Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in Hardy type spaces

Lemma 2.1

Let 1 < p < ∞ and

Then^∁L^p,φ(I) ↪ L¹(I).

Proof

The proof is straightforward: use dyadic decomposition, which is the standard tool when working with Morrey type norm (see, for instance, [6, 31]) for the interval I and apply Hölder inequality. We have (ℤ₊ ≡ ℕ ∪ {0}):

We refer, for instance, to Lemma 3.2 from [6] for the estimation of the infinite sum via integral. □

Note that in the classical case of complementary Morrey space when φ(t) = t^λ the condition (2.2) is λ < p−1. If there exists β ∈(0,p−1) such that φ(t)tβ decreases on I, then the condition (2.2) is satisfied for such φ.

Remark 2.1

In what follows when considering the Morrey space L^p,φ(I) and complementary Morrey space ^∁L^p,φ(I), we always assume the mentioned above natural assumptions on φ: the function φ: I → ℝ₊ is increasing on I and φ(t) > 0 for t > 0, φ(0) = 0. Additionally, for the Morrey space we suppose that φ(t) ⩾ Ct, when t → 0, and for the complementary Morrey space ^∁L^p,φ(I) we assume validity of (2.2).

Remark 2.2

In the definition of the Morrey space one may want to use a more classical way, writing φ(4rh) instead of φ(h), since the measure of the interval (r−h,r+h) with respect to the 2rdr equals to 4rh. Such introduced mixed norm space, let us denote it as L∗q;φ(𝔻), will be different from 𝓛^q;φ(𝔻). However, the corresponding subspaces of analytic functions coincide up to norm equivalence. For instance, the proof of the Theorem 6.3 explicitly shows that we will have similar results in that another setting. We leave the details for the reader. Also note that the usual definition of the Morrey space deals with the supremum over all r,h ∈ I and intervals I ∩ (r−h, r+h). For our goals, for the definition of L^p,φ(I) we admit only intervals (r−h, r+h) ⊂ I.

Lemma 2.2

Let Φ satisfy the doubling Δ₂condition for t > 0 with the constant C₍₂₎. Then given any β ⩾ log₂C₍₂₎we have

1. t^−βΦ(t) is almost decreasing for t > 0;

2. Φ(At) ⩽ A^βC₍₂₎ Φ(t), for any constant A ⩾ 1, for t > 0.

Proof

The statements (1) and (2) are equivalent up to the constants in the inequalities. Let us prove the second one. Naturally we assume that C₍₂₎ > 1. Fix N = [log₂ A] so that 2^N ⩽ A < 2^N+1. Then Φ(At)⩽Φ(2N+1t)⩽C(2)N+1Φ(t)=C(2)[log2⁡A]+1Φ(t)⩽C(2)log2⁡A+1Φ(t)=C(2)log2⁡AC(2)Φ(t)=Alog2⁡C(2)C(2)Φ(t). Hence it follows that given any β ⩾ log₂C₍₂₎ we have Φ(At) ⩽ A^βC₍₂₎ Φ(t), t > 0. □

Theorem 3.1

The space 𝓛^q;X(𝔻), 1 ⩽ q < ∞, is complete.

Proof

Let f^k ∈ 𝓛^q;X(𝔻), k = 0,1,…, be a Cauchy sequence. Then for each n ∈ℤ the sequence of Fourier coefficients fnk, k = 0,1,…, is Cauchy sequence in X(I) and converges in X(I) to some element f_n ∈ X(I). From the inequality

it follows that the numerical sequence of elements {∥ fnk∥_X(I)}_{n ∈ℤ}, k = 0,1,…, is Cauchy sequence in l^q, 1 ⩽ q < ∞, and, hence, converges to some numerical sequence {a_n}_{n ∈ℤ} ∈ l^q. Obviously, a_n = ∥f_n∥_X(I), and therefore the distribution f = ∑_{n ∈ℤ}f_n(r)e^inα is the limit of the sequence f^k ∈ 𝓛^q;X(𝔻), k = 0,1,…, and its distributional Fourier coefficients are nothing but f_n, by Lemma 7.1. Hence, f ∈ 𝓛^q;X(𝔻) and the space 𝓛^q;X(𝔻) is complete. □

