∈ ∂U, and λ ∈ (0, 1], then F has a fixed point in U.

Henryk Hudzik also contributed to the theory of partial differential equations. His joint paper with Baoxiang [H160] is a study of the Cauchy problem for the nonlinear Schrödinger equation, the nonlinear Klein–Gordon equation

iu _t − ∆u − f (u) = 0, u(0) = u ⁰ , and the nonlinear Klein–Gordon equation

u _{t t} + (I − ∆)u + f (u) = 0, u(0) = u 0 , u _t (0) = u 1 , where u = u(t, x) is a complex-valued function of (t, x) ∈ R × R ⁿ , i = √

−1, u ^t = ∂/∂t,

u t t = ∂ ² /∂ ² t, ∆ = ∂ ² /∂ ² x 1 + . . . + ∂ ² /∂ ² x n , f is nonlinear scalar function on C, and u ⁰ ,

u ₁ ∶ R ⁿ → C. The authors proved a theorem on the existence and uniqueness of global

solutions for sufficiently small Cauchy data in the so-called modulation spaces. For more

details on modulation spaces we refer to [11].

During his scientific activity Henryk Hudzik received numerous invitations to uni- versities in Barcelona, Bangkok, Berlin, Chiang Mai, Caracas, Harbin, Karlsruhe, Madrid, Mainz, Luleå, Narvik, Paris, Petersburg, Saarbrücken, Valencia, and Zaragoza. He also stayed at Memphis University where he held a visiting professorship.

Henryk Hudzik is a member of the editorial board of 15 international mathematical journals. He received many scientific awards, including the Stefan Banach Main Prize of the Polish Mathematical Society (1986) and the Award of the Minister of National Educa- tion (5 times).

Mieczysław Mastyło

Doctoral theses written under Henryk Hudzik’s supervision

– Haifeng Ma, Construction of some generalized inverses of operators between Banach spaces and their selections, perturbations, and applications, 2012

– Radosław Kaczmarek, Moduli and characteristics of monotonicity of some spaces of scalar- and vector-valued functions, 2007 (in Polish)

– Topological and geometric structures of Cesàro–Orlicz sequence spaces, 2007 (in Polish) – Xinbo Liu, Local monotonicity and rotundity structure of Musielak–Orlicz spaces, 2005 – Lucjan Szymaszkiewicz, Geometry of some Köthe spaces, 2005 (in Polish)

– Karol Wlaźlak, Geometry of normed spaces defined via sublinear operators, 2004 (in Po- lish)

– Yuwen Wang, Some geometric properties of Banach spaces and their applications in the operator theory, 2003

– Agata Narloch, Structure of some real and complex Banach lattices, 2002 (in Polish) – Wojciech Kowalewski, On local and global geometry of Musielak–Orlicz spaces, 2001

(in Polish)

– Lifang Liu, On some local and global geometric properties of Musielak–Orlicz spaces, 2001

– Paweł Foralewski, On topological and geometric structure of generalized Calderon–

–Lozanovski˘ ı spaces, 1997 (in Polish)

– Yunan Cui, On some geometric coefficient of Orlicz spaces, 1995

– Zenon Zbąszyniak, Smoothness in Musielak–Orlicz spaces, 1994 (in Polish)

– Małgorzata Doman, Weak uniform rotundity of Orlicz and Musielak–Orlicz spaces, 1994 (in Polish)

– Ghassan Alherk, On some geometric properties of Musielak–Orlicz spaces, 1991

The complete list of Henryk Hudzik’s papers

[H1] H. Hudzik, On generalized Orlicz–Sobolev space, Funct. Approximatio Comment. Math. 4 (1976), 37–51.

[H2] H. Hudzik, A generalization of Sobolev spaces. I, Funct. Approximatio Comment. Math. 2 (1976), 67–73.

[H3] H. Hudzik, A generalization of Sobolev spaces. II, Funct. Approximatio Comment. Math. 3 (1976), 77–85.

