We construct a quantity in terms of Lp integral of the Jacobian of a conformal self-map on the unit ball of ℝn+1. Then, we characterize the fractional Carleson measures on the unit ball by the quantity.

1. Introduction

Let 𝔻 be the unit disc of the complex plane and let ∂𝔻 be the boundary of 𝔻. For any arc I⊆∂𝔻, let |I|=∫Idζ/2π be the normalized length of I. The Carleson square based on an arc I is the set
(1)S(I)={z∈𝔻:|z|>1-|I|,z|z|∈I}.
We set S(I)=𝔻 when I=∂𝔻. Then, for s>0, a nonnegative measure ν on 𝔻 is called an s-Carleson measure if there exists a constant C>0 such that
(2)ν(S(I))≤C|I|s,∀I⊂∂𝔻.νis said to be a compact s-Carleson measure if ν is an s-Carleson measure and
(3)lim|I|→0ν(S(I))|I|s=0.
The 1-Carleson measure is the classical Carleson measure (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 ν on 𝔻 is a 1-Carleson measure if and only if
(4)supa∈𝔻∫𝔻1-|a|2|1-a¯z|2dν(z)<∞
(see [2]). The s-Carleson measures can be characterized by modifying (4) (see [4, 7, 10–12]). For instance, in [4], for τ,s>0, it is shown that a nonnegative measure ν on 𝔻 is an s-Carleson measure or a compact s-Carleson measure if and only if
(5)supa∈𝔻∫𝔻(1-|a|2)τ|1-a¯z|τ+sdν(z)<∞orlim|a|→1-∫𝔻(1-|a|2)τ|1-a¯z|τ+sdν(z)=0.

Recently, Wu [13] provided a different way for the characterization of s-Carleson measures in terms of the Lp estimates instead of L1 in (5). Suppose that τ,s>0, 1/s≤p<∞, and ν is a nonnegative measure on 𝔻. For a∈𝔻, ζ∈∂𝔻, write
(6)𝒜ν(a,τ,s,p)=∥∫Γ(ζ)(1-|a|2)τ|1-a-z|τ+sdν(z)1-|z|∥Lp(∂𝔻),
where Γ(ζ)={z∈𝔻:|ζ-z|<2(1-|z|)} is the cone in 𝔻 with the vertex ζ. In [13], it is shown that

supa∈𝔻𝒜ν(a,τ,s,p)<∞ if and only if ν is an (s+1-1/p)-Carleson measure;

𝒜ν(a,τ,s,p)→0 as |a|→1- if and only if ν is a compact (s+1-1/p)-Carleson measure.

The relations among the Carleson measures, quantities 𝒜ν(a,τ,s,p), 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 ℝn+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 ω∈𝔹∖{0}, set
(7)E(ω)={z∈𝔹:|z-ω|ω||<1-|ω|},
and set E(0)=𝔹. For t>0, a nonnegative measure μ on 𝔹 is called a t-Carleson measure or a compact t-Carleson measure if and only if
(8)supω∈𝔹μ(E(ω))(1-|ω|)t<∞orlim|ω|→1-μ(E(ω))(1-|ω|)t=0
(see [14, 15]). For z0∈𝔹, a conformal self-map Tz0 of 𝔹 is defined by
(9)Tz0(ω)=(1-|z0|2)(ω-z0)-|ω-z0|2ω1+|ω|2|z0|2-2ω·z0.
Let Tz0′(ω) stand for the Jacobian matrix of Tz0 at ω∈𝔹. Then the Jacobian of Tz0 is
(10)|Tz0′(ω)|=1-|z0|21+|ω|2|z0|2-2ω·z0.
For more details about this conformal self-map, one can refer to [16–19]. Then, it is shown in [14] that μ is a t-Carleson measure or a compact t-Carleson measure on 𝔹 if and only if
(11)supz0∈𝔹∫𝔹|Tz0′(ω)|tdμ(ω)<∞orlim|z0|→1-∫𝔹|Tz0′(ω)|tdμ(ω)=0,
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 Tz0 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.

