CC BY
47Ученые записки Таврического национального университета им. В. И. Вернадского
Серия «Физико-математические науки» Том 27 (66) № 1 (2014), с. 234-246.
УДК 517.98
О. S. Kisel, J. S. Pashkova
COMPARISON OF ORLICZ, LORENTZ AND ORLICZ-LORENTZ SPACES
In this paper we investigate conditions imposed on Orlicz functions and Lorentz functions such that one Orlicz-Lorentz space is embedded to another or these spaces coincide. Similar results are showed for general rearrangement invariant spaces and, in particularly, for Orlicz and Lorentz spaces.
Keywords: rearrangement spaces, Orlicz-Lorentz spaces, comparison
Introduction
The theory of symmetric spaces dates back to the classical Lp spaces, 1 < p < ж. The theory was developed intensively during the last century; it contains many interesting and deep results that have important applications in various areas of the theory of functions and functional analysis. It applies in particularly, to the areas of interpolation of linear operators, ergodic theory, harmonic analysis and mathematical physics.
Last years much attention has been given to Orlicz-Lorentz spaces. From a general point of view It help us to investigate such rearrangement invariant spaces as Orlicz spaces, Lorentz spaces and Lp,q spaces. There are several ways of defining these spaces.
We can refer to [14], [15], [3], [4], [5], [6], [9] and to the references cited therein as well.
Our definition of Orlicz-Lorentz spaces is differ from another which is the basis in cited papers. In our opinion, it is more convenient to deal with issues related to the question embedding one Orlicz-Lorents space to another.
1. Preliminaries
Let ц be the Lebesgue measure on the positive semiaxis [0, ж), Lo the space of all ц,-measurable almost everywhere finite functions f on (0, ж), Lp, 1 < p < +ж — Banach space of functions from L0, an integrable p degree.
A Banach space E C Lo is called rearrangement invariant if
f e Lo ,g e E, f* < g* f e E , \\f ||E < \\g\\E •
Here f * denotes the decreasing right-continuous rearrangement of \f |. It can be defined as the right-continuous generalized inverse
f *(x) := inf{y e [0, +o): nf (y) < x}, x e [0, o)
of the distribution function nf of \f \, which is
nf (x) = n {u e (0, o): \ f (u)\ > x} ,
It is known (see [10], Ch. II, §4.1 or [12], Ch. 2.a), that for every rearrangement invariant space E there exist continuous inclusions
Li n L^ C E C Li + L^ C Lo-
Enclosed symmetric space. Let E1 and E2 — are two symmetric spaces.
Theorem 1. Let E1 C E2. Then the embedding
i : Ei 9 f ^ f e E2
is continuous (bounded).
Proof. Let {fn} is the sequence in the space E1, such that
\\fn - f \\ei ^ 0 and \\fn - g\\E2 ^ 0.
Then we have fn ^ f and fn ^ g in measure and f = i(f) = g. Thus the graph is closed for embedding operator i. In accordance with the theorem on a closed operator, the operator i is bounded (continuous). □
Remark 1. The continuity of the embedding E1 C E2 means that
\\f \\e2 < c\\f \\E!, f e E1.
for some c> 0.
As follows from the open mapping theorem, the space E1 is closed in the space E2 if and only if this embedding is open, i.e.
\\f \\E! < C1\f\\e2,f e E1.
for some c1 > 0.
2. Comparison of Orlicz spaces
Let $: [0, +to) ^ [0, +to] be an Orlicz function, i.e., $(0) =0, $ is increasing left-continuous and convex. Assume also that $ is nontrivial, i.e., $(x) > 0 and $(y) < to for some x,y > 0. The derivative $' exists a.e., and it is assumed to be left-continuous with $'(x) = +to iff $(x) = +to.
The Orlicz space L, is the set defined as follows
( 7 1
:= If € Lo: / $(f/a) d/ < to for some a > 0,
equipped with the norm
\\f \\l* :=inf|a> 0: j $(|f|/a) d/ < l| , f € Lo , where inf 0 := to.
Notice that this "slightly generalized" definition includes the spaces Li, L7 and also Li n L7 , L1 + L7 as the smallest and largest Orlicz spaces ( see , [8], Ch. 2, [2], Ch. 2, §2.1 and also [16], [17], [18]).
The fundamental function of Orlicz space. Now we turn to the fundamental function (l* of the Orlicz space L$.
