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13
ISSN 2074-1871 Уфимский математический журнал. Том 10. № 3 (2018). С. 135-145.
NEW CHARACTERIZATIONS OF BLOCH SPACES, BERS-TYPE AND ZYGMUND-TYPE SPACES AND RELATED QUESTIONS
M. GARAYEV, H. GUEDIRI, H. SADRAOUI
Abstract. In terms of Berezin symbols, we give new characterizations of the Bloch spaces B and B0, Bers-tvpe and the Zygmund-tvpe spaces of analytic functions on the unit disc D in the complex plane C. We discuss some properties of Toeplitz operators on the Bergman space ¿„(D). We provide a new characterization of certain function space with variable
exponents. Namely, given a function f(z) = ^ f(n)zn G Hol(D) with a bounded sequence
ra=0
j/(n)j of Taylor coefficients f(n) = ^, (n = 0,1,2,...), we have f G Hp(-),q(-),j(-) if and only if
1/2* \ fit) f 1 f ~ ^ P(t) \ 7 (t)P(t)-g(t)
J I 2^J D(f(n)e^t)(V^ dt I (1 - r) dr < + «,.
oo
Here D(ttn) denotes the associate diagonal operator on the Hardv-Hilbert space H2 defined by the formula D(an)zn = anzn (n = 0,1, 2,...).
Keywords: Bers-tvpe space, Zvgmund-tvpe space, Bloch spaces, Berezin symbol. Mathematics Subject Classification: 47B33, 30H30
1. Introduction
Let D be the unit disc in complex plane C, D = {z G C : |z| < 1} and let Hol(D) be the class of functions analytic in D. We denote by H= H(D) the space of bounded analytic functions on D. Recall that a function f G Hol(D) belongs to the Bloch space B = B(D) if
:= sup (l - ^|2) /(z)
zero
< +oo.
Being equipped with the norm ||/||B = |/(0)| + ||/||6, B is a Banach space. Let B0 = B0(D) be the space consisting of all / G B satisfying
lim(1 -|z|2) If'(z)| = 0. This space is called a little Bloch space.
M. Garayev, H. Guediri, H. Sadraoui, New Characterizations of Bloch spaces, Bers-type
and zygmund-type spaces and related questions.
© Garayev M., Guediri H., Sadroui H. 2018.
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no RGP-VPP-323. Поступила 29 июня 2017 г.
ь
Let a ^ 0. The Bers-type space, denoted by H« = H«(D), is a Banach space consisting of all f e Hol(D) such to
«.a = SUP(1 f)" If(Z)l < +<X>.
zero
Clearly, H« = H«.
Let 0 < p, q < j > -1. If a function f e Hol(D) is such that
, „ 9
1/2 -K \ P
HP,q„ = I I ^ I \f )\Pdd I (1 - rydr <
0 \ 0
we say that f belongs to a mixed norm space denoted by Hpqri = Hpqri v
Let 3 > 0. The Zygmund-type space denoted by Z13 consist of all f e Hol(D) obeying
:= I/(0)| + I/(0)| + sup (1 -M2)" f"(z)
zero
The space Z13 becomes a Banach space with the above norm . Let 3 = 1. Then Z1 = Z is the classical Zygmund space. For more information on the Bers-type, Zygmund-type and Bloch spaces on the unit disc D, see, e.g., K. Zhu [14, 15], X. Zhu [16], S. Stevic [11], [12], Y. Ren [9], P. Duren [2], J. Shi [10].
Recall that the following old problem for most of functional spaces X of analytic functions in D, including the Hardy space HP(D), is open (see Privalov [8] and Duren [2]): How a function in X can be recovered by the behavior of its Taylor coefficients? Ideally, one would like to find a condition on the f(n) := (Taylor coefficient), which is both necessary and sufficient for f to be in X Of course, for X = Hp, p =2, the problem
« \ 2
is completely solved: f e H2 if and only if ^^ \/(n) < For p = to, the problem of
n=0
coefficients was solved by I. Schur in 1919 (see, Privalov [8, Ch. 2]). Some classical results on the Taylor coefficients of functions in Hardy and Bergman spaces are also known (see, for instance, [2, 8, 14]). Some recent results about Taylor coefficients of H1 functions and entire functions in the Fock spaces Fa have been obtained, respectively, by Pavlovic [6] and Tung [13]. But the general situation is much more complicated, and no complete answer is available (more informations about this are contained in [1] and references therein.)
