ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I (1968)
A. Wa s z a k (Poznań)
Strong summability of functions in Orlicz metrics
1. In papers [16] and [17] I considered strong summability of se
quences to zero, applying a definition which is a generalization of that of Musielak and Orlicz [8], [9 ], [10], [13]. Here, I shall deal with the same kind of problems as in [16] but concerning strong summability of functions to zero by integral methods. Although the problems and the ideas of proofs are similar to those of [16], new difficulties arise when we apply integral transforms of functions in place of matrix transforms of sequences. The assumptions imposed on the kernel a(t, r) of the in
tegral transform are of importance. In this paper the assumptions on a(t, r) are chosen in such a manner that they imply continuity of the transforms and are satisfied for a number of particular kernels well known in the theory of summability.
I am glad to express my regards to Professor W. Orlicz for his valuable suggestions and numerous kind remarks.
1.1. The following notation will be used in this paper. X will denote the space of real, finite, measurable functions in < 0 , o o ) (strictly speaking, the space of classes of equivalence of such functions with respect to the relation of equality almost everywhere). Ш stands for the space of func
tions measurable and bounded in < 0 , o o ), X f for the space of measurable functions in < 0 , o o ) equal to zero up starting with a tfe< 0 , o o ). Measurable functions in < 0 , o o ) will be denoted x(t), y(t), z(t), ...; for the sake of brevity they will be written also as x , y , z, ... Instead of sup#(£), where t runs over the interval < 0 , o o ) , it will be written sup ж. Pointwise or uni
form convergence of xn(t) to x(t) in < 0 , o o ) as n -> o o will be denoted by xn -> x or xn x, respectively. If x , y e X , x < у means that x(t)
^ y ( t ) almost everywhere in < 0 , o o ). The relation < defines a partial order in X, and X is a linear order-complete lattice with respect to this order relation. The supremum or infimum of two elements x , y e X will be denoted by x v у or x л у, respectively; the symbols хг v x2 v ... v xn,
\Jxn, etc., are defined analogously. Here, (x v y)(t) — mp(x(t), y(t)), (x л y)(t) = mi(x(t), y{t)), etc.
116 A. W a s z a k
1.2. A 99-function is defined as a continuous, non-decreasing function
<p(u) for и > 0 such that 99(0) = 0, 9o(u) > 0 for u > 0 and <p(u) 00 as и -> со. 99-functions will be denoted by 99, гр, ..., and their inverse functions by <p_i, y>_i, . .., respectively. A 99-function <p is said to satisfy the condition (A 2) for large и if there are numbers Тс > 1 and w0 > 0 such that <p(2u) < Tccp{u) for и ^ щ (see [5]). The symbol cp{\x\) means the function 9?(l®(<)l)-
a(t, r) will mean always kernels of integral methods of summabil- ity. Throughout this paper it will be supposed that:
(1) The kernel a (t,r ) is defined for tfe<0, 00) and r > r* with r*
fixed for a given kernel, and is a measurable function of the variable t in <0, 00) for every r > r*,
(2) a(t, r) > 0 for almost every t and for every r > r*.
Besides the above two assumptions, we shall need from time to time also the following hypotheses:
(I) For every t0 > 0 there exist constants r0 and d > 0 such that r0 > t0 and a(t, r0) ^ d for ie<0, t0>.
(II) If arbitrary t ^ 0 and r > r* are given, then
a) there exists a constant c > 0 such that a (t,r ) < c for r > r
^ r*, t > t > 0,
b) for te <0, ty the kernel a (t,x) satisfies the Lipschitz-Holder condition with a constant L and an exponent a, 0 < a < 1, i.e. \a(t, r) —a(t, r0)\ < L |t—r0|a for r , r 0 < t , r , r 0e<r*, r> and for r, r0 ^ tj t , Tq c <(t , t)> i the constant L is independent of
t e <0, £> but may depend on t and r.
00
(III) If x(t) is a measurable function such that j a(t, r)cp[\x(t)\]dt < 00 0
OO
for r ^ r0 and j a(t, r)<p(\x(t)\)dt -» 0 as r - > o o ? then for every
0 OO
e > 0 and xx > r* there exists an In > 0 such that J u(£, t) 99 ( ( #) |) dt
h
< e for all re<r*, тх>.
СЮ
(I V) There exists a constant A > 0 such that j a(t, r)dt < A for т > r*.
0
(V) For an arbitrary finite interval <a, /3) <= <0 , 00), there holds J a(ź, r)dt -> 0 as r —> 00.
a
(VI ) a(t, r) = 0 for t > r ^ t* .
E e m a r k 1. Obviously, condition (У1) implies condition (III).
