Differences of the weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces
We characterize the boundedness and compactness of the differences of weighted differentiation composition operators $D^n_{\varphi_1, u_1} - D^n_{\varphi_2, u_2}$, where $n \in N_0, u_1, u_2 \in H(D)$, and $\varphi_1, \varphi_2 \in S(D)$, from mixed-norm spaces $H(p, q, \phi)$, where $0 <...
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Institute of Mathematics, NAS of Ukraine
2016
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507766145155072 |
|---|---|
| author | Chen, Cui Zhou, Ze-Hua Чень, Цуй Чжоу, Цзи-Хуа |
| author_facet | Chen, Cui Zhou, Ze-Hua Чень, Цуй Чжоу, Цзи-Хуа |
| author_sort | Chen, Cui |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:30:37Z |
| description | We characterize the boundedness and compactness of the differences of weighted differentiation composition operators
$D^n_{\varphi_1, u_1} - D^n_{\varphi_2, u_2}$, where $n \in N_0, u_1, u_2 \in H(D)$, and $\varphi_1, \varphi_2 \in S(D)$, from mixed-norm spaces $H(p, q, \phi)$, where $0 < p,\; q < \infty$ and \phi is normal, to weighted-type spaces $H^{\infty}_v$. |
| first_indexed | 2026-03-24T02:14:31Z |
| format | Article |
| fulltext |
UDC 517.9
Cui Chen (Tianjin Univ. Finance and Economics, China),
Ze-Hua Zhou (Tianjin Univ., China)
DIFFERENCES OF THE WEIGHTED DIFFERENTIATION COMPOSITION
OPERATORS FROM MIXED-NORM SPACES TO WEIGHTED-TYPE SPACES*
РIЗНИЦI ЗВАЖЕНИХ ДИФЕРЕНЦIАЛЬНИХ ОПЕРАТОРIВ КОМПОЗИЦIЇ
З ПРОСТОРIВ IЗ МIШАНОЮ НОРМОЮ У ПРОСТОРАХ ЗВАЖЕНОГО ТИПУ
We characterize the boundedness and compactness of the differences of weighted differentiation composition operators
Dn
\varphi 1,u1
- Dn
\varphi 2,u2
, where n \in \BbbN 0, u1, u2 \in H(\BbbD ), and \varphi 1, \varphi 2 \in S(\BbbD ), from mixed-norm spaces H(p, q, \phi ), where
0 < p, q < \infty and \phi is normal, to weighted-type spaces H\infty
v .
Проаналiзовано обмеженiсть i компактнiсть рiзниць зважених диференцiальних операторiв композицiї Dn
\varphi 1,u1
-
Dn
\varphi 2,u2
, де n \in \BbbN 0, u1, u2 \in H(\BbbD ) та \varphi 1, \varphi 2 \in S(\BbbD ), iз просторiв iз мiшаною нормою H(p, q, \phi ), де 0 < p, q < \infty ,
а \phi є нормальним, у просторах зваженого типу H\infty
v .
1. Introduction. Let \BbbN 0 denote the set of all nonnegative integers, H(\BbbD ) and S(\BbbD ) represent the
class of analytic functions and analytic self-maps on the unit disk \BbbD of the complex plane of \BbbC ,
respectively.
A positive continuous function \phi is called normal [13] if there exist \delta \in [0, 1) and s, t (0 < s < t)
such that
\phi (r)
(1 - r)s
is decreasing on [\delta , 1) and \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 1
\phi (r)
(1 - r)s
= 0,
\phi (r)
(1 - r)t
is increasing on [\delta , 1) and \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 1
\phi (r)
(1 - r)t
= \infty .
For 0 < p, q < \infty and a normal weight \phi , the mixed-norm space denoted by H(p, q, \phi ) is the
space of all functions f \in H(\BbbD ) satisfying
\| f\| pH(p,q,\phi ) :=
1\int
0
Mp
q (f, r)
\phi p(r)
1 - r
dr < \infty ,
where
Mq(f, r) =
\left( 2\pi \int
0
| f(rei\theta )| qd\theta
\right) 1/q
, 0 \leq r < 1.
