# The Product-Type Operators from Hardy Spaces intonth Weighted-Type Spaces

Authors: Ebrahim Abbasi
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

The main goal of this paper is to investigate the boundedness and essential norm of a class of product-type operatorsTu,v,φm,m∈ℕ from Hardy spaces into nth weighted-type spaces. As a corollary, we obtain some equivalent conditions for compactness of such operators.


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## Body

## 1. Introduction

LetD denote the open unit disc of the complex plane ℂ and HD denotes the space of all holomorphic functions on D. The space of bounded holomorphic functions on D is denoted by H∞; it is a Banach space with the equipped norm
(1)gH∞=supz∈Dgz.Let0<p<∞. A Hardy space Hp consists of all g∈HD such that
(2)gHp=sup0<r<112π∫02πgreiθpdθ1/p<∞.When1≤p<∞, Hp is a Banach space with the norm ·Hp. If 0<p<1, Hp is a nonlocally convex topological vector space and it is a complete metric space (see [1]).Letn∈ℕ0=0,1,2,⋯ and μz be a weight, continuous, and positive function on D. The nth weighted-type space Wμn, consists of all g∈HD such that
(3)bWμng=supz∈Dμzgnz<∞.It is a Banach space with the following norm(4)gWμn=∑i=0n−1gi0+bWμng.The littlenth weighted-type space Wμ,0n is a closed subspace of Wμn such that for any g∈Wμ,0n(5)limz⟶1μzgnz=0.For more information aboutnth weighted-type spaces, see [2–4]. Let α>0. Then, W1−z2a0=H−α(growth space), W1−z2a1=Bα(Bloch-type space), and W1−z2a2=Zα(Zygmund-type space). Also Wμ0=Hμ(weighted-type space), Wμ1=Bμ(weighted Bloch space), Wμ2=Zμ(weighted Zygmund space), and W1−z2log2/1−z21 coincide with the logarithmic Bloch space Blog.Letn,k∈ℕ0 such that k≤n; the partial Bell polynomials are triangular
(6)Bn,kx1,⋯,xn−k+1=∑n!j1!⋯jn−k+1!x11!j1⋯xn−k+1n−k+1!jn−k+1,where the sum is taken over all nonnegative integers j1,⋯,jn−k+1 such that
(7)j1+⋯+n−k+1jn−k+1=n,j1+⋯+jn−k+1=k.More information about Bell polynomials can be found in ([5], p 134).Letm∈ℕ0, u,v∈HD and φ∈SD be the set of all holomorphic self-map of D. In [6], Stevic′, Sharma and Krishan defined a new product-type operator Tu,v,φm as follows:
(8)Tu,v,φmgz=uzgmφz+vzgm+1φz,g∈HD,z∈D.Whenm=0, we obtain the Stevic′-Sharma-type operator, and for v≡0, we get the generalized weighted composition operators Du,φm. Product-type operators on some spaces of analytic functions on the unit disc have become a subject of increasing interest in the recent years. We refer the reader to [6–10] and the references therein.Liu and Yu have considered boundedness and compactness of operatorTu,v,φ0 from Hardy spaces and H∞ into the logarithmic Bloch space in [11, 12]. Also, Zhang and Liu have found some characterizations for boundedness and compactness of operator Tu,v,φ0 from Hardy spaces into the weighted Zygmund space in [10]. Recently, the boundedness, compactness, and norm of operator Tu,v,φ0:Hp⟶Wμn are considered in [13].Motivated by previous works, the results found in them will be generalized for operatorTu,v,φm. For this purpose in the second section of this paper, we give some characterizations for boundedness of operator Tu,v,φm:Hp⟶Wμn where m,n∈ℕ and 0<p≤∞. In the third section, some new estimates are obtained for the essential norm of such operators. As a corollary, some equivalent conditions are acquired for compactness of such operators.Throughout this paper, if there exists a constantc such that a≥cb, we use the notation a⪰b. The symbol a≈b means that a⪰b⪰a.