CO
E e m a r k 2. If f n{r) — J a(t, r)(p[\x(t)\)dt are continuous functions П
of г for r > r * , n — 0 , 1 , 2 , . . . , then (III) is satisfied, because f n(r) > / n+i(r),
f n[x) 0 as n -> oo, and it suffices to apply the Dini theorem; com
pare 1.2 .3.
R e m a rk 3 . It is clear that the condition (III) implies the integral remainders to converge to 0 uniformly in <t*, oo).
Let the kernel a(t, x) be given and let <p be a ^-function. The follow
ing notation will be used:
OO
<гф(т,ж) = f a(t, r)<p(\x(t)\)dt, о
X v — {x e X : а<р{х, x) < o o for r > r*; x) -> 0 for r -> o o}, X v = {x eX : hceXy for arbitrary Я > 0},
X* — {x eX : AxeXv for a certain Я > 0 } .
This notation does not point out the dependence on the kernel a(t, r) and the number t* explicitely, since they are both fixed in the sequel.
1.2.1. The characteristic function of the interval <a, /5) will be denoted %<щ3>. In order that x ^ e X * it is necessary and sufficient that
P ’/3
J a(t, r)dt -> 0 as r -> oo and f a(t, x)dt < o o for r > r*. In particular,
a a
a sufficient condition in order that every characteristic function of an interval belongs to X * is that a(t, x)z£ 0 as r -> o o for an arbitrary in
terval <a, /?> c <0 , o o ) and that (IY) holds. It is easily seen that step functions equal to 0 starting with some t0 > 0 belong to X* if and only
to h
if J a(t, x)dt < o o for x > г and J a(t, x)dt 0 as r -> o o in each in-
o о
terval <0 , t±} <= <0 , o o ). Moreover, a sufficient condition in order that such step functions belong to X* is that a(t, x) tend to 0 almost uniformly in <0, o o ) as x -> o o (i.e. uniformly in each finite subinterval of <0, o o ))
and that the condition (IV) holds.
1.2.2. If the kernel a(t, x) has the property (I) and if xeXy , then x is locally 99-integrable, i.e. j <p[\x{t)\) dt < 00 in each interval <0,/0>.
0
To prove this let us remark that
00 <0 *0
0<р(хо, x) = J a(t, x0)<p(\x(t)\) dt-^t J a(t, x0)<p(\x(t)\) dt ^ d J <p(\x(t)\)dt.
0 0 0
1.2.3. If the kernel a(t, x) has the properties (I), (II), (III), and if x e X ^ , then <rv(r, x) is a continuous function of x for r > x*.
Let x > x* and let т > r > r0 > x*. Given an s > 0, let us choose
0° __
l = In where J [a(t, x0) — a ( t , r)]<p(\x(t)\) dt < e for т0, те<т*, т ); by (III)
118 A. W a s z a k
this is possible. We may suppose Ji > r. The following estimations hold
T o
M ro> ®) — x )\ < / \a(t, r0) — a(t, x)\(p(\x(t)\)dt +
о
t h
+ | J (a(t,T0) — a(t,T))<p(\x(t)\)dt\+f \a{t, r0) — a(t, x)\<p(\x(t)\) dt +
T0 r
o o T 0
+ 1 / (a (t, ro) — T))ę>(|a?($)|) dt I < L\t— T0|a j <p(\x(t)\)dt +
h " о
t h oo
+ 2c J q>(\x{t)\)dt + L\T— r0JaJ <p(\x{t)\)dt + \j (a(t, r0) — a(t, x)) <p(\x(t)\) dt\.
t q t h
h
By 1.2.2, J <p(\x(t)\)dt < oo. Hence the right-hand side of the last in- o
equality is less than 3e for x sufficiently near to r0.
1.2.4. Let a(t, r) = g{r)f(tlx), where f{tjx) > 0, g{r) > 0, and g(r) is a bounded function in each finite interval <r*, r>.
If f(tjx) is for every t a non-decreasing function of the variable r for r > t*, then the condition (III) is satisfied.
If t < r 1? we have f(tfx) < / ( i / r 1). Moreover, the assumptions imply
< A7 for те<т*, r x>, where N = const. Thus, the following inequality holds, for sufficiently large h:
OO OO
J a(t, x)<p(\x(t)\) dt < N J fitjrjy^xit)]) dt < e
h h
for те<т*, Tj) and Ji ^ Tiq(s).
1.2.5. If a(t, r) is for every t > 0 a non-decreasing function of the variable r for x ^ r*, then the condition (III) holds.
If t* < г < tx, then a(t, x) < a(t, Tq). Hence, given e > 0 , the follow
ing inequality holds for sufficiently large h and for те<т*, r x>:
[ OO OO
j a(t, x)y(\x(t)\) dt < J a(<, T1)9>(|a?(/)|) < £.