We introduce the mixed norm Bergman space 𝓐^q;X(𝔻), 1 ⩽ q < ∞, as the space of functions from 𝓛^q;X(𝔻) which are analytic in 𝔻. Hence, the norm of a function f ∈ 𝓐^q;X(𝔻) is given by

Remark 3.1

From the definition of the space 𝓐^q;X(𝔻), it can be derived that the Fourier coefficients f_n = f_n(r), n ∈ℤ, of a function f in Bergman space 𝓐^q;X(𝔻) may be represented as

where {a_n}_{n ∈ ℤ+} ∈ l+q, |a_n| = ∥f_n∥_X(I), n ∈ℤ₊, moreover, ∥f∥_{𝓐^q;X(𝔻)} = ∥{a_n}_{n ∈ ℤ+}∥_l+q.

It is evident that the multipliers ∥rn∥X(I)−1, n ∈ ℤ₊, in (3.2) characterize the functions in 𝓐^q;X(𝔻). Their behaviour when n → ∞ is a crucial point in the whole study. This behavior depends only on the choice of the space X(I). So to characterize the introduced space in each particular case of X(I) we should examine the asymptotic behavior of the numbers ∥rn∥X(I)−1. This is a quite difficult issue for spaces of functions with special norm, such as, for instance, variable exponent Lebesgue norm, Morrey space norm, etc.

Remark 3.2

Our main goal is the study of the Bergman space of analytic functions 𝓐^q;X(𝔻). The space 𝓛^q;X(𝔻) plays only a background role: the space 𝓐^q;X(𝔻) will be obtained from 𝓛^q;X(𝔻) via the Bergman projection. In the definition of the space 𝓛^q;X(𝔻) in view of the property of the Bergman projection (see Lemma 3.1) negative entries may be replaced for instance by ∥f_n∥_Y(I) with an arbitrary Banach space Y(I), or even more generally ∥{f_n}_{n ∈ℤ∖ ℤ+}∥_Z(I), where Z(I) is an arbitrary Banach space of sequences of functions. The resulting subspace of analytic functions 𝓐^q;X(𝔻) will be the same independently what kind of norm is used for negative entries. Theorem 3.1 remains also true under such changes if the space Y(I) or Z(I) is complete. For simplicity of presentation we keep the definition of the space 𝓛^q;X(𝔻) as given in (3.1).

Lemma 3.1

Given a function f in L¹(𝔻) let f_n = f_n(r), n ∈ℤ, denote the Fourier coefficients of the function f. Then the Fourier coefficients of the function B_𝔻f are

whereϑ_n(f) = (n+1) ∫_Iτⁿf_n(τ) 2τdτ, n ∈ℤ₊.

Let S0X(𝔻) denote the set of functions f(z)=f(r,eiα)=∑n=−NNfn(r)einα,f_n ∈ X(I), where N ∈ℤ₊ is arbitrary. Since X(I) ⊂ L¹(I), then S0X(𝔻) ⊂ L¹(𝔻) and therefore the Bergman projection B_𝔻 is well defined on functions of such type as integral operator (2.1). It is evident that S0X(𝔻) is a dense subset in 𝓛^q;X(𝔻), 1 ⩽ q < ∞. The Bergman projection B_𝔻 on 𝓛^q;X(𝔻) is understood as a continuous extension from this dense subset (see the proof of Theorem 3.2).

