Diana Caponetti University of Palermo, Italy Joint work with G. Lewicki Integration Vector Measures and Related Topics VI Bȩdlewo, June 15-21, 2014
Aim of this talk is to introduce modular function spaces as defined by Kozlowoski, to present some examples and prove admissibility of the spaces.
Introduction Orlicz spaces generalize Lebesgue spaces by considering spaces of functions with some growth properties different from the power type growth control provided by the L p -norms.
Introduction Orlicz spaces generalize Lebesgue spaces by considering spaces of functions with some growth properties different from the power type growth control provided by the L p -norms. Further development: The theory of Banach function spaces [Luxemburg, Zaanen]. The theory of modular spaces [Nakano] [Orlicz, Luxemburg] obtained by replacing the given integral form of the nonlinear functional which controls the growth of the functions, by an abstract functional, the modular ρ.
Introduction Orlicz spaces generalize Lebesgue spaces by considering spaces of functions with some growth properties different from the power type growth control provided by the L p -norms. Further development: The theory of Banach function spaces [Luxemburg, Zaanen]. The theory of modular spaces [Nakano] [Orlicz, Luxemburg] obtained by replacing the given integral form of the nonlinear functional which controls the growth of the functions, by an abstract functional, the modular ρ. Here we consider a class of modular spaces given by modulars not of a particular form but having much more convenient properties than the abstract modulars can possess.
Definitions and notations [W.M. Kozlowski, Modular function spaces, 1988 ] X nonempty set - P nontrivial δ-ring of subsets of X - Σ the smallest σ-algebra of subsets of X s.t. P Σ. Assume E A P for E P and A Σ, and X = n=1 X n where X n X n+1 and X n P for any n N.
Definitions and notations [W.M. Kozlowski, Modular function spaces, 1988 ] X nonempty set - P nontrivial δ-ring of subsets of X - Σ the smallest σ-algebra of subsets of X s.t. P Σ. Assume E A P for E P and A Σ, and X = n=1 X n where X n X n+1 and X n P for any n N. (W, ) Banach space. E the linear space of all P-simple functions. M(X, W ) the set of all measurable functions. f : X W measurable if s n (x) f (x) for any x X, with {s n } P-simple functions
A functional ρ : E Σ [0, + ] is called a function modular if it satisfies the following properties:
A functional ρ : E Σ [0, + ] is called a function modular if it satisfies the following properties: (1) ρ(0, E) = 0 for every E Σ, (2) ρ(f, E) ρ(g, E) whenever f (x) g(x) for all x E and any f, g E (E Σ); (3) ρ(f, ) : Σ [0, + ] is a σ-subadditive measure for every f E; (4) ρ(α, A) 0 as α R + decreases to 0 for every A P, where ρ(α, A) = sup{ρ(rχ A, A) : r W, r α}; (5) there is α 0 0 such that sup β>0 ρ(β, A) = 0 whenever sup α>α0 ρ(α, A) = 0; (6) ρ(α, ) is order continuous on P for every α > 0, that is ρ(α, A n ) 0 for any sequence {A n } P decreasing to.
A functional ρ : E Σ [0, + ] is called a function modular if it satisfies the following properties: (1) ρ(0, E) = 0 for every E Σ, (2) ρ(f, E) ρ(g, E) whenever f (x) g(x) for all x E and any f, g E (E Σ); (3) ρ(f, ) : Σ [0, + ] is a σ-subadditive measure for every f E; (4) ρ(α, A) 0 as α R + decreases to 0 for every A P, where ρ(α, A) = sup{ρ(rχ A, A) : r W, r α}; (5) there is α 0 0 such that sup β>0 ρ(β, A) = 0 whenever sup α>α0 ρ(α, A) = 0; (6) ρ(α, ) is order continuous on P for every α > 0, that is ρ(α, A n ) 0 for any sequence {A n } P decreasing to. For f M(X, W ) ρ(f, E) = sup{ρ(g, E) : g E, g(x) f (x) x E}.
A set E Σ is said to be ρ-null if and only if ρ(α, E) = 0 for every α > 0.
A set E Σ is said to be ρ-null if and only if ρ(α, E) = 0 for every α > 0. Then the functional ρ : M(X, W ) [0, + ] defined by ρ(f ) = ρ(f, X ) is a semimodular, that is ρ(λf ) = 0 for any λ > 0 iff f = 0 ρ-a.e.; ρ(αf ) = ρ(f ) if α = 1 and f M(X, W ); ρ(αf + βg) ρ(f ) + ρ(g) if α + β = 1 (α, β 0) and f, g M(X, W ).