Л ft
1.3.1. X* is a linear space and X ę a convex set in X *, assuming usual definitions of addition of functions and of multiplication of a func
tion by a scalar. In particular, if Х0х е Х ч for a Л0> 0 , then ЛхеХ^ for
o < i < ;.0.
1.3.2. The space X*v is a linear lattice with the same order-relation as X ; indeed, it is a sublattice of the lattice X . The lattice X* is c-order complete (conditionally complete in Birkhoff’s terminology), in fact, it is order-complete.
We may restrict ourselves to the case у ^ 0, Xi ^ 0 for i = 1 , 2 , ...
Let %i < у for i = 1, 2 , xif y eX *. It is to be shown that there exists x0 = \ZxieX*. There holds Xi(t) < y(t) for almost every t. Let x0 — sup#*,
i then #0 < y. Since there exists a A > 0 such that ХуеХщ, we have
00 00
a(p{r,Xy) = f a(t, r)cp[X\y{t)\] dt > J a{t, r)cp{X | iu0(/)|) dt = av{r, Xxf).
о 0
Thus we obtained <tv(t, Xx0) -> 0 as r -> o o , because a v { r , Xy) -> 0 as т -> o o .
Hence x0eX* and x0 is the least majorant in X *.
1.4.1. I f the kernel a (t ,r) satisfies the condition (IV), then for any two cp-functions 9о and ip there holds X* гл M = X* r> 31.
The proof is similar to that given in [13] and is based on the inequality
ip(my)
(i) f(u ) <4>(*l)3---7-r-fp(u)
<PW
where у > 0, 0 < и < my, and m is a positive number. The proof will be concluded if we show V* ^ M а X* r\ 31. If x e X * гл 31, we have XqXcX ^ for some X0 > 0. Moreover, given у > 0, we have X0\x{t)\ < ym for sufficiently large m and for almost all t. By (i), we have
r OO ip {my)
a { r ,X 0x )^ ip {y ) a(t, r)dt-\--- — - or (r, A0a?)
J <p(v)
^ jr Г \ I V(mrl)
< Kip{y)Ą--- ~— a9{x, X0x).
c p { y )
Hence ayj{r, X0x) < o o for т ^ r*, limsupor^fr, X0x) ^ Kip{y),
r~>oo
lim ov{r, X0x) = 0.
T—^00
1.4.2. I f the kernel a(t, r) satisfies the condition (IV), then X* r\ 31
= X ę гл 31 for an arbitrary cp-function cp.
It is to prove that V* ^ 31 c= X ę r\ 31. Let x e X * n 31, i.e. X0x e X ęQ for a X0 > 0, and xe3I. Writing ip(u) — cp{ujX0), Lemma 1.4.1 gives aw{ r , XQx) -> 0 as r -> oo. But av{r, X0x) — av{r, x) -> 0 as r -> oo; hence XeX<p r\ 31.
1.5. A function (p is called non-weaker than a function ip for large и (we write ip -§ cp) if there are constants c , b , l , к, щ > г 0 such that
cip{lu) < b(p{ku) for и > u0.
120 A. W a s z a k
A function cpis called equivalent to a function ip(we write ip ~ q>) if there are constants а, Ъ, с, kx, k2, l, u0 > 0 such that
a<p(kxu) < cip(lu) < b(p(k2u) for и > u0.
It is clear that cp rL у if and only if ip t? (p and p Ą ip, simultaneously (see [4], [5]).
1.6. Let us suppose the kernel a(t, r) satisfies the condition (IV). I f ip -4 <p, then X* <= X* and X 9 <= X w.
By the assumption, there exist constants k ,b , u0 > 0 such that ip(u) ^ Ъ<р{ки) for и ^ u0. Let x e X * , i.e. l^xeX ^ for some A0 > 0. Hence if t satisfies the inequality A0k~1x(t) > u0, then 1р[Л0к~гх (*)) < bp{f0x («)- We define the function x x as follows
Xi (t) x(t) for t satisfying XQk 1a?(<) < u0,
0 elsewhere.
Obviously, x xeX* and by 1.4.1, xxeX* гл M — X* гл M, i.e. x xeX *.
Moreover, the inequality
(i)
00
J a(t, r)ip[XQk~l \x{t)\) dt 0
00
a(t, r)(p(A0\x(t)\) dt о
implies x 2 = x — x xeX *. Finally, x = x x-\-x2eX *. The inclusion X v cz X v follow from (i) in a similar way.