[H4] H. Hudzik, Density of C

( R

) in generalized Orlicz–Sobolev space W

( R

), Funct. Approx. Com- ment. Math. 7 (1979), 15–21.

[H5] H. Hudzik, The problems of separability, duality, reflexivity and of comparison for generalized Orlicz–

–Sobolev spaces W

(Ω), Comment. Math. Prace Mat. 21 (1980), no. 2, 315–324.

[H6] H. Hudzik, J. Musielak, and R. Urbański, Interpolation of compact sublinear operators in generalized Orlicz spaces of nonsymmetric type, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978), Colloq.

Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, 625–638.

[H7] A. Kamińska and H. Hudzik, Some remarks on convergence in Orlicz space, Comment. Math. Prace Mat. 21 (1980), no. 1, 81–88.

[H8] H. Hudzik, J. Musielak, and R. Urbański, Some extensions of the Riesz-Thorin theorem to generalized Orlicz spaces L

(T ), Comment. Math. Prace Mat. 22 (1980), no. 1, 43–61.

[H9] H. Hudzik, W. Orlicz, and R. Urbański, Interpolation of sublinear, bounded operators in sequence spaces h

(X ) of nonsymmetric type, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), no. 1-2, 45–54 (1981) (English, with Russian summary).

[H10] H. Hudzik, J. Musielak, and R. Urbański, Riesz-Thorin theorem in generalized Orlicz spaces of nonsymme- tric type, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 678-715 (1980), 145–158 (1981) (English, with Serbo-Croatian summary).

[H11] H. Hudzik and A. Kamińska, Equivalence of the Orlicz and Luxemburg norms in generalized Orlicz spaces L

(T ), Funct. Approx. Comment. Math. 9 (1980), 29–37.

[H12] H. Hudzik, Strict convexity of Musielak-Orlicz spaces with Luxemburg’s norm, Bull. Acad. Polon. Sci.

Sér. Sci. Math. 29 (1981), no. 5-6, 235–247 (English, with Russian summary).

[H13] H. Hudzik, Musielak-Orlicz spaces isomorphic to strictly convex spaces, Bull. Acad. Polon. Sci. Sér. Sci.

Math. 29 (1981), no. 9-10, 465–470 (English, with Russian summary).

[H14] H. Hudzik, J. Musielak, and R. Urbański, Linear operators in modular spaces. An application to approxi- mation theory, Approximation and function spaces (Gdańsk, 1979), North-Holland, Amsterdam-New York, 1981, 279–286.

[H15] H. Hudzik, Intersections and algebraic sums of Musielak-Orlicz spaces, Portugal. Math. 40 (1981), no. 3, 287–296 (1985).

[H16] H. Hudzik, J. Musielak, and R. Urbański, Linear operators in modular spaces, Comment. Math. Prace Mat. 23 (1983), no. 1, 33–40.

[H17] H. Hudzik, Uniform convexity of Musielak-Orlicz spaces with Luxemburg’s norm, Comment. Math. Pra- ce Mat. 23 (1983), no. 1, 21–32.

[H18] H. Hudzik, On some equivalent conditions in Musielak-Orlicz spaces, Comment. Math. Prace Mat. 24 (1984), no. 1, 57–64.

[H19] H. Hudzik, J. Musielak, and R. Urbański, A totally nonatomic set-valued measure, Comment. Math.

Prace Mat. 24 (1984), no. 1, 65–67.

[H20] H. Hudzik, Flat Musielak-Orlicz spaces under Luxemburg’s norm, Bull. Polish Acad. Sci. Math. 32 (1984),

no. 3-4, 203–208 (English, with Russian summary).

[H21] H. Hudzik, A criterion of uniform convexity of Musielak-Orlicz spaces with Luxemburg norm, Bull. Po- lish Acad. Sci. Math. 32 (1984), no. 5-6, 303–313 (English, with Russian summary).

[H22] H. Hudzik, Convexity in Musielak-Orlicz spaces, Hokkaido Math. J. 14 (1985), no. 1, 85–96, DOI 10.14492/hokmj/1381757691.