For 1 \leq p < \infty , H(p, q, \phi ) is a Banach space equipped with the norm \| \cdot \| H(p,q,\phi ). But when
0 < p < 1, \| \cdot \| H(p,q,\phi ) is just a quasinorm on H(p, q, \phi ), and then H(p, q, \phi ) is a Fréchet space but
not a Banach space. If 0 < p = q < \infty , H(p, q, \phi ) becomes a Bergman-type space, and moreover
if \phi (r) = (1 - r)
\alpha +1
p for \alpha > - 1, H(p, q, \phi ) is equivalent to the classical weighted Bergman space
Ap
\alpha defined by
* The paper was supported in part by the National Natural Science Foundation of China (Grant Nos. 11371276;
11301373; 11201331).
c\bigcirc CUI CHEN, ZE-HUA ZHOU, 2016
842 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
DIFFERENCES OF THE WEIGHTED DIFFERENTIATION COMPOSITION OPERATORS . . . 843
Ap
\alpha =
\left\{ f \in H(\BbbD ) : \| f\| p
Ap
\alpha
= (\alpha + 1)
\int
\BbbD
| f(z)| p(1 - | z| 2)\alpha dm(z) < \infty
\right\} ,
and the norms \| f\| Ap
\alpha
and \| f\| H(p,q,\phi ) are equivalent in this case. Recently there has been a great
interest in studying mixed norm spaces and operators on them on various domains in the complex plane
or in the n-dimensional complex vector space \BbbC n (see, for example, [4, 7, 9, 14 – 16, 18, 19, 21, 23, 24]
and the related references therein).
Let v be a strictly positive continuous and bounded function (weight) on \BbbD . The weighted-type
space H\infty
v is defined to be the collection of all functions f \in H(\BbbD ) that satisfy
\| f\| v := \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
v(z)| f(z)| < \infty .
With this norm the weighted-type space becomes a Banach space.
Let \varphi be a holomorphic self-map of \BbbD , the composition operator C\varphi induced by \varphi is defined by
(C\varphi f)(z) = f(\varphi (z)), f \in H(\BbbD ), z \in \BbbD .
Let D = D1 be the differentiation operator, i.e., Df = f \prime . If n \in \BbbN 0 then the operator Dn is
defined by D0f = f, Dnf = f (n), f \in H(\BbbD ). Some of the first product-type operators studied in the
literature were products of the composition and differentiation operators (see, e.g., [3, 5 – 7, 17, 20, 25]
and the related references therein).
The weighted differentiation composition operator, denoted by Dn
\varphi ,u, is defined by (Dn
\varphi ,uf)(z) =
= u(z)f (n)(\varphi (z)), which was studied in some recent papers such as [8, 10, 22 – 24].
Recently, there have been an increasing interest in studying the compact difference of operators
acting on different spaces of holomorphic functions. Motivated by some recent papers such as
[1, 7, 21, 23, 25, 26], here we characterize the boundedness and compactness of the operators
Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v .
Our results involve the pseudohyperbolic metric. For a \in \BbbD , let \varphi a be the automorphism of \BbbD
exchanging 0 and a, that is, \varphi a(z) =
a - z
1 - az
. For z, w \in \BbbD , the pseudohyperbolic distance between
z and w is given by \rho (z, w) = | \varphi z(w)| .
Throughout this paper, we will use the symbol C to denote a finite positive number, and it may
differ from one occurrence to the other. The notation A \asymp B means that there is a positive constant
C such that B/C \leq A \leq CB.
2. Background and some lemmas. Now let us state a couple of lemmas, which are used in the
proofs of the main results in the next sections. The first lemma is taken from [15] and [23].
Lemma 2.1. Assume that 0 < p, q < \infty , \phi is normal and f \in H(p, q, \phi ). Then for every
n \in \BbbN 0, there is a constant C independent of f such that
| f (n)(z)| \leq C
\| f\| H(p,q,\phi )
\phi (| z| )(1 - | z| 2)
1
q+n
, z \in \BbbD . (2.1)
The next lemma can be found in [13].