Given the semimodular ρ we consider the modular space L ρ = {f M(X, W ) : lim ρ(λf ) = 0}, λ 0 + and we endowed it by the F -norm { ( ) } f f ρ = inf λ > 0 : ρ λ. λ
Given the semimodular ρ we consider the modular space L ρ = {f M(X, W ) : lim ρ(λf ) = 0}, λ 0 + and we endowed it by the F -norm { ( ) } f f ρ = inf λ > 0 : ρ λ. λ Let E ρ the closed subspace of L ρ defined by E ρ = {f M(X, W ) : ρ(αf, ) is order continuous for every α > 0}. For any set S in E ρ we denote by cls the closure of S with respect to ρ. We recall that E ρ = cle.
E ρ = L ρ if and only if the function modular ρ satisfies the 2 -condition:
E ρ = L ρ if and only if the function modular ρ satisfies the 2 -condition: sup ρ(2f n, A k ) 0 k, n whenever {f n } n M(X, W ), A k Σ, A k and sup n ρ(f n, A k ) 0 as k.
Examples Musielak-Orlicz spaces.
Examples Musielak-Orlicz spaces. (X, Σ, µ) measure space. Let φ : X R + R + be a φ 1 -function 1. φ(x, ) is continuous for every x X 2. φ(x, 0) = 0 for every x X 3. φ(x, u) as u 4. φ(, u) is locally integrable for every u 0 5. There exists u 0 0 such that φ(, u) > 0 for u > u 0
Examples Musielak-Orlicz spaces. (X, Σ, µ) measure space. Let φ : X R + R + be a φ 1 -function 1. φ(x, ) is continuous for every x X 2. φ(x, 0) = 0 for every x X 3. φ(x, u) as u 4. φ(, u) is locally integrable for every u 0 5. There exists u 0 0 such that φ(, u) > 0 for u > u 0 ρ(f, E) = E φ(x, f (x) )dµ
The function modular ρ satisfies the 2 -condition iff
The function modular ρ satisfies the 2 -condition iff Orlicz 2 -condition holds: There exists K > 0 and an integrable g : X R + such that for every u 0 φ(x, 2u) Kφ(x, u) + g(x) ρ-a.e.
A generalization of Musielak-Orlicz spaces. M family of measures on (X, Σ). ρ(f, E) = sup φ(x, f (x) )dµ µ M E
A generalization of Musielak-Orlicz spaces. M family of measures on (X, Σ). ρ(f, E) = sup φ(x, f (x) )dµ µ M E Orlicz 2 condition + the set function g dµ is order continuous. sup µ M ( )
Admissibility The notion of admissibility, introduced by Klee, allows one to approximate the identity on compact sets by finite dimensional maps. Locally convex spaces are admissible. Not all nonlocally convex spaces are admissible [Cauty has provided an example of a metric linear space in which the admissibility fails].
Admissibility The notion of admissibility, introduced by Klee, allows one to approximate the identity on compact sets by finite dimensional maps. Locally convex spaces are admissible. Not all nonlocally convex spaces are admissible [Cauty has provided an example of a metric linear space in which the admissibility fails]. Definition Let E be a Haudorff topological vector space. The space E is said to be admissible if for every compact subset K of E and for every neighborhood V of zero in E there exists a continuous mapping H : K E such that dim(span [H(K)]) < + and f Hf V for every f K.
Main result Theorem The space E ρ is admissible.
Main result Theorem The space E ρ is admissible. K a compact set in E ρ and ε > 0. We consider H : K E ρ H = H ε P k T a F n. 1. Set F n f = f χ Xn, where X = n=1 X n. Find n N such that 2. Find a > 0 satisfying sup{ F n f f ρ : f K} ε. sup{ T a F n f F n f ρ : f K} ε.
3. Let Π k = {E 1,...E l }, Π n = {F 1,..., F m } and Π k Π n. Define P k,n : S Πn S Πk P k,n s = l i=1 mi j=1 w j m i χ Ei. where s = m j=1 w jχ Fj, and E i = m i j=1 F ij, with F ij Π n, for j = 1,..., m i.
3. Let Π k = {E 1,...E l }, Π n = {F 1,..., F m } and Π k Π n. Define P k,n : S Πn S Πk P k,n s = l i=1 mi j=1 w j m i χ Ei. where s = m j=1 w jχ Fj, and E i = m i j=1 F ij, with F ij Π n, for j = 1,..., m i. Q = {Π n } sequence of partitions of X with Π 1 = {X } and Π 1 Π 2...Π n... Then for any k N, define P k : S(Q) = j=1 S Π j S Πk P k s = { s if s k j=1 S Π j P k,n s if s j=k+1 S Π j.
