AN N ALES SOCIETATIS M A T H E M A TIC A E P O L O N A E Series I: C O M M ENT ATION ES M A T H E M A TIC A E X X IX (1989) R O C Z N IK I P O LSK IE G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O
Séria I: PRACE M A T E M A T Y C ZN E XXIX (1989)
Jan Szajkowski (Zielona Gôra)
On linear functionals in Hardy-Orlicz spaces in the half-plane. I
Abstract. In this paper we give the representation of linear functionals norm continuous on the space of finite elements Н0ф in the Hardy-Orlicz space Н*ф.
The linear functionals in Hardy-Orlicz spaces in the unit disc were considered by Lesniewicz ([4] and [5]).
Strômberg in [8] considers dual spaces of the Hardy spaces making use of bounded mean oscillation with Orlicz norms.
This paper can be regarded as a continuation of papers [9]—[13], which contain the study of Hardy-Orlicz spaces of analytic functions in the half-plane. Some results of papers [9]—[13] and other papers will be needed here. We collect them in the first section.
I would like to thank Dr. R. Lesniewicz for his help in the preparation of this paper.
1. Let i]/: <0, oo)-><0, oo) be an N-function, i.e., ф is increasing and convex and satisfies the following conditions:
(0 . ) l i m ^ = 0 and (ood r lim --- = oo.«Ж И-0 +
ф satisfies the condition (A2) if for some constant d > 1 the inequality ф(2и) ^ йф(и) for и ^ 0 holds. For any N-function the function
ф*(p) = sup {up — ф(и): и ^ 0} for v ^ 0 is also an N-function. Moreover, (i^*)* = ф ([3], Chapter I).
1.1. If S denotes the space of all complex-valued functions, defined and measurable on the interval ( — oo, oo), then the functional
«?*(/)= J *(i/W D *
is a convex modular in the sense of Musielak and Orlicz. We define an Orlicz space 1?ф as
L** = {fe S : еф(.¥) < oo for some к > 0}, whereas a space of finite elements 1?ф of Ь*ф as
L°* = {fe S : вфт < oo for every к > 0}.
Two equivalent norms can be defined in the space Ь*ф ll/ll* = inf{£ > 0: e*(//e) < 1} for feL **, and
oo
Il/IU) = sup{| f f(t)g(t)dt\: gr {g) < 1, g e l f } for f еЬ*ф
—00 ([3], [6], [7]).
1.2. If v4(&) denotes the space of analytic functions F in the half-plane Q = (weC: Rew > 0} (in the sense of [9]), then the functional
00
e*(F) = sup{ f ÿ(\F{x + iy)\)dy: x > 0}
— oo
defined on the space A(Q) is a convex modular.
We define the following classes of functions:
H* = {FeA(Q): g^{F) < oo},
Н*ф = {FeA{Q): kFeH* for some к > 0}, Н0ф = {F eA(Q): kFeH ф for every к > 0}.
The class Нф is an absolutely convex set in A(Q) and the classes Н*ф and Н0ф are linear subspaces of A(Q). Obviously,
Н0ф с Н ф а Н*ф <= A{Q).
The class Нф we call Hardy-Orlicz class in Q, Н*ф Hardy-Orlicz space in Q and Н0ф the space of finite elements in Н*ф. In the space Н*ф we introduce the norm by the formula
l|f II* = inf{e > 0: e*(F/e) < 1} for FeH**.
In Н*ф we can introduce another norm by the formula 00
\\F\\m = sup{| J F(x + iy)g(y)dy\: gr (g) ^ 1 ,д е Ь ф\ x > 0}
—00
for FsH **. The norms || ■ !|^ and Ц-Ц^ are equivalent on Я**; namely Ilf IU « Ilf II W « 2||F||, for every FsH**.
Linear functionals in Hardy-Orlicz spaces in the half-plane. I 87 If FeH*^, then F has non-tangential limits in almost every point of the imaginary axis and the boundary function F(i-) belongs to the space Ь*ф ([9]).
For FeH*ф we have
l|i% = \т-)\\ф and II F || w = ||F(F)||W ([10]).
In Н*ф, similarly as in Ь*ф, we can define two convergences: a norm convergence and a modular convergence. A sequence {Fn} «= Н*ф is convergent in norm to F еН *фif ||F„ — F||^->0 as «-► oo; this holds iff дф(А(Рп — F))-*0 as и->оо for any A > 0. Moreover, a sequence {Fn} а Н*ф is convergent in modular to FeH*ф if £^(2,(F„ — F))->0 as n-»oo for some A > 0 (in general, dependent on (F„ —F}). In the case where ф satisfies the condition (d2), Н*ф = Н0ф and norm and modular convergences are equivalent. Otherwise, we have Н0ф а Н*ф only and the norm convergence implies the modular convergence ([11]).