2. Preliminaries

Throughout this paper, C denotes a positive constant that may change from one step to the next. For fixed z∈𝔹, we call the set
(12)σ(z|z|,1-|z|)={x∈𝕊n:|x-z|z||<2(1-|z|)}
a spherical cap centered on z/|z| with radius 1-|z|. It is easy to see that σ(z/|z|,1-|z|) is the projection of E(z) on the unit sphere. We always write a spherical cap as σ without pointing out its center and radius, if there are no confusing cases. For a spherical cap σ, we also denote the radius of σ by r(σ) and the Lebesgue measure of σ by |σ|. Clearly, there is the estimate
(13)|σ|≍(r(σ))n,
where we say that F and G are equivalent, denoted by F≍G, if there are two positive constants c and C such that cF≤G≤CF. The Carleson box based on a spherical cap σ is defined by
(14)S(σ)={z∈𝔹:z|z|∈σ,1-r(σ)<|z|≤1}.
The tent based on σ is defined by
(15)T(σ)={z∈𝔹:σ(z|z|,1-|z|)⊂σ}.
The cone Γ(x) in 𝔹 with the vertex x∈𝕊n is defined by
(16)Γ(x)={z∈𝔹:x∈σ(z|z|,1-|z|)}.
For any fixed z∈𝔹, set
(17)σ(z)={x∈𝕊n:z∈Γ(x)}.
Clearly, if z≠0, σ(z) is just the spherical cap in 𝕊n with center z/|z| and radius 1-|z|.

For z∈𝔹 and a measurable function f defined on 𝕊n, we denote by
(18)P(f)(z)=∫𝕊nf(x)p(x,z)dx=∫𝕊nf(x)1-|z|2|𝕊n||x-z|n+1dx
the Poisson extension of f onto 𝔹. The nontangential maximal function of P(f) is the function
(19)P*(f)(x)=supz∈Γ(x)|P(f)(z)|
defined on 𝕊n.

For g∈L1(𝕊n) and any z∈𝔹, we write
(20)T(g)(z)=1|σ(z)|∫σ(z)g(x)dx,
which is also an extension of g onto 𝔹.

For x∈𝕊n, we call the function
(21)M(f)(x)=sup0<r<11|σ(x,r)|∫σ(x,r)|f(y)|dy
the centered Hardy-Littlewood maximal function of f defined on 𝕊n.

The following two lemmas give a lower estimate and an upper estimate for the Poisson integral on 𝔹. In the case of n=1, one can refer to [20, Theorems 2.4 and 2.5].

Lemma 1.

There exists a constant C, such that
(22)T(g)(z)<CP(g)(z),∀z∈𝔹
holds for all g≥0 and g∈L1(𝕊n), where P(g)(z) is the Poisson extension of g.

Proof.

It is sufficient to prove that the estimate
(23)1|σ(z)|χσ(z)(x)≤C1-|z|2|x-z|n+1,∀x∈𝕊n
holds for any z∈𝔹. Since 1-|z|≥C|x-z| as x∈σ(z), by the estimate in (13), it is easy to obtain the conclusion.

Lemma 2.

There exists a constant C, such that
(24)P(g)(z)<CT(P*(g))(z),∀z∈𝔹
holds for all g≥0 and g∈L1(𝕊n).

Proof.

Clearly, it is the consequence of the definitions of T(P*(g))(z) and Γ(x). Noticing that P*(g)(x)≥P(g)(z) as x∈σ(z), we have
(25)T(P*(g))(z)=1|σ(z)|∫σ(z)P*(g)(x)dx≥P(g)(z)1|σ(z)|∫σ(z)dx
for any z∈𝔹. Using (13), it implies the required conclusion.

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]).

Lemma 3.

Let f be a measurable function defined on 𝕊n.

If f∈Lp(𝕊n) and 1≤p≤∞, M(f) is finite almost everywhere.

If f∈L1(𝕊n), then for any α>0,
(26)|{x∈𝕊n:M(f)(x)>α}|≤Cα∫𝕊n|f(x)|dx.

If f∈Lp(𝕊n) and 1<p≤∞, then M(f)∈Lp(𝕊n) and
(27)∥M(f)∥Lp(𝕊n)≤Ap∥f∥Lp(𝕊n),
where Ap depends only on C and p.

Lemma 4.

If f∈Lp(𝕊n), p≥1, then P*(f)(x)≤CM(f)(x) holds for almost every x∈𝕊n.

For the convenience, we define Carleson measures on 𝔹 in terms of Carleson boxes or tents.

Definition 5.

Let s>0 and σ a spherical cap in 𝕊n. A nonnegative measure μ on 𝔹 is called an s-Carleson measure if there exits a constant C such that
(28)μ(S(σ))≤C|σ|s,∀σ⊂𝕊n.μ is called a compact s-Carleson measure if μ is an s-Carleson measure and
(29)lim|σ|→0μ(S(σ))|σ|s=0.

