Applications of Littlewood-Paley Theory for 𝐵̇_𝜎-Morrey Spaces to the Boundedness of Integral Operators

Abstract

The boundedness of the various operators on 𝐵̇_𝜎-Morrey spaces is considered in the framework of the Littlewood-Paley decompositions. First, the Littlewood-Paley characterization of 𝐵̇_𝜎-Morrey-Campanato spaces is established. As an application, the boundedness of Riesz potential operators is revisted. Also, a characterization of 𝐵̇_𝜎-Lipschitz spaces is obtained: and, as an application, the boundedness of Riesz potential operators on 𝐵̇_𝜎-Lipschitz spaces is discussed.

1. Introduction

One of the main aims of the present paper is to investigate the structure of the 𝐵_𝜎-spaces 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) by proving Theorem 1 below. To state the main result, we need the following setup. We write 𝑄(𝑥,𝑟)∶=𝑦=𝑦₁,𝑦₂,…,𝑦_𝑛∈ℝ^𝑛∶max_{𝑗=1,…,𝑛}𝑥_𝑖−𝑦_𝑖<𝑟 (1) for 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛 and 𝑟>0. We abbreviate 𝑄(0,𝑟) to 𝑄_𝑟. For a set 𝐸⊂ℝ^𝑛, we denote by 𝜒_𝐸 its indicator function. The space 𝐶^∞_comp(ℝ^𝑛⧵{0}) denotes the set of all compactly supported 𝐶^∞(ℝ^𝑛)-functions supported on ℝ^𝑛⧵{0}. Now we formulate our main result in the simplest form.

Theorem 1. Let 1<𝑝<∞, 𝜎>0 and 𝜆∈[−𝑛/𝑝,−𝜎). Let 𝜑∈𝐶^∞_{𝑐𝑜𝑚𝑝}(ℝ^𝑛⧵{0}) be a function such that ^∞_𝑗=−∞𝜑_𝑗(𝜉)=^∞_𝑗=−∞𝜑2^−𝑗𝜉=𝜒_{ℝ𝑛⧵{0}}(𝜉)𝜉∈ℝ^𝑛. (2) If a measurable function 𝑓 belongs to 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛), that is, 𝑓 satisfies 𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}∶=sup_𝑟>01𝑟^𝜎sup_{𝑥∈ℝ𝑛𝑄(),𝑠>0𝑥,𝑠⊂𝑄𝑟}1(2𝑠)^𝜆1𝑠^𝑛_{𝑄(𝑥,𝑠)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝<∞, (3) then 𝑓∈𝒮^′(ℝ^𝑛) and 𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}∼sup_𝑟>01𝑟^𝜎×sup_{𝑧∈ℝ𝑛𝑄,𝑘∈ℤ𝑧,2−𝑘⊂𝑄𝑟}2^𝑘𝜆×2^𝑘𝑛_{𝑄(𝑧,2−𝑘)}^∞_{𝑗=𝑘−1}ℱ⁻¹𝜑_𝑗∗𝑓𝑦^2𝑝/2𝑑𝑦^1/𝑝, (4) where ℱ⁻¹ denotes the inverse Fourier transform and the implicit constant in ∼ does not depend on 𝑓.

When 𝜎=0, then this is due to Mazzucato [1]. See [2] as well.

The space 𝐵̇_𝜎(𝐿_𝑝,𝜆) appearing in Theorem 1 is a special case of 𝐵_𝜎 spaces. Theory of 𝐵_𝜎 spaces stems from the Beurling work [3]. Beurling introduced the space 𝐵^𝑝(ℝ^𝑛) together with its predual 𝐴^𝑝(ℝ^𝑛) so-called the Beurling algebra [3]. Later, to extend Wiener’s ideas [4, 5] which describe the behavior of functions at infinity, Feichtinger [6] gave an equivalent norm on 𝐵^𝑝(ℝ^𝑛), which is a special case of norms to describe nonhomogeneous Herz spaces 𝐾^𝛼_𝑝,𝑟(ℝ^𝑛) introduced in [7]. On the other hand, Lu and Yang [8, 9] introduced the central bounded mean oscillation space CBMO^𝑝(ℝ^𝑛) with the norm 𝑓_CBMO𝑝=sup_𝑟>01𝑄_𝑟𝑄𝑟𝑓(𝑥)−𝑓_𝑄𝑟^𝑝𝑑𝑥^1/𝑝. (5) Recently, in [10], 𝐵_𝜎-Morrey-Campanato spaces have been introduced to unify central Morrey spaces, central bounded mean oscillation spaces, and usual Morrey-Campanato spaces. Using 𝐵_𝜎-spaces, we can study both local and global regularities of functions simultaneously. For example, when we consider 𝐵̇_𝜎(𝐿_𝑝,𝜆), the underlying norm given by sup_{𝑥∈ℝ𝑛𝑄(),𝑠>0𝑥,𝑠⊂𝑄𝑟}1(2𝑠)^𝜆1𝑠^𝑛_{𝑄(𝑥,𝑠)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝 (6) measures local regularities of functions, and the parameter 𝜎 plays the role of global regularity. Moreover, the proof of Theorem 13 reveals this aspect of our studying both local and global regularities of functions simultaneously.

