The Fractional Carleson Measures on the Unit Ball of R n + 1

The 1-Carlesonmeasure is the classical Carlesonmeasure (see [1, 2]). Carleson measures are related to certain holomorphic function spaces, such as BMO, Morrey spaces, and Q spaces in a natural way (see [2–5]). Besides these, much research has been done about the characterizations of Carleson measures (see [6–10]). A well-known result is that a nonnegative measure ] on D is a 1-Carleson measure if and only if


Introduction
Let D be the unit disc of the complex plane and let D be the boundary of D. For any arc  ⊆ D, let || = ∫  /2 be the normalized length of .The Carleson square based on an arc  is the set We set () = D when  = D.Then, for  > 0, a nonnegative measure ] on D is called an -Carleson measure if there exists a constant  > 0 such that ] ( ()) ≤ ||  , ∀ ⊂ D. ( ] is said to be a compact -Carleson measure if ] is an -Carleson measure and lim The 1-Carleson measure is the classical Carleson measure (see [1,2]).
The relations among the Carleson measures, quantities A ] (, , , ), and some function spaces defined on D are also displayed, which are applied to characterizing the boundedness and compactness of Volterra-type operators from Hardy spaces to some holomorphic spaces.One can refer to [13] for more details.
In order to show that the Jacobian of a conformal self-map of the unit ball B in R +1 obeys the weak Harnack inequality, Kotilainen et al. introduced an integral form of the fractional Carleson measures on the unit ball B (see [14]).For  ∈ B \ {0}, set and set (0) = B.For  > 0, a nonnegative measure  on B is called a -Carleson measure or a compact -Carleson measure if and only if (see [14,15]).For  0 ∈ B, a conformal self-map   0 of B is defined by Let    0 () stand for the Jacobian matrix of   0 at  ∈ B. Then the Jacobian of   0 is For more details about this conformal self-map, one can refer to [16][17][18][19].Then, it is shown in [14] that  is a -Carleson measure or a compact -Carleson measure on B if and only if sup which is the analogue of (4) and ( 5).
Pursuing the above, in this paper, analogically to (6), we will construct a quantity on the unit ball B by using the Jacobian of   0 and establish the connections between the fractional Carleson measures on B and the quantity.In Section 2, we give some preliminaries, which contain the fractional Carleson measure defined in terms of tents or Carleson boxes.In Section 3, we state our main results and their proof.The results are the extension of the ones in [13], and the real analysis techniques used in this paper should have an application in studying the operators on the function spaces defined on the unit sphere in future.

Preliminaries
The tent based on  is defined by The cone Γ() in B with the vertex  ∈ S  is defined by For any fixed  ∈ B, set Clearly, if  ̸ = 0, () is just the spherical cap in S  with center /|| and radius 1 − ||.
For  ∈ B and a measurable function  defined on S  , we denote by the Poisson extension of  onto B. The nontangential maximal function of () is the function defined on S  .
For  ∈  1 (S  ) and any  ∈ B, we write which is also an extension of  onto B.
For  ∈ S  , we call the function holds for all  ≥ 0 and  ∈  1 (S  ).
The following two lemmas are well known.One is the generalized maximal theorem; the other is the fact that the nontangential maximal function can be pointwise controlled by maximal function (see [21]).(2) For any  ⊆ S  , we can replace () with () in Definition 5.
For 0 <  ≤  < ∞, a well-known result which is due to Carleson in [1] for  =  and Duren in [22] for  <  says that a nonnegative measure  on the unit disc D of the complex plane C is a bounded (/)-Carleson measure if and only if By Definition 5 and using the real analysis techniques, we obtain the following extension of this result on B.