Remark 6.

(1) Comparing Definition 5 with the definition in (8), one can see that the s in Definition 5 is equal to nt in (8).

For any σ⊆𝕊n, we can replace S(σ) with T(σ) in Definition 5.

For n=1, it goes back to the one in [2].

For 0<p≤q<∞, a well-known result which is due to Carleson in [1] for p=q and Duren in [22] for p<q says that a nonnegative measure μ on the unit disc 𝔻 of the complex plane ℂ is a bounded (q/p)-Carleson measure if and only if
(30)∫𝔻|f(z)|qdμ(z)≤∥f∥Hpq,∀f∈Hp(𝔻).
By Definition 5 and using the real analysis techniques, we obtain the following extension of this result on 𝔹.

Theorem 7.

Let μ be a nonnegative measure on 𝔹.

For 0<p≤q<∞, let ϕ(z) be a μ-measurable function defined on 𝔹 and let ϕ* be the nontangential maximal function of ϕ(z). If μ is a (q/p)-Carleson measure on 𝔹, then
(31)∫𝔹|ϕ(z)|qdμ(z)≤C∥ϕ*∥Lp(𝕊n)q.

For 1<p≤q<∞, μ is a (q/p)-Carleson measure on 𝔹 if and only if
(32)∫𝔹|P(f)(z)|qdμ(z)≤C∥f∥Lp(𝕊n)q
for any f∈Lp(𝕊n), and here P(f) is the Poisson integral of f.

Proof.

It is sufficient to (31) if
(33)∫𝔹|φ(z)|dμ(z)≤C∥φ*∥Lp/q(𝕊n).
Indeed, write φ(z)=|ϕ(z)|q, and above if the inequality holds, then
(34)∫𝔹|ϕ(z)|qdμ(z)≤C(∫𝕊n|φ*|p/qdx)q/p=C(∫𝕊n|ϕ*|pdx)q/p,
which is what we need.

Suppose that μ is a (q/p)-Carleson measure on 𝔹. Write Ω={x∈𝕊n:φ*(x)>α}. Recalling the Whitney decomposition (see [21]), we know that there exists a disjoint collection of spherical caps {σk} such that ⋃kσ*k=Ω, and here σ*k is the spherical cap with the same center as σk but radius C times. Now, we claim that
(35){z∈𝔹:φ(z)>α}⊆⋃kT(σ*k).
Indeed, let z∈𝔹 so that φ(z)>α. By the definition of φ*, we have that φ*(x)>α for all x∈𝕊n satisfying x∈σ(z/|z|,1-|z|). Thus, σ(z/|z|,1-|z|)⊆Ω=⋃kσ*k, and clearly z∈⋃kT(σ*k).

Clearly, we have
(36)μ({z∈𝔹:φ(z)>α})≤∑kμ(T(σ*k))≤C∑k|σ*k|q/p≤C∑k|σk|q/p≤C|Ω|q/p.
Thus, we have
(37)∫𝔹|φ(z)|dμ(z)=∫0∞μ{z∈𝔹:φ(z)>α}dα≤C∫0∞|{x∈𝕊n:φ*(x)>α}|q/pdα≤C∫0∞(∫𝕊nχφ*(x)>αdx)q/pdα.
By Minkowski’s inequality acting on the last inequality, we have
(38)∫𝔹|φ(z)|dμ(z)≤C(∫𝕊n|φ*|p/qdx)q/p=C∥φ*∥Lp/q(𝕊n),
Then, (31) follows.

Now, turn to the proof of the “only if” part of (b). If μ is a (q/p)-Carleson measure, we can obtain
(39)∫𝔹|P(f)(z)|qdμ(z)≤C∥P*(f)∥Lp(𝕊n)q
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 σ in 𝕊n, if z∈T(σ), we have σ(z)⊆σ. Let f=χσ(x). Since the simple fact |x-z|≤C(1-|z|) as x∈σ(z) and the estimate in (13), we obtain the useful estimate
(40)P(f)(z)=∫𝕊nχσ(x)p(x,z)dx≥C∫σ(z)1(1-|z|)ndx≥C.
Now, it is easy to see
(41)μ(T(σ))≤C∫T(σ)|P(f)(z)|qdμ(z)≤C(∫𝕊n|χσ(x)|pdx)q/p=C|σ|q/p,
which is to say that μ is a (q/p)-Carleson measure. The proof of (b) is completed.