Proposition 1. The fundamental function (l* of the Orlicz space L, is:
(l*(x) = ($-1 (x"1))"1, x > 0. (1)
Proof. For all x > 0 and a > 0 we have:
7 X
x, (a. i[o,x])=/ $ Q • i[oxi)dm=/ $ (a)dm=x $ ( a) •
oo
Therefore
(L*(x) = \\1[o,x]\\l* =inf ja> 0: x$ ^^ < 1 j =
a
= f a > 0: a > ($-( i))"' } = ($-( i))"\
□
Corollary 1. The Orlicz function $ is uniquely reconstructed from the fundamental function (l* of Orlicz space L, that is
$"1(x) = ((L* (, x> 0,
and $ = ($_1)_1 is the inverse function of the function $_1.
The embedding L$1 C L$2 of Orlicz space L$1 in the Orlicz space L$2 can be described in terms of corresponding Orlicz functions $1 and $2.
Recall, that the embedding space L$1 in the space L$2 is always bounded, i.e.
< c\\f ||l4i, f £ L$1, for some c> 0 (Theorem 1).
Definition 1. Let $1 h $2 are Orlicz functions. It say that
1). $1 is majorizes the $2 in 0 ($1 y0 $2), if there are exist positive numbers a, b, x0 such that the inequality
$2(x) < b $1(ax)
is satisfied for all 0 < x < x0.
2). $1 is majorizes $2 on oo ($1 y^ $2), if there are exist positive numbers a, b, x0 such that the inequality
$2(x) < b $1(ax)
is satisfied for all x > x0.
3). $1 is majorizes $2 ($1 y $2), if $1 y0 $2 and $1 y^ $2.
Remark 2. 1). We can take b = 1 in the conditions 1) and 2). 2). The condition y $2 has the form
$2(x) < b $1(ax), x > 0
for some b > 0 and a> 0.
Theorem 2. Let $1 and $2 are Orlicz functions, the functions and are
fundamental function of corresponding Orlicz spaces L$1 u L$2. Then the following conditions are equivalent:
$1 y $2; L$1 c L^2;
\\ • \\l$2 < a\\ • \\l41 for some a > 0; <Pl$2 < for some a > 0;
$2(x) < $1(ax) for some a > 0 and for all x > 0.
Proof. 1) 2). The condition $1 y $2 has the form:
$2(x) < b $1(ax), x > 0
for some b > 0 and a> 0.
If f € , then for some c> 0 we have
oo
( C)=/$1 ( f)dm <
0
Therefore
oo oo
£)=/$■ ( 0^)dm < bi* (f )dm=b C) <
00
i.e. f € L$2, whence L$1 Ç L$2.
2) 3). It is following from Proposition 1.
3) 4). We use the function f = 1[0^] in the condition 3). Thus we have 4).
4) 5). We use the formula (1) for the fundamental function of the Orlicz space. Thus
1 (x"1))"1 < a ($r (x"1))"1, x > 0. Suppose x"1 = $2(y), then
($2(y)) < a($2(y)) = ay, y > 0
or
$2(y) < $1(ay), y> 0.
5) 1). It is obvious. □
Definition 2. Two Orlicz functions $1 and $2 are called equivalent ($1 « $2), if the conditions $1 y $2 and $2 y $1 are hold.
Corollary 2. Let $1 and $2 are Orlicz functions. Then the following conditions are equivalent:
1). $1 « $2/
2). L$1 = L$2 (as sets);
3). Norms || • ||l$1 and || • ||l^2 are equivalent, i.e.
a1\\f ||l41 <wf ||l*2 < a2|f||l41 for all f and some a1 > 0, a2 > 0;
4). Fundamental functions and are equivalent, i.e.
ai^Lj, (x) < <£l42 (x) < (x)
for all f and some ai > 0, a2 > 0; 5). Functions and $2 are equivalent in the next sense:
$i(aix) < $2(x) < $i(a2x), x > 0
for some ai > 0 and a2 > 0.
Remark 3. Constants a1 and a2 in all conditions 3), 4) and 5) of previous Theorem 2 are the same. For example, the condition 5) is equivalent of the condition
-$2%) < ^21(v) < -^(y), y > 0
a2 ai
(by y = $2(x)). So
"I (^ (X))2i < (-1 (X))2i < «2 (X))21..> 0,
i.e.
(x) < (x) < a2^l$1 (x), x> 0.
3. Comparison of Lorentz spaces
Let W be an increasing function on [0, +to) such that: W(0) =0, W is concave on (0, +to), and W(x) > 0 for some x > 0. Then W is absolutely continuous on the open interval (0, to) with the decreasing density function W'(x), x > 0, while W(0+) may be positive.
The Lorentz space AW is defined as
Aw := {f € Lo: \\f ||aw. < +to}
with the norm
Av
œ œ
:=y f * (x) dW (x) = f *(0)W (0+)^ f * (x) W'(x) dx < to
where +to-0 = 0 (see [10], Ch. II, §5.1, and also [12], Ch. 2, [13] and references therein.
oo
The Stieltjes integral / f*(x) dW(x) has an atomic part f*(0) W(0+) in the case o
W(0+) > 0. The Lorentz spaces are maximal rearrangement invariant spaces with respect to the norm \\ • \\Aw.