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
844 CUI CHEN, ZE-HUA ZHOU
Lemma 2.2. For \beta > - 1 and m > 1 + \beta , one has
1\int
0
(1 - r)\beta
(1 - \rho r)m
dr \leq C(1 - \rho )1+\beta - m, 0 < \rho < 1. (2.2)
Lemma 2.3. Assume that 0 < p, q < \infty , \phi is normal and n \in \BbbN 0. Then for each f \in H(p, q, \phi ),
there is a constant C independent of f such that\bigm| \bigm| \bigm| \phi (| z| )(1 - | z| 2)
1
q+n
f (n)(z) - \phi (| \omega | )(1 - | \omega | 2)
1
q+n
f (n)(\omega )
\bigm| \bigm| \bigm| \leq
\leq C\| f\| H(p,q,\phi )\rho (z, \omega ). (2.3)
Proof. For f \in H(p, q, \phi ), let u(z) = \phi (| z| )(1 - | z| 2)
1
q+n
, by Lemma 2.1, we obtain f (n) \in
\in H\infty
u , so from Lemma 3.2 in [2] and Lemma 2.1, there is a constant C > 0 such that
| u(z)f (n)(z) - u(\omega )f (n)(\omega )| \leq C\| f (n)\| u\rho (z, \omega ) \leq C\| f\| H(p,q,\phi )\rho (z, \omega )
for all z, \omega \in \BbbD .
Remark. From the proof of Lemma 2.3, it is not difficult to see that for any z, \omega \in r\BbbD = \{ z \in \BbbD :
| z| < r < 1\} , then\bigm| \bigm| \bigm| \phi (| z| )(1 - | z| 2)
1
q+n
f (n)(z) - \phi (| \omega | )(1 - | \omega | 2)
1
q+n
f (n)(\omega )
\bigm| \bigm| \bigm| \leq
\leq C\rho (z, \omega ) \mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in r\BbbD
\phi (| \zeta | )(1 - | \zeta | 2)
1
q+n| f (n)(\zeta )| (2.4)
for any f \in H(p, q, \phi ).
The next Schwartz-type lemma can be proved in a standard way [12].
Lemma 2.4. Suppose n \in \BbbN 0, 0 < p, q < \infty , u1, u2 \in H(\BbbD ), \varphi 1, \varphi 2 \in S(\BbbD ) and \phi is normal.
Then the operator Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v is compact if and only if Dn
\varphi 1,u1
- Dn
\varphi 2,u2
:
H(p, q, \phi ) \rightarrow H\infty
v is bounded and for any bounded sequence (fk)k\in \BbbN in H(p, q, \phi ) which converges
to zero uniformly on compact subsets of \BbbD , we have \| (Dn
\varphi 1,u1
- Dn
\varphi 2,u2
)fk\| v \rightarrow 0, as k \rightarrow \infty .
The following result is well-known. It can be proved by a slight modification of the proof of
Theorem 2 in [4].
Lemma 2.5. Assume that 0 < p, q < \infty , \phi is normal and n \in \BbbN 0. Then for each f \in H(p, q, \phi ),
1\int
0
Mp
q (f, r)
\phi p(r)
1 - r
dr \asymp
n - 1\sum
j=0
| f (j)(0)| +
1\int
0
Mp
q (f
(n), r)
\phi p(r)
1 - r
(1 - r)npdr. (2.5)
3. Boundedness of \bfitD \bfitn
\bfitvarphi \bfone ,\bfitu \bfone
- \bfitD \bfitn
\bfitvarphi \bftwo ,\bfitu \bftwo
: \bfitH (\bfitp , \bfitq , \bfitphi ) \rightarrow \bfitH \infty
\bfitv . In this section we will charac-
terize the boundedness of the operator Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v . For the purpose, we list
the following three conditions which we will use below:
M1 = \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
v(z)| u1(z)| \rho (\varphi 1(z), \varphi 2(z))
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
< \infty , (3.1)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
DIFFERENCES OF THE WEIGHTED DIFFERENTIATION COMPOSITION OPERATORS . . . 845
M2 = \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
v(z)| u2(z)| \rho (\varphi 1(z), \varphi 2(z))
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
< \infty , (3.2)
M3 = \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| v(z)u1(z)
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
- v(z)u2(z)
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < \infty . (3.3)
Theorem 3.1. Suppose n \in \BbbN 0, 0 < p, q < \infty , u1, u2 \in H(\BbbD ), \varphi 1, \varphi 2 \in S(\BbbD ) and \phi is
normal. Then the following statements are equivalent:
(i) Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v is bounded.
(ii) The conditions (3.1) and (3.3) hold.
(iii) The conditions (3.2) and (3.3) hold.