Let σ be a spherical cap in 𝕊n with center x0=z0/|z0| and radius r(σ)=1-|z0|. Let j be a nonnegative integer and, N be the greatest integer less than log2(1/r(σ)). Denote the spherical cap with the same center as σ but radius 2jr(σ) by σj. Then, σ0=σ and σN+1=𝕊n. Moreover, for any spherical cap σ⊆𝕊n, we have
(42)𝔹=S(σ)⋃(⋃j=0N(S(σj+1)∖S(σj))).

Lemma 8.

For fixed z0∈𝔹, let σ be a spherical cap in 𝕊n with center x0=z0/|z0| and radius r(σ)=1-|z0|. Then, one has the following estimates:

1+|ω|2|z0|2-2ω·z0≍(r(σ))2, if ω∈S(σ),

1+|ω|2|z0|2-2ω·z0≍22j(r(σ))2, if ω∈S(σj+1)∖S(σj), j∈[0,N].

Proof.

If ω∈S(σ), then we have the estimates
(43)1+|ω|2|z0|2-2ω·z0≥1+|ω|2|z0|2-2|ω||z0|=(1-|ω||z0|)2≥(1-|z0|)2,1+|ω|2|z0|2-2ω·z0=1+|ω|2|z0|2-(|z0|2+|ω|2-|z0-ω|2)=(1-|ω|2)(1-|z0|2)+|z0-ω|2≤C(1-|z0|)2.
Part (i) is yielded by the above.

For part (ii), if ω∈S(σj+1)∖S(σj), we have
(44)1+|ω|2|z0|2-2ω·z0=1+|ω|2|z0|2-(|z0|2+|ω|2-|z0-ω|2)=(1-|ω|2)(1-|z0|2)+|z0-ω|2≍22j(r(σ))2.
The proof is completed.

Combining together Definition 5, the decomposition of 𝔹 in (42), and Lemma 8, we can obtain the following characterization for s-Carleson measures on 𝔹, which is similar to [14, Theorems 2.3 and 2.4].

Theorem 9.

Let s>0 and 0<τ<∞. A nonnegative measure μ on 𝔹 is an s-Carleson measure or a compact s-Carleson measure if and only if
(45)supz0∈𝔹∫𝔹((1-|z0|2)τ(1+|ω|2|z0|2-2ω·z0)(1+τ)/2)nsdμ(ω)<∞,
or
(46)lim|z0|→1-∫𝔹((1-|z0|2)τ(1+|ω|2|z0|2-2ω·z0)(1+τ)/2)nsdμ(ω)=0.
In particular, when τ=1, μ is an s-Carleson measure or a compact s-Carleson measure on 𝔹 if and only if
(47)supz0∈𝔹∫𝔹|Tz0′(ω)|nsdμ(ω)<∞,orlim|z0|→1-∫𝔹|Tz0′(ω)|nsdμ(ω)=0.

3. The Carleson Measures Characterized by <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M315"><mml:mrow><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> Behaviors

By the estimate in (13) and Theorem 9, we observe that
(48)∫𝔹|Tz0′(ω)|nsdμ(ω)≍∫𝔹∫SnχΓ(x)(ω)(1-|z0|21+|ω|2|z0|2-2ω·z0)ns×dx(1-|ω|)ndμ(ω)≍∫𝕊n∫Γ(x)(1-|z0|21+|ω|2|z0|2-2ω·z0)nsdμ(ω)(1-|ω|)ndx=∥∫Γ(x)|Tz0′(ω)|nsdμ(ω)(1-|ω|)n∥L1(𝕊n).
For z0∈𝔹, s∈ℝ, and 0<p≤∞, write
(49)ℜμ(z0,s,p)=∥∫Γ(x)|Tz0′(ω)|nsdμ(ω)(1-|ω|)n∥Lp(𝕊n).
By Theorem 9, it is clear that μ is an s-Carleson measure on 𝔹 if and only if supz0∈Bℜμ(z0,s,1)<∞.

For a spherical cap σ⊆𝕊n and 0<p≤∞, define
(50)ℑμ(S(σ),p)=∥∫Γ(x)∩S(σ)dμ(ω)(1-|ω|)n∥Lp(𝕊n).
It is also clear that μ is an s-Carleson measure on 𝔹 if and only if ℑμ(S(σ),1)≤C|σ|s for all σ⊆𝕊n.