Theorem 1 concerns the Littlewood-Paley theory. The Littlewood-Paley theory is one of the most powerful tools in harmonic analysis. Roughly speaking, this is a technique of transforming functions into good ones in order to measure the norms. Another side of Littlewood-Paley theory is that the functions are broken into good pieces of functions. We shall illustrate that the Littlewood-Paley theory is very useful by applying this to the fractional integral operator 𝐼_𝛼 of order 𝛼. The fractional integral operator 𝐼_𝛼(0<𝛼<𝑛), which is given by 𝐼_𝛼𝑓(𝑥)∶=_ℝ𝑛𝑓𝑦𝑥−𝑦^𝑛−𝛼𝑑𝑦𝑓∈𝐿¹ℝ^𝑛, (7) plays an important role not only in harmonic analysis but also in partial differential equations. It is well known that 𝐼_𝛼 can be seen as the inverse operator of (−Δ)^𝛼/2 modulo a multiplicative constant, and hence 𝐼_𝛼 has the smoothing effect. However, due to this smoothing effect, 𝐼_𝛼 still seems to have a lot to be investigated.

To describe the related function spaces and formulate Theorem 1 in the full statement, we now fix some more notations. Since cubes play the central role, let us fix our notation of cubes. We denote by 𝒬 the set of all cubes of the form 𝑄(𝑥,𝑟) with 𝑥∈ℝ^𝑛 and 𝑟>0. Also, we denote by 𝒟 the set of dyadic cubes: 𝒟∶=2^−𝜈𝑚+2^−𝜈[0,1)^𝑛∶𝜈∈ℤ,𝑚∈ℤ^𝑛, (8) where to define the right-hand side we used the Minkowski sum. Given 𝑄∈𝒟∪𝒬, we denote by 𝑐(𝑄) the center of a cube 𝑄 and by |𝑄| its volume. The notation ℓ(𝑄) stands for |𝑄|^1/𝑛. Given a set 𝐸⊂ℝ^𝑛, we define 𝒟(𝐸){},𝒬(𝐸){}.∶=𝑄∈𝒟∶𝑄⊂𝐸∶=𝑄∈𝒬∶𝑄⊂𝐸 (9)

With these notations in mind, let us recall the definition of the function space 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) and related function spaces defined in [10]. Here we redefine 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) in terms of dyadic cubes. However, a geometric observation shows that the definition of 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) through dyadic cubes and that through cubes are equivalent.

Definition 2. Let 1≤𝑝≤∞, 𝜎≥0, and −𝑛/𝑝≤𝜆≤0. Define 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) as the set of all 𝑓∈𝐿^𝑝_loc(ℝ^𝑛) such that ‖𝑓‖_{̇𝐵𝜎(𝐿𝑝,𝜆)}<∞, where 𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}∶=sup_𝑟>01𝑟^𝜎sup_{𝑄𝑄∈𝒟𝑟}1ℓ(𝑄)^𝜆1|𝑄|_𝑄𝑓𝑦^𝑝𝑑𝑦^1/𝑝. (10) Define 𝐵̇_𝜎(𝑊𝐿_𝑝,𝜆)(ℝ^𝑛) in analogy with 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛). The norm is given by 𝑓_{̇𝐵𝜎(𝑊𝐿𝑝,𝜆)}∶=sup_𝑟>01𝑟^𝜎sup_{𝑄𝑄∈𝒟𝑟}1ℓ(𝑄)^{𝜆+𝑛/𝑝}sup_𝜆>0𝜆𝑧∈𝑄∶𝑓(𝑥)>𝜆^1/𝑝. (11)