2.1. Let the kernel a(t, r) satisfy the conditions (I), (II), (III). Then av( r ,x ) are continuous functions of r for r > r* (see 1.2.3). The condi
tion t, x) -> 0 as t o o implies sup av (r, x) < o o for x e X n . We define the following functional in X*:
Qę {x) = sup f a{t, r)(p[\x{t)\) dt
T > T * J
OO
for for
X e X<P 0 ’
XeX*(p\ X<Po'
This definition and 1.2.3 implies that qv{x) — oo in the following cases only: if av{r, x) = oo for some т > r*; if av(r, x) < oo for every т > т*, but sup a9(r, x) — oo and <t¥(t, x) does not tend to 0 as r -> oo; if sup crv(r, ж) < o o but av(t, x) does not tend to zero as т -> o o . Thus,
0v(a?) < oo if and only if x e X <PQ. By the assumptions (I), (II), (III) on the kernel a{t, r), the functional qv(x) is a modular in X* in the sense of
the definition given by W. Orlicz [12], [14], i.e. it satisfies the following conditions:
A. qv{x) = 0 if and only if x = 0, B. qv(xx) < qv{x2) if |aq| < |a?2|,
C. v x 2) < ev(aq) + Qy{x2) if x x ^ 0, x 2 > 0, D. qv{ Ix) -> 0 if Я -> 0.
We limit ourselves to proofs of C and D.
In order to prove C let us take x x, a?2eX*, x x > 0, x 2 > 0. The in
equality <p(xx v x z) ^ <р(х1) + (р(х2) implies
OO OO OO
(i) J a(t, r)<p[xx(t) v x 2(t)) dt < j a(t, r)<p[xx(t)) dt-\- J a(t, r)cp[x2{t)) dt.
0 0 0
If e.g. x X€X*\X,p , C is satisfied obviously. Now, let x 1, x 2ęX <PQ. Then the above inequality implies aę {r , x x v x 2) -> 0 as r -> 00. Hence Q(p{xx v x 2)
OO
— sup j a{t, x)(p{xx(t) v x 2(t))dt, and (i) implies the inequality C.
T > T * 0
In order to prove 33 let us suppose that x e X * , i.e. for some Я0 > 0. Hence there exists r0 such that a ^ r, l 0x) < e for т > т0, and so- Gy (т, Яа?) < £ for 0 < Я < Я0. But we have
a,p(r, lx) < £+ sup о-Дт, Яа?)
T Q ^ O T *
and so in order to prove D it is sufficient to show that sup ^ ( r , lx) < e
T * < r < T Q
for sufficiently small Я. For every те<т*, т0> there holds the inequality
«(<, т)9>(Я|я?С<)|) ^ , т)ср{1$ |#(f)|), where a(f, т)у[1 |a?(tf)|) 0 as Я -> 0, and the function at the right-hand of the above inequality is integrable- in <0,<0> for an arbitrary t0. Thus J a(t, r)q>[l\x(t)\) dt h 0 as Я -> (L
fo о
Choosing Я„ I 0, the integrals J a(t, т) 9c(Ям |o?(^) |) ^ form a decreasing;
0
*0
sequence of continuous functions. By 33ini theorem, sup J a ( t ,r ) x
* * < T < T 0 0
*0
X rp{ln \x(t)\) dt ->■ 0 as l n \ 0. Hence sup j a(t, r)(p{l\x(t)\) dt - > 0 as
T * < r < T Q 0
Я -> 0 for an arbitrary tQ. By the assumption (III), the integral remainders
OO OO
j a(t, T)(p(l0\x(t)\) dt < £ for г* < г < r0. Hence J a(t, r)99(Я |a?(<)|) dt < e
ft ft
for 0 ^ Я ^ Я(,. Taking Ti — tfj it is seen that sup o*^(t, lx) 2£ fon sufficiently small Я.
B e m ark 1. The condition C implies
Ma®i+/?®2) < e*(®i)+GvM for a, /5 ^ 0, a + fi = 1.
1 2 2 A. W a s z a k
This follows from the relation
+ < ev(a(|a?i|v|a?2|H-/S(|a?1|v|a?2|)) = ^ (K | v |a?2|).
R em ark 2. Let ns suppose that a 99-function 99 is s-convex, i.e.
9o{au-\- fiv) < a <p(u) + q>{v) for a , / 3 ^ 0 , a s + /3s = l , 0 < s < l , and ar
bitrary u , v > 0 . Then a modular may be defined assuming the condi
tions A, В and the following condition in place of C:
Cs. qv{oxx-\-(ix^ < a Qyioci ) + /f qv(x2).
Here, if а — 0 and qv(xx) = 0 0, the respective expression is assumed to be equal to 0. Condition Cs implies D, immediately.
2.2. In the sequel we shall always suppose that the kernel a(t, r) satisfies the conditions (I), (II), (III).
It follows from 2.1 that A* is a semiordered modular-space in the sense of the definition of [11]. The following functional Ц-Ц,, may be de
fined in X *:
2.2.1. It is known [9] that |NI»> < 1 implies qv(x) < \\x\\ę and that the following two properties are equivalent:
Moreover, it is known [9], [12], [14] that the functional (*) is an F-norm, i.e.
