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Владикавказский математический журнал 2016, Том 18, Выпуск 1, С. 21-25
yffK 517.98
TWO MEASURE-FREE VERSIONS OF THE BREZIS-LIEB LEMMA
E. Yu. Emelyanov, M. A. A. Marabeh
Dedicated to Professor A. E. Gutman on the occasion of his 50th anniversary
We present two measure-free versions of the Brezis-Lieb lemma for uo-convergence in Riesz spaces.
Mathematics Subject Classification (2010): 28A20, 46E30, 46B42.
Key words: Brezis-Lieb lemma, uniformly integrable sequence, Riesz space, uo-convergence, almost order bounded set, auo-continuous mapping.
1. Introduction
The Brezis-Lieb lemma [2, Theorem 2] has numerous applications mainly in calculus of variations (see, for example [3, 6]). We begin with its statement. Let j : C ^ C be a continuous function with j(0) = 0. In addition, let j satisfy the following hypothesis: for every sufficiently small e > 0, there exist two continuous, nonnegative functions ip£ and such that
|j(a + b) - j(a)| < ei£(a) + ^(b) (1)
for all a, b e C. The following result has been stated and proved by H. Brezis and E. Lieb
in [2].
Theorem 1.1 (Brezis-Lieb lemma [2, Theorem 2]). Let (fi, X, be a measure space. Let the mapping j satisfy the above hypothesis, and let fn = f + gn be a sequence of measurable functions from fi to C such that:
(i) gn 0;
(ii) jo f e L1;
(iii) f ip£ o gnd^ ^ C < to for some C independent of e and n;
(iv) / o fdy < to for all e > 0. Then, as n ^ to,
J (j (f + gn) - j(gn) - j (f ))d^ ^ 0. (2)
Here we reproduce its proof from [2, Theorem 2] with several simple remarks. < Fix e > 0 and let W£,n = [| j o f,n - j o g,n - j o f | - ei o gn]+. As n ^ to, W£,n -- 0. On the other hand,
|j o fn - j o gn - j o f | < |j o fn - j o gn| + |j o f | < ei o gn + o f + |j o f |.
© 2016 Emelyanov E. Yu., Marabeh M. A. A.
1 This research was funded by Middle East Technical University BAP, research project № BAP-01-01-
2016-001.
and thus
Therefore, 0 ^ W£;n ^ o f + |j o f | g L1. By dominated convergence,
lim / = 0. (3)
n—d J '
However,
|j o fn - j o gn - j o f | ^ We ,n + o gn (4)
In := J |j o fn - j o gn - j o f |du ^ J[We,n + £<£>£ o gnjd^u.
Consequently, lim sup In ^ eC. Now let e ^ 0. >
Remark 1.1. (i) The conditions (3) and (4) mean that the sequence |j o fn - j o gn| lies eventually in the set [—|j o /1, |j o /1] + ^f-BLi, where is the unit ball of L1. In other words, the sequence j o fn - j o gn is almost order bounded.
(ii) The superposition operator Jj : L0 ^ L0, Jj(f ) := j o f induced by the mapping j in the proof above can be replaced by a mapping J : L0 ^ L0 satisfying some reasonably mild conditions for keeping the statement of the Brezis-Lieb lemma.
(iii) Theorem 1.1 is equivalent to its partial case when the C-valued functions are replaced by R-valued ones.
The following proposition is motivated directly by the proof of [2, Theorem 2].
Proposition 1.2 (Brezis-Lieb lemma for mappings on L0). Let (fi, X,^) be a measure space, fn = f + gn be a sequence in L0 such that gn 0, and J : L0 ^ L0 be a mapping satisfying J(0) = 0 and such that the sequence J(fn) - J(gn) is almost order bounded. Then
lim f (J(f + gn) - (J(gn) + J(f))) d^ = 0. (5)
n—J
< As in the proof of the Brezis-Lieb lemma above, denote /n := / | J(f + gn) — (J(f) + J(gn))| d^. By the conditions, the sequence
J (f + gn) — (J (f ) + J (gn)) = ( J (fn) — J (gn ))) — J (f )
a.e.-converges to 0 and is almost order bounded. Therefore, by the generalized dominated convergence, lim In = 0. >
Since almost order boundedness is equivalent to uniform integrability in finite measure spaces, the following corollary is immediate.