## 2. Boundedness

In this section, some equivalent conditions are found for the boundedness of operatorTu,v,φmm∈ℕ from Hp0<p≤∞ into nth weighted-type spaces. Firstly, we state some lemmas.Lemma 1 (see [14], Propositions 8).
Letα>0. Then, for any g∈Bα and n∈ℕ,
(9)∑i=0n−1gi0+sup1−z2α−1+nz∈Dgnz≈gBα.Lemma 2 (see [15], Lemma 2.1).
Letα>0 and B0α=W1−z2a,01. The sequence jα−1zj1∞ is bounded in B0α and
(10)limj⟶∞jα−1zjBα=2αea.From Lemma1, Proposition 5.1.2 [16] and [1], the next lemma is obtained.Lemma 3.
Let0<p≤∞, n∈ℕ0 and g∈Hp. Then,
(11)gnz⪯gHp1−z21/p+n,z∈D.Lemma 4 (see [4]).
Letφ∈SD and u,g∈HD. Then, for any m,n∈ℕ0(12)uzgmφzn=∑i=0ngm+iφz∑l=innlun−lzBl,iφ′z,φ′′z,⋯,φl−i+1z.
In this paper, we set(13)fi,az=1−a2i1−a¯z1/p+i,0≠a,z∈D,i∈ℕ,Ii,φn,uz=∑l=innlun−lzBl,iφ′z,φ′′z,⋯,φl−i+1z,In+1,φn,uz=I−1,φn,vz=0,pjz=zj.By using the functionsfi,a, we obtain the following lemma. Since the proof of it resembles the proof of Lemma 1 [2], therefore, it is omitted.Lemma 5.
Letδik be Kronecker delta. For any 0≠a∈D, m∈ℕ0, and i∈0,⋯,n+1, there exists a function gi,a∈Hp such that
(14)gi,am+ka=δika¯m+k1−a2m+k+1/p.In this case,gi,az=∑j=1n+2cjifj,az, where cji are independent of choice a.Theorem 6.
Letm,n∈ℕ, 0<p≤∞, u,v∈HD, μ be a weight and φ∈SD. Then, the following statements are equivalent
(a)
The operatorTu,v,φm:Hp⟶Wμn is bounded(b)
The operatorTu,v,φm:B1+1/p⟶Wμn is bounded(c)
The operatorTu,v,φm:B01+1/p⟶Wμn is bounded(d)
supj≥1j1/pTu,v,φmpjWμn<∞(e)
For eachi∈0,1,⋯,n+1,supa∈DTu,v,φmfi+1,aWμn<∞ and supz∈DμzIi,φn,u+Ii−1,φn,vz<∞(f)
For eachi∈0,⋯,n+1,(15)supz∈DμzIi,φn,u+Ii−1,φn,vz1−φz2m+i+1/p<∞Proof.
b⟹c Since B01+1/p⊂B1+1/p, we get c.
c⟹d It follows from Lemma 2.
d⟹e For each i∈0,⋯,n+1 and a∈D,
(16)fi+1,az=1−a2i+1∑j=0∞Γi+1+1/p+jj!Γi+1+1/pa¯jzj.