Proof. First, we prove the implication (i) \Rightarrow (ii). Assume that Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow
\rightarrow H\infty
v is bounded. Fix w \in \BbbD , we consider the function fw defined by
fw(z) =
z\int
0
tn\int
0
\cdot \cdot \cdot
t2\int
0
(1 - | \varphi 1(w)| 2)t+1
\phi (| \varphi 1(w)| )(1 - \varphi 1(w)t1)
1
q+t+1+n
\varphi \varphi 2(w)(t1)dt1dt2 \cdot \cdot \cdot dtn. (3.4)
Next we show that fw \in H(p, q, \phi ). Notice that
f (n)
w (z) =
(1 - | \varphi 1(w)| 2)t+1
\phi (| \varphi 1(w)| )(1 - \varphi 1(w)z)
1
q+t+1+n
\varphi \varphi 2(w)(z),
according to Lemma 1.4.10 in [11],
Mp
q (f
(n)
w , r) =
\left( 2\pi \int
0
(1 - | \varphi 1(w)| 2)q(t+1)
\phi q(| \varphi 1(w)| )| 1 - \varphi 1(w)rei\theta | 1+q(t+1+n)
| \varphi \varphi 2(w)(re
i\theta )| qd\theta
\right) p/q
\leq
\leq (1 - | \varphi 1(w)| 2)p(t+1)
\phi p(| \varphi 1(w)| )
\left( 2\pi \int
0
d\theta
| 1 - \varphi 1(w)rei\theta | 1+q(t+1+n)
\right) p/q
\asymp
\asymp (1 - | \varphi 1(w)| 2)p(t+1)
\phi p(| \varphi 1(w)| )(1 - r| \varphi 1(w)| )p(t+1+n)
. (3.5)
By using Lemma 2.5, (3.5), the fact that f (j)
w (0) = 0, j = 1, . . . , n - 1, the normality of \phi , Lemma
1.4.10 in [11] and Lemma 2.2, we have
\| fw\| pH(p,q,\phi ) \leq C
1\int
0
(1 - | \varphi 1(w)| 2)p(t+1)
\phi p(| \varphi 1(w)| )(1 - r| \varphi 1(w)| )p(t+1+n)
\phi p(r)
1 - r
(1 - r)npdr \leq
\leq C
1\int
0
(1 - | \varphi 1(w)| 2)p(t+1)
\phi p(| \varphi 1(w)| )(1 - r| \varphi 1(w)| )p(t+1)
\phi p(r)
1 - r
dr \leq
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
846 CUI CHEN, ZE-HUA ZHOU
\leq C
\left( | \varphi 1(w)| \int
0
(1 - | \varphi 1(w)| 2)p(t+1)
\phi p(| \varphi 1(w)| )(1 - r| \varphi 1(w)| )p(t+1)
\phi p(r)
1 - r
dr+
+
1\int
| \varphi 1(w)|
(1 - | \varphi 1(w)| 2)p(t+1)
\phi p(| \varphi 1(w)| )(1 - r| \varphi 1(w)| )p(t+1)
\phi p(r)
1 - r
dr
\right) \leq
\leq C(1 - | \varphi 1(w)| 2)p
| \varphi 1(w)| \int
0
(1 - r)pt - 1
(1 - r| \varphi 1(w)| )p(t+1)
dr+
+C(1 - | \varphi 1(w)| 2)p
1\int
| \varphi 1(w)|
(1 - r)ps - 1
(1 - r| \varphi 1(w)| )p(s+1)
dr \leq C.
Therefore fw \in H(p, q, \phi ), and moreover \mathrm{s}\mathrm{u}\mathrm{p}w\in \BbbD \| fw\| H(p,q,\phi ) \leq C. Note that
f (n)
w (\varphi 1(w)) =
\rho (\varphi 1(w), \varphi 2(w))
\phi (| \varphi 1(w)| )(1 - | \varphi 1(w)| 2)
1
q+n
and f (n)
w (\varphi 2(w)) = 0.
So by the boundedness of Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v , we obtain
\infty > \| (Dn
\varphi 1,u1
- Dn
\varphi 2,u2
)fw\| v =
= \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
v(z)| u1(z)f (n)
w (\varphi 1(z)) - u2(z)f
(n)
w (\varphi 2(z))| \geq
\geq v(w)| u1(w)f (n)
w (\varphi 1(w)) - u2(w)f
(n)
w (\varphi 2(w))| =
=
v(w)| u1(w)| \rho (\varphi 1(w), \varphi 2(w))
\phi (| \varphi 1(w)| )(1 - | \varphi 1(w)| 2)
1
q+n
. (3.6)
Since w \in \BbbD is an arbitrary element, (3.1) comes from (3.6).