Note that in his book [6], Pavlovic proved the following characterization of functions belonging to the Hardy space H1 :
i 1 7
H1 = If e Hol(D) : sup — \ f(r elt)\dt < +to
1 0<r< 1 2^J
Theorem 1.1. For a function f analytic in D, the following statements are equivalent: (a) f e H1;
n
(P) sup^0 7+1||s( Ww < ;
3=0
(C) supra^0 llPnf11H1 <
Here Pnf := ^^^J+i'Sj(f), where an = ^^—+y, n = 0,1, 2,..., and Sj(/) are the partial
j
an
3=0 j=0
sums of the Taylor series for .
Popa [7] gave some interesting generalization of this result of Pavlovic by proving a similar characterization of upper triangular trace class matrices. Recently, Karaev [3] gave some characterizations of Hardy and Besov classes of analytic functions on D with the variable exponents.
In this paper, we use the so-called Berezin symbols technique to characterize the Bers-type, Zygmund-type spaces, and the Bloch spaces B and B0. We also consider some spaces defined by the variable exponent. We discuss Toeplitz operators on the Bergman space L2
2. Berezin symbols and characterizations of the Bers-type, Zygmund-type
and Bloch spaces
In this section, we characterize the spaces H^, Z13, B and Bo in terms of Berezin symbols of diagonal operators (associated with the Taylor coefficients of the functions from the spaces 2P, B and B0) acting on the Hardy space H2 = H2(D). Recall that Hardy-Hilbert space H2 = H2(D) is the collection of analytic functions in D which satisfy the inequality
i
2* \ 2
h2 .= | sup — | f(reu) | 2 de I < 0<r<1 2lT J
It is well-known that H2 is a reproducing kernel Hilbert space with the orthonormal basis en(z) = zn (n = 0,1, 2,...), and consequently with the reproducing kernel
ro ro
kx(z) = YM*)en(z) = ^AV = A e D.
„ „ 1 — Az
n=0 n=0
The reproducing property means that (f,k\)H2 = /(A) for all f e H2 and A e D. For any bounded sequence (an)n^0 of complex numbers an, we denote by D(an) the associate diagonal operator on the Hardy-Hilbert space H2 which is defined by the formula
D{an) zn = anzn, n = 0,1,2,... (1)
For any bounded linear operator T on H2, its Berezin symbol T is the following bounded complex-valued function in D .
T(A) := (TK,\, Kx) (A e D),
where
M*) := ^ = (1 - |A|2)2(1 - Az)-1
\\k\(z)\\H2
is the normalized reproducing kernel of H2, and ber(T) := supAeD T(A) of T.
The following lemma is well-known.
is the Berezin number
Lemma 2.1. The Berezin symbol of the diagonal operator D(an) on the Hardy space H2 is the following radial function:
<x
12
DK)(|A|} = (1 -|A|2)J]ak |A|2fc (A e D) (2)
Proof. Indeed, by using that
k=0
kx(z) = -^ = Y, Akzk
1- A
k=0
is the reproducing kernel of H2, we have:
DK)(A) = <DK)JCx(z), Kx(z))
D,
(a„)~
kx(z) kx(z)
txtfllS ||kx(z)l
/ « «
(1 -I A\2){Dia,)Y, , £
\ k=0 k=0 «
(1 -|A|2)£afc |A|2fc.
« « \
(1 -|A|2)(£ Xkakzk, £ Akzk\
k=0 /
,k=0
k=0
Hence,
DK)(A)= (1 -|A|2)^afc |A|2fc , A e D,
k=0
which proves formula (2). □
Our next result characterizes the spaces Ha«, Z13, B and B0 in terms of behavior of the
Berezin symbols of the corresponding diagonal operators mentioned above.