As we remarked above, (3) and (10) are equivalent definitions, and the same can be said for weak type spaces. Namely, 𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}∼sup_𝑟>01𝑟^𝜎sup_{𝑄𝑄∈𝒬𝑟}1ℓ(𝑄)^𝜆1|𝑄|_𝑄𝑓𝑦^𝑝𝑑𝑦^1/𝑝,𝑓_{̇𝐵𝜎(𝑊𝐿𝑝,𝜆)}∼sup_𝑟>01𝑟^𝜎sup_{𝑄𝑄∈𝒬𝑟}1ℓ(𝑄)^{𝜆+𝑛/𝑝}sup_𝜆>0𝜆𝑧∈𝑄∶𝑓(𝑥)>𝜆^1/𝑝 (12) for any measurable function 𝑓 on ℝ^𝑛. In view of (12), we identify the right-hand side and the left-hand side in these formulas. Note that from the definition of norms, we have 𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}∼sup_{𝑥∈ℝ𝑛,𝑟>0}1(|𝑥|+𝑟)^𝜎1𝑟^{𝜆+𝑛/𝑝}_{𝑄(𝑥,𝑟)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝,𝑓_{̇𝐵𝜎(𝑊𝐿𝑝,𝜆)}∼sup_{𝑥∈ℝ𝑛,𝑟>0}1(|𝑥|+𝑟)^𝜎×1𝑟^{𝜆+𝑛/𝑝}sup_𝜆>0𝜆𝑧∈𝑄(𝑥,𝑟)∶𝑓(𝑥)>𝜆^1/𝑝 (13)(14) for any measurable function 𝑓 on ℝ^𝑛.

We need to pay attention to the word “local,” when we discuss 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛). Burenkov and Guliyev, together with their successors, investigated local Morrey-type spaces in [11]. By “local” in [11], they meant that they defined the “local Morrey-type” norm by sup_𝑟>0𝑟^−𝜆1𝑄_𝑟𝑄𝑟𝑓𝑦^𝑝𝑑𝑦^1/𝑝, (15) which indicates that sup appearing in the definition is restricted to all balls centered at the origin. However, by “local” in the present paper we meant that we measure the regularity of functions by (6). Nowadays there are many different definitions related to classical Morrey spaces, so we need to carefully distinguish the different definitions and the names given to the definitions. See [12] for related usage of the word “local” such as central mean oscillation, central BMO, 𝜆-central bounded mean oscillation, and 𝜆-central Morrey spaces.

The goal of the present paper is to show that these function spaces fall under the scope of the Littlewood-Paley theory. As an application of this fact we show the boundedness property of 𝐼_𝛼 and singular integral operators. The Littlewood-Paley theory is a powerful tool to investigate the boundedness property of 𝐼_𝛼. To consider the connection between 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) and the Littlewood-Paley theory, we present definitions. Here and below we use the definition of the Fourier transform below for definiteness: (𝜉)1ℱ𝑓∶=(2𝜋)^{𝑛/2𝐑𝑛}𝑓(𝑥)𝑒^{−𝑖𝑥⋅𝜉}ℱ𝑑𝑥,₋₁𝑓(𝑥)1∶=(2𝜋)^{𝑛/2𝐑𝑛}𝑓(𝜉)𝑒^{𝑖𝑥⋅𝜉}𝑑𝜉. (16) Let 𝜑∈𝐶^∞_comp(ℝ^𝑛⧵{0}) be a function such that ^∞_𝑗=−∞𝜑2^−𝑗𝜉=𝜒_{ℝ𝑛⧵{0}}(𝜉)𝜉∈ℝ^𝑛. (17) Following [13], we define 𝜏_𝑗∶=𝜏2^−𝑗⋅𝑗∈ℤ,𝜏∈𝒮ℝ^𝑛,𝜏(𝐷)𝑓∶=ℱ⁻¹𝜏⋅ℱ𝑓𝜏∈𝒮ℝ^𝑛,𝑓∈𝒮^′ℝ^𝑛. (18) It may be helpful to observe that ℱ⁻¹𝜏_𝑗(𝑥)=2^𝑗𝑛ℱ⁻¹𝜏2^𝑗𝑥𝑥∈ℝ^𝑛,𝑗∈ℤ (19) for each 𝜏∈𝒮(ℝ^𝑛).