1° INI? = 0 if and only if x = 0, 2° \\x + y\\9 < Nl„+||0||v,
3° |И|„ = \\-x\\9,
4° if an a and \\xn— x ^ -> 0, then \\anxn— сигЦ^-^О.
Applying 2.1 В it is also easily seen that
5° \x\ < \y\ implies INI* < \\yWy, and in particular, \\x\\v = |||fl?|||v.
2.2.2. If 99 is an s-convex 99-function (see Remark 2 to 2.1), a norm may be defined in X * by means of the formula
This norm is s-homogeneous, i.e. it satisfies the condition ||atf||S9, = |a|s||a?|[S(?, (see [6], [1 1]).
The norm (**) is equivalent to the norm (*), and the following in
equalities hold:
(a) INk> < (INI*)® if INI», <1,
(b) (INI,)1+e< IN k if INI., < 1 •
(*)
(b)
(a) \Ы<р -> 0 as n 0 0,
Q<p(Xxn) -> 0 as n -> 00 for an arbitrary real X.
Indeed, if ||ж||,, < e < 1, then Qę{x{es) 1/s) < e < 1. Hence ||ж||8(р < es and
SO ||®||.„ < (Pll«p)s. If INU < e < 1, then
^ £ " 1/(1+s)) = ^ (£1/S(1+SW - 1/S) < e1!^ Q<P(xe-lls) <
Hence pilę, < e1/(1+8) and this implies (b) (compare [11]).
If s = 1, i.e. if cp is a convex ^-function, the modular qv{x) is a convex functional in X *. In this case the norm (**) will be denoted by ||-||8; it is a homogeneous norm. From 2.1.В the monotonicity of ||-||Sę, follows immediately; this means that \x±\ < \x2\ implies \\хг\\8Ч) < ||#2IU>.
2.2.3. X v is a linear subspace of X * closed with respect to the norm (*).
This follows from the equivalence of the conditions (a) and (b) in 2.2.1 and from the inequality qv(Ax0) < д(р{2Л(хп—а?0))+ дч>(2?,Хп) which assures us that \\xn— ^ollę, 0, xn€X ę , implies x ^ X ę .
In the sequel, X *, X v, etc. will always mean the respective spaces provided with topology defined by the norm (*) (called the norm gene
rated by cp).
2.3. Let the kernel a(t, r) satisfy also conditions (IY) and (V). We denote A <atP>— sup j a(t, r)dt. Supposing cpfi is an s-convex 99-fune-
t^ t* a
tion, cpis strictly increasing and one may easily calculate the s-norm of the characteristic function y<a>j3> of an interval <a, /5) <= <0, 0 0) which evidently belongs to X *. This norm is given by the formula
2.4. Let a function x(t) be given and let
\x(t) for t < h, X(k){t) =
[0 for t > Tc.
I f xeXy, then Wx — x^Wy -> 0 as к -> со.
Let xeX y and let e > 0 be given. There exists a r0 such that
CO
for t > t0. By (III), there exists a t0 satisfying' the inequality
OO
J a(t, r)ę:(e_1 |a?(£)|)dt < \e for те< т*, т0>. Hence we obtain for к > /0:
Q<p
X(k) sup
T^T* ■ dt < sup I aft, x)cp 11 - ) dt f-
£ j тътл J \ £
f a{t, r ) v ( ^ \
Г l И * ) | \ ,
+ sup a ( t , r ) c p \--- \ dt
r*<T<r0 J \ e j < £.
124 A. W a s z a k
2 .5 . In 2.5 and 2.6 we suppose that the 99-functions 99 is convex.
Then the formula
q\ {x) = super,,(t, x) for XeX*
T^T*
defines a modular satisfying the conditions 2 .1 A, B, D and Cx. This modular defines the norm
INIS = inf |£ > 0: qI < 1J.
Let us remark that ||a?||° < \\x\\cv, but the sign of equality does not need to hold. However, ||a?||° = \\x\\cę for xeX^ because qI (x) = q^x) for xeX^.
Let us see that the following situation may appear. It may happen that av(r, lx) -> 0 as t - > o o for 0 < l < 1 and sup огДт, x) < 1, but aę {r, x)
r^r*
is not convergent to 0 as т -> o o . Then qqv{x) < 1 but q^x) — o o , q^ Ix) < oo for 0 < Я < 1.
2.6. Let cpbe a convex 99-function satisfying the conditions
( 0 , ) <P(U ) n
--- > 0
и as и -> 0 +,
(o c q )
( p ( u )
•--- > 0 0 as и — o o . и
The function 99* complementary to 99 is defined by the formula 99* (v) = sup(w — 99 (w));
W>0
it is also a convex 99-function satisfying the conditions (0X), (оох). The following Young inequality holds (see e.g. [3 ]):
uv < q>(u)-Ą-(p*(v) for u , v ^ 0 .