Next we prove (3.3). Fix w \in \BbbD , let
gw(z) =
z\int
0
tn\int
0
. . .
t2\int
0
(1 - | \varphi 2(w)| 2)t+1
\phi (| \varphi 2(w)| )(1 - \varphi 2(w)t1)
1
q+t+1+n
dt1dt2 . . . dtn.
Similarly as for the test functions in (3.4), we obtained that gw \in H(p, q, \phi ) with g
(n)
w (\varphi 2(w)) =
=
1
\phi (| \varphi 2(w)| )(1 - | \varphi 2(w)| 2)
1
q+n
. Then
\infty > \| (Dn
\varphi 1,u1
- Dn
\varphi 2,u2
)gw\| v \geq
\geq v(w)| u1(w)g(n)w (\varphi 1(w)) - u2(w)g
(n)
w (\varphi 2(w))| = | I(w) + J(w)| , (3.7)
where
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
DIFFERENCES OF THE WEIGHTED DIFFERENTIATION COMPOSITION OPERATORS . . . 847
I(w) = \phi (| \varphi 2(w)| )(1 - | \varphi 2(w)| 2)
1
q+n
g(n)w (\varphi 2(w))
\left[ v(w)u1(w)
\phi (| \varphi 1(w)| )(1 - | \varphi 1(w)| 2)
1
q+n
-
- v(w)u2(w)
\phi (| \varphi 2(w)| )(1 - | \varphi 2(w)| 2)
1
q+n
\right]
and
J(w) =
v(w)u1(w)
\phi (| \varphi 1(w)| )(1 - | \varphi 1(w)| 2)
1
q+n
\Bigl[
\phi (| \varphi 1(w)| )(1 - | \varphi 1(w)| 2)
1
q+n
g(n)w (\varphi 1(w)) -
- \phi (| \varphi 2(w)| )(1 - | \varphi 2(w)| 2)
1
q+n
g(n)w (\varphi 2(w))
\Bigr]
.
By Lemma 2.3 and (3.1), we conclude that | J(w)| < \infty . From this along with (3.7) we get
| I(w)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| v(w)u1(w)
\phi (| \varphi 1(w)| )(1 - | \varphi 1(w)| 2)
1
q+n
- v(w)u2(w)
\phi (| \varphi 2(w)| )(1 - | \varphi 2(w)| 2)
1
q+n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < \infty
for all w \in \BbbD , thus (3.3) holds.
(ii) \Rightarrow (iii). Assume that (3.1) and (3.3) hold, we only need to show that (3.2) holds. In fact,
v(z)| u2(z)| \rho (\varphi 1(z), \varphi 2(z))
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\leq v(z)| u1(z)| \rho (\varphi 1(z), \varphi 2(z))
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
+
+
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| v(z)u1(z)
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
- v(z)u2(z)
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \rho (\varphi 1(z), \varphi 2(z)).
From which, using (3.1) and (3.3), the desired condition (3.2) holds.
(iii) \Rightarrow (i). Assume that (3.2) and (3.3) hold. By Lemma 2.1 and Lemma 2.3, for any f \in
\in H(p, q, \phi ), we have
v(z)| u1(z)f (n)(\varphi 1(z)) - u2(z)f
(n)(\varphi 2(z))| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
f (n)(\varphi 1(z))
\left[ v(z)u1(z)
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
-
- v(z)u2(z)
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\right] +
\Bigl[
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
f (n)(\varphi 1(z)) -
- \phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
f (n)(\varphi 2(z))
\Bigr] v(z)u2(z)
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq C\| f\| H(p,q,\phi ) + C
v(z)| u2(z)| \rho (\varphi 1(z), \varphi 2(z))
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\| f\| H(p,q,\phi ) \leq
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
848 CUI CHEN, ZE-HUA ZHOU
\leq C\| f\| H(p,q,\phi )
for each z \in \BbbD . From which it follows that Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v is bounded.
Theorem 3.1 is proved.