Proposition 1.3 (Brezis-Lieb lemma for uniform integrable sequence J(fn) — J(gn)). Let (fi, X, be a finite measure space, fn = f + gn be a sequence in L0 such that gn 0, and J : L0 ^ L0 be a mapping satisfying J(0) = 0 and such that the sequence J(fn) — J(gn) is uniformly integrable. Then
lim f(J(f + gn) — (J(gn) + J(f)))d^ = 0.
(6)
2. Two variants of the Brezis—Lieb lemma in Riesz spaces
Recall that a sequence xn in a Riesz space E is order convergent (or o-convergent, for short) to x £ E if there is a sequence zn in E satisfying zn | 0 and |xn — x| ^ zn for all n £ N (we write xn — x). In a Riesz space E, a sequence xn is unbounded order convergent (or uo-convergent, for short) to x £ E if |xn — x| A y — 0 for all y £ E+ (we write xn —— x).
Here we give two variants of the Brezis-Lieb lemma in Riesz space setting by replacing a.e.-convergence by uo-convergence, integral functionals by strictly positive functionals and the continuity of the scalar function j (in Theorem 1.1) by the so called a-unbounded order continuity of the mapping J : E — F between Riesz spaces E and F. As standard references for basic notions on Riesz spaces we adopt the books [1, 7, 8] and on unbounded order convergence the papers [4, 5].
It is well known that if (fi, be a a-finite measure space, then in Lp (1 ^ p ^ to), uo-convergence of sequences is the same as the almost everywhere convergence (see, for example [5]). Therefore, in order to obtain versions of Brezis-Lieb lemma in Riesz spaces, we shall replace the a.e.-convergence by the uo-convergence.
A mapping f : E — F between Riesz spaces is said to be a-unbounded order continuous (in short, auo-continuous) if xn ^ x in E implies f (xn) —> f (x) in F. Clearly this definition is parallel to the well-known notion of a-order continuous mappings between Riesz spaces.
Let F be a Riesz space and l be a strictly positive linear functional on F. Define the following norm on F:
II x ||j:= l(|x|). (7)
Recall that a Banach lattice E is said to be order continuous if every order null net is norm null, and a subset A of E is said to be almost order bounded if for any e > 0 there exists ue £ E+ such that A c [—ue, ue] + eBE, where is the closed unit ball in E. We say that a net xa is almost order bounded if the set of its members is almost order bounded.
The next lemma will be used to prove a version of Brezis-Lieb lemma for arbitrary strictly positive linear functionals.
Lemma 2.1 (See [5, Proposition 3.7]). Let X be an order continuous Banach lattice. If a net xa is almost order bounded and uo-convergent to x, then xa converges to x in norm.
Suppose that F is a Riesz space and l is a strictly positive linear functional on F, then the || ■ ||i-completion (F;, || . |) of (F, || ■ ||j) is an AL-space, and so it is order continuous Banach lattice. The following result is a measure-free version of Proposition 1.2.
Proposition 2.2 (A Brezis-Lieb lemma for strictly positive linear functionals). Let E be a Riesz space and F; be the AL-space constructed above. Let J : E — F; be auo-continuous with J(0) = 0, and xn be a sequence in E such that:
(i) xn —- x in E;
(ii) the sequence (J(xn) — J(xn — x))n is almost order bounded in Fj. Then
lim || J(xn) — J(xn — x) — J(x) ||j = 0. (8)
< Since xn —— x and J is auo-continuous, then J(xn) —— J(x) and J(xn — x) —— J(0) = 0. Thus, J(xn) — J(xn — x) —— J(x). It follows from Lemma 2.1 that lim || J(xn) — J(xn — x) — J(x) ||j = 0. >
In the following Brezis-Lieb type lemma, the auo-continuity of mappings between Riesz spaces is used.