«
Theorem 2.1. Let a function f(z) = ^^f(n)zn e Hol(D) have a bounded sequence
n=0
(n) > of Taylor coefficients f(n) J n^0 and only if
sup (1 - )
0<r<1
0^e<2n
f (n)(0)
(n = 0,1, 2,...). Then f e H« (a ^ 0) if
a—1
D
(f(n)ein 0)
(V )
< + OO.
Proof. Indeed, let a function f be as in the statement of the theorem. Then, rewriting f and using Lemma 2.1, we obtain:
f(z) = m ^) = f(re») = J]f(n)(re»)n
n=0 oo
(1 - r)Y^ f(n)ein9rn
E/(n)
n) ein&rn
n=0
D
(f(n)ein0)
(V )
1 - 1 -Hence, for any z = r e%e, with r = |z| and 9 = arg(z), 0 ^r < 1, 0 ^9 < 2n, we have
n=0
je
( )
D
(f(n)ein0)
(V)
1
(3)
Using (3), we get
(1 -|z|2)a |/(z)| = (1 - r2)
2n aD (f(n)ein0)
(V )
1
(1 - r)a-1(1 + r)c
D
(f(n)ein 0)
in 0) (VO (4)
for all r, 0 ^ r < 1 and all 9 e [0, 2n). Hence, we obtain
(1 - )
a1
D
(f(n)ein 0)(
(v^) ^ (1 -M2)a | /(*)| ^ 2a(1 - r)
a1
D
(f(n)ein0)
(V)
for all z = re% e D. In particular, by (5) we conclude that that f e H'(« if and only if
sup (1 - )
0^r<1
ee[0.2^)
a1
D
(f(n)ein0)
(V)
< +00.
This completes the proof.
□
n\
An immediate corollary of inequalities (5) is as follows.
Corollary 1. If f G H£ and a > 0, then
sup (1 — r)
0^r <1 0<2TT
a— 1
D
(f(n) ein )
(^ )
P,a
^ 2a sup (1 — r)
0^r <1 0^ 0<2TT
a 1
D
( f(n) ein )
(^ )
Theorem 2.2.
Let a function f(z) = ^^/(n)zn G Hol(D) have the sequence j/(n)| of
n=0
Taylor coefficients such that /(n) = O (n) as n ^ œ. Then (a) f belongs to Bloch space B if and only if
(b) f G B0 if and only if
sup
0^r <1 0^ 0<2TT
lim 1
D
((n+1)/(n+1) ein e )
)
< +oo;
D
((n+1)f(n+1)eine)
(^ )
for allé G [0, 2n).
Proof. It follows from the condition f(n) = O (-) (n ^ œ) that (nf(n))n is bounded, and
hence, the diagonal operator D^f^)) is bounded on the Hardy-Hilbert space H2. This implies immediately that D((n+1)^n+1)eine) is also bounded in H2 for every fixed 9 e [0, 2n). Therefore, by formula (2) in Lemma 2.1, we have
oo
(1 — M2) f(z)
1 — r2)£ f(n)zn = (1 — r2) J>/(n)
n=0 n=1
<x
1 — r2)£(n +1) Î(n +1)^n
n=0
<x
1 — r)(1 + r)£(n + 1) f(n + 1) e mern 1 + )
n=0 oo
(1 — ((n + 1) f(n + 1)emrr
n=0
1 + r)D ((n+1) f(n+1)e ™ e
Hence,
(1 - M2) f(Z) = (1 + r)D((n+1)/(n+1)eine)(W (6)
for all z = re%e e D, where r e [0,1) and 9 e [0, 2n). Now formula (6) shows that f e B if and only if I I
sup
0^r <1 0^ 0<2TT
D
((n+1)f (n+1)ein e )
in e )
<+
and this proves (a).
On the other hand, it follows also immediately from (6) that f e B0 if and only if
lim 1
D
((n+1)/(n+1)ein e)
(V~T )
0
for all 9, 0 ^9 < 2-k. This proves (b).
Formula (6) implies also the following results.