Using (3.3) we get the following expression for the X(I) - norm for (B_𝔻f)_n, n ∈ℤ₊ when f ∈ L¹(𝔻): ∥(B_𝔻f)_n∥_X(I) = |ϑ_n(f)| ∥rⁿ∥_X(I), n ∈ℤ₊. This allows us to formulate the following condition on the space X(I) that ensures boundedness of the corresponding Bergman projection as a projection from 𝓛^q;X(𝔻) onto 𝓐^q;X(𝔻): there exists C₀ > 0 such that

Theorem 3.2

Let 1 ⩽ q < ∞, and let condition (3.4) be satisfied. The operator B_𝔻is bounded as a projection from 𝓛^q;X(𝔻) onto 𝓐^q;X(𝔻).

Proof

For f ∈ S0X(𝔻) we obtain

where the constant C₀ comes from condition (3.4) and does not depend on f. Making use of the Banach-Steinghaus theorem we finish the proof. □

Corollary 3.1

Let 1 ⩽ q < ∞, and let the condition (3.4) be satisfied. The space 𝓐^q;X(𝔻) is a closed subspace of 𝓛^q;X(𝔻).

Theorem 5.1

Let Φ be a Young function, Φ ∈ Δ₂ ∩ ∇₂. Then for an analytic in 𝔻 function f(z) = ∑_{n ∈ℤ+}c_nzⁿ ∈ 𝓐^q;Φ(𝔻), z ∈𝔻 the norm ∥f∥_{𝓐^q;Φ(𝔻)}is equivalent to∑n∈Z+|cn|Φ−1(n)q1q.

Theorem 5.2

Under the conditions of Theorem 5.1, each function f ∈𝓐^q;Φ(𝔻) has the formf = 𝓓_Xg with some analytic function g(z) = ∑_{n ∈ℤ+}g_nzⁿ, such that {g_n}_{n ∈ℤ+} ∈ l+q, where 𝓓_Xg(z) = ∑_{n ∈ℤ+} Φ⁻¹(n) g_nzⁿ, z ∈ 𝔻.

Theorem 5.2 follows form Theorem 5.1. The later, in view of Remark 3.1, is the corollary of the following result which will be also crucial for the proof of the boundedness of the Bergman projection in the next section.

Theorem 5.3

Let Φ be a Young function. Then

If, in addition, Φ satisfies the Δ₂doubling condition with the constant C₍₂₎, then

whereγ=C(2)12log2⁡C(2)+1s.If moreover Φ ∈ ∇₂, then

Proof

Due to the convexity of Φ:

so the estimate (5.1) follows by the definition of the norm in L^Φ(I).

Put A=r−s,β=log2⁡C(2),γ=C(2)12log2⁡C(2)+1s. We assume C₍₂₎ > 1. Use the second statement of Lemma 2.2 (s > 0):

Again by the definition of the norm in L^Φ(I) we obtain the estimate (5.2). To prove (5.3) we have to show that γs⩽Φγ2Φ−1(s). Setting Φ⁻¹(s) = t, we get γΦ(t)⩽Φγ2t, which is exactly the ∇₂ condition for Φ. □

We find it convenient for future references to outline the following bilateral estimate which is proved in Theorems 5.3: if a Young function Φ ∈ Δ₂ ∩ ∇₂, then

Remark 5.1

Note that Theorems 5.1, 5.2 are obtained under the condition: Φ ∈Δ₂ ∩∇₂, which eliminates such Young functions which have exponential type growth. This case undoubtedly presents a great interest and we formulate as an open problem: to prove similar results for the mixed norm Bergman-Orlicz type spaces 𝓐^q;Φ(𝔻) when the condition Φ ∈Δ₂ ∩∇₂ is either omitted or at least weakened.

Theorem 5.4

Let Φ be a Young function. Then the condition (3.4) withX(I) = L^Φ(I) is satisfied.

Proof

Due to the Hölder inequality,

Therefore,

The ultimate inequality follows by (2.3). □

As an immediate consequences of Theorem 3.2 we obtain the following results.

Theorem 5.5

Let Φ be a Young function. The operatorB_𝔻is bounded as a projection from 𝓛^q;Φ(𝔻) onto 𝓐^q;Φ(𝔻), 1 ⩽ q < ∞.

