Opuscula Math. 38, no. 6 (8), 89 839 https://doi.og/.7494/opmath.8.38.6.89 Opuscula Mathematica ESSENTIAL NORM OF AN INTEGRAL-TYPE OPERATOR FROM ω-bloch SPACES TO µ-zygmund SPACES ON THE UNIT BALL Juntao Du and Xiangling Zhu Communicated by Stevo Stević Abstact. In this pape, we give an estimate fo the essential nom of an integal-type opeato fom ω-bloch spaces to µ-zygmund spaces on the unit ball. Keywods: essential nom, integal-type opeato, ω-bloch space, µ-zygmund space. Mathematics Subject Classification: 47B33, 3H.. INTRODUCTION Let µ be a positive continuous function on [, ). We say that µ is nomal if thee exist positive numbes a and b, < a < b, and δ [, ) such that (see [9]), µ() µ() is deceasing on [δ, ) and lim ( ) a ( ) a =, µ() µ() ( ) b is inceasing on [δ, ) and lim ( ) b =. Let B be the open unit ball of C n, H(B) be the space of all holomophic functions in B. When n =, B is the open unit disk D of the complex plane and H(D) is the holomophic function space on D. Fo f H(B), the adial deivative and complex gadient of f at z will be denoted by Rf(z) and f(z), espectively, that is, Rf(z) = n j= ( f f z j (z) and f(z) = (z), f (z),..., f ) (z). z j z z z n c Wydawnictwa AGH, Kakow 8 89
83 Juntao Du and Xiangling Zhu Let ω be nomal. An f H(B) is said to belong to ω-bloch space (o Bloch type space), denoted by B ω = B ω (B), if f Bω = f() + sup ω( z ) Rf(z) <. The little ω-bloch space B ω, = B ω, (B), consists of all f B ω such that lim ω( z ) Rf(z) =. B ω is a Banach space with the nom Bω. When < α < and = ( t ) α, we get the α-bloch space (often also called Bloch type space), denoted by B α = B α (B). In paticula, when = t, we get the Bloch space, denoted by B = B(B). See [3, 36] fo moe infomation of the Bloch space B and ω-bloch space B ω on the unit ball. Suppose µ is nomal on [, ). We say that an f H(B) belongs to Z µ = Z µ (B) if f Zµ = f() + sup R f(z) <, whee R f(z) = R(Rf)(z). Unde the above nom, Z µ becomes a Banach space. Z µ will be called the µ-zygmund space o the Zygmund type space. The little µ-zygmund space, denoted by Z µ,, is the space consisting of all f Z µ such that lim R f(z) =. When µ() =, the µ-zygmund space becomes the Zygmund space Z. See [6,4,36] fo moe infomation of the Zygmund space on the unit ball. Thee has been some ecent inteest in studying of vaious concete opeatos, including integal-type ones, fom o to the Zygmund type spaces, see, fo example, [,, 4, 5, 7, 8, 4, 7]. Some genealizations of the Zygmund type spaces and opeatos on them, can be found in [9, 3]. Assume that g H(B), g() = and ϕ is a holomophic self-map of B. In [], Stević intoduced the following integal-type opeato P g ϕ on H(B): P g ϕf(z) = f(ϕ(tz))g(tz), f H(B), z B. t Some esults on the opeato has been got in [7, 3, 5, 6, 8, 3, 38]. Hee we conside a paticula case of P g ϕ when ϕ(z) = z, that is the opeato P g f(z) = f(tz)g(tz), f H(B), z B. t Fo all g H(B), P Rg is just the extended Cesào opeato (o Riemann-Stieltjes opeato), which was intoduced in [3], and studied in [, 3 5, 8,, 3, 6, 3, 34]. Some elated integal-type opeatos can be found in [, 7, 3, 37, 38].