So,(17)Tu,v,φmfi+1,aWμn⪯1−a2i+1∑j=0∞jia¯jj1/pTu,v,φmpjWμn≤supj≥1j1/pTu,v,φmpjWμn1−a2i+1∑j=0∞jia¯j,⪯supj≥1j1/pTu,v,φmpjWμn1−a2i+1∑j=0∞Γi+1+jj!Γi+1a¯j,⪯2i+1supj≥1j1/pTu,v,φmpjWμn.
Hence,supa∈DTu,v,φmfi+1,aWμn<∞. It is remained to show that for each i∈0,1,⋯,n+1, supz∈DμzIi,φn,u+Ii−1,φn,vz<∞. Applying the operator Tu,v,φm to pmz, by using Lemma 4, we have
(18)supz∈DμzI0,φn,uz+I−1,φn,vz⏟0≤Tu,v,φmpjWμnm!<∞.
Now, assume that we have the following inequalities for0≤i≤j−1,
(19)supz∈DμzIi,φn,u+Ii−1,φn,vz<∞,where j≤n+1. By applying the operator Tu,v,φm for pj+mz and using Lemma 4, we get
(20)supz∈Dμz∣j+m!j!φzjI0,φn,uz+∑k=1jj+m!j−k!φzj−kIk,φn,u+Ik−1,φn,vz∣≤Tu,v,φmpjWμn<∞.
Sinceφ≤1, so from the triangle inequality, we have
(21)supz∈DμzIj,φn,uz+Ij−1,φn,vz<∞.
e⇒f For any φa≠0 and i∈0,⋯,n+1, by using Lemmas 4 and 5, we obtain
(22)μaφam+iIi,φn,u+Ii−1,φn,va1−φa2m+i+1/p≤supa∈DTu,v,φmgi+1,aWμn≤∑j=1n+2cjisupa∈DTu,v,φmfi+1,aWμn<∞.
Therefore from the last inequality,(23)supφa>1/2μaIi,φn,u+Ii−1,φn,va1−φa2m+i+1/p⪯∑j=1n+2cjisupa∈DTu,v,φmfi+1,aWμn<∞.
On the other hand, frome, we have
(24)supφa≤1/2μaIi,φn,u+Ii−1,φn,va1−φa2m+i+1/p⪯supφa≤1/2μaIi,φn,u+Ii−1,φn,va<∞.
f⟹b For any f∈B1+1/p, by using Lemmas 1 and 4, we obtain
(25)μz∣Tu,v,φmfnz≤μz∑i=0n+1fm+iφzIi,φn,u+Ii−1,φn,vz∣≤fB1+1/p∑i=0n+1supz∈DμzIi,φn,u+Ii−1,φn,vz1−φz2m+i+1/p.
Also for each0≤k<n, we have
(26)Tu,v,φmfk0≤fB1+1/p∑i=0k+1Ii,φk,u+Ii−1,φk,v01−φ02m+i+1/p.
So, the operatorTu,v,φm:B1+1/p⟶Wμn is bounded.
b⟹a From Lemma 3, Hp⊂B1+1/p, so we obtain a.
a⟹e It is clear that fi,a∈Hp and supa∈Dfi,aHp<∞. Hence, for each i∈0,⋯,n+1,
(27)supa∈DTu,v,φmfi+1,aWμn≤Tu,v,φmHp⟶Wμnsupa∈Dfi+1,aHp<∞.
The proof of the second part ofe is similar to the proof d⟹e, so it is omitted. The proof is complete.☐