[H23] H. Hudzik and A. Kamińska, On uniformly convexifiable and B-convex Musielak-Orlicz spaces, Com- ment. Math. Prace Mat. 25 (1985), no. 1, 59–75.

[H24] H. Hudzik, Orlicz spaces of essentially bounded functions and Banach-Orlicz algebras, Arch. Math. (Ba- sel) 44 (1985), no. 6, 535–538, DOI 10.1007/BF01193994.

[H25] H. Hudzik, Some class of uniformly nonsquare Orlicz–Bochner spaces, Comment. Math. Univ. Carolin.

26 (1985), no. 2, 269–274.

[H26] H. Hudzik, Uniformly non-l

Orlicz spaces with Luxemburg norm, Studia Math. 81 (1985), no. 3, 271–284.

[H27] H. Hudzik, Locally uniformly non-l

Orlicz spaces, Proceedings of the 13th winter school on abstract analysis (Srní , 1985), 1985, 49–56 (1986).

[H28] R. Grząślewicz, H. Hudzik, and W. Orlicz, Uniform non-l

property in some normed spaces, Bull.

Polish Acad. Sci. Math. 34 (1986), no. 3-4, 161–171 (English, with Russian summary).

[H29] H. Hudzik, On some renorming problems, Arch. Math. (Basel) 48 (1987), no. 6, 505–510, DOI 10.1007/BF01190357.

[H30] H. Hudzik, An estimation of the modulus of convexity in a class of Orlicz spaces, Math. Japon. 32 (1987), no. 2, 227–237.

[H31] H. Hudzik, A. Kamińska, and J. Musielak, On some Banach algebras given by a modular, A. Haar me- morial conference, Vol. I, II (Budapest, 1985), Colloq. Math. Soc. János Bolyai, vol. 49, North-Holland, Amsterdam, 1987, 445–463.

[H32] H. Hudzik, Musielak-Orlicz algebras, Proceedings of the 14th winter school on abstract analysis (Srní, 1986), 1987, 335–338.

[H33] H. Hudzik, A. Kamińska, and W. Kurc, Uniformly non-l

Musielak-Orlicz spaces, Bull. Polish Acad.

Sci. Math. 35 (1987), no. 7-8, 441–448 (English, with Russian summary).

[H34] H. Hudzik, A. Kamińska, and J. Musielak, On the convexity coefficient of Orlicz spaces, Math. Z. 197 (1988), no. 2, 291–295, DOI 10.1007/BF01215197.

[H35] S. Chen and H. Hudzik, On some convexities of Orlicz and Orlicz–Bochner spaces, Comment. Math.

Univ. Carolin. 29 (1988), no. 1, 13–29.

[H36] H. Hudzik, On some renorming problems. II, Arch. Math. (Basel) 52 (1989), no. 4, 365–366, DOI 10.1007/BF01194412.

[H37] H. Hudzik, On smallest and largest Orlicz spaces, Math. Nachr. 141 (1989), 109–115, DOI 10.1002/ma- na.19891410113.

[H38] G. Alherk and H. Hudzik, Uniformly non-l

Musielak-Orlicz spaces of Bochner type, Forum Math. 1 (1989), no. 4, 403–410, DOI 10.1515/form.1989.1.403.

[H39] H. Hudzik, Orlicz spaces containing a copy of l

, Math. Japon. 34 (1989), no. 5, 747–759.

[H40] H. Hudzik, On some convexity properties of Orlicz spaces of vector valued functions, Congress on Func- tional Analysis (Madrid, 1988), 1989, 137–144.

[H41] H. Hudzik and L. Maligranda, An interpolation theorem in symmetric function F -spaces, Proc. Amer.

Math. Soc. 110 (1990), no. 1, 89–96, DOI 10.2307/2048246.

[H42] H. Hudzik and Y. N. Ye, Support functionals and smoothness in Musielak-Orlicz sequence spaces endo-

wed with the Luxemburg norm, Comment. Math. Univ. Carolin. 31 (1990), no. 4, 661–684.