4. Compactness of \bfitD \bfitn
\bfitvarphi \bfone ,\bfitu \bfone
- \bfitD \bfitn
\bfitvarphi \bftwo ,\bfitu \bftwo
: \bfitH (\bfitp , \bfitq , \bfitphi ) \rightarrow \bfitH \infty
\bfitv . In this section, we turn
our attention to the problem of the compactness of the operator. Here we consider the following
conditions:
M4 =
v(z)u1(z)\rho (\varphi 1(z), \varphi 2(z))
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
\rightarrow 0 as | \varphi 1(z)| \rightarrow 1, (4.1)
M5 =
v(z)u2(z)\rho (\varphi 1(z), \varphi 2(z))
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\rightarrow 0 as | \varphi 2(z)| \rightarrow 1, (4.2)
M6 =
v(z)u1(z)
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
- v(z)u2(z)
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\rightarrow 0
as | \varphi 1(z)| \rightarrow 1 and | \varphi 2(z)| \rightarrow 1. (4.3)
Theorem 4.1. Suppose n \in \BbbN 0, 0 < p, q < \infty , u1, u2 \in H(\BbbD ), \varphi 1, \varphi 2 \in S(\BbbD ) and \phi is
normal. Then Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v is compact if and only if Dn
\varphi 1,u1
- Dn
\varphi 2,u2
:
H(p, q, \phi ) \rightarrow H\infty
v is bounded and the conditions (4.1) – (4.3) hold.
Proof. First we suppose that Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v is bounded and the conditions
(4.1) – (4.3) hold. It is clear that the conditions (3.1) – (3.3) hold by Theorem 3.1. From (4.1) – (4.3),
it follows that for any \varepsilon > 0, there exists 0 < r < 1 such that
v(z)| u1(z)| \rho (\varphi 1(z), \varphi 2(z))
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
\leq \varepsilon for | \varphi 1(z)| > r, (4.4)
v(z)| u2(z)| \rho (\varphi 1(z), \varphi 2(z))
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\leq \varepsilon for | \varphi 2(z)| > r, (4.5)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| v(z)u1(z)
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
- v(z)u2(z)
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \varepsilon
(4.6)
for | \varphi 1(z)| > r, | \varphi 2(z)| > r.
Now, let (fk)k\in \BbbN be a bounded sequence in H(p, q, \phi ) with \| fk\| H(p,q,\phi ) \leq 1 and fk \rightarrow
\rightarrow 0 uniformly on compact subsets of \BbbD . By Lemma 2.4 we need only to show that \| (Dn
\varphi 1,u1
-
- Dn
\varphi 2,u2
)fk\| v \rightarrow 0 as k \rightarrow \infty . A direct calculation shows that
v(z)| u1(z)f (n)
k (\varphi 1(z)) - u2(z)f
(n)
k (\varphi 2(z))| = | Ik(z) + Jk(z)| , (4.7)
where
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
DIFFERENCES OF THE WEIGHTED DIFFERENTIATION COMPOSITION OPERATORS . . . 849
Ik(z) = \phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
f
(n)
k (\varphi 2(z))
\left[ v(z)u1(z)
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
-
- v(z)u2(z)
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\right]
and
Jk(z) =
v(z)u1(z)
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
\Bigl[
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
f
(n)
k (\varphi 1(z)) -
- \phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
f
(n)
k (\varphi 2(z))
\Bigr]
.
We divide the argument into four cases:
Case 1: | \varphi 1(z)| \leq r and | \varphi 2(z)| \leq r.
By the assumption, note that fk converges to zero uniformly on E = \{ w : | w| \leq r\} as k \rightarrow \infty ,
and using (3.3), it is easy to check that Ik(z) \rightarrow 0, k \rightarrow \infty uniformly for all z with | \varphi 2(z)| \leq r.
On the other hand, from (2.4), (3.1) and since fk converges to zero uniformly on E, we have that
| Jk(z)| \leq C
v(z)| u1(z)| \rho (\varphi 1(z), \varphi 2(z))
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
\mathrm{s}\mathrm{u}\mathrm{p}
| \zeta | \leq r
\phi (| \zeta | )(1 - | \zeta | 2)
1
q+n| f (n)(\zeta )| \leq C\varepsilon .
Case 2: | \varphi 1(z)| > r and | \varphi 2(z)| \leq r.
As in the proof of Case 1, Ik(z) \rightarrow 0 uniformly as k \rightarrow \infty . On the other hand, using Lemma
2.3 and (4.4) we obtain | Jk(z)| \leq C\varepsilon .
Case 3: | \varphi 1(z)| > r and | \varphi 2(z)| > r.
For k sufficiently large, by Lemma 2.1 and (4.6) we obtain that | Ik(z)| \leq C\varepsilon . Meanwhile,
| Jk(z)| \leq C\varepsilon by Lemma 2.3 and (4.4).