## 3. Essential Norm

In this section, we find some approximations for the essential norm of operatorTu,v,φm from Hardy spaces into nth weighted type-spaces. As a corollary, we give some equivalent conditions for compactness of such operators.LetX and Y be Banach spaces and T:X⟶Y be the continuous linear operator. The essential norm of T is the distance from T to the compact operators, that is,
(28)Te,X⟶Y=infT−K:K:X⟶Yiscompact.It is clear thatT is compact if and only if Te,X⟶Y=0.Theorem 7.
Letm,n∈ℕ, 0<p≤∞, u,v∈HD, φ∈SD, and μ be a weight such that Tu,v,φm:B1+1/p⟶Wμn be bounded. Then
(29)Tu,v,φme,B1+1/p⟶Wμn≈maxAii=0n+1≈maxBii=0n+1,where
(30)Ai=limsup∣a∣⟶1Tu,v,φmfi+1,aWμn,Bi=limsupφz⟶1μzIi,φn,u+Ii−1,φn,vz1−φz2m+i+1/p.Proof.
For eachi∈0,⋯,n+1, supa∈Dfi+1,aB1+1/p<∞ and fi+1,a⟶0 uniformly on compact subsets of D as a⟶1. Applying Lemma 2.10 from [17], for any compact operator K from B1+1/p into Wμn, we have
(31)lima⟶1Kfi+1,aWμn=0.
Hence, for anyi∈0,⋯,n+1,(32)Ai=limsupa⟶1Tu,v,φmfi+1,aWμn−lima⟶1Kfi+1,aWμn≤limsupa⟶1Tu,v,φm−Kfi+1,aWμn⪯Tu,v,φm−KB1+1/p⟶Wμn.
So,(33)maxAii=0n+1⪯infKTu,v,φm−KB1+1/p⟶Wμn=Tu,v,φme,B1+1/p⟶Wμn.
Now, we prove that(34)maxBii=0n+1⪯Tu,v,φme,B1+1/p⟶Wμn.
Letzjj∈ℕ be a sequence in D such that limj⟶∞φzj⟶1. Since Tu,v,φm:B1+1/p⟶Wμn is bounded, by using Lemmas 4 and 5 for any compact operator K:B1+1/p⟶Wμn and i∈0,⋯,n+1, we obtain
(35)Tu,v,φm−KB1+1/p⟶Wμn⪰limsupj⟶∞Tu,v,φmgi,φzjWμn−limj⟶∞Kgi,φzjWμn⪰limsupj⟶∞μzjφzjm+iIi,φn,u+Ii−1,φn,vzj1−φzj2m+i+1/p.
So, from the definition of the essential norm, we get (34).
Forr∈0,1, we define Krfz=frz=frz. It is apparent that Kr is a compact operator on B1+1/p. Let rj⊂0,1 be a sequence such that rj⟶1 as j⟶∞. Since fr⟶f uniformly on compact subsets of D as r⟶1, then, for any positive integer j, the operator Tu,v,φmKrj:B1+1/p⟶Wμn is compact. Based on the definition of the essential norm, we obtain
(36)Tu,v,φme,B1+1/p⟶Wμn≤limsupj⟶∞Tu,v,φm−Tu,v,φmKrj.
So, it is sufficient to show that(37)limsupj⟶∞Tu,v,φm−Tu,v,φmKrj⪯minmaxAii=0n+1,maxmaxBii=0n+1.
Letf∈B1+1/p such that fB1+1/p≤1 and for all j≥N, rj≥3/4, therefore,
(38)Tu,v,φm−Tu,v,φmKrjfWμn≤∑t=0n−1Tu,v,φmf−frjt0⏟St+supφz≤rNμz∑k=0n+1f−frjk+mφzIk,φn,u+Ik−1,φn,vz⏟H1+supφz>rNμz∑k=0n+1f−frjk+mφzIk,φn,u+Ik−1,φn,vz⏟H2.
For anys∈ℕ0 and compact subset of D, f−frjs⟶0 uniformly, hence, from Theorem 6, we obtain
(39)limsupj⟶∞H1=0,limsupj⟶∞St=0t=0,⋯,n−1.
On the other hand(40)H2≤∑k=0n+1supφz>rNμzfk+mφzIk,φn,u+Ik−1,φn,vz⏟Mk+∑k=0n+1supφz>rNμzrjkfk+mrjφzIk,φn,u+Ik−1,φn,vz⏟Nk.
Now, estimate forMk is obtained. Employing Lemmas 1 and 5,