1.3. By Hl in the disc D = {zeC : |z| < 1} we denote the class of all functions G analytic in D for which
2n
Il G !| x = sup { J \G(reie)\d6: 0 ^ r < 1} < oo holds, о
By H1 in the half-plane Q we denote the class of all functions F analytic in Q for which the integrals
00
J \F(x + it)\dt
—00
are uniformly bounded for x > 0 ([1], [2]).
1.3.1. If the function F belongs to H1 in Q, then J F(it)dt = 0.
Proof. Let F belong to Я1 in Q. Then by (Theorem 1.9, [12]), the function
G(z) = - - - ~2f( 1 ^ ] (zeD), (1 — Z) VI —z
belongs to H1 in D. We denote
Mz) = |l _ | ' i ± f V W — for zeD . 1 — z } \ 1 —z
Since |z| ^ 1, so \h(z)\ ^ 2|G(z)| for z eD. Hence we deduce that h belongs to H1 in D. From above, in virtue of the fact that z = (w— l)/(w + 1) is the
homographie transformation of the half-plane Q into the disc D, in view of the Poisson’s integral formula for H1 in D, we get
00 00
i J F(it)dt = ^ J (l-(it )2)F(ityj^-p
—00
о
2я
1 14
2к h(eie)dO = h{ 0) = 0.
2. Since every function F e H** has the boundary function defined off a set of measure zero, so if each F eH** is identified with its boundary function, H**
can be regarded as a subspace of L**. Besides we have ||F||^ = ||F(F)||^ and
||F||W = ||F(r)||w . Similarly, H°* is a subspace of I?'*'.
2.1. For every function geL*** the formula
00
(*) ((F) = f F{it)g(t)dt for FeH°*
— 00
defines a linear functional, norm continuous on FI0*; besides, H b = sup{|£(F)|: F e H 0*, ||F||, < 1} < Ы|(^
and
U\\m = sup{|£(F)|: F eH °*, ||F||W ^ 1} ^ \\g\\r .
Proof. This is obvious if g = 0. Let us suppose that the function g eL *** is different from zero. By the Holder’s inequality (see [3], Chapter II), we have
00
s \F(it)g(t)\dt ||F(!-)iy0||№.) = ||ЛМЫ|,о
— 00
and
î IF (it)g W t « l|F(i-)llw)ll0lU- = l|F||w ll9lU-
— 00
for any function F e H 0*, and we get
\Z(F)\ ^ IIF b \\д\\(Г) and |£(F)| ^ ||F||w \\g\\r .
Hence we deduce that the linear functional £ is norm continuous on H°*;
moreover,
U b < M tr) and и\\(ф) ^ \]g\\r .
Linear functionals in Hardy-Orlicz spaces in the half-plane. I 89
2.2. For every linear functional Ç, norm continuous on H0*, there exists a function gEb*'1'* such that
(*)
00
UF) = f F(it)9(t)dt
—00 for FeH°+;
moreover,
II €11* = II0 !!<*•> and J II «Г
Proof. Since Н0ф can be embedded isometrically in L0^, then by the Hahn-Banach theorem there exists a linear functional /, norm continuous on L0*, such that Ç(F) = l(F(i •)) for FeH°* and ||/||, = М\\ф, \\l\\w « ||€l|w . We know ([3], Chapter II) that for a functional l there exists a function деЬ*ф*
such that
ПЛ = ? f(t)g(t)dt for /e L °*
— 00
and
Il0ll(**> = \\Ц\ф> \\д\\ф*= Pll(^)*
There holds formula (*) and equalities for norms.
2.3. Let L*0r denote the class of all functions g e L such that
00
J F(it)g(t)dt = 0
— 00
for any function FeH0'1'.
2.4. L*0r = FI*'1'* (isomorphically).
Proof. Let GeH*'1'*. Then G(i')eL*^. We take an arbitrary function F eH0*. By the Holder inequality we state that FGeH1(Q). Hence and from 1.3.1 we deduce that
J F(it)G(it)dt = 0.
— 00
This fact proves the inclusion Н*ф* c= ЦУ*.