The norm \\f \\Aw of function f e Aw can be written as
\aw = j W (nif |(x))dx = y W o Vf * dm. (2)
0 0 Indeed, using the substitution x = nf * (y), y = f *(x), we get:
Aw
0 0
œ
= J f *dW = f * (x)W (x)iœ - j W (x)df *(x) =
= ^ W (x)df*(x) = j W (nf* (y))dy = j W o nf* dm.
œ 0 0
By this definition Aw C Lœ if W(0+) > 0, and Aw D Lœ if W(+to) := lim W (x) < +to. Whence Aw = L7 if both the conditions W (0+) > 0 and W(+to) < +to hold.
We calculate the fundamental function of Lorentz space AW:
œ x
PAW (x) = J( 1 [o,x])*dW = J dW = W(x) 00 for all x > 0 and pAw(0) = W(0) = 0, i.e.,
PAW = W. (3)
Theorem 3. Let W1 u W2 are two Lorentz function. Then the following conditions are equivalent:
1). AWi C aw2 ;
2). W2(x) < cWi(x) for all x > 0 and some c> 0.
Proof. 1 ) 2). Let AWl C AW2. Then, there exists c> 0 such that
\\f \\aw2 < c\\f \\aw1 ,
(from Proposition 1). Therefore
W2(x) = PAw2 (x) = \\1m\\aw2 < c\ 1[0,x] \\Aw1 = cPAw1 (x) = cW1(x).
2) 1). Backwards, let W2(x) < cW1(x) for all x > 0 and some c > 0 and
f e Aw1. Then
œ
\\f \\aw1 = J W1 (n|f |(x))dx < to, 0
(from formula (2)).
Therefore
oo oo
\\f ||aW2 = J W2(n\f\{x))dx < cj Wi(n\f\{x))dx < to, 0 0 and f € Aw2. Thus Awi C Aw2.
□
Corollary 3. Let W1 and W2 are two Lorentz functions. Then, the following conditions are equivalent
1). Awi = AW2;
2). cl Wl(x) < W2(x) < c2 Wl(x) for all x > 0 and some cl, c2 > 0.
4. Comparison of Orlicz-Lorentz spaces The rearrangement invariant spaces A*,w, can be defined by
A*,w := {f € L0: I*,w(f/a) < to for some a > 0}
with the norm
\\f \\a4iW := inf {a > 0: I*,w(f/a) < 1} ,
where
oo
w (f) := J Ф^ *(x)) dW (x) , f & Lq.
0
The functions $ and W is Orlicz and Lorentz functions consequently.
We can refer to [14], [15], [3], [4], [5], [6], [9] and to the references cited therein as well.
Remark 4. As f € A*, w if and only if when exist such a> 0 that
f
(£) = j.(m)dw (x) = ||.( m)
(a)
1ф , W — = / Ф
a ) J \ a
q
< то,
Aw
so
' f *
f & ЛФ, w ^^ Ф ( a ) & Аф,w для некоторого a > 0.
We obtain the fundamental function of Orlicz-Lorentz space A*, w.
Proposition 2. The fundamental function w of Orlicz-Lorentz space A* ,w is:
W(x)
1 ^
^..w(x) = (ф-1 (W^)) , x> (4)
Proof. For any x > 0 and a > 0 we have:
œ x
I* w w ( a • 1[o, x]) = / $ ( a • 1[0, x](t)) dW (t) = j $ ( ^ dW (t) = W (x) $ ( ^.
So
1
PA*,W (x) = \\1[0, x]\a4,w = inf { a> 0 : W (x) $ < 1 j
f a>0:a >(-1 ( wW ))2] =(-1 ( wW ))
□
Theorem 4. Let and $2 are Orlicz functions, W is Lorentz functions, the functions w and Pa$2 w are fundamental functions of corresponding Orlicz-Lorentz spaces A$1, w and A$2)w ■ Then the following conditions are equivalent:
1). $i ^ $2;
A$i,w q a$2, w;
\\ • wa^w < c\\ • \\A4i,W for some c >0;
Pa$2,w < cPa$1,w for some c > 0;
$2(x) < $i(cx) for some c > 0 and for all x > 0.
Proof. 1) 2). The condition $1 >- $2 has the form:
$2(x) < b $1(ax), x > 0
for some b > 0 and a> 0.
If f e A*1 w w, then for some c> 0 we have
00
w (f^j = / $1 (ç) dw < to.
