Zygmund inequality of the conjugate function on Morrey-Zygmund spaces

This paper aims to extend the celebrated Zygmund inequality to theMorrey-Zygmund space on the unit circle T = {eiθ : −π < θ 6 π}. The classical Zygmund inequality gives the borderline behavior of the conjugate function operator on L1. It shows that the conjugate function operator is a bounded linear mapping from the Zygmund space L log L to L1 [1]. The importance of the conjugate function operator stems from its role in the study of Fourier series. Roughly speaking, the boundedness of the conjugate function operator on a rearrangement-invariant Banach function space X on T yields the convergence of the Fourier series on a subspace of X. The reader is referred to [2, Chapter 3, Theorem 6.10] for the details and the precise statement of this result. Since the introduction of the classical Morrey spaces on Rn in [3], several important results in Lebesgue spaces have been extended to Morrey spaces. These include results on the boundedness of the HardyLittlewood maximal operator, the singular integral operators, the fractional integral operators [4–8] and the two-weight norm inequalities [9, 10] had been extended to Morrey spaces. Inspired by the recent developments of the studies of Morrey spaces, we investigate the extension of the Zygmund inequality on Morrey spaces. Since we study the conjugate function operator, we consider the Morrey type spaces defined on the unit circle [11]. The main result of this paper establishes the boundedness of the conjugate function operator as a mapping from the Morrey spaces built on Zygmund space to Morrey spaces. The main result of this paper is also related with the results from [12]. The results in [12] consider the Hardy-Littlewood maximal function on the case q = 1, while we consider the Hilbert transform for Morrey spaces on T. This paper is organized as follows. Section 2 contains the definitions and some basic properties of the Zygmund space. The Zygmund inequalities on Morrey-Zygmund spaces are established in Section 3.


Introduction
This paper aims to extend the celebrated Zygmund inequality to the Morrey-Zygmund space on the unit circle T = {e iθ : −π < θ π}. The classical Zygmund inequality gives the borderline behavior of the conjugate function operator on L 1 . It shows that the conjugate function operator is a bounded linear mapping from the Zygmund space L log L to L 1 [1]. The importance of the conjugate function operator stems from its role in the study of Fourier series. Roughly speaking, the boundedness of the conjugate function operator on a rearrangement-invariant Banach function space X on T yields the convergence of the Fourier series on a subspace of X. The reader is referred to [2, Chapter 3, Theorem 6.10] for the details and the precise statement of this result.
Since the introduction of the classical Morrey spaces on R n in [3], several important results in Lebesgue spaces have been extended to Morrey spaces. These include results on the boundedness of the Hardy-Littlewood maximal operator, the singular integral operators, the fractional integral operators [4][5][6][7][8] and the two-weight norm inequalities [9,10] had been extended to Morrey spaces. Inspired by the recent developments of the studies of Morrey spaces, we investigate the extension of the Zygmund inequality on Morrey spaces. Since we study the conjugate function operator, we consider the Morrey type spaces defined on the unit circle [11].
The main result of this paper establishes the boundedness of the conjugate function operator as a mapping from the Morrey spaces built on Zygmund space to Morrey spaces. The main result of this paper is also related with the results from [12]. The results in [12] consider the Hardy-Littlewood maximal function on the case q = 1, while we consider the Hilbert transform for Morrey spaces on T.
This paper is organized as follows. Section 2 contains the definitions and some basic properties of the Zygmund space. The Zygmund inequalities on Morrey-Zygmund spaces are established in Section 3.

Definitions and preliminaries
In this section, we present the definitions and some basic properties of the Zygmund spaces. Let T be the unit circle {e iθ : −π < θ π} endowed with the measure 1 2π dθ, where dθ is the Lebesgue measure on T. We write where log + x = max(log x, 0). We endow the Zygmund space with the norm The associate space of Lexp is L log L and Lexp is also a rearrangement-invariant Banach function space. Furthermore, Lexp is the dual space of L log L (up to equivalence of norms). Therefore, according to the definition of associate spaces, for any f ∈ L log L and g ∈ Lexp We have the following results for the norms of characteristic functions of Lebesgue measurable sets E ⊂ T in L log L and Lexp.