(41)Mk=supφz>rNμz1−φz2m+k+1/pfm+kφzφzm+k×φzm+kIk,φn,u+Ik−1,φn,vz1−φz2m+k+1/p⪯fB1+1/psupφz>rNTu,v,φmgk,φzWμn⪯∑i=0n+1ci+1ksupa>rNTu,v,φmfi+1,aWμn.
Taking the limit whenN⟶∞, we get
(42)limsupj⟶∞Mk⪯∑i=0n+1limsupa⟶1Tu,v,φmfi+1,aWμn⏟Ai⪯maxAii=0n+1,limsupj⟶∞Mk⪯Bk.
Likewise, we have(43)limsupj⟶∞Nk⪯∑i=0n+1limsupa⟶1Tu,v,φmfi+1,aWμn⏟Ai⪯maxAii=0n+1,limsupj⟶∞Nk⪯Bk.
Thus, by using (38), (39), (40), (42) and (43), we obtain
(44)limsupj⟶∞Tu,v,φm−Tu,v,φmKrjB1+1/p⟶Wμn=limsupj⟶∞supfB1+1/p≤1Tu,v,φm−Tu,v,φmKrjfWμn⪯maxAii=0n+1,limsupj⟶∞Tu,v,φm−Tu,v,φmKrjB1+1/p⟶Wμn⪯maxBii=0n+1.
Hence, from (36),
(45)Tu,v,φmB1+1/p⟶Wμn⪯minmaxAii=0n+1,maxBii=0n+1.
Consequently,(46)Tu,v,φmB1+1/p⟶Wμn≈maxAii=0n+1≈maxBii=0n+1.
The proof is complete.☐Theorem 8.
Letm,n∈ℕ, 0<p≤∞, u,v∈HD, φ∈SD, μ be a weight. If Tu,v,φm:Hp⟶Wμnbe bounded then
(47)Tu,v,φme,Hp⟶Wμn≈Tu,v,φme,B1+1/p⟶Wμn.Proof.
It is evident that(48)Tu,v,φme,Hp⟶Wμn⪯Tu,v,φme,B1+1/p⟶Wμn.
On the other hand, sincefi,az=1−a2i/1−a¯zi+1/p∈Hp, for any compact operator K:Hp⟶Wμn, from Lemma 2.10 in [17], for any i∈0,⋯,n+1, we get
(49)Tu,v,φm−KHp⟶Wμn⪰limsupa⟶1Tu,v,φm−Kfi+1,aWμn≥limsupa⟶1Tu,v,φmfi+1,aWμn−lima⟶1Kfi+1,aWμn=Ai.
So, from the last inequality and Theorem7(50)Tu,v,φme,Hp⟶Wμn⪰maxAii=0n+1≈Tu,v,φme,B1+1/p⟶Wμn.
The proof is complete.☐Theorem 9.
Letm,n∈ℕ, 0<p≤∞, u,v∈HD, μ be a weight and φ∈SD such that Tu,v,φm:B1+1/pB01+1/p⟶Wμn be bounded. Then,
(51)limsupj⟶∞j1/pTu,v,φmpjWμn≈Tu,v,φme,B01+1/p⟶Wμn≈Tu,v,φme,B1+1/p⟶Wμn.Proof.
Letj be any positive integer and hjz=j1/ppjz. It is clear that hjB1+1/p≈1, hj∈B01+1/p, and hjj∈ℕ converge to 0 uniformly on compact subsets of D. By using Lemma 2.10 in [17], for any compact operator K from B01+1/p into Wμn, we get
(52)limj⟶∞KhjWμn=0.
Hence,(53)Tu,v,φm−KB01+1/p⟶Wμn⪰limsupj⟶∞Tu,v,φm−KhjWμn≥limsupj⟶∞Tu,v,φmhjWμn∥−limsupj⟶∞KhjWμn=limsupj⟶∞j1/pTu,v,φmpjWμn.
So,Tu,v,φme,B01+1/p⟶Wμn⪰limsupj⟶∞j1/pTu,v,φmpjWμn. Now, we prove that
(54)limsupj⟶∞j1/pTu,v,φmpjWμn⪰Tu,v,φme,B1+1/p⟶Wμn.
From Theorem6, for any fixed positive integer k≥m and 0≤i≤n+1, we have
(55)Tu,v,φmfi+1,aWμn≤Ci+11−a2i+1∑j=0∞jiajj1/pTu,v,φmpjWμn=Ci+11−a2i+1∑j=0k−1jiajj1/pTu,v,φmpjWμn+∑j=k∞jiajj1/pTu,v,φmpjWμn≤2QCi+1k−1i+11−ak1−a2i+2i+1Ci+1Ci+1′supj≥kj1/pTu,v,φmpjWμn,where Q=supj≥mj1/pTu,v,φmpjWμn. Letting a⟶1, we obtain
(56)Ai+1=limsup∣a∣⟶1Tu,v,φmfi+1,aWμn⪯supj≥kj1/pTu,v,φmpjWμn.
Applying Theorem7, we get
(57)Tu,v,φme,B1+1/p⟶Wμn≈maxAii=0n+1⪯limsupj⟶∞j1/pTu,v,φmpjWμn.
It is clear thatTu,v,φme,B01+1/p⟶Wμn≤Tu,v,φme,B1+1/p⟶Wμn; so, from the last inequalities, we have
(58)limsupj⟶∞j1/pTu,v,φmpjWμn≈Tu,v,φme,B01+1/p⟶Wμn≈Tu,v,φme,B1+1/p⟶Wμn.
The proof is complete.☐


