Abstract. Let G be a locally compact Hausdorff group with Haar measure, and let L

POLONICI MATHEMATICI LXIII.2 (1996)

Approximation by nonlinear integral operators in some modular function spaces

by Carlo Bardaro (Perugia), Julian Musielak (Pozna´ n) and Gianluca Vinti (Perugia)

Abstract. Let G be a locally compact Hausdorff group with Haar measure, and let L

(G) be the space of extended real-valued measurable functions on G, finite a.e. Let

% and η be modulars on L

(G). The error of approximation %(a(T f − f )) of a function f ∈ (L

(G))

∩Dom T is estimated, where (T f )(s) = R

G

K(t−s, f (t)) dt and K satisfies a generalized Lipschitz condition with respect to the second variable.

1. Let G be a locally compact Hausdorff group with neutral element θ and with the family U of open neighbourhoods of θ in G. For the sake of simplicity from now on we will assume G to be abelian. Let Σ be the Borel σ-field of G, let |A| be the Haar measure of a measurable set A ⊂ G and let R

G

f (t) dt denote the Haar integral of f .

We shall denote by M

(G) the space of all extended real-valued measur- able functions f : G → R = R∪{∓∞}, and by L

(G) ⊂ M

(G) its subspace of functions f finite almost everywhere (a.e.), both provided with equality a.e.

Let % : L

(G) → R

and η : L

(G) → R

be two modulars in L

(G), and let (L

(G))

and (L

(G))

be the respective modular spaces (for termi- nology, see e.g. [4]).

We make the following assumptions:

1

% and η are monotone, i.e. if f, g ∈ L

(G) and |f | ≤ |g|, then %(f ) ≤

%(g) and η(f ) ≤ η(g);

2

% is J -convex, i.e. for any two measurable functions p : G → R

and

1991 Mathematics Subject Classification: 46E30,46A80,41A46,45P05.

Key words and phrases: modular space, nonlinear integral operator, generalized Lip- schitz condition, approximation by singular integrals.

This paper was written when the second author was a Visiting Professor at the Di- partimento di Matematica, Universit` a degli Studi di Perugia.

[173]

F : G × G → R with R

G

p(t) dt = 1,

%

 R

G

p(t)|F (t, ·)| dt



≤ R

G

p(t)%(F (t, ·)) dt;

3

η is τ -bounded, i.e. there are a number c ≥ 1 and a measurable, bounded function h : G → R

such that h(t) → 0 as t → θ and

η(f (t + ·)) ≤ η(cf ) + h(t), t ∈ G,

for all f ∈ L

(G) such that η(f ) < ∞; we shall write h

= sup

h(t).

We may extend both modulars % and η to M

(G), putting %(f )

= η(f ) = ∞ for f ∈ M

(G) \ L

(G).

Let ψ : G × R

→ R

be such that for all t ∈ G, the function ψ(t, ·) is continuous and nondecreasing for u ≥ 0, ψ(t, 0) = 0, ψ(t, u) > 0 for u > 0, ψ(t, u) → ∞ as u → ∞, and such that for every u ≥ 0, ψ(t, u) is a measurable function of t.

The following connection between both modulars % and η and the func- tion ψ will be assumed:

(I) there is a set G

⊂ G with |G \ G

| = 0 such that for every λ ∈ ]0, 1[

there exists a C

∈ ]0, 1[ satisfying the inequality

%[C

ψ(t, |F (·)|)] ≤ η(λF (·)) for all t ∈ G

and F ∈ L

(G).

A condition of this type was introduced in special cases in [3].

Let us still remark that we may choose C

in such a manner that C

& 0 as λ & 0. Condition (I) implies immediately the following inequality:

%[C

ψ(t, F

(·))] ≤ η(λF

(·))

for every t ∈ G

and for any family (F

(·))

of functions F

∈ L

(G).

A function K : G × R → R will be called a kernel function if K(t, 0) = 0 for t ∈ G and K(·, u) ∈ L

(G) for all u ∈ R. Let L : G → R

, L ∈ L

(G).

We say that a kernel function K satisfies the (L, ψ)-Lipschitz condition if

|K(t, u) − K(t, v)| ≤ L(t)ψ(t, |u − v|) for t ∈ G, u, v ∈ R (see [1], p. 10).

In the following we shall write L = R

G

L(t) dt, p(t) = L(t)/L.

Let us remark that if K is an (L, ψ)-Lipschitz kernel function and f ∈ L

(G), then the superposition K(t, f (t + s)) is a measurable function of t ∈ G for all s ∈ G.