□
0
Corollary 2. Assume that f(z) = ^^f(n)zn satisfies the condition of Theorem 2.2. If f e B, then
n=0
inf
0^r< 1 Q^0<27T
D
((n+1) f (n+1)ein9 )(
(V~r) + If(0)1 ^ \\f\\B ^ |/(0)1 + 2 sup
r <1 q^ e<2ir
D
((n+1) f(n+1) ein 9 )
(^ )
Corollary 3. If f is the same as in Corollary 2, then we have:
(a) ||/||B ^ |/(0)| + 2Qsup^ber(D({n+1)T{n+1)ein0)).
(b) HfHB ^ |/(0)| + 2supn^(n + 1) \/(n +1) Clearly, Statement (a) implies Statement (b) since for each arbitrary fixed 9 e [0, 2n)
( D(
(n+1) f (n+1) e™
<
D
((n+1) f(n+1)ein 9)
= sup(n + 1) f(n + 1)
n>0
Corollary 4. Each function f(z) = ^^ f(n)zn e Hol(D) obeying f(n) = o(n)
,-) as n —> oo
n
n=0
belongs to the little Bloch space B0.
Proof. Indeed, since by assumption limra^« n
f(n)
0, we see that D
((n+1)f(n+1) ein° )
is
a compact diagonal operator, and since H2 is the standard reproducing kernel Hilbert
(1_| \ |2 ) 2
space (which means that the normalized reproducing kernel lC\(z) = ( 1 _j-J weakly tends to zero as A approaches any boundary point of the unit disc D), we obtain that
limr comple
D
((n+1)f(n+1) é'
e.
0 for all 9 e [0, 2tt), which implies that f e B0. The proof is
□
Theorem 2.3. Let 3 > 0 and f(z) = ^f(n)zn e Hol(D) be a function such
n=0
that f(n) = O ^ (n 11)n j as n ^ <x>. Then f e Z3 if and only if
sup (1 — r)13 1
0^r <1
o^e<2ir
D
((n+1)(n+2) f(n+2) ein 9 )
(^ )
< +00.
Proof. Since by assumption ((n - 1)n/(n))ra^2 is a bounded sequence, the diagonal operator D((n- 1)nf(n)ein0) is a bounded operator on the Hardy space H2 for all 9 e [0, 2n). Then, by Lemma 2.1 we have:
(1 — M2)31 №)I
1 — r2) 1 — r2)3 1 — r2)3
„n-2
J2(n — 1)n/(n)
n=2 <x
£(n + 1)(n + 2) f(n + 2) z
n=0 <x
£(n + 1)(n + 2) f(n + 2) emer
n=0
1 + r)3 (1 — r)3-1 1 + r)3 (1 — r)3-1
<x
(1 — [(n + 1)(n + 2)f(n + 2)e
D
— r> 2-^i + 1''n
n=0
((n+1)(n+2) f(n+2)ein9) V^"
n
n
and hence,
(1 - MY l№l = (1 + rf (1 - rf-1 d((ra+1)(ra+2)/(ra+2)el„,) W
for all z = re%e G D. This equality shows that f G Z3 if and only if
(7)
sup (1 — rf 1 <1
o^e<2ir
D
((n+1)f'(n+1)ei«s )
(^ )
which proves the theorem. □
3. Generalized subharmonicity and Toeplitz operators on Bergman space
In this section we apply representations (4), (6) and (7) for studying the boundedness, compactness and belonging to Schatten-Neumann class for Toeplitz operators acting in the Bergman space Ll = Ll(D).
Let dA(z) be the area measure on D normalized so that the area of D is 1. In terms of Cartesian and polar coordinates is reads as
d A(z) = — dx dy = —r drd6 ■n n
For1 ^ p < the usual Lebesgue space Lp(D,dA) denote the Banach space of Lebesgue
measurable functions on D with the norm
I f(z)lP dA(z)
The Bergman space Lpl = Ll (D) is defined to be the subspace of LP(D, dA) consisting of analytic functions. For p = 2, L2a is a reproducing kernel Hilbert space with the reproducing kernel
k(z, w)
1
(1 — zw)2
Recall that P : L2(D,dA) ^ L2a is the Bergman projection and this is an integral operator given by the formula
Pf (z) = k(z ,w)f(w)dA(w)
( w) -——dA(w).