Lemma 2.1. Let E be a Lebesgue measurable set on T. We have
Proof. For any Lebesgue measurable set E on T, we have (χ E ) * = χ [0,|E|) . Therefore, The identity ‖χ E ‖ Lexp = 1 1−log |E| follows from the above result and (2.3). Next, we present the celebrated Zygmund inequality. We first recall the definition of the conjugate function operator from [2, Chapter 3, (6.11)]. The conjugate function operator f →f is defined as the principal value integralf The conjugate function operator can be considered as the periodic analogue of the Hilbert transform on R.
The following is the Zygmund inequality for the conjugate function operator.

Theorem 2.2.
There exists a constant C > 0 such that for any f ∈ L log L, we have For the proof of the above result, the reader is referred to [2, Chapter 4, Corollary 6.8] and [1].

Main result
In this section, we obtain the main result of this paper. Namely, the extension of the Zygmund inequality to Morrey-Zygmund spaces. Let β, t ∈ [−π, π], and write I(β, t) = {e iθ : β − t < θ < β + t} and I = {I(β, t) : β, t ∈ (−π, π]}. Note that T ∈ I. For any j ∈ N and I = I(β, t) ∈ I∖{T}, write 2 j I = I(β, 2 j t). Let N I ∈ N be the smallest positive integer such that 2 N I I = T. We now give the definition of the Morrey-Zygmund space on T.
The above definition is related to the generalized Morrey spaces of the third kind given in [14]. In view of (2.1), we have M u L log L ˓→ M u 1 . When we replace the L 1 norm from the above definition by the L p norm with 1 < p < ∞ and take u(I) = |I| λ p with 0 < λ < 1, we have the Morrey spaces on T studied in [11]. In addition, for the duality theory of Morrey spaces on T, the reader may consult [11].
Since T ∈ I, the above definition yields Therefore,

2)
then for any I ∈ I, χ I ∈ M u L log L .
Proof. It suffices to verify that χ T ∈ M u L log L . Then (3.2) yields Therefore, χ T ∈ M u L log L .
In particular, if ω is continuous and ω(t) > 0 when t > 0, then (3.2) can be relaxed to
Since L log L ⊂ L 1 and (2.1) assures that Mū L log L ⊂ Mū 1 , Mū L log L is a proper nontrivial subset of L log L and Mū 1 is also a proper nontrivial subset of L 1 . The embedding (3.1) also guarantees that the conjugate function operator is well defined on M u L log L . In view of the results in [17][18][19][20][21][22], the action of the singular integral operators on Morrey type spaces on R n cannot directly be defined by the principal value integral. It shows another difference between the Morrey spaces and the Morrey-Zygmund spaces on T and R n .
The reader is referred to [23] for the study of the weak type estimate for maximal commutator and commutator of maximal function on the Morrey-Zygmund spaces defined on R n .
We are now ready to present and establish the main result of this paper, the Zygmund inequalities on Morrey-Zygmund spaces. These inequalities give the boundedness of the conjugate function operator as a mapping from the Morrey-Zygmund space M u L log L to the Morrey space M u 1 .
In view of the above inequalities, we have where we use (2.2) to establish the last inequality.
By applying the norm ‖ · ‖ L 1 and multiplying 1 u(I) on both sides of the above inequality, we obtain 1 u(I) for some C > 0 independent of j and I. As a result of (3.4), (3.6) and (3.7), we have 1 u(I) for some C > 0 independent of I ∈ I. Finally, by taking the supremum over I ∈ I, we establish The result in Theorem 3.2 sharpens the classical Zygmund inequality in the sense that when we consider the function in the subspace M w L log L of L log L, the image of the conjugate function belongs to M u 1 which is a proper subspace of L 1 . The assumption (3.4) is related with [24, (1.3)].