2. Following [1–4] we shall deal with nonlinear integral operators T of

the form

(T f )(s) = R

G

K(t − s, f (t)) dt = R

G

K(t, f (t + s)) dt.

We denote by Dom T the set of all functions f ∈ L

(G) such that (T f )(s) exists for a.e. s ∈ G and T f is a measurable function on G.

Proposition 1. Let f ∈ (L

(G))

∩Dom T and let λ ∈ ]0, 1[ be so small that η(cλf ) < ∞, where c ≥ 1 is the constant from 3

. Suppose that K is an (L, ψ)-Lipschitz kernel function and the condition (I) is satisfied. Then, for every ε > 0, there exists a U ∈ U such that

%  C

L T f



≤ η(cλf ) + h

R

G\U

p(t) dt + ε.

Consequently, %((C

/L)T f ) < ∞.

P r o o f. Applying monotonicity of %, the (L, ψ)-Lipschitz condition, J -convexity of % and the condition (I), we obtain

%  C

L T f



≤ %  C

L

R

G

|K(t, f (t + ·)|) dt



≤ %  R

G

p(t)C

ψ(t, |f (t + ·)|) dt 

≤ R

G

p(t)%[C

ψ(t, |f (t + ·)|)] dt ≤ R

G

p(t)η(λ|f (t + ·)|) dt.

Since η(λf ) < ∞, by τ -boundedness of η, we get η(λ|f (t+·)|) ≤ η(cλf )+h(t) for t ∈ G. Consequently, since R

G

p(t) dt = 1, we obtain

(1) %  C

L T f



≤ η(cλf ) + R

G

p(t)h(t) dt.

However, since h(t) → 0 as t → θ, for any ε > 0 there is a U ∈ U such that h(t) < ε for t ∈ U . Since h(t) ≤ h

for t ∈ G \ U , the required inequality follows from (1).

3. The map ω

: L

(G) × U → R

= [0, ∞] defined by ω

(f, U ) = sup

t∈U

η(f (t + ·) − f (·))

for f ∈ L

(G), U ∈ U , is called the η-modulus of continuity (see [4], p. 85).

We shall apply the following notation:

r

= sup

1/k≤|u|≤k

    1 u

R

G

K(t, u) dt − 1    

, A

= {t ∈ G : |f (t)| > k}, B

= {t ∈ G : |f (t)| < 1/k}, C

= G \ (A

∪ B

), f ∈ L

(G),

k = 1, 2, . . . , and r = sup

k

r

.

We shall give an estimate of the modular error of approximation %(a(T f −f )) for sufficiently small a > 0.

Theorem 1. Let f ∈ (L

(G))

∩ Dom T . Let λ ∈ ]0, 1[ and a ∈ ]0, C

/(16L)[ be so small that η(2cλf ) < ∞ and %(16af ) < ∞. Then, for every U ∈ U , k = 0, 1, 2, . . . and S ∈ Σ, we have

(2) %(a(T f − f )) ≤ ω

(λf, U ) + [2η(2cλf ) + h

] R

G\U

p(t) dt + R

, where R

is given by

R

= %(2arf ),

R

= η(λf χ

) + %(16af χ

) + η(λf χ

)

+ %(16af χ

) + η(λf χ

) + %(16af χ

) + %(8ar

f ), k = 1, 2, . . .

P r o o f. We have %(a(T f − f )) ≤ J

+ J

, where J

= % n

2a R

G

|K(t, f (t + ·)) − K(t, f (·))| dt o ,

J

= % n 2a

R

G

K(t, f (·)) dt − f (·)    o

(see [4], p. 88). By the (L, ψ)-Lipschitz condition, by J -convexity of % and taking into account the condition (I) we have

J

≤ R

G

p(t)%[C

ψ(t, |f (t + ·) − f (·)|] dt

≤ R

U

p(t)η(λ|f (t + ·) − f (·)|) dt

+ R

G\U

p(t)η(λ|f (t + ·) − f (·)|) dt = J

+ J

. But

J

≤ R

G

p(t)ω

(λf, U ) dt ≤ ω

(λf, U ).

Now,

J

≤ R

G\U

p(t)η(2λf (t + ·)) dt + η(2λf ) R

G\U

p(t) dt.

By τ -boundedness of η we obtain

R

G\U

p(t)η(2λf (t + ·)) dt ≤ [η(2cλf ) + h

] R

G\U

p(t) dt

whence, by monotonicity of η,

J

≤ [2η(2cλf ) + h

] R

G\U

p(t) dt.

Consequently,

J

≤ ω

(λf, U ) + [2η(2cλf ) + h

] R

G\U

p(t) dt.