0
Therefore
o o
I*2, w ^a^J = J $2 ^^ dW < b j $i ^f^ dW = b I®-,, w ^ fj < to, oo i.e. f € A$2 ,w , whence A$1, W Q A$2, W.
2) 3). It is follows from 1.
3) 4). Let \\f waj2,w < c\\f ||aj1,w for some c > 0 and for any f € A$i,w. Using the function f = 1[0,x], we get:
Pa$2,W(x) = w1[0,^]\a$2,W < cWl[0,x]WAsl,w = cPA$1,w(x)
4) 5). We use the formula (4) for the fundamental function of Orlicz-Lorentz
space A$)W, we have:
M wm))" < cM wW))" ■ x> 0
Suppose —= t, then W (x)
^(t))-1 < c (i-1 (t))-1, t> 0, therefore
$-1(t) < c$-1(t).
Suppose t = $2(y). Then
($2(y)) < c $-1 ($2(y)) = cy, y > 0
or
$2(y) < $1(cy), y> 0. 5) 1). It is obvious. □
Corollary 4. Let $1 and $2 are Orlicz functions, W is Lorentz function. the functions W and w are fundamental functions of corresponding Orlicz-Lorentz spaces )w and A$2)w ■ Then the following conditions are equivalent:
1). $1 « $2/
2). A$1 ,w = A$2,w (as sets);
3). Norms || • ||a$1 w u || • || a$2 w are equivalent, 'i.e.
a1||f ||a®1 ,W < Wf ||a®2W < a2||f ||a®1 , W
for all f and some a1 > 0, a2 > 0;
4). Fundamental functions w and w are equivalent, i.e.
a1(PA$1, w (x) < <PA®2 ,w (x) < a2^A$1 ,w (x)
for all f and some a1 > 0, a2 > 0;
5). Functions $1 and $2 are equivalent in the next sense:
$1(a1x) < $2(x) < $1(a2x), x > 0 for some a1 > 0 and a2 > 0.
Theorem 5. Let W1 and W2 are Lorentz functions and $ is Orlicz function.
1). If W2(x) < cWi(x) for all x > 0 and some c > 0, then A$)Wl C A$)W2;
2). If the space A$)w1 is normally embedded in the space A$)w2 with embedded constant c < 1, mo W2(x) < c W1(x) for all x > 0.
Proof■ 1) 2). As the space A$,Wl is normally embedded in the space A$,W2, so
\\f \a ,w2 < c wf was
where c 1 . Therefore
PA$,w2 (x) = PMUSw < cWl[0,^]|Aj,wl = cPA$,wl (x),
i.e.
($-I(
1
-i
<c[ $
H
1
Consequently,
,W2(x)
\Wi(x)J " \W2(x)J
The function $ is increasing convex on (0, to). So
Wi(x) 1
-i
$ ^$-i ^
1
Wi(x)
$
(c$-I( W«)) < -H
W2(x)
i.e.,
11
< c-
Whence
for all x 0.
Wi(x) _ W2(x)' W2 (x) < cWi(x)
2) 1). Conversely, let W2(x) < cWi(x) for all x > 0 and some c > 0 and let
f € A$,Wl. So $ ^a^J € AWl for some a > 0 and, using the formula (2),
Therefore
f 11», = ! wi( M a )(x0 ^
o o
(a) |aw = / W2 (M if ){x))< c/wi (M a )(x0dx <
a
20
whence f € A$,W2. Thus, A$,Wl Q A$,W2.
□
i
1
Corollary 5. Let $i u $2 are Orlicz functions, and Wi u W2 are Lorentz functions. If
1). $i ^ $2;
2). ciWi(x) < W2(x) < c2 Wi(x) for all x > 0 and some ci,c2 > 0, so a$1,w1 = a$2,w2;
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Сравнение пространств Орлича, Лоренца и Орлича-Лоренца
В работе рассматриваются условия на функции Орлича и функции Лоренца, при выполнении которых одно пространство Орлича-Лоренца содержится в другом или когда эти пространства совпадают. Приведены аналогичные результаты для общих перестановочно инвариантных пространств и, в частности, для пространств Орлича и Лоренца.
Ключевые слова: перестановочно инвариантные пространства, пространства Орлича-Лоренца, сравнение пространств.
Пор1вняння простор1в Орл1ча, Лоренца та Орл1ча-Лоренца
В роботг розглядаються умови на функцгг Орлгча та на функцгг Лоренца, при виконаннг яких один простгр Орлгча-Лоренца е у гншому або коли цг простори спгвпадають. Наведено аналоггчнг результати для загальних перестановочно гнваргантних просторгв та, зокрема, для просторгв Ор-лгча та Лореца.
Ключов1 слова: перестановочно швар1антш простори, простори Орл1ча-Лоренца, пор1вняння простор1в.