In 2 .6 -2 .6.5 we shall suppose that besides (I)-(III), the kernel a(t, r) satisfies the condition (V).
Let # eY* , then the functional OO
||#||* = sup sup f a(t, t)x(t)y(t)dt\,
у 0
where sup is taken over all y e X ę* satisfying the inequality Q<p*(y) < l r
<p
is a monotone В -norm in X *.
In all the considerations of 2 .6 -2 .6.5 we suppose that x ,y e X are non-negative.
Let XQx e X n , y e X v*. Multiplying Young inequality with и = l Qx{t),
v = y{t) by a(t, t) and integrating over (0, oo) we obtain
0 0 O O 0 0
f a(t, r)x(t)y(t)dt ^ — f a(t, r)cp(^x{t))dt-\-— f aft, T)<p*(y{t)) dt.
J An J /In J
0 U 0 U 0
Hence |NI* < oo. Homogeneity and subadditivity of ||-||* in X * is easily shown and obviously, x — 0 implies INI* = 0. In order to show that INI* = 0 implies x = 0 we take у = x<a>jS>. Evidently, y e X ę* and Q A h w f )
/3
< 1 if X = Since 0 = |N|* > A f a(t, x)x{t)dt and a ( t , r ) > 0
a
for almost every t and for r > t*, we get x(t) = 0 for every t e f a, /?>.
2 . 6 . 1 . Let x c X * . For every yeX^* and every t > r * the following
inequality holds:
г t* ^ *f Q v 'iy X 1 !
J a(t, r)x(t)y(t)dt <\
о \M\<pQ<p*(y) Q<p*iy) > 1 •
Let xeX*ę and y e X ę*. The first of the inequalities is an immediate consequence of the definition of ||-||*. To prove the second one let us remark that Qg,*(y) > 1 implies
9 У <P (y)
Qq>*{y) ! 6g>*(y)
Hence
OO 00
f a(t, r)<p* [ ) dt < — 1—- f a(t, T)<p*(y{t))dt x \ Qg>*(y)! Qy*(y) J
< 1 for every r > r*. This implies
us
/ a(t, r)x(t) dt < ||a?||J.
qA v)
(a)
(b)
2 . 6 . 2 . Le m m a. The following formulae hold:
lim INNlJ = N lJ if ®*X%, k—>oo
lim N (&)||*
k—>oo if ХеХф.
Since 0 < ж(1) < ж(2) < ... < x, the sequence is non-decreasing and bounded from above by |NlJ- Let lim = Y ^ £« -> ||%)llj.
and en > ||л?(Л)||®, then
J \ £«.
k—>oo
dt < 1,
126 A. W a s z a k
By Fatou lemma,
J кa(t, x)<p I•
л '
%)(<) \ 0
Taking к -> oo we have
dć < 1 for т ^ r*, & — 1, 2, ..
f° ( x (t)\
I a (i, т)д?|--- J 6Й < 1
n \ ^? /
for any fixed £0. Consequently, о^Дт, x/rj) < 1 for all т > r*, Hence INI” < ??, and this shows (a).
To prove (b) let us see that \\x(k)\\* is also non-decreasing and bounded from above and we may denote lim ||a?(fe)||* = y. Choosing y e X 9* so that
к—>oo
Q<p*{y) < 1 ? the inequality
j a(t, r)x{t)y(t)dt
holds for every к and x > x*. Taking к -> oo we get r\ ^ J a{t, x)x(t)y(t)dt
oo 0
for x ^ x* and consequently, у ^ sup J a(t, x)x(t)y(t)dt. Thus П > INC T^sT* 0
and we proved (b).
2.6.3. I f INC ^ 1 and X€X * гл X u then q\(x) < ||ж||*.
It is known that given x > 0, there exists a non-negative measurable function i/ such that x(t)y(t) = 9>(®(0) + 9>*(у(0) for every t. Let ||a?|C < 1, xeX^ X f. We state that Qę*(y) < 1. We denote
xn(t) x(t) if a?(£) _ (?/(/) if ж(£) < n , Vn(t) = I
0 if x(t) > n , [0 if x(t) > n.
Then
(i) xn(t)yn(t) = <p(®n(t)) + <p*{yn(t)).