It remains to prove that J

≤ R

, k = 0, 1, 2, . . . For k = 0 this is obvious.

Suppose k > 0. Then, taking any set S ∈ Σ, we have J

≤ % n

8a   

R

G

K(t, f (·)χ

(·)) dt − f (·)χ

(·)    o

+ % n

8a   

R

G

K(t, f (·)χ

(·)) dt − f (·)χ

(·)    o

+ % n 8a

R

G

K(t, f (·)χ

(·)) dt − f (·)χ

(·)    o

+ % n 8a

R

G

K(t, f (·)χ

(·)) dt − f (·)χ

(·)    o

.

By (L, ψ)-Lipschitz condition, monotonicity and J -convexity of % and by condition (I), for every P ∈ Σ we get

% n

8a   

R

G

K(t, f (·)χ

(·)) dt − f (·)χ

(·)    o

≤ % n 16a R

G

|K(t, f (·)χ

(·))| dt o

+ %(16af χ

)

≤ R

G

p(t)%[C

ψ(t, f (·)χ

(·))] dt + %(16af χ

)

≤ η(λf χ

) + %(16af χ

).

Thus, for P = G \ S, P = S ∩ A

, P = S ∩ B

and by the definition of r

, we obtain

J

≤ η(λf χ

) + %(16af χ

) + η(λf χ

) + %(16af χ

) + η(λf χ

) + %(16af χ

) + %(8ar

f )

and so the assertion follows.

4. Let W be a nonempty, abstract set of indices and let W be a filter of subsets of W.

A family K = (K

)

of kernel functions will be called a kernel.

Let L = (L

)

be a family of nonnegative functions L

∈ L

(G). We say that the kernel K satisfies the (L, ψ)-Lipschitz condition if the kernel functions K

satisfy the (L

, ψ)-Lipschitz condition, and L = sup

L

= sup

R

G

L

(t) dt < ∞. Set p

(t) = L

(t)/L

(see [1], pp. 12–13). The kernel K will be called singular if for every U ∈ U,

R

G\U

p

(t) dt −→ 0

and

r

(w) = sup

1/k≤|u|≤k

    1 u

R

G

K

(t, u) dt − 1    

−→ 0

for k = 1, 2, . . . If, moreover,

r(w) = sup

k=1,2,...

r

(w) −→ 0,

the kernel K will be called strongly singular.

Let us define a family T = (T

)

of operators by (T

f )(s) = R

G

K

(t − s, f (t)) dt = R

G

K

(t, f (t + s)) dt.

Set Dom T = T

w∈W

Dom T

. We shall deduce from Theorem 1 a theorem on convergence %(a(T

f − f )) −→ 0 for small a > 0. We need some additional

notions, namely of absolute finiteness and absolute continuity of modulars (see [4], p. 84, [2], p. 4).

Definition 1. A modular η on L

(G) is called finite if for every mea- surable set A ⊂ G such that |A| < ∞ we have χ

∈ (L

(G))

.

Definition 2. A modular η on L

(G) is called absolutely finite if it is finite and if for every ε > 0 and for every λ

> 0, there is a δ > 0 such that η(λ

χ

) < ε for every measurable set B ⊂ G of measure |B| < δ .

Definition 3. A modular η on L

(G) is called absolutely continuous (with respect to the measure in G) if there exists an α > 0 such that for every f ∈ L

(G) with η(f ) < ∞ the following two conditions are satisfied:

(α) for every ε > 0 there exists a measurable set A ⊂ G such that

|A| < ∞ and η(αf χ

) < ε;

(β) for every ε > 0 there exists a δ > 0 such that η(αf χ

) < ε for all measurable sets B ⊂ G of measure |B| < δ.

Let us remark that if η is monotone, τ -bounded, absolutely finite and

absolutely continuous, then for every f ∈ (L

(G))

there is a λ

> 0 such

that for every ε > 0 there exists a U

∈ U such that ω

(λ

f, U

) < ε (see

[4], Theorem 1, p. 85; the condition (P) mentioned there is always satisfied, as was kindly shown to us by Prof. D. Candeloro).

Theorem 2. Let K = (K

)

be a singular kernel and let the modular

% be monotone and J -convex , and the modular η-monotone, τ -bounded , absolutely finite and absolutely continuous. Let f ∈ (L

(G))

∩ Dom T.

Finally, let one of the following conditions hold : (a) K is strongly singular ;

(b) % is finite and absolutely continuous.

Then %(a(T

f − f )) −→ 0 for sufficiently small a > 0 (depending on f ).