Now, we are going to state our main arguments in this section, which are devoted to establishing the connections between s-Carleson measures on 𝔹 and
(51)supz0∈𝔹ℜμ(z0,s,p)<∞orℑμ(S(σ),p)≤C|σ|s.
Here, we emphasiz that the following results have been achieved when n=1 (see [13]).

Theorem 10.

Let s>0, 0<p≤∞, and μ a nonnegative measure on 𝔹.

supz0∈𝔹ℜμ(z0,s,p)<∞ if and only if ℑμ(S(σ),p)≤C|σ|s for all spherical cap σ⊂𝕊n.

lim|z0|→1-ℜμ(z0,s,p)=0 if and only if ℑμ(S(σ),p)=o(|σ|s) for all spherical cap σ⊂𝕊n.

Remark 11.

(1) In Theorem 10, the Carleson box S(σ) can be replaced by the tent T(σ).

(2) The results in the above theorem hold for s∈ℝ on 𝔻 (see [13, Theorem 1]), but they do only for s>0 here.

Proof.

For any spherical cap σ, there must exist z0∈𝔹 such that σ=σ(z0/|z0|,1-|z0|). By Lemma 8, we have
(52)(1-|z0|21+|ω|2|z0|2-2ω·z0)ns≍|σ|-s,ω∈S(σ).
Thus,
(53)∫Γ(x)∩S(σ)dμ(ω)(1-|ω|)n≤C|σ|s∫Γ(x)(1-|z0|21+|ω|2|z0|2-2ω·z0)nsdμ(ω)(1-|ω|)n.
The “only if” parts of (i) and (ii) are concluded from the above inequality.

To prove the “if” part of (i), let σ=σ(z0/|z0|,1-|z0|) and σ-1=ϕ. By the decomposition of 𝔹 in (42), it is clear that
(54)Γ(x)=Γ(x)∩𝔹=⋃j=0N+1(Γ(x)∩(S(σj)∖S(σj-1))).
By Lemma 8, we have
(55)∫Γ(x)(1-|z0|21+|ω|2|z0|2-2ω·z0)nsdμ(ω)(1-|ω|)n=∑j=0N+1∫Γ(x)∩(S(σj)∖S(σj-1))(1-|z0|21+|ω|2|z0|2-2ω·z0)ns×dμ(ω)(1-|ω|)n≤C∑j=0N+1∫Γ(x)∩S(σj)|σ|s22jns|σ|2sdμ(ω)(1-|ω|)n.
Taking Lp norm on both sides of the above inequality, if 1≤p≤∞, we have
(56)ℜμ(z0,s,p)≤C∑j=0N+1|σ|s22jns|σ|2sℑμ(S(σj),p)≤C∑j=0N+1|σ|s22jns|σ|2s|σj|s≤C∑j=0N+12-jns≤C.
If 0<p<1, then
(57)(ℜμ(z0,s,p))p≤C∑j=0N+1(|σ|s22jns|σ|2sℑμ(S(σj),p))p≤C∑j=0N+1(|σ|s22jns|σ|2s|σj|s)p≤C∑j=0N+12-jnps≤C.
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≤p≤∞. We observe first that for ε>0, there must exist an integer m>0 such that
(58)C∑j=mN+112jns<ε2.
Then, we have
(59)ℜμ(z0,s,p)≤C∑j=0N+1|σ|s22jns|σ|2sℑμ(S(σj),p)≤C∑j=0m-1|σ|s22jns|σ|2sℑμ(S(σm),p)+ε2≤C1-2-2ns2mns|σm|sℑμ(S(σm),p)+ε2.
By the assumption of the “if” part of (ii), there must exist δ>0 such that if |σm|<δ(60)ℑμ(S(σm),p)<1-2-2nsC2mnsε2(|σm|s).
If (1-|z0|)n≍|σ|<2-mnδ, then we have
(61)ℜμ(z0,s,p)<ε,
which is the desired result. With the same process, we can obtain the results when 0<p<1. The proof of Theorem 10 is completed.

Theorem 12.

Suppose that s>0, 1≤p<∞, and μ is a nonnegative measure on 𝔹.

supz0∈𝔹ℜμ(z0,s,p)<∞ or ℑμ(S(σ),p)≤C|σ|s for all spherical cap σ⊂𝕊n if and only if |P(f)(z)|dμ(z) is an s-Carleson measure for all f∈Lp/(p-1)(𝕊n).

lim|z0|→1-ℜμ(z0,s,p)=0 or ℑμ(S(σ),p)=o(|σ|s) for all spherical cap σ⊂𝕊n if and only if |P(f)(z)|dμ(z) is a compact s-Carleson measure for all f∈Lp/(p-1)(𝕊n).