Corollary 5.1

Let Φ be a Young function. The space 𝓐^q;Φ(𝔻) is the closed subspace of 𝓛^q;Φ(𝔻), 1 ⩽ q < ∞.

Remark 5.2

Note that Theorem 5.5 holds for an arbitrary Young function, i.e. it admits, in particular, exponential growth of Φ, like Φ(t) = e^t−1. In this case the norms ∥rⁿ∥_L^Φ(I) will be bounded by log-type estimate: 2ln⁡(n+2).

Theorem 6.1

For an analytic in 𝔻 function f(z) = ∑_{n ∈ℤ+}c_n zⁿ, z ∈𝔻 the following statements hold:

1. Let 1 ⩽ p < ∞. Let the function φ be concave on I, the functionφ(t)tbe decreasing on I, lim_{t → 0}φ(t)t = ∞, and the functiontφ(1t)be concave for t > 1. Then the norm ∥f∥_{𝓐^q;p,φ(𝔻)}of f ∈ 𝓐^q;p,φ(𝔻) is equivalent to∑n∈Z+|cn|qnpvarphi1np−qp1q.

2. Let 1 < p < ∞. Let there exists β ∈ (0, p−1) such thatφ(t)tβis decreasing on I. Then the norm ∥f∥_{^∁𝓐^q;p,φ(𝔻)}off ∈ ^∁𝓐^q;p,φ(𝔻) is equivalent to∑n∈Z+|cn|q1nφ1nqp1q.

Theorem 6.2

Under the conditions of Theorem 6.1, each function f ∈𝓐^q;p,φ(𝔻) is represented as f = 𝓓_Xgwith some analytic function g(z) = ∑_{n ∈ℤ+}g_nzⁿ, such that {g_n}_{n ∈ℤ+} ∈ l+q, where 𝓓_Xg(z) = ∑n∈Z+npφ1np1pgnzn,z∈D.Analogously, each functionf ∈^∁𝓐^q;p,φ(𝔻) is represented as f = 𝓓_Xg with some analytic function g(z) = ∑_{n ∈ℤ+}g_n zⁿ, such that {g_n}_{n ∈ℤ+} ∈ l+q, whereDXg(z)=∑n∈Z+1nφ1n−1pgnzn,z∈D.

Theorem 6.2 follows form Theorem 6.1. The latter, in view of Remark 3.1, is the corollary of the estimates obtained below for the norms ∥rⁿ∥_L^p,φ(I)and ∥rⁿ∥_{^∁L^p,φ(I)}, which will be also crucial for the proof of the boundedness of the Bergman projection in the next section.

Theorem 6.3

Let 1 ⩽ p < ∞. If the functionφ(t)tis decreasing onI, lim_{t → 0}φ(t)t = ∞, and the functiontφ(1t)is concave for t > 1, then∥rn∥Lp,φ(I)⩽C1npφ1np−1p,n→∞.If the function φ is concave onI, thenC2npφ1np−1p⩽∥rn∥Lp,φ(I),n→∞.Here C₁, C₂are some positive constants which do not depend onn.

Proof

Apply the Hölder inequality (2.4) with Young function Φ which will be determined below (h>0,r±h∈I):1φ(h)∫r−hr+htnp2tdt⩽2φ(h)∥tnp∥LΦ(I)∥χr,h∥LΨ(I), where χ_r,h is the characteristic function of (r−h,r+h) and Ψ is the conjugate of Φ. Using (2.5), (2.3) and (5.1) we obtain

By assumption, the function tφ(1t) is concave and increasing for t > 1, so for t > 1 we set now Φ⁻¹(t) = tφ(1t), t > 1. We have to take care about a possibility of concave continuation of Φ⁻¹ to the interval I = (0,1) with the condition Φ⁻¹(0) = 0. Note that the behavior of φ when t approaches t = 1 is not of importance for the Morrey space up to the equivalence of norms. Therefore, if needed, we may modify the function φ in a left-sided neighbourhood of t = 1 so that the function Φ⁻¹ will always have a concave continuation for t ⩽ 1. To this end, it suffices to change the function φ so that φ ∈ C¹(1−δ, 1], and φ′(1) > 0.