P r o o f. Choose an arbitrary ε > 0. Since η is monotone, τ -bounded, absolutely finite and absolutely continuous, there is a U ∈ U such that ω

(λ

f, U ) < ε/4 for sufficiently small λ

> 0. Taking λ

∈ ]0, λ

[ small enough, we get η(2cλ

f ) < ∞. Due to singularity of K, keeping the above U ∈ U fixed, we have R

G\U

p

(t) dt −→ 0. Hence there exists a W

∈ W such that

[2η(2cλf ) + h

] R

G\U

p

(t) dt < ε/4 for λ ∈ ]0, λ

], w ∈ W

and the above U ∈ U .

Thus, for a fixed λ ∈ ]0, λ

[ let C

be the corresponding constant in (I), and for a ∈ ]0, C

/(16L)[ , we have

%(a(T

f − f )) < ε/2 + R

for w ∈ W

and k = 0, 1, 2, . . . , where we have applied (2) with T

and p

in place of T and p.

Assuming that (a) holds, we apply (3) with k = 0, obtaining R

=

%(2ar(w)f ). However, since f ∈ (L

(G))

, there is a W

∈ W such that

%(2ar(w)f ) < ε/2 for w ∈ W

. This gives %(a(T

f − f )) < ε for w ∈ W

∩ W

∈ W, which implies our assertion.

Now suppose (b). We apply Theorem 1 with a given S ∈ Σ with |S| < ∞.

Since A

⊃ A

⊃ . . . , we have S ∩ A

⊃ S ∩ A

⊃ . . . , and |S ∩ A

| < ∞.

Hence lim

|S ∩ A

| = |S ∩ T

k=1

A

|. But f ∈ L

(G) whence there exists a set G

⊂ G, G

∈ Σ, |G

| = 0, such that |f (t)| < ∞ for t ∈ G \ G

. From the inclusion T

k=1

A

⊂ G

, we deduce lim

|S ∩ A

| = 0.

Now applying absolute continuity of η and %, we may choose λ and a so small that

η(λf χ

) + %(16af χ

) < ε/12 for a suitable set S ∈ Σ, |S| < ∞.

Keeping S fixed, we may find an index k such that

η(λf χ

) + %(16af χ

) < ε/12.

Moreover, η(λf χ

) + %(16af χ

) ≤ η((λ/k)χ

) + %((16a/k)χ

) and since χ

∈ (L

(G))

we may find k such that η(λf χ

)+%(16af χ

)

< ε/12, which gives R

< ε/4 + %(8ar

(w)f ). Taking w ∈ W

we obtain by (3), %(a(T

f − f )) < 3ε/4 + %(8ar

(w)f ). But f ∈ (L

(G))

, whence there is a W

∈ W such that %(8ar

(w)f ) < ε/4 for w ∈ W

. This gives

%(a(T

f − f )) < ε for w ∈ W

∩ W

∈ W, which implies our statement.

5. We give some examples of modulars % and η satisfying the assumptions of Theorems 1 and 2.

Examples. 1. Let Φ : R

→ R

be such that Φ(0) = 0, Φ(u) > 0 for u > 0, Φ nondecreasing in R

, and Φ(u) → ∞ as u → ∞. Then Φ generates a modular

σ(f ) = I

(f ) = R

G

Φ(|f (t)|) dt

in L

(G), and the respective modular space (L

(G))

is the Orlicz space L

(G).

The modular σ is monotone, absolutely finite, absolutely continuous and τ -bounded (with c = 1, h(t) ≡ 0). If Φ is convex on R

, then σ is J -convex.

Thus, I

satisfies the assumption of Theorem 2, (b).

Finally, if we take two functions Φ

and Φ

and we put % = I

, η = I

, then (I) is certainly satisfied with λ = C

if we assume the concavity of the function ψ with respect to the second variable and that (Φ

◦ ψ)(u) ≤ Φ

(u) for u ≥ 0.

2. Let V be a nonempty set of indices filtered by a set W of its subsets.

Let a

: [a, b[ → R

, v ∈ V, be such that 1

R

a

a

(x) dx ≤ 1 for all v ∈ V;

2

if g : [a, b[ → R

is such that 0 ≤ g(x) % s < ∞ as x → b−, then

b

R

a

a

(x)g(x) dm −→ s;

3

for every Lebesgue measurable set C ⊂ [a, b[ of measure m(C) > 0 there exists a Lebesgue measurable subset C

of measure m(C

) > 0 and an index v ∈ V such that a

(x) > 0 m-almost everywhere in C

.