(1 — zw)2
Given a function p G L°
we define the Toeplitz operator Tv on L2a by Tpf = P(ipf),
f G L2a. Since the Bergman projection has the unit norm, we clearly get ||T^|| ^ IMloo. More information about Bergman space Toeplitz operators can be found, for example, in the book by K.Zhu [14].
Definition 3.1 ([14]). Suppose f is a nonnegative function on D. We say that f has a generalized subharmonic property if there exists a constant C > 0 such that
f(z) ^ TDT^ i f(w)dA(w) | D( , )|
D(z ,r)
for all z G D. Here for each r > 0 and a G D
D(a, r) := {z G D : p(z, a) < r]
is the Bergman disc with the Bergman metric
Of N ^ 1 + P(z ,w) p (z,w) = - log ■
2 1 — p(z, w)'
p
p
where p(z,w) = \ f—^ \ (z,w e D) is the pseudo-hyperbolic distance on D) and ^(a, r)| is the normalized area of D(a, r).
Before proving a next theorem, we mention the following known result [14].
Lemma 3.1. If p ^ 0 possesses the generalized subharmonic property, then
(1) Tv is bounded on L2a if and only if p is bounded as |z| ^ 1-;
(2) Tv is compact on L2a if and only if p(z) ^ 0 as |z| ^ 1-;
(3) Tv is in Sp(L2a) if and only if p e Lp(D,dA), where dA(z) = ^^fiL is the Mobius invariant measure on D.
Theorem 3.1. Let a function f(z) = ^^f(n)zn G Hol(D) have a bounded sequence
n=0
nU of Taylor coefficients. Let a ^ 0. Then
n>0
(a)T
D (f(n)e™ arg(z)
(1 -|z|)a-1
is bounded as \ z\ ^ 1 ;
( ) T
(i+|zr(i-M)«-i
D(f(n)e.™ arg(z))^VW\)
is bounded on L2a if and only if
D(f(n)ein arg«) ( V\z\)
is compact on L2a if and only if
(1 -W)
a—1
D
(f(n)ein arg( z))
,(>/W)
0
as 1-
(c) T belongs to the Schatten-Neumann class Sp := Sp (L2a) if and only if
/(1 + \Z\r(l - \Z\)P(a-1) |D (Hn)ei„ arg(z) )(^\Ä)\dX(Z) < + TO
D.
where d\(z) = (f—z^ is the Mobius invariant measure on
Proof. Indeed, one can prove that if f is analytic, p > 0, and a is real, then the function (1 - |z|)a | f(z)f possesses the generalized subharmonic property (see K.Zhu [14]). In particular, the function (1 - |z|2)a | f (z)| possesses the same property. On the other hand, by formula (4),
(1 -M2)a \/(*)| = (1 + W)a(1 -\z\)
a1
D
( f(n)ein arg( z))
,(>/W)
for all z G D. Now the statement of theorem is immediately implied by Lemma 3.1.
□
By using formulae (6) and (7), the nextg two results can be proved by the same method as in the above proof, and therefore the proof is omitted.
Theorem 3.
.2. Let a function f(z) = ^^/(n)zn G Hol(D) have the sequence j/(n)| of
n=0
Taylor coefficients such that /( n) = O Q) as n ^ to. Then
(a) T(1-|*|2)|f' (*)| is bounded on L2 if and only if D ((n+1)f(n+1)e™ zrgi^VW^
M^ 1-;
(b) T(1-i^2)if,(z)i is compact on L2a if and only if
is bounded as
D
((n+1) f(n+1)ein arg( z))
0
as \z\ ^ 1 ;
(c) T(i-\z\2)\f(z) | is m Sp(L2a) if and only if
(1 + W)p D(( n+1)/(n+1)e™ arg(z))^\/Rl)l d\(z) < +œ.