Obviously, yn(t) are bounded, and since yneXf we have y ^ X ^ . Let us suppose qv*(y) > 1, then Qv*{yn^ > 1 for a certain nQ, because yn(t) -» y(t) as n -> oo. By (i), there holds the inequality
<P*(yn0(t)) < <P*(yn0(t)) + (p(xnQ(t)) = 00(t)yno(t)
and since (i) implies xn (t) > 0 on a set of positive measure, multiplying the above inequality by a{t, x) and integrating over (0, oo) we obtain
OO OO OO
f a(t, x)(p*(yno{t))dt < j a(t, x)(p*{yno{t))dt+ J a(t, x)<p(xnQ(t)) dt
0 0 0
oo
= / a (h r)x no(t)ynQ(t)dt.
Taking into account 2.6.1 we obtain
oo oo
(ii) j a(t, r)<p*(yno(t))dt < f a(t, r) xnQ(t)yno(t) dt < ||a?«0||Jev(y«0(<))-
0 0 oo
Since there exists a r0 such that д9*(уПо) = j x0)q){yno(t)) dt, the in- o
equality (ii) gives finally Qę*{yno) < ll®n0llj e9*(^»0)- Hence 1 < N„0||* < INlJ»
a contradiction to the assumption ||a?||* < 1. How, we shall prove the inequality q\{x) < ||ж||*. We have
0 0 OO 0 0
J a{t, r)(p[x(t)) dt< J a(t, x)(p[x{t)) dt-\- j a(t, r)<p*(y(t)) dt
o o o
OO
= j a(t, r)x{t)y(t)dt о
for t > r*. Since ^ * ( y ) < l , yeX^*, the last inequality implies д\{х)
< IWIJ.
Eem ark . From 2.6.3 it follows that (iii)
e r ( im i; ) < l .
2.6.4. The following inequalities are satisfied:
(a) I K < 2 i r t for XeX\
(b) in i; < INI* < 2 INI? for XeX^
By the Eemark to 2.6.3 x e X * ^ X f implies (iii). Hence it follows from the definition of the norm ||-||° that ||®||J < ||a?||J. In particular, ll%)llj> < ll®(ft)llj» and 2.6.2 yields the left of the inequalities (a). To prove the right one it is sufficient to note that given any s > 0, Young inequality leads to the following inequalities
00 00 00
f a ^ ’ T) T dt ^ f a ( h * ) (P\ || f a(t,r)(p*(y{t))dt
J INI„ + e j \IN Iv+ e / J
n I x (t) \
< 4 m Pm ) + !vW
if Qv*(lf) < 1 . Thus ||a?||J <2||ж||°.
The inequalities (b) follow from the remark to 2.5 that ||ж||£ = ||ж||°
for XeXę .
2.6.5. The following inequality holds for arbitrary x e X * , у eXg,*:
00
j a(t, r)x{t)y(t)dt < INlJlNI**-
о
128 A. W a s z a k
Indeed, by the definition of the norm \\y\\°v we have qv [уК\\у\\1* -f e)] < 1.
00
Hence f a(t, r)x(t)y(t)dt < ||^||*(||y||®* +e). Taking e -> 0 and applying the о
inequality (a) 2.6.4 to the 99-function y* we obtain the required inequality.
3. Throughout this section we shall also suppose that the kernel
■a(t, r) satisfies the conditions (I), (II), (III).
3.1. The space X * is complete with respect to the norm Ц-Ц,,.
We shall apply the the following theorem ([14], p. 307). If the axiom
€ . II is satisfied in a modular space X (q), then this space is complete witch respect to any norm generated by q. Thus it is sufficient to show that X* satisfies С. II, i.e. if xn eX * , xn > 0 for n = 1 , 2 , . . . and if qv(x1)4-
+ (^2) + • • • < °°? then there exists x0 = \/xneX *.
We consider the sequence yk = aqv ... vxk — sup(aq, . .. , xk). Since
<р{\Ук\) <9?(H il)+ ••• +<р(Ы) we get
к oo oo
{i) Gq,(t, yk) < JT* <rv(r, Xi) < £ a<p(r ’ °°1) < £ < 00
1= i i=i i=i
for any t. But yk(t) is an increasing sequence for every t. Hence we have yk{t) -> x0(t) for all t, where xQ = \J xk. We show that aqeX*. From the above it follows that lim 0^(7, yk) — av(r, x0) < 00 for every r > r*.
k—>oo
By (i), we get
OO
ffp O b ®o) < Л ® i)-
7=1 Hence
aę i r J Жо) ^ а<р{Г 1 ®l) + • ' • + 0 ’9)(Т ') ^ f c - l ) + Qq> (®k) + Qę (%k+1) + • • •
lor every 7. By the assumption,
OO
Q<p{xi) < e l=k
for sufficiently large 7c, i.e. av(r, x0) < 0^(7, х г)-\-.. сг9(т, xk_i)-\-e for every г and for sufficiently large 7c. Since av(t, хг) -> 0 as r -> 0 0, we may choose t0 so large that 0^(7, aq)+...-|-0^(7, # *_ !)< e for r > r0.