Now, we take an arbitrary function дЕЬ*ф*. Then we have
00
f Un(it)g(t)dt = 0,
— 00
where Un (n — 1, 2, . . . ), denote functions from Lemma 1.4 ([13]). We con
sider the homographie transformation z = (w—l)/(w+1) of the half-plane Q into the disc D. Since this transformation has on the boundary the form eie = (it— l)/(it+1) therefore t = cot|0 (0 < в < 2n) and hence 0 = 2*arc cott.
If t-> oo, then 0->0; whereas if f - > — oo, then G-^2n. We have ( — 2df)/(l + f2) = dO, also. From above, we obtain
00
0 = U„(it)g(t)dt
— 00
00
—00
(it — l)"-1 (it+ 1)" + 1g(t)dt
( i t -1)" - 2<fr (it+\)n + l '9it)'T + ?
2n
— 12 eineg(9)d0, о
where g(0) = gf(cot^0). The function g is integrable on the interval [0, 2n), because for и ^ 1 we have ш//*(1) ^ ф*(и); and
2я
«о
^ 2к Т"
ф*(1) Ф*(Ш*)\ ) j ^ ^ 2 n + 2 + t‘ <A*(i)
From the above we deduce ([2], Chapter III) that g is a boundary function of the some analytic function G in D and that G can be represented by the Poisson integral of g, i.e.,
2 я
Since the reciprocal map w = (1 +z)/(\—z) of disc D into the half-plane Q is conformal and
1 — \z\2 _ x (l-K 2)
\z — eie\2 x2 + (y — t)2’
therefore the function G, corresponding to function G in D, is analytic in Q and
G(x + iy) = G(w) = G (W
4W+1/ 71
because g(t) = <7 (2-arc cot t). Hence we get oo
il/*(\G(x + iy)\)^~ f ij/*(\g(t)\)
9(t)x 2 + ( y - t ) :îdt,
71 J ' 'X2+ ( y - t ) 2dt.
00
Linear functionals in Hardy-Orlicz spaces in the half-plane. I 91 Integrating both sides of this inequality with respect to y on the interval ( — 00, oo) and changing the order of integration at the right-hand side we obtain for x > 0
ij/*(\G(x + iy)\)dy ^ ijj*(\g(t)\)dt.
Hence and from the fact that g e L we deduce that дф*(С) < оо. This means, in virtue of the analyticity of G, that Therefore the inclusion I * r Ç- H* r holds.
2.5. We denote by (H0^)* the space of all norm continuous linear functionals on Я 0*^ equipped with the norms
= sup{|£(F)|: F e H0*, -Ü- /Л £e(Я °*)*
MI,*> = sup{|£(F)|: F e H 0*, ^4 ? /Л ^ ( Я 0*)*
Let us note that norms ||-||^ and |j • || satisfy the inequalities imu for £ е ( я ° т .
2.6. In the quotient space L^/H*'1'* we identify functions in Я *** with their boundary functions. Let g denote the element of L*^*/#*^* determined by the element деЬ*ф*, i.e.,
0 = 9 + H*r = {g + f: feH ***}, 9 ^ . Now the quotient space /H*^* is equipped with two norms
\\g\\r = mf{\]g+f\\r : f e H * r }, geL*r , or
Ш<Г> = inf{ll9+/ll<«: /6Н***}. eeV*-
These norms are equivalent; namely,
\\g\\r < ll^ll(^) <211^11^ for geL*r /H*r .
2.7. The space (Я°^)# is isometrically isomorphic to the quotient space Ь*ф*/Н*ф* (the space Ь*'1'*/Н*ф* is equipped with the norm || • [|| • Ц^*] if the space (Н0ф)* is equipped with the norm Ц-Ц^ [||-||w ]).
This isomorphism establishes formula (*) from 2.1.
Proof. Two functions gt , д2еЬ*ф* represent (according to formula (*)) the same functional £,е{Н0ф)* , i.e.,
00 00
£(F) = J F ÿ tjg ^ d t = J F(it)g2(t)dt for Р еН 0ф
— 00 — 00
iff the difference д1 — д2еЬ*ф* satisfies the condition
00
J F(it)(g1( t ) - g 2(t))dt = 0 for Р еН оф;
— 00
which by 2.4 is equivalent to the property g1 — д2еН*ф*. In this case formula (*) establishes the isomorphism of the space 1*^*/Н*ф* with the space (Н0ф) *.
By 2.1 and 2.2, this isomorphism is an isometry.
References
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