For f cl(s(q)) such that sup{ f (x) : x X } a <, for any k N, define P k f = lim n P k s n, {s n } S(Q), f s n ρ 0, the limit in the ρ -norm.
Lemma For f cl(s(q)) such that sup{ f (x) : x X } a <, for any k N, define P k f = lim n P k s n, {s n } S(Q), f s n ρ 0, the limit in the ρ -norm. Let {f n } be a sequence in cl(s(q)) and f cl(s(q)). If sup{ f n (x) : x X, n N} a <, sup{ f (x) : x X } a < and f f n ρ 0, then lim n (sup{ P k f n P k f ρ : k N}) = 0.
Theorem Let K a be a compact subset of E ρ such that sup{ f (x) : x X, f K a } a <. Then there exists a sequence Π 1 Π 2...Π n..., which will be denoted by Q, of partitions of X such that Π 1 = {X } and K a cl(s(q)). Moreover, for any ε > 0 there exists k N such that sup{ P k f f ρ : f K a }) ε, being {P k } be the sequence of operators corresponding to the sequence of partitions Q.
4. The space of P-simple functions generated by a given partition of X is admissible.
4. The space of P-simple functions generated by a given partition of X is admissible. Proposition Let Π = {A 1,, A n } be a partition of X. Then the subspace S Π = { s E ρ : s = n w i χ Ai, i=1 } w i W of E ρ is admissible.
Proof. K a compact set in E ρ and ε > 0.
Proof. K a compact set in E ρ and ε > 0. 1. 2. sup{ F n f f ρ : f K} ε. sup{ T a F n f F n f ρ : f K} ε.
Proof. K a compact set in E ρ and ε > 0. 1. 2. sup{ F n f f ρ : f K} ε. sup{ T a F n f F n f ρ : f K} ε. 3. Considering (T a F n )(K) we have sup{ P k T a F n f T a F n f ρ : f K} ε. 4. V = (P k T a F n )(K) is a compact subset of S Πk. Let H ε : V E ρ such that span[h ε (V )] is finite-dimensional and sup{ H ε P k T a F n f P k T a F n f ρ : f K} ε.
Now we consider H = H ε P k T a F n. We have dim[span[h(k)] <. Moreover, by the above facts, for any f K f Hf ρ ε. and the admissibility of E ρ is proved.
Corollary Let T : L ρ E ρ be a compact and continuous mapping. Then there exists f E ρ such that Tf = f.
Some references D.C. and G. Lewicki, A note on the admissibility of modular function spaces, J. Nonlinear Conv. Anal., to appear O. Hadic, Fixed point theorems in not necessarily locally convex topological vector spaces. Functional analysis, 118-130 (Dubrovnik, 1981), Lecture Notes in Math., 948, Springer, Berlin-New York (1982). J. Ishii, On the admissibility of function spaces, J. Fac. Sci. Hokkaido Univ. Series I, 19, 49-55 (1965). M. A. Khamsi and W. M. Koz lowski, On asymptotic pointwise nonexpasive mappings in modular function spaces, J. Math. Anal. Appl., 380 no. 2, 697-708 (2011). M. A. Khamsi, W. M. Koz lowski and S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal., 14 no. 11, 935-953 (1990). W.M. Koz lowski, Modular function spaces, Series of Monographs and Textbooks in Pure and Applied Mathematics, Vol. 122, Dekker, New York/Basel 1988.
V. Klee, Leray-Schauder theory without local convexity, Math. Ann., 141, 286-296 (1960). C. Krauthausen, Der Fixpunktsatz von Schauder in nicht notwendig konvexen Räumen, sowie Anwendungen auf Hammersteinsche Gleichungen, Dissertation, Aachen (1976). J. Mach, Die Zulässigkeit und gewisse Eigenshaften der Funktionenräume L φ,k und L φ, Ber. Ges. f. Math. u. Datenverarb. Bonn, 1, 38 pp. (1972). T. Riedrich, Der Räum S(0, 1) ist zulässig, Wiss. Z. Techn. Univ. Dresden, 13, 1-6 (1964). T. Riedrich, Die Räume L p (0, 1) (0 < p < 1) sind zulässig, ibid. 12, 1149-1152 (1963). T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, 3, Berlin (1996).