Hence <7,, (r, a?0) < 2e for 7 > 70. Consequently, 0^(7, ж0) -> 0 as 7 -> 0 0. R em ark 1. It follows from the above that
Q<p(x o) ^ £4( ^1) + Q ę { ^ 2 ) J r - > -
B em ark 2. The space X v is also complete with respect to the norm ||*||v because it is closed with respect to this norm (see 2.2.3).
3.2. I f X* <= X* (or X q cz X v), then ||aq]|,, -> 0 implies \\Xi]\v -> 0 for arbitrary aqeX* (or X i t X ę , respectively).
The easy proof is based upon the closed graph theorem and will be omitted here.
3.3. Let the kernel a(t, r) satisfy the conditions (I)-(Y). We define 2/+1
A y — A ,vy+ly = sup f a(t, r)dt.
Or* /
It follows from (IV) that A y is finite for every у > 0.
3.3.1. A y is a continuous function of у for у > 0.
Let у be a given positive number and let у ^ у > y0 > 0, y — y0 < 1.
We denote
2/0 + 1 ' 2/+1
1Уо(т) ~ f ti(tf r)d tf I y(r) = j a (t,r)d t.
Уо у
Then we have
2/+1 V
(i) 1у( г ) - 1 щ(т) = J a ( t ,T )d t — j a (t,r)d t.
2/0+l 2/0
By the condition (V),
2/+1 2/+1
j a(t, r)dt < f a(t, r)dt < e for т ^ r0,
2/0+l о
2/ 2 / + 1
J a(t, r)dt < I* a(t, r)dt < e for т > r0.
2/0 6
Moreover, by (II) a), a ( t ,r ) for re (r* , r0> and Je<0, i/+l>. Hence
2 / + 1 2/
c ( y — y 0), ( a(t, r)dt < c(y — y0).
2/0 + 1 2/0
Finally, we obtain |7„(т)— (r)| < 2e for те<т*, r°> if y — y0 < <5, where
<5 > 0 is sufficiently small. By (ii), |Iy(r) —1Уо(т)| < 2e for r > r0. Hence -^2/(T) ^ 2 e + Jj,0(r) ^2e-)-Myo for every r ^ r*. Consequently, ^2e-|- + А Уо. The inequality < 2 e + J .„ is obtained analogously. Thus we get \АУ—А Щ\ < 2e for y - y 0 < <5, 0 < y0 < у < у.
3.4. Let us suppose that the kernel a(t, r) satisfies the conditions (I)-(V) and that the function A y defined in 3.3 tends to 0 as у oo. I f -> 0 implies ||a^||v -> 0 for an arbitrary sequence ą e l / , then f i cp.
To prove this statement we shall apply the following remark con
cerning the function A y:
(+) Let us denote к = sup A y. If A>l / fc, then there exists y0 such 2/Ss0
that (Ащ)*1 = X.
Roczniki PTM — P race M atem atyczne X II 9
130 A. Waszak
This remark is an immediate consequence of 3.3. Now, we shall prove 3.4 indirectly. Let us suppose y> i <p does not hold. This means that for every system of constants Jc, b , #0 > 0 there exists § > $0 such that 'ip(M) > Ъ(р{&). Let s > 0 be given. We choose a number и satisfying the inequality
(i) e~1p(u) — X > /с-1,
where /с is defined by (+), and the inequality
(ii) ip(eu) > е~г(р(и) .
Let y0 be such that X — (Ayfff~l. Inequalities (i) and (ii) imply
2/+1
Qv>(eUX<v,v+1>) = sup f a(t, x)y(eu)dt = Avy{eu) v
> Aj/99('^)e_1 = 1.
By the definition of the norm (*), we see that \\ещ<у>у+1у\\у, ^ 1. However,
( eu \ y+ir e
Qv \----X<v,v+i> ) = SUP a{t,r)<p{u)dt = A y< p (u )= A y— = e
\ G I J -A.у
and it follows from the definition of the norm (*) that ||eMjf<y,y+1>||v < e.
Since e is arbitrary one may define a sequence ą e l ; such that 11#*)),, -» 0 but \\%i\\v ^ 1, a contradiction.
3.5. The following theorem is obtained from 1.6, 3.2 and 3.4.
By the same assumptiovis as in 3.4, the following conditions are equi- valent:
i V -S <P, (a)
m x ; c x ; ,
(Y) -Z^, C -Z^,,
(8) \\щ\\<р -> 0 implies -*• 0 for an arbitrary sequence xi eX 1
Co r o l l a r y. By the same assumptions as in 3.4, the following condi
tions are equivalent:
(a) i
(P) К = K >
(y) z v = x ,,,
(S) the norms ||*||^ and Ц-||v are equivalent in the space X f.