Proof.

In the situation of p=1, the results are deduced by Theorems 9 and 10 and the estimate
(62)ℜμ(z0,s,1)=∫𝕊n∫Γ(x)(1-|z0|21+|ω|2|z0|2-2ω·z0)nsdμ(ω)(1-|ω|)ndx≍∫𝔹(1-|z0|21+|ω|2|z0|2-2ω·z0)nsdμ(ω).

For 1<p<∞, it is well known that P*(f)∈Lp/(p-1)(𝕊n) if f(x)∈Lp/(p-1)(𝕊n). By Lemma 2, we have
(63)P(f)(z)≤T(P*(f))(z),∀z∈𝔹.
Now, by the estimate in (13) and Hölder’s inequality, we have
(64)∫S(σ)P(f)(z)dμ(z)≤∫S(σ)T(P*(f))(z)dμ(z)≤C∫S(σ)1|σ(z)|∫σ(z)P*(f)(x)dxdμ(z)≤C∫𝕊nP*(f)(x)∫S(σ)⋂Γ(x)dμ(z)(1-|z|)ndx≤C∥P*(f)∥Lp/(p-1)(𝕊n)ℑμ(S(σ),p).
Combining with Theorem 10, we complete the proof of “only if” parts.

Turn to the proof of “if” parts. For f∈Lp/(p-1)(𝕊n) and f≥0, suppose that |P(f)(z)|dμ(z) is an s-Carleson measure or a compact s-Carleson measure. By Lemma 1, we know that
(65)T(f)(z)≤CP(f)(z),∀z∈𝔹,
which implies that T(f)(z)dμ(z) is an s-Carleson measure or a compact s-Carleson measure. By the estimate in (13), we have
(66)∫𝕊nf(x)∫Γ(x)⋂S(σ)dμ(z)(1-|z|)ndx≍C∫S(σ)T(f)(z)dμ(z).
Using the above estimate, the Lp/(p-1) duality, and also Theorem 10, we deduce the desired results.

Now, we are going to consider the case of p=∞.

Theorem 13.

Let s>0 and μ a nonnegative measure on 𝔹. Then, the following are equivalent.

supz0∈𝔹ℜμ(z0,s,∞)<∞ or ℑμ(S(σ),∞)≤C|σ|s for all spherical cap σ⊆Sn.

|T(f)(z)|dμ(z) is an s-Carleson measure for all f∈L1(𝕊n).

Moreover, the following are equivalent.

supz0∈𝔹ℜμ(z0,s,∞)<∞ or ℑμ(S(σ),∞)≤C|σ|s for all spherical cap σ⊆𝕊n.

|P(f)(z)|dμ(z) is an s-Carleson measure for all f∈H1(𝕊n).

μ is an (s+1)-Carleson measure.

The equivalence above holds for the compact case also.

Proof.

For f(x)∈L1(𝕊n) and f≥0, we have
(67)∫𝕊nf(x)∫Γ(x)⋂S(σ)dμ(z)(1-|z|)ndx≍C∫S(σ)T(f)(z)dμ(z),
which implies the equivalence of (a1) and (b1) by duality theorem and Theorem 10.

Invoking Lemma 2, we observe that “(a2)⇒(b2)” is a direct consequence of the equivalence of (a1) and (b1).

To prove “(b2)⇒(c2),” let z0 be any fixed point in 𝔹 and let
(68)f(x)=(1-|z0|21+|x|2|z0|2-2x·z0)n
be a function defined on 𝕊n. For z∈T(σ(z0)), one should observe the fact σ(z)⊆σ(z0) and
(69)P(f)(z)=∫𝕊nf(x)p(x,z)dx≥C1|σ(z0)|∫σ(z0)p(x,z)dx≥C1|σ(z0)|∫σ(z)1-|z||x-z|n+1dx≥C1|σ(z0)|
by Lemma 8 and the estimate (13). Then, the assumption that |P(f)(z)|dμ(z) is an s-Carleson measure implies that
(70)1|σ(z0)|∫T(σ(z0))dμ(z)≤C∫T(σ(z0))|P(f)(z)|dμ(z)≤C|σ(z0)|s
holds for any z0∈𝔹, which is to say that μ is an (s+1)-Carleson measure.