Essential nom of an integal-type opeato... 83 In [7], Li and Stević studied the boundedness and compactness of the opeato P g : B ω (B ω, ) Z µ (Z µ, ). Motivated by [7], in this pape we investigate the essential nom of P g : B ω (B ω, ) Z µ. Recall that the essential nom of P g : X Y, denoted by P g e,x Y, is defined by P g e,x Y = inf{ P g K X Y ; K is a compact opeato fom X to Y }. In this pape, constants ae denoted by C, they ae positive and may diffe fom one occuence to the next. We say that A B if thee exists a constant C such that A CB. The symbol A B means A B A.. AUXILIARY RESULTS In this section, we give some auxiliay esults which will be used in poving the main esults of this pape. They ae incopoated in the lemmas which follow. Lemma. ([6]). Suppose ω is nomal on [, ). Then thee exists ω H(D) such that (i) ω (t) is positive and inceasing on [, ), (ii) fo all z D, ω (z) ω ( z ), (iii) < inf ω()ω () sup ω()ω () <. << << In the est of the pape we will always use ω to denote the analytic function elated to ω in Lemma.. To study the compactness, we need the following lemma, which can be get by Lemma. in [33]. Lemma.. Suppose ω and µ ae nomal. If T : B ω (B ω, ) Z µ is bounded, then T is compact if and only if wheneve {f k } is bounded in B ω (B ω, ) and f k unifomly on compact subsets of B, lim k T f k Zµ =. Lemma.3 ([]). Assume g H(B) and g() =. Then, fo evey f H(B), RP g f(z) = f(z)g(z). By some calculations, we have the following lemma. Lemma.4 ([, Lemma ]). Suppose ω is nomal. Then the following statements hold. (i) Thee exists a δ (, ), such that ω is deceasing on [δ, ) and lim t =. (ii) Fix α >, β (, ). When t (, ), s (β, ), ω(t α ) ω (t), sα s ω ( t). (iii) Fo any z D, z ω (η)dη ω (t). If η z, ω( z ) ω (η) < C.
83 Juntao Du and Xiangling Zhu Lemma.5 ([35, Lemmas. and.4]). Suppose ω is nomal. Then the following statements hold. ( (i) If f B ω, then f(z) + ) z f Bω. (ii) If (iii) If = and f B ω,, then lim f(z) =. <, {f k} is bounded in B ω, and conveges to unifomly on compact subsets of B, then lim k sup f k (z) =. The following lemma gives some folkloe estimates, but we will pove it fo the completeness. Lemma.6. Suppose ω is nomal, <, s < and f H(B). Then, fo all z s, Rf(z) n s max n( ) f(z) and f(z) f(z) max f(z). z +s s z +s Poof. Set z = (z, z,..., z n ) B such that z s. Fo i =,,..., n, let { Γ z,i = η D : η z i = s }, and λ(z, i, η) = (z,..., z i, η, z i+,..., z n ), η Γ z,i. f Since f H(B), z i H(B). Taking f as a one complex vaiable function about the i-th component of z, by Cauchy s integal fomula, we have f (z) z i = f(λ(z, i, η)) π (η z i ) dη Γ z,i π ( ( = π( s) f λ z, i, z i + s eiθ)) e iθ dθ s max f(z). z +s Hee we use the change η = z i + s eiθ ( θ π). Then, When z s, f(z) f(z) = The poof is complete. Rf(z) df(tz) n s max f(z). z +s = ( ) sup f(z) ( f)(tz), z n( ) s max f(z). z +s
Essential nom of an integal-type opeato... 833 3. MAIN RESULTS AND PROOFS Theoem 3.. Assume that µ, ω ae nomal and g H(B) such that g() = and =. If P g : B ω Z µ is bounded, then P g e,bω, Z µ lim sup Rg(z) Poof. The following inequality is obvious. P g e,bω, Z µ. + lim sup ω( z ) g(z). Fist, we find the lowe estimate of P g e,bω, Z µ. Suppose K : B ω, Z µ is compact and {z k } B such that z k and lim k z k =. Let f k (z) = p3 k (z) p k (z k) 3p k (z) p k (z k ) whee p k (z) = z,z k ω (t). By Lemmas. and.4, we have and h k (z) = p3 k (z) p k (z k) p k (z) p k (z k ), p k (z) z,z k ω (t) z z k ω (t), and Rp k (z) = z, z k ω ( z, z k ) ω ( z z k ). When j =, 3, Rp j k and.4, we have the following statements. (z) = jpj k (z)rp k (z). Since (i) f k B ω,, f k Bω, Rf k (z k ) =, f k (z k ) z k =, by Lemmas.. (ii) h k B ω,, h k Bω, h k (z k ) =, Rh k (z k ) ω(z k ). (iii) Both {f k } and {h k } convege to unifomly on compact subsets of B as k. By Lemma.3, P g K Bω, Z µ (P g K)f k Zµ µ( z k ) Rg(z k ) f k (z k ) µ( z k ) g(z k ) Rf k (z k ) Kf k Zµ µ( z k ) Rg(z k ) z k Kf k Zµ. (3.)