Theorem 3.3. Let ft > 0 and f(z) = ^f(n)zn G Hol(D) be a function such
n=0
that f(n) = 0(((n — 1)n) 1) as n ^ x>. Then
(a) T{1-\z\2)ti | f is bounded onLl ifand onlV if (1 — M)/3-1 V ((n+1)(n+2)/(n+2)e- arg(,))(V^^)
is bounded as |z| ^ 1-;
T(! |_z|2)3is compact on L2a if and only if
(1 -My-1
D
( (n+1)(n+2) /(n+2) ein arg( z) )
(VW)
->• 0
as 1-
(c) T(i-\z\2y| f is in sp(l2) ifand onlyif
(1 + |z|)3(1 - |z|)3-1 |D((ra+i)(ra+2)/>+2)e»arg( ,))(V^T)| G LP(D, dA).
4. Characterization of mixed norm space H^t)^)^) with variable exponents
Let T = <9D and let p = p(t) and q(t), t G T, be bounded positive measurable functions defined on T, and let j(t) > —1 on T. Following by Kokilashvili and Paatashvili [4, 5] (see also Karaev [2]), we say that the analytic function / in D belongs to the Hardy class Hp(i) if
sup ^ f | f(relt) | P(t)dt <
0<r<1
where p(t) = p(e%t), t G [0, 2k).
Similarly, we say that a function f G Hol(D) belongs to the mixed norm space Hp(t),q(t),-y(t) with the variable exponents if
( )
H.
p(t),q(t),1(t)
1 / 2tT
U l f(re«)\p(t)dt 00
q(t) p(t)
(1 - r)l(t)dr <
where p(t) = p(e%t), q(t) = q(e%t) and ^(t) = 7(e%t), t G [0, 2n).
For p(t) = p = co n s t > 0, q(t) = q = con s t > 0 and 7 (t) = 7 = const > —1, the class Hp^)tg(•),7(^) coincides with the class Hp q>1.
The following theorem characterize the spaces Hp^)tq(j,7(j in terms of Berezin symbols and Taylor coefficients.
Theorem 4.1. Let f(z) = ^^f(n)zn G Hol(D) be a function with the bounded sequence
n=0
\ f(n)\ of Taylor coefficients f(n) = f {r),(0), (n = 0,1, 2,...). Then f G Hp(), q()n() if and
I J n^0 ' ' ' only if
1 / 2tT 1
D
(f(n)ein )
in t )
q(t) . p(t)
P(i) \ l(t)p(t)-q(t)
dt I (1 — r) p(i) dr < +œ.
Proof. Using formula (2) in Lemma 2.1, we rewrite the function f(z) = ^^/(n)zn as follows:
n=0
it N it it\ D(T(n)eint) , .
f(z) = f(r eu) = —)--(8)
1-
for each z = reu e D. Now the statement of the theorem follows the definition of the space Hp(.u(-),7(-). Namely, in view of (8), we obtain
1/2tt \ p(t)
g(t) ^ q(t)
1 i \f(re*)\pW dt I (1 - r^dr I D(^)eint)(v^
2irJ 1 v n 1-r
(f(n) ein 1)
0
This yields that f e Hp(.).q(j.7(j if and only if
p(t) \ p(t)
dt I (1 - r)l(t)dr
q(t) p(t)
- p(t) \ u\ <!(*)
D (f(n)en )(V?) dt I (1 - ry(t)~ ^dr.
g(t)
1/2^ \ p(t)
1
~ p(t) \ (+)_ aW
D{f(n)eint^Vr) dt I (1 - r)1(t) pwdr<
The proof is complete. □
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Mubariz Garayev,
Department of Mathematics, College of Science, King Saud University,
P.O. Box 2455, Riyadh 11451, Saudi Arabia E-mail: [email protected]
Hocine Guediri,
Department of Mathematics, College of Science, King Saud University,
P.O. Box 2455, Riyadh 11451, Saudi Arabia E-mail: [email protected]
Houcine Sadraoui,
Department of Mathematics, College of Science, King Saud University,
P.O. Box 2455, Riyadh 11451, Saudi Arabia E-mail: [email protected]