To prove “(c2)⇒(a2),” assume that μ is an (s+1)-Carleson measure. For any z0∈𝔹, let σ be the sphere cap σ=σ(z0/|z0|,1-|z0|). Let i be a nonnegative integer. For some x∈𝕊n, write σ*=σ(x,2(1-|z0|)). Let σi* be the spherical cap with the same center as σ* but radius 2-ir(σ*). We choose a number ε small enough such that
(71)Γ(x)⋂S(σ*)⊂⋃i=0∞S(σi*)^ε.
Here S(σi*)^ε={z∈S(σi*):1-|z|≥εr(σi*)} is the top of S(σi*). We observe the fact that
(72)Γ(x)⋂S(σ)={z∈B:x∈σ(z|z|,1-|z|),z∈S(σ)}⊆{z∈B:z|z|∈σ(x,1-|z0|),|z0|≤|z|}⊆S(σ*)⋂Γ(x)
holds for any x∈𝕊n. Now, it is easy to see that for any x∈𝕊n(73)∫Γ(x)⋂S(σ)dμ(z)(1-|z|)n≤∫S(σ*)⋂Γ(x)dμ(z)(1-|z|)n≤∑i=0∞∫S(σi*)^εdμ(z)(1-|z|)n≤C∑i=0∞∫S(σi*)^ε|σi*|-1dμ(z)≤C∑i=0∞|σi*|s≤C|σ*|s∑i=0∞2-isn,
which is to say that ℑμ(S(σ),∞)≤C|σ|s. The proof is completed.

Theorem 14.

Let s>0, max(1/s,1)≤p<∞, and μ a nonnegative measure on 𝔹.

supz0∈𝔹ℜμ(z0,s,p)<∞ or ℑμ(S(σ),p)≤C|σ|s for all spherical cap σ⊆𝕊n if and only if μ is an (s+1-1/p)-Carleson measure.

lim|z0|→1-ℜμ(z0,s,p)=0 or ℑμ(S(σ),p)=o(|σ|s) for all spherical cap σ⊆𝕊n if and only if μ is a compact (s+1-1/p)-Carleson measure.

Proof.

To prove the “if” parts, suppose that μ is an (s+1-1/p)-Carleson measure or a compact (s+1-1/p)-Carleson measure. By Theorem 10, it is sufficient to prove that
(74)ℑμ(S(σ),p)≤C(μ(S(σ)))s/(s+1-1/p)
holds for all σ⊆𝕊n.

For p=1, the results are trivial because of the fact of ℑμ(S(σ),1)≍μ(S(σ)).

For p>1, let g∈Lp′(𝕊n) and g≥0 (here p′ is the conjugate of p, that is, 1/p+1/p′=1). By Lemma 1, we have
(75)T(g)(z)≤CP(g)(z),∀z∈𝔹.
Let q=p′(s+1-1/p). It is easy to see that q>p′>1 and q′=(s+1-1/p)/s (here q′ is the conjugate of q). Now, by the previous assumption, we have that μ is a (q/p′)-Carleson measure. Using Hölder’s inequality and Theorem 7, we have
(76)∫𝕊ng(x)∫Γ(x)⋂S(σ)dμ(z)(1-|z|)ndx≤C∫S(σ)T(g)(z)dμ(z)≤C(∫S(σ)|P(g)(z)|qdμ(z))1/q(∫S(σ)dμ(z))1/q′≤C∥g∥Lp′(𝕊n)μ(S(σ))s/(s+1-1/p).
By the duality theorem, we conclude that
(77)ℑμ(S(σ),p)≤Cμ(S(σ))s/(s+1-1/p),
which completes the proof of “if” parts.

We now turn to the “only if” parts. Assume that ℑμ(S(σ),p)≤C|σ|s or ℑμ(S(σ),p)=o(|σ|s) for all spherical cap σ⊆𝕊n. For a spherical cap σ⊆𝕊n, we have
(78)σ(z|z|,1-|z|)⊂σ,∀z∈T(σ).
Then, by the estimate in (13) and Fubini’s theorem, we obtain that
(79)1|σ|∫T(σ)dμ(z)≤C1|σ|∫T(σ)∫σ∩σ(z)dx(1-|z|)ndμ(z)≤C∫σ∫T(σ)⋂Γ(x)dμ(z)(1-|z|)ndx|σ|.
Since p≥1, Jensen’s inequality and the above inequality imply that
(80)1|σ|∫T(σ)dμ(z)≤C|σ|-1/p(∫𝕊n(∫T(σ)⋂Γ(x)dμ(z)(1-|z|)n)pdx)1/p≤C|σ|-1/pℑμ(T(σ),p).
Moreover, by the assumption and the remarks below Definition 5 and Theorem 10, we have
(81)ℑμ(T(σ),p)≤C|σ|sorℑμ(T(σ),p)=o(|σ|s).
Then, we conclude that μ is an (s+1-1/p)-Carleson measure or a compact (s+1-1/p)-Carleson measure. The proof of Theorem 14 is completed.