We claim that 4rhΦ−114rh⩽4hΦ−11h. Indeed, for 4r < 1 we use the fact that φ increases on I, and for 4r > 1 we apply the fact that φ(t)t decreases on I.

Taking into account the made change of the function φ, up to equivalence of norms we have ∥rn∥Lp,φ(I)p⩽Cnpφ1np−1,n∈N.

Now we prove the estimate from below. Again, let Φ be a Young function and Ψ – its conjugate. Denote

Applying (2.3), we have

The function φ is concave and increasing on I, so we set now Φ⁻¹(t) = φ(t), 0 < t < 1. Again, we need to guarantee the existence of a concave continuation of Φ⁻¹ for t ⩾ 1 with the condition Φ⁻¹(∞) = ∞. As in the previous case we free to modify φ in a left sided neighbourhood of t = 1 so that φ ∈ C¹(1−δ, 1], and φ′(1) > 0.

Since 4rh > h on E_n, and 4np+2>1np,n≠0, then

This finishes the proof. □

Theorem 6.4

Let 1 < p < ∞. ThenC11nφ1n1p⩽∥rn∥∁Lp,φ(I),n →∞. If there exists β ∈ (0, p−1) such thatφ(t)tβis decreasing on I, then∥rn∥∁Lp,φ(I)⩽C21nφ1n1p,n→∞.HereC₁, C₂are some positive constants which do not depend on n.

Proof

In is obvious that

Since φ is increasing on I, then sup0<h<1nφ(h)1−hnp+2np+2⩽C11nφ1n, and since there exists β ∈ (0, p−1) such that φ(t)tβ is decreasing on I, then

□

We find it convenient for further references to outline the following bilateral estimates which are proved in Theorems 6.3, 6.4:

1. Let 1 ⩽ p < ∞. Let the function φ be concave on I, the function φ(t)t be decreasing on I, lim_{t → 0}φ(t)t = ∞, and the function tφ(1t) be concave for t > 1. Then C1npφ1np−1p⩽∥rn∥Lp,φ(I)⩽C2npφ1np−1p,n → ∞, where C₁, C₂ are some positive constants which do not depend on n.

2. Let 1 < p < ∞. Let there exists β ∈ (0, p−1) such that φ(t)tβ is decreasing on I. Then C11nφ1n1p⩽∥rn∥∁Lp,φ(I)⩽C21nφ1n1p,n → ∞, where C₁, C₂ are some positive constants which do not depend on n.

Theorem 6.5

Let 1 ⩽ p < ∞. Let the functionφ(t)tbe decreasing onI, lim_{t → 0}φ(t)t = ∞, and the functiontφ(1t)be concave for t > 1. Then the condition (3.4) is valid with X(I) = L^p,φ(I).

Proof

Passing to the dyadic decomposition over the intervals I_k = (1−2^−k,1−2^−k−1), k ∈ℤ₊, we have: ∫Iτng(τ)2τdτ=∑k∈Z+∫Ikτng(τ)2τdτ. Let as usual 1p+1p′=1. Below we will proceed with the case p > 1. Using the Hölder inequality we obtain

For each k ∈ℤ₊ we have I_k = (1−2^−k,1−2^−k−1) = (r_k−h_k, r_k+h_k), where r_k = 1−2^−k−1−2^−k−2, h_k = 2^−k−2. Since φ increases on I, we get

It is convenient to introduce the notation φ_*(t) = φ(t)t, t ∈ I. Direct estimation gives

Hence, ∫Iτng(τ)2τdτ⩽4∥g∥Lp,φ(I)∫I(1−t)nφ1p(t)t−1pdt. Further,