## 4. Some Applications

For0<p<∞, by using Lemma 3, we have Hp⊂B01+1/p. Also, for p=∞, H∞⊈B0 and H∞∩B0 are a Banach space with the norm ·H∞. In this case, we get the following corollary.Corollary 10.
Letm,n∈ℕ, u,v∈HD, and μ be a weight and φ∈SD. The operator Tu,v,φm:H∞⟶Wμn is bounded if and only if the operator Tu,v,φm:H∞∩B0⟶Wμn be bounded.Corollary 11.
Letm,n∈ℕ, u,v∈HD, φ∈SD, and μ be a weight. If Tu,v,φm:H∞⟶Wμnbe bounded, then,
(59)Tu,v,φme,H∞⟶Wμn≈Tu,v,φme,H∞∩B0⟶Wμn.Proof.
It is clear thatTu,v,φme,H∞∩B0⟶Wμn⪯Tu,v,φme,H∞⟶Wμn and fi,az=1−a2i/1−a¯zi∈H∞∩B0. So, for any compact operator K:H∞∩B0⟶Wμn, from Lemma 2.10 in [17], for any i∈0,⋯,n+1, we obtain
(60)Tu,v,φme,H∞∩B0⟶Wμn⪰limsupa⟶1Tu,v,φm−Kfi+1,aWμn≥limsupa⟶1Tu,v,φmfi+1,aWμn−lima⟶1Kfi+1,aWμn=Ai.
Hence, from the last inequality and Theorem7,
(61)Tu,v,φme,H∞∩B0⟶Wμn⪰maxAii=0n+1≈Tu,v,φme,B01+1/p⟶Wμn.
The proof is complete.☐From Theorems7, 8 and 9 and Corollary 11, the next corollaries are obtained.Corollary 12.
Letm,n∈ℕ, 0<p<∞, u,v∈HD, φ∈SD, and μ be a weight such that Tu,v,φm:Hp⟶Wμn be bounded. Then, the following statements are equivalent.
(a)
The operatorTu,v,φm:Hp⟶Wμn is compact(b)
The operatorTu,v,φm:B1+1/p⟶Wμn is compact(c)
The operatorTu,v,φm:B01+1/p⟶Wμn is compact(d)
limj⟶∞j1/pTu,v,φmpjWμn=0(e)
For eachi∈0,⋯,n+1, limsup∣a∣⟶1Tu,v,φmfi+1,aWμn=0(f)
For eachi∈0,⋯,n+1,(62)limsupφz⟶1μzIi,φn,u+Ii−1,φn,vz1−φz2m+i+1/p=0Corollary 13.
Letm,n∈ℕ, u,v∈HD, φ∈SD, and μ be a weight such that Tu,v,φm:H∞⟶Wμn be bounded. Then, the following statements are equivalent.
(a)
The operatorTu,v,φm:H∞⟶Wμn is compact(b)
The operatorTu,v,φm:H∞∩B0⟶Wμn is compact(c)
The operatorTu,v,φm:B⟶Wμn is compact(d)
The operatorTu,v,φm:B0⟶Wμn is compact(e)
limj⟶∞Tu,v,φmpjWμn=0(f)
For eachi∈0,⋯,n+1, limsup∣a∣⟶1Tu,v,φmfi+1,aWμn=0(g)
For eachi∈0,⋯,n+1(63)limsupφz⟶1μzIi,φn,u+Ii−1,φn,vz1−φz2m+i=0Remark 14.
By puttingv≡0 in Theorems 6, 7, 8, and 9 and Corollaries 12 and 13, some characterizations are acquired for boundedness, essential norm, and compactness of the generalized weighted composition operator from Hardy spaces 0<p≤∞ into nth weighted-type spaces.
Since(64)I0,φ1,u+I−1,φ1,vz=u′z,I1,φ1,u+I0,φ1,vz=uzφ′z+v′z,I2,φ1,u+I1,φ1,vz=vzφ′z,we obtain the next remark.Remark 15.
Letα>0. Setting n=1μz=1−z2a,1−z2log2/1−z in Theorems 6, 7, 8, and 9 and Corollaries12 and 13 and using (64) we get similar results for operator Tu,v,φm:Hp⟶BμTu,v,φm:Hp⟶Bα,Tu,v,φm:Hp⟶Blog(see [11, 12]).
(65)I0,φ2,u+I−1,φ2,vz=u′′z,I1,φ2,u+I0,φ2,vz=2u′zφ′z+uzφ′′z+v′′z,I2,φ2,u+I1,φ2,vz=uzφ′2z+2v′zφ′z+vzφ′′z,I3,φ2,u+I2,φ2,vz=vzφ′2z.Remark 16.
Puttingn=2μz=1−z2a in Theorems 6, 7, 8, and 9 and Corollaries 12 and 13 and applying (65), similar results are achieved for operator Tu,v,φm:Hp⟶ZμTu,v,φm:Hp⟶Zα (generalizing Theorems 7 and 9 [10]).



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*Source: 5556275-2021-07-16.xml*