Theorems 10 and 14 imply the following characterizations for s-Carleson measures on 𝔹 as s≥1.

Corollary 15.

Let s≥1, 1≤p<∞ and μ a nonnegative measure on 𝔹.

μ is an s-Carleson measure on 𝔹 if and only if
(82)supz0∈𝔹ℜμ(z0,s-1+1p,p)<∞orℑμ(S(σ),p)≤C|σ|s-1+1p∀σ⊆𝕊n.

μ is a compact s-Carleson measure on 𝔹 if and only if
(83)sup|z0|→1-ℜμ(z0,s-1+1p,p)=0orℑμ(S(σ),p)=o(|σ|s-1+1/p)∀σ⊆𝕊n.

Remark 16.

By Theorem 13, the above corollary holds if s>1 and p=∞.

For 0<s<1, we have the following results.

Theorem 17.

Let 0<s<1 and μ a nonnegative measure on 𝔹.

If 0<p≤1 and μ is an s-Carleson measure on 𝔹, then
(84)supz0∈𝔹ℜμ(z0,s-1+1p,p)<∞,ℑμ(S(σ),p)≤C|σ|s-1+1/p∀σ⊆𝕊n.

If 1≤p<1/(1-s) and supz0∈𝔹ℜμ(z0,s-1+1/p,p)<∞ or ℑμ(S(σ),p)≤C|σ|s-1+1/p for all σ⊆𝕊n, then μ is an s-Carleson measure.

If 1/(1-s)≤p<∞, then ℑμ(S(σ),p)≤C|σ|s-1+1/p for all σ⊆𝕊n if and only if
(85)∫𝔹|P(f)(z)|p-1dμ(z)≤C∥f∥Lp(𝕊n)p-1,∀f∈Lp(𝕊n).

Proof.

To prove (a), it is sufficient to prove
(86)ℑμ(T(σ),p)≤C|σ|1/p-1μ(T(σ))
for any spherical cap σ⊆𝕊n. For 0<p≤1 and any spherical cap σ, by Hölder’s inequality and Fubini’s theorem, it is clear that
(87)(ℑμ(T(σ),p))p=∫𝕊n(∫Γ(x)⋂T(σ)dμ(z)(1-|z|)n)pdx=∫σ(∫Γ(x)⋂T(σ)dμ(z)(1-|z|)n)pdx≤C(∫σdx)1-p(∫σ∫Γ(x)⋂T(σ)dμ(z)(1-|z|)ndx)p≤C|σ|1-p(∫T(σ)dμ(z))p≤C|σ|1-p(μ(T(σ)))p.
Then, by the assumption that μ is an s-Carleson measure, it is easy to see that ℑμ(T(σ),p)≤C|σ|s-1+1/p.

For (b), let f(x)=χσ(x), and then ∥f∥Lp′(𝕊n)=|σ|1/p′ (here p′ is also the conjugate of p). Noticing the fact that σ(z/|z|,1-|z|)⊂σ for any z∈T(σ), we obtain
(88)μ(T(σ))=∫T(σ)dμ(z)≤C∫T(σ)1|σ(z)|∫σ(z)f(x)dxdμ(z)≤C∫𝕊nf(x)∫Γ(x)⋂T(σ)dμ(x)(1-|z|)ndx≤C∥f∥Lp′(𝕊n)ℑμ(T(σ),p)≤C|σ|s.
By Theorem 10, we complete the proof of (b).

For (c), since condition ℑμ(S(σ),p)≤C|σ|s-1+1/p for all σ⊆𝕊n is equivalent to ℑμ(𝔹,p)≤C, it is the direct consequence of the estimate for g≥0(89)∫𝔹T(g)(z)dμ(z)≍∫𝕊ng(x)∫Γ(x)dμ(z)(1-|z|)ndx
and the duality theorem with appropriate choices of g.

Acknowledgments

The authors are partially supported by grants from the NSF of China (11271162) and the NSF of Zhejiang province (Y6100810).

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