Proposition 2.3 (A Brezis-Lieb lemma for auo-continuous linear functionals). Let E, F be Riesz spaces, l a auo-continuous linear functional on F, J : E ^ F a auo-continuous mapping with J(0) = 0, and xn x in E. Then
lim l(J(xn) - J(xn - x) - J(x)) = 0 . (9)
n—d
< Since xn -^ x and J is auo-continuous, then J(xn) -J(x) and J (xn - x) -^ J (0) = 0. Thus, (J (xn ) - J (xn - x) - J (x)) 0. But l is auo-continuous, so l( J (xn) - J (xn - x) - J (x)) 0. Since in R the uo-convergence, the o-convergence, and the standard convergence are all equivalent, then lim l(J(xn) - J(xn - x) - J(x)) = 0. >
n—d
Note that in opposite to Proposition 2.3, in Proposition 2.2 we do not suppose the functional l to be auo-continuous.
References
1. Aliprantis C. D., Burkinshaw O. Positive operators.—Orlando, Florida: Acad. Press, Inc., 1985.— xvi+367 p.—(Pure and Appl. Math. Vol. 119).
2. Brezis H., Lieb E. A relation between pointwise convergence of functions and convergence of functionals // Proc. Amer. Math. Soc.-1983.-Vol. 88, № 3.-P. 486-490.
3. Brezis H., Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents // Comm. Pure Appl. Math.-1983.-Vol. 36, № 4.-P. 437-477.
4. Gao N., Troitsky V., Xanthos F. Uo-convergence and its applications to Ceskro means in Banach lattices.—Preprint, arXiv:1509.07914.
5. Gao N., Xanthos F. Unbounded order convergence and application to martingales without probability // J. Math. Anal. Appl.-2014.-Vol. 415, № 2.-P. 931-947.
6. Lieb E. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities // Ann. of Math.— 1983.—Vol. 118, № 2.-P. 349-374.
7. Luxemburg W. A. J., Zaanen A. C. Riesz Spaces. Vol. I.—Amsterdam: North-Holland Publ. Comp., 1971.—viii+514 p.
8. Zaanen A. C. Riesz Spaces II // North-Holland Mathematical Library.—Amsterdam: North-Holland Publ. Comp., 1983.—xi+720 p.
Received January 11, 2016.
Eduard Yu. Emelyanov
Middle East Technical University,
Department of Mathematics, Prof.
TURKEY, 06800, Ankara, Dumlupinar Bulvari, 1
E-mail: eduardOmetu. edu. tr;
Sobolev Institute of Mathematics,
Laboratory of Functional Analysis, leading researcher
4 Koptyug Avenue, Novosibirsk, 630090, Russia
E-mail: emelanov@math.nsc.ru
Mohammad A. A. Marabeh Middle East Technical University, Department of Mathematics, Ph.D. student TURKEY, 06800, Ankara, Dumlupinar Bulvari, 1 E-mail: mohammad. marabehOmetu. edu. tr
ДВА ВАРИАНТА ЛЕММЫ БРЕЗИСА - ЛИВА БЕЗ ИСПОЛЬЗОВАНИЯ СХОДИМОСТИ ПОЧТИ ВСЮДУ
Емельянов Э. Ю., Мараби М. А. А.
Рассматриваются две версии леммы Брезиса — Либа для -мо-сходимости в пространствах Рисса.
Ключевые слова: лемма Брезиса — Либа, равномерно интегрируемая последовательность, пространство Рисса, -мо-сходимость, почти порядково ограниченное множество, ст-мо-сходимость.