Let Φ : [a, b[ × R

→ R

satisfy

1) Φ(x, u) is a nondecreasing continuous function of u ≥ 0, for every x ∈ [a, b[;

2) Φ(x, 0) = 0, Φ(x, u) > 0 for u > 0, and Φ(x, u) → ∞ as u → ∞, for

every x ∈ [a, b[;

3) the limit lim

Φ(x, u) = e Φ(u) < ∞ exists for every u ≥ 0;

4) Φ(x, u) is a Lebesgue measurable function of x in [a, b[, for every u ≥ 0.

Moreover, suppose Φ(x, u) to be of monotone type in a subinterval [c, b[ ⊂ [a, b[ and equicontinuous in [a, b[ at u = 0 (for these notions see [2], Sec- tion 4).

Let m be a measure on [a, b[ defined on all Lebesgue measurable subsets of [a, b[. Then

A

(f ) = sup

v∈V b

R

a

a

(x)J

(x, f ) dm(x), where

J

(x, f ) = R

G

Φ(x, |f (t)|) dt,

is a modular in the subspace L

(G) ⊂ L

(G) of functions f for which J

(x, f ) is a Lebesgue measurable function on [a, b[.

In [2] sufficient conditions are obtained in order that A

be absolutely finite and absolutely continuous. Evidently A

is monotone. If Φ(x, u) is a convex function of u ≥ 0 for all x ∈ [a, b[, then for all measurable functions F : G × G → R and p : G → R

with R

G

p(t) dt = 1 we have Jensen’s inequality

Φ

 x, R

G

p(t)F (t, s) dt



≤ R

G

p(t)Φ(x, |F (t, s)|) dt

for x ∈ [a, b[, s ∈ G. Hence it follows that, in this case, A

is J -convex. It is easily observed that J

(x, f (t + ·)) = J

(x, f ) for every t ∈ G, whence A

is τ -bounded with c = 1, h(t) ≡ 0.

The theory developed in [2] for the modular A

contains as a particular case the discrete modulars of the type

A

(f ) = sup

n∈N

∞

X

i=1

a

I

(f ) (for details see [2], Section 5).

Moreover, it is possible to prove that the more general modulars of type

A e

(f ) = sup

v∈V b

R

a

J

(x, f ) dm

(x), f ∈ L

(G),

where {m

} is a family of measures, satisfy the conditions of Theorem 1

and Theorem 2, under the assumptions of Section 6 of [2].

We remark that among these modulars there are those studied in [6], namely

I

(f ) = sup

x∈ [a,b[

J

(x, f ) (see also Section 6 of [2]).

Hence we may state the following

Corollary 1. Let K = (K

)

be a singular kernel and let %, η be any of the modulars defined in Examples 1 or 2, satisfying (I). Suppose the function Φ generating % is convex. In case any of the modulars is as in Example 2, suppose that the respective function Φ satisfies the assumptions of Theorem 1 of [2]. Then, for any function f ∈ (L

(G))

∩ Dom T,

%(a(T

f − f )) −→ 0

for sufficiently small a > 0.

References

[1] C. B a r d a r o, J. M u s i e l a k and G. V i n t i, Modular estimates and modular conver- gence for a class of nonlinear operators, Math. Japon. 39 (1994), 7–14.

[2] —, —, —, On absolute continuity of a modular connected with strong summability , Comment. Math. Prace Mat. 34 (1994), 21–33.

[3] C. B a r d a r o and G. V i n t i, Modular approximation by nonlinear integral operators on locally compact groups, ibid., to appear.

[4] J. M u s i e l a k, Nonlinear approximation in some modular function spaces. I , Math.

Japon. 38 (1993), 83–90.

[5] —, On the approximation by nonlinear integral operators with generalized Lipschitz kernel over a locally compact abelian group, Comment. Math. Prace Mat. 34 (1995), 153–164.

[6] —, On some linearly indexed families of submeasures, to appear in Atti del Convegno

“Real Analysis and Measure Theory” (Ischia July 1–6, 1994) and in Atti Sem. Mat.

Fis. Univ. Modena.

DIPARTIMENTO DI MATEMATICA FACULTY OF MATHEMATICS

UNIVERSIT `A DEGLI STUDI AND COMPUTER SCIENCE

VIA VANVITELLI, 1 ADAM MICKIEWICZ UNIVERSITY

06123 PERUGIA, ITALY MATEJKI 48/49

E-mail: MATEVIN@IPGUNIV.BITNET 60-769 POZNA ´N, POLAND

Re¸ cu par la R´ edaction le 1.12.1994

R´ evis´ e le 26.4.1995