Distance from Bloch-Type Functions to the Analytic Space F ( p , q , s )

and Applied Analysis 3 Proof. We can use the reproducing formula for f󸀠 to get that f 󸀠 (z) = C∫ D (1 − |w| 2 ) b−1 f 󸀠 (w) (1 − wz) b+1 dA (w) (18) for some constant C, where b is a real number greater than 1 + (q + s)/p; see, for example, [14, page 55]. Let 0 < α < 2 + q. If p > 1, denote p󸀠 = p/(p − 1); it follows from the Hölder’s inequality and (15) that 󵄨󵄨󵄨󵄨 f 󸀠 (z) 󵄨󵄨󵄨󵄨 ≲ ∫ D (1 − |w| 2 ) (q+s)/p (1 − |z| 2 ) α/p 󵄨󵄨󵄨󵄨 f 󸀠 (w) 󵄨󵄨󵄨󵄨 |1 − wz| (s+α)/p × (1 − |w| 2 ) b−1−(q+s)/p dA (w) (1 − |z| 2 ) α/p |1 − wz| b+1−(s+α)/p ≲ (∫ D (1 − |z| 2 ) α |1 − wz| s+α |f 󸀠 (w)| p (1 − |w| 2 ) q+s dA(w)) 1/p ×(∫ D (1 − |w| 2 ) p 󸀠 (b−1−(q+s)/p) dA(w) (1 − |z| 2 ) p 󸀠 (α/p) |1 − wz| p 󸀠 (b+1−(s+α)/p) ) 1/p 󸀠 ≲ 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩F(p,q,s),α


Introduction
Let D denote the unit disc { ∈ C : || < 1} of the complex plane C and let T = { ∈ C : || = 1} be its boundary.As usual, (D) denotes the space of all analytic functions on D.
The little Bloch-type space B 0  is the subspace of all  ∈ B  with lim || → 1 ()      = 0. ( It is well known that B  is a Banach space under the norm          * B  =      (0)     +         B  .
In particular, when  = 1, B  becomes the classic Bloch space B, which is the maximal Möbius invariant Banach space that has a decent linear functional; see [1,2] for more details on the Bloch spaces.
For 0 <  < ∞, we say that a nonnegative Borel measure  defined on D is an -Carleson measure if where the supremum ranges over all subarcs  of T, || denotes the arc length of , and is the Carleson square based on a subarc  ⊆ T. We write CM  for the class of all -Carleson measures.Moreover,  is said to be a vanishing -Carleson measure if lim For  an analytic function on D, we define It was proved in [3] The following result is obtained by Zhao in [9].
Theorem 1. Suppose 1 ≤  < ∞, 0 <  ≤ 1, and  ∈ B. The following two quantities are equivalent: (1) dist B (, (,  − 2, )); When  = 2 and  = 1, the above characterization is Peter Jone's distance formula from a Bloch function to BMOA (Peter Jone never published his result but a proof was provided in [10]).Also, similar type results can be found in [11][12][13].Specifically, distance from Bloch function to   -type space is given in [11]; to the little Bloch space is obtained in [12], and to the   space of the ball is characterized in [13].All these spaces are Möbius invariant.
This paper is dedicated to characterize the distance from  ∈ B (+2)/ to (, , ), which extends Zhao's result.The main result is following. where The strategy in this paper follows from Theorem 3.1.3in [14].The distance from a B  function to Campanato-Morrey space was given in [15] with similar idea.
Notation.Throughout this paper, we only write  ≲  (or  ≳ ) for  ≤  for a positive constant , and moreover  ≈  for both  ≲  and  ≲ .

Preliminaries
We begin with a lemma quoted from Lemma 3.1.1 in [14].Lemma 3. Let ,  ∈ (0, ∞), and  be nonnegative Radon measures on D. Then,  ∈ CM  if and only if According to Lemma 3 and the fact that  ∈ (, , ) if and only if d  is an -Carleson measure, we can easily get the following corollary.

Corollary 4.
Let  be an analytic function on D.  ∈ (, , ) if and only if there exists an  > 0 such that We will also need the following standard result from [16].
The following lemma, quoted from Lemma 1 in [9], is an extension of Lemma 5. See also [17].
Next, we see that (, , ) is contained in B (2+)/ .We thank Zhao for pointing out that the following result is firstly proved in [3].Here, we give another proof with a different approach.
Recall that  > 1 +  +  and 0 <  < 2 + .We can easily use (4) to check that sup ,∈D Again, the above inequality follows from Lemma 5.This completes the proof.
Our strategy relies on an integral operator preserving the -Carleson measures.For ,  > 0, we define the integral operator  , as The following lemma is similar to Theorem 2.5 in [18].Indeed, Qiu and Wu proved the case 1 <  < ∞.Specially, the  = 2 case is just Lemma 3.1.2in [14].Proof.We firstly prove the case  = 1 and then sketch the outline argument of the case 1 <  < ∞ modified from [18] for the completeness.

Proof of the Main Result
Proof of Theorem 2. For  ∈ B (+2)/ , it is easy to establish the following formula (see, e.g., [19, (1.1)] or [14, page 55].Notice that it is a special case of the -order derivative of , as  = 0 in [14], which holds for all holomorphic  on D).Consider Define, for each  > 0,