Case 4: | \varphi 1(z)| \leq r and | \varphi 2(z)| > r. We rewrite
v(z)| u1(z)f (n)
k (\varphi 1(z)) - u2(z)f
(n)
k (\varphi 2(z))| = | Pk(z) +Qk(z)| ,
where
Pk(z) = \phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
f
(n)
k (\varphi 1(z))
\left[ v(z)u1(z)
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
-
- v(z)u2(z)
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\right]
and
Qk(z) =
v(z)u2(z)
\phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
\Bigl[
\phi (| \varphi 1(z)| )(1 - | \varphi 1(z)| 2)
1
q+n
f
(n)
k (\varphi 1(z)) -
- \phi (| \varphi 2(z)| )(1 - | \varphi 2(z)| 2)
1
q+n
f
(n)
k (\varphi 2(z))
\Bigr]
.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
850 CUI CHEN, ZE-HUA ZHOU
The desired result follows by an argument analogous to that in the proof of Case 2. Thus, together
with the above cases, we conclude that
\| (Dn
\varphi 1,u1
- Dn
\varphi 2,u2
)fk\| v =
= \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
v(z)| u1(z)f (n)
k (\varphi 1(z)) - u2(z)f
(n)
k (\varphi 2(z))| \leq C\varepsilon (4.8)
for sufficiently large k. Employing Lemma 2.4 combining with the arbitrariness of \varepsilon , we obtain the
compactness of Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v .
For the converse direction, we suppose that Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v is compact.
From which we can easily obtain the boundedness of Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v . Next we
only need to show that (4.1) – (4.3) hold.
Let (zk)k\in \BbbN be a sequence of points in \BbbD such that | \varphi 1(zk)| \rightarrow 1 as k \rightarrow \infty . Define the functions
fk(z) =
z\int
0
tn\int
0
. . .
t2\int
0
(1 - | \varphi 1(zk)| 2)t+1
\phi (| \varphi 1(zk)| )(1 - \varphi 1(zk)t1)
1
q
+t+1+n
\varphi \varphi 2(zk)(t1)dt1dt2 . . . dtn. (4.9)
Clearly, fk \in H(p, q, \phi ) with \mathrm{s}\mathrm{u}\mathrm{p}k\in \BbbN \| fk\| H(p,q,\phi ) \leq C, and fk converges to 0 uniformly on compact
subsets of \BbbD as k \rightarrow \infty . Moreover,
f
(n)
k (\varphi 1(zk)) =
\rho (\varphi 1(zk), \varphi 2(zk))
\phi (| \varphi 1(zk)| )(1 - | \varphi 1(zk)| 2)
1
q+n
and f
(n)
k (\varphi 2(zk)) = 0. (4.10)
Then
\| (Dn
\varphi 1,u1
- Dn
\varphi 2,u2
)fk\| v = \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
v(z)| u1(z)f (n)
k (\varphi 1(z)) - u2(z)f
(n)
k (\varphi 2(z))| \geq
\geq v(zk)| u1(zk)f
(n)
k (\varphi 1(zk)) - u2(zk)f
(n)
k (\varphi 2(zk))| =
=
v(zk)| u1(zk)| \rho (\varphi 1(zk), \varphi 2(zk))
\phi (| \varphi 1(zk)| )(1 - | \varphi 1(zk)| 2)
1
q+n
. (4.11)
On the other hand, since Dn
\varphi 1,u1
- Dn
\varphi 2,u2
: H(p, q, \phi ) \rightarrow H\infty
v is compact, by Lemma 2.4, it follows
that \| (Dn
\varphi 1,u1
- Dn
\varphi 2,u2
)fk\| v \rightarrow 0, k \rightarrow \infty . Letting k \rightarrow \infty in (4.11), it follows that (4.1) holds.
The condition (4.2) holds for the similar arguments.
Now it remains to show that condition (4.3) holds. Assume that (zk)k\in \BbbN is a sequence in \BbbD such
that | \varphi 1(zk)| \rightarrow 1 and | \varphi 2(zk)| \rightarrow 1 as k \rightarrow \infty . Define the function
gk(z) =
z\int
0
tn\int
0
. . .
t2\int
0
(1 - | \varphi 2(zk)| 2)t+1
\phi (| \varphi 2(zk)| )(1 - \varphi 2(zk)t1)
1
q+t+1+n
dt1dt2 . . . dtn.