834 Juntao Du and Xiangling Zhu Letting k, by Lemma., we have P g K Bω, Z µ Since K and {z k } ae abitay, we obtain P g e,bω, Z µ lim sup µ( z k ) Rg(z k ) k z k. lim sup Rg(z). Replacing f k by h k in (3.), we similaly obtain that Theefoe P g e,bω, Z µ P g e,bω, Z µ lim sup lim sup Rg(z) ω( z ) g(z). + lim sup ω( z ) g(z). Next, we find the uppe estimate of. Fo < < and f B ω, let f (z) = f(z) and T g, f = P g f. Since f Bω f Bω, we see that T g, f Zµ = P g f Zµ P g Bω Z µ f Bω. So T g, : B ω Z µ is bounded. Using the test functions h(z) = and q(z) = z j (j =,,..., n), we have M := sup By Lemma.3, T g, f Zµ = sup Rg(z) < and M := sup g(z) <. (3.) R T g, f(z) M sup By Lemmas. and.6, T g, is compact. Assume f Bω and s (, ). By Lemma.3, f(z) + M sup (Rf)(z). P g f T g, f Zµ = P g (f f ) Zµ sup g(z) R(f f )(z) + sup Rg(z) f(z) f(z) + sup g(z) R(f f )(z) s< z < + sup Rg(z) f(z) f(z). s< z < (3.3)
Essential nom of an integal-type opeato... 835 Since and =, by Lemmas.5 and.6, we have n( ) sup f(z) f(z) s n( ) max f(z) z +s s n( ) sup R(f f )(z) = sup (Rf)(z) (Rf)(z) s Since f B ω and by Lemma.5, we get and +s max z +s g(z) sup g(z) R(f f )(z) sup s< z < s< z < ω( z ) sup Rg(z) f(z) f(z) s< z < By (3.) (3.7), we have Letting s, we get lim sup P g T g, Bω Z µ sup Rg(z) s< z <, (3.4) ω( z ). (3.5) (3.6). (3.7) g(z) z sup + sup Rg(z) s< z < ω( z ) s< z <. lim sup Rg(z) as desied. The poof is complete. + lim sup g(z), (3.8) ω( z ) Theoem 3.. Assume that µ, ω ae nomal and g H(B) such that g() = and <. If P g : B ω Z µ is bounded, then P g e,bω, Z µ lim sup g(z). (3.9) ω( z ) Poof. We begin with finding the lowe estimate of P g e,bω, Z µ. Suppose K : B ω, Z µ is compact and {z k } B such that lim z k = and k z k >. Let z,z k q k (z) = ω( z k ) ω (η)dη.
836 Juntao Du and Xiangling Zhu By Lemmas. and.4, {q k } ae bounded in B ω, and convege to unifomly on compact subsets of D. By Lemmas. and.5, we see that lim Kq k Zµ = and lim k P g K Bω, Z µ (P g K)q k Zµ sup k q k (z) =. (3.) µ( z k ) R (P g q k )(z k ) Kq k Zµ µ( z k ) g(z k ) Rq k (z k ) µ( z k ) Rg(z k ) q k (z k ) Kq k Zµ. Since B ω, Z µ is bounded and f(z) = B ω,, we get (3.) Letting k in (3.), by (3.) and (3.), we obtain P g K Bω, Z µ Since K and {z k } ae abitay, we see that P g e,bω, Z µ sup Rg(z) <. (3.) z D lim sup k lim sup µ( z k ) ω( z k ) g(z k). Next, we find the uppe estimate of. Since <, fo any given ε >, thee is a ξ ( we have Thus < ε. g(z). (3.3) ω( z ), ), when ξ < <, Fix, s ( ξ, ). Fo all f B ω with f Bω, when z > s, we have f(z) f(z) = df(tz) = (Rf)(tz) t ε. ω(t z )t sup s< z < Rg(z) f(z) f(z) ε sup Rg(z). (3.4) Then similaly to have (3.8), just instead (3.7) with (3.4), we have Since ε is abitay, by (3.), lim sup ω( z ) lim sup g(z) + ε sup Rg(z). g(z). (3.5) ω( z )
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Essential nom of an integal-type opeato... 839 [34] J. Xiao, Riemann-Stieltjes opeatos on weighted Bloch and Begman spaces of the unit ball, J. London. Math. Soc. 7 (4), 99 4. [35] X. Zhang, J. Xiao, Weighted composition opeatos between µ-bloch spaces on the unit ball, Sci. China 48 (5), 349 368. [36] K. Zhu, Spaces of Holomophic Functions in the Unit Ball, Spinge, New Yok, 5. [37] X. Zhu, Genealized composition opeatos and Voltea composition opeatos on Bloch spaces in the unit ball, Complex Va. Elliptic Equ. 54 (9), 95. [38] X. Zhu, Voltea composition opeatos on logaithmic Bloch spaces, Banach J. Math. Anal. 3 (9), 3. Juntao Du jtdu7@63.com Faculty of Infomation Technology Macau Univesity of Science and Technology Avenida Wai Long, Taipa, Macau Xiangling Zhu jyuzxl@63.com School of Compute Engineeing Zhongshan Institute Univesity of Electonic Science and Technology of China 584, Zhongshan, Guangdong, P. R. China Received: Januay, 8. Revised: Mach, 8. Accepted: Mach, 8.