Lemma 1. Suppose 1 < p < ∞. Then, there exists a positive constant C such that

and $$ for every $$.

Proof. Suppose r > 0 and $$. Then, there exists a constant C > 0 such that

which implies that

The inequalities in (8) hold. Here, D(z, r) is the hyperbolic disk (see [3]). From (8), we see that $$ are contained in the disk algebra for p > 1. Hence, we get that $$.

Lemma 2. Let 1 < p < ∞. If $$, then for all t ∈ (0, 1) and $$, there exists a positive constant C such that

Proof. Fix $$. Let t ∈ (0, 1) and $$. By Lemma 1,

as desired.

Lemma 3. Let 1 < p < ∞. Every sequence in $$ bounded in norm has a subsequence which converges uniformly in $$ to a function in $$.

Lemma 4 (see [5].)Let X be a Banach space that is continuously contained in the disk algebra, and let Y be any Banach space of analytic functions on $$. Suppose that

• (i)

  The point evaluation functionals on Y are continuous

• (ii)

  For every sequence {f_n} in the unit ball of X that exists an f ∈ X and a subsequence $$ such that $$ uniformly on $$

• (iii)

  The operator T : X⟶Y is continuous if X has the supremum norm and Y is given by the topology of uniform convergence on compact sets

•  

  Then, T is a compact operator if and only if, given a bounded sequence {f_n} in X such that f_n⟶0 uniformly on $$, then the sequence $$ as n⟶∞.

Lemma 5. Let 1 < p < ∞ and μ be a weight. If $$ is bounded, then T is compact if and only if $$ as k⟶∞ for any sequence {f_k} in $$ bounded in norm which converge to 0 uniformly in $$.

Theorem 6. Let v be a radial, nonincreasing weight tending to zero at the boundary of $$. Let 1 < p < ∞, $$, and $$. Then, the following statements are equivalent.

• (i)

  The operator $$ is bounded

• (ii)

  $$

  $$ (13)
  and
  $$ (14)

• (iii)

  $$

$$ and $$

Proof. (ii)⇒(i). For any $$ and $$, by Lemma 1, we have

Hence,

Therefore, $$ is bounded.

(i)⇒(ii). Applying the operator ψC_φ to z^j with j = 0, 1, 2 and using the boundedness of ψC_φ, we get that $$, $$, and $$. Hence, we obtain

For any $$, set

It is easy to check that

Therefore, by the boundedness of $$ and arbitrary of $$, we get

For $$, we get

From (21) and (22), we obtain

From (24), we get

On one hand, from (25), we obtain

On the other hand, from the fact that $$, we get

From (26) and (27), we see that P is finite. Using similar arguments, we see that Q is also finite.

(ii)⇔(iii). From [19], we see that the inequality in is equivalent to the operator $$ is bounded. By [20], the boundedness of (2ψ^′φ^′ + ψφ^′′)C_φ is equivalent to

From [21], we get $$, which together with (28) imply that

Similarly, the inequality in is equivalent to

The proof is complete.

Theorem 7. Let v be a radial, nonincreasing weight tending to zero at the boundary of $$. Let 1 < p < ∞, $$, and $$. Suppose that $$ is bounded. Then,

Proof. First we show that $$ Let $$ be a sequence in the unit disk such that |φ(z_j)|⟶1 as j⟶∞. Define

After a calculation, we get all k_j and m_j belong to $$ and

Moreover, k_j and m_j converge to 0 uniformly on $$ as j⟶∞. Hence, for any compact operator $$, by Lemma 5, we get

Hence,

as desired.

Next, we show that

Let r ∈ [0, 1). Define $$ by

It is clear that K_r is compact on $$ and $$. Moreover, f_r − f⟶0 uniformly on compact subsets of $$ as r⟶1. Let {r_j} ⊂ (0, 1) such that r_j⟶1 as j⟶∞. Then, $$ is compact for each j ∈ ℕ. Hence,

Thus, we only need to prove that

For any $$ with $$, by the facts that

we have where t ∈ ℕ is large enough such that r_j ≥ 1/2 for all j ≥ t. Since $$, $$, and $$ uniformly on compact subsets of $$ as j⟶∞, by Lemma 3, we obtain

Using Lemma 1 and $$, we obtain

Taking the limit as t⟶∞, we get

Similarly,

Taking the limit as t⟶∞, we get

Hence, by (43), (44), (45), (46), (48), and (50), we get

which with (40) implies the desired result.

Finally, we prove that

On one hand, by the proof of Theorem 6, we see that the boundedness of $$ is equivalent to the boundedness of $$ and $$. From [19, 20], we have

Hence,

On the other hand, from [19, 21], we have

Therefore,

The proof is complete.

Corollary 8. Let v be a radial, nonincreasing weight tending to zero at the boundary of $$. Let 1 < p < ∞, $$, and $$. Suppose that $$ is bounded. Then, the following statements are equivalent:

• (i)

  The operator $$ is compact

• (ii)

  $$ and $$

• (iii)

  $$

Similarly to the above proof, we can get the characterizations of the boundedness, compactness, and essential norm of the weighted composition operator $$ as follows. The details are left to the interested readers.

Theorem 9. Let v be a radial, nonincreasing weight tending to zero at the boundary of $$. Let 1 < p < ∞, $$, and $$. Then, the following statements are equivalent.

• (i)

  $$ is bounded

• (ii)

  $$ and

• (iii)

  $$ and $$

Theorem 10. Let v be a radial, nonincreasing weight tending to zero at the boundary of $$. Let 1 < p < ∞, $$, and $$. Suppose that $$ is bounded. Then,

Corollary 11. Let v be a radial, nonincreasing weight tending to zero at the boundary of $$. Let 1 < p < ∞, $$, and $$. Suppose that $$ is bounded. Then, the following statements are equivalent:

• (i)

  The operator $$ is compact

• (ii)

  $$

• (iii)

  $$