It is easy to check that gk converges to 0 uniformly on compact subsets of \BbbD as k \rightarrow \infty and
gk \in H(p, q, \phi ) with \| gk\| H(p,q,\phi ) \leq C for all k \in \BbbN . It follows from Lemma 2.4 that \| (Dn
\varphi 1,u1
-
- Dn
\varphi 2,u2
)gk\| v \rightarrow 0, k \rightarrow \infty . On the other hand, we have
\| (Dn
\varphi 1,u1
- Dn
\varphi 2,u2
)gk\| v \geq v(zk)| u1(zk)g
(n)
k (\varphi 1(zk)) - u2(zk)g
(n)
k (\varphi 2(zk))| =
= | I(zk) + J(zk)| , (4.12)
where
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
DIFFERENCES OF THE WEIGHTED DIFFERENTIATION COMPOSITION OPERATORS . . . 851
I(zk) = \phi (| \varphi 2(zk)| )(1 - | \varphi 2(zk)| 2)
1
q+n
g
(n)
k (\varphi 2(zk))
\left[ v(zk)u1(zk)
\phi (| \varphi 1(zk)| )(1 - | \varphi 1(zk)| 2)
1
q+n
-
- v(zk)u2(zk)
\phi (| \varphi 2(zk)| )(1 - | \varphi 2(zk)| 2)
1
q+n
\right] ,
J(zk) =
v(zk)u1(zk)
\phi (| \varphi 1(zk)| )(1 - | \varphi 1(zk)| 2)
1
q+n
\biggl[
\phi (| \varphi 1(zk)| )(1 - | \varphi 1(zk)| 2)
1
q+n
g
(n)
k (\varphi 1(zk)) -
- \phi (| \varphi 2(zk)| )(1 - | \varphi 2(zk)| 2)
1
q+n
g
(n)
k (\varphi 2(zk))
\biggr]
.
By Lemma 2.3 and the condition (4.1) that has been proved, we get J(zk) \rightarrow 0, k \rightarrow \infty .
This along with (4.12) shows that I(zk) \rightarrow 0, k \rightarrow \infty . Hence (4.3) is true since g
(n)
k (\varphi 2(zk)) =
=
1
\phi (| \varphi 2(zk)| )(1 - | \varphi 2(zk)| 2)
1
q+n
.
Theorem 4.1 is proved.
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ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 6
|
| id | umjimathkievua-article-1883 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:14:31Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/aa/d01be031f0f1e340dedd5a4ff54135aa.pdf |
| spelling | umjimathkievua-article-18832019-12-05T09:30:37Z Differences of the weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces Рiзницi зважених диференцiальних операторiв композицiї з просторiв iз мiшаною нормою у просторах зваженого типу Chen, Cui Zhou, Ze-Hua Чень, Цуй Чжоу, Цзи-Хуа We characterize the boundedness and compactness of the differences of weighted differentiation composition operators $D^n_{\varphi_1, u_1} - D^n_{\varphi_2, u_2}$, where $n \in N_0, u_1, u_2 \in H(D)$, and $\varphi_1, \varphi_2 \in S(D)$, from mixed-norm spaces $H(p, q, \phi)$, where $0 < p,\; q < \infty$ and \phi is normal, to weighted-type spaces $H^{\infty}_v$. Проаналiзовано обмеженiсть i компактнiсть рiзниць зважених диференцiальних операторiв композицiї $D^n_{\varphi_1, u_1} - D^n_{\varphi_2, u_2}$, де $n \in N_0, u_1, u_2 \in H(D)$ та $\varphi_1, \varphi_2 \in S(D)$, iз просторiв iз мiшаною нормою $H(p, q, \phi)$, де $0 < p,\; q < \infty$, а $\phi$ є нормальним, у просторах зваженого типу $H^{\infty}_v$. Institute of Mathematics, NAS of Ukraine 2016-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1883 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 6 (2016); 842-852 Український математичний журнал; Том 68 № 6 (2016); 842-852 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1883/865 Copyright (c) 2016 Chen Cui; Zhou Ze-Hua |
| spellingShingle | Chen, Cui Zhou, Ze-Hua Чень, Цуй Чжоу, Цзи-Хуа Differences of the weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces |
| title | Differences of the weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces |
| title_alt | Рiзницi зважених диференцiальних операторiв композицiї з просторiв iз мiшаною нормою у просторах зваженого типу |
| title_full | Differences of the weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces |
| title_fullStr | Differences of the weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces |
| title_full_unstemmed | Differences of the weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces |
| title_short | Differences of the weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces |
| title_sort | differences of the weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1883 |
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