Theorem 7.
Let  be a nonnegative measure on B.
(a) For 0 <  ≤  < ∞, let () be a -measurable function defined on B and let  * be the nontangential maximal function of ().If  is a (/)-Carleson measure on B, then as a direct sequence of (a).Noting Lemmas 3 and 4, we complete the proof of the "only if " part of (b).
The proof of the "if " part of (b) is easy.For any spherical cap  in S  , if  ∈ (), we have () ⊆ .Let  =   ().Since the simple fact | − | ≤ (1 − ||) as  ∈ () and the estimate in (13), we obtain the useful estimate Now, it is easy to see which is to say that  is a (/)-Carleson measure.The proof of (b) is completed.
Let  be a spherical cap in S  with center  0 =  0 /| 0 | and radius () = 1 − | 0 |.Let  be a nonnegative integer and,  be the greatest integer less than log 2 (1/()).Denote the spherical cap with the same center as  but radius 2  () by   .Then,  0 =  and  +1 = S  .Moreover, for any spherical cap  ⊆ S  , we have Proof.If  ∈ (), then we have the estimates Part (i) is yielded by the above.
Combining together Definition 5, the decomposition of B in (42), and Lemma 8, we can obtain the following characterization for -Carleson measures on B, which is similar to [ or In particular, when  = 1,  is an -Carleson measure or a compact -Carleson measure on B if and only if

The Carleson Measures Characterized by 𝐿 𝑝 Behaviors
By the estimate in (13) and Theorem 9, we observe that Here, we emphasiz that the following results have been achieved when  = 1 (see [13]).(2) The results in the above theorem hold for  ∈ R on D (see [13,Theorem 1]), but they do only for  > 0 here.
To prove the "if " part of (i), let  = ( 0 /| 0 |, 1 − | 0 |) and  −1 = .By the decomposition of B in (42), it is clear that By Lemma 8, we have Taking   norm on both sides of the above inequality, if 1 ≤  ≤ ∞, we have The proof of the "if " part of (i) is completed.
With the same technique used in the proof of "if " part of (i), one can obtain the "if " part of (ii).Suppose that 1 ≤  ≤ ∞.We observe first that for  > 0, there must exist an integer  > 0 such that Then, we have By the assumption of the "if " part of (ii), there must exist  > 0 such that if |  | < which is the desired result.With the same process, we can obtain the results when 0 <  < 1.The proof of Theorem 10 is completed.
Proof.In the situation of  = 1, the results are deduced by Theorems 9 and 10 and the estimate For 1 <  < ∞, it is well known that  * () ∈  /(−1) (S  ) if () ∈  /(−1) (S  ).By Lemma 2, we have Journal of Function Spaces and Applications 7 Now, by the estimate in ( 13) and Hölder's inequality, we have Combining with Theorem 10, we complete the proof of "only if " parts.Turn to the proof of "if " parts.For  ∈  /(−1) (S  ) and  ≥ 0, suppose that |()()|() is an -Carleson measure or a compact -Carleson measure.By Lemma 1, we know that which implies that ()()() is an -Carleson measure or a compact -Carleson measure.By the estimate in (13), we have
Moreover, the following are equivalent.
The equivalence above holds for the compact case also.
To prove "(b2)⇒(c2), " let  0 be any fixed point in B and let holds for any  0 ∈ B, which is to say that  is an ( + 1)-Carleson measure.
For  > 1, let  ∈    (S  ) and  ≥ 0 (here   is the conjugate of , that is, 1/ + 1/  = 1).By Lemma 1, we have Let  =   ( + 1 − 1/).It is easy to see that  >   > 1 and   = ( + 1 − 1/)/ (here   is the conjugate of ).Now, by the previous assumption, we have that  is a (/  )-Carleson measure.Using Hölder's inequality and Theorem 7, we have By the duality theorem, we conclude that which completes the proof of "if " parts.We now turn to the "only if " parts.Assume that I  ((), ) ≤ ||  or I  ((), ) = (||  ) for all spherical cap  ⊆ S  .For a spherical cap  ⊆ S  , we have Then, by the estimate in ( 13) and Fubini's theorem, we obtain that Since  ≥ 1, Jensen's inequality and the above inequality imply that  and the duality theorem with appropriate choices of .
For the convenience, we define Carleson measures on B in terms of Carleson boxes or tents.Definition 5. Let  > 0 and  a spherical cap in S  .A nonnegative measure  on B is called an -Carleson measure if there exits a constant  such that  ( ()) ≤ ||  , ∀ ⊂ S  .