In the present paper, the following function space of Littlewood-Paley type will play a key role. Here and below we denote 𝐽_𝑄∶=−log₂ℓ(𝑄) for a cube 𝑄∈𝒟. Observe that 2^𝐽𝑄=ℓ(𝑄)⁻¹(𝑄∈𝒟). (20)

Definition 3. Let 1<𝑝<∞,𝜎>0,−𝑛/𝑝≤𝜆≤0. Given 𝑓∈𝒮^′(ℝ^𝑛), set 𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}∶=sup_𝑟>01𝑟^𝜎sup_{𝑄𝑄∈𝒟𝑟}1ℓ(𝑄)^𝜆×1|𝑄|_𝑄^∞_{𝑗=𝐽𝑄}𝜑_𝑗(𝐷)𝑓𝑦^2𝑝/2𝑑𝑦^1/𝑝. (21) The function space 𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆)(ℝ^𝑛) denotes the set of all tempered distributions 𝑓∈𝒮^′(ℝ^𝑛) for which the quantity ‖𝑓‖_{̇𝐵𝜎(𝐿}^𝐷_𝑝,𝜆) is finite.

Note that Definition 3 is closely related to the space 𝐹̇^𝑠,𝜏_𝑝,𝑞(ℝ^𝑛) defined by Yang and Yuan and the function space 𝐹̇^𝑠,𝜏_𝑝,𝑞(ℝ^𝑛) is investigated in [14–21].

In Theorem 1, we did not mention what happens if 𝑓∈𝒮^′(ℝ^𝑛), and the right-hand side of (4) is finite. Here including this problem, we reformulate and reinforce Theorem 1.

Theorem 4. Let 1<𝑝<∞,𝜎>0,−𝑛/𝑝≤𝜆<−𝜎. Then the spaces 𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆)(ℝ^𝑛) and 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) are isomorphic. More precisely, we have the following.(a)𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛)↪𝒮^′(ℝ^𝑛) in the sense of continuous embedding. (b)𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆)(ℝ^𝑛)↪𝐿^𝑝_loc(ℝ^𝑛) in the sense of continuous embedding. (c)Let 𝑓∈𝒮^′(ℝ^𝑛)∩𝐿^𝑝_loc(ℝ^𝑛). If 𝑓∈𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛), then 𝑓∈𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆)(ℝ^𝑛). Conversely, if 𝑓∈𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆)(ℝ^𝑛), then there exists a polynomial 𝑃 such that 𝑓−𝑃∈𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛), and in this case the norms ‖𝑓−𝑃‖_{̇𝐵𝜎(𝐿𝑝,𝜆)} and ‖𝑓‖_{̇𝐵𝜎(𝐿}^𝐷_𝑝,𝜆) are equivalent. (d)Different choices of 𝜑 satisfying (17) will yield the equivalent norms in the definition of ‖⋅‖_{̇𝐵𝜎(𝐿}^𝐷_𝑝,𝜆).

Instead of Theorem 1 itself, we shall prove Theorem 4 in the present paper.

Let 1<𝑝<∞ and 𝜆≥−𝑛/𝑝. The Morrey space 𝐿_𝑝,𝜆(ℝ^𝑛), which can be realized as 𝐵̇₀(𝐿_𝑝,𝜆)(ℝ^𝑛), admits two different Littlewood-Paley characterizations. In [1], Mazzucato established 𝑓_{𝐿𝑝,𝜆}∼sup_𝑄∈𝒟1ℓ(𝑄)^𝜆1|𝑄|_𝑄^∞_𝑗=−∞𝜑_𝑗(𝐷)𝑓(𝑥)^2𝑝/2𝑑𝑥^1/𝑝. (22) Meanwhile, combining this equivalence with what we proved in [15], we can say that 𝑓_{𝐿𝑝,𝜆}∼sup_𝑄∈𝒟1ℓ(𝑄)^𝜆1|𝑄|_𝑄^∞_{𝑗=𝐽𝑄}𝜑_𝑗(𝐷)𝑓(𝑥)^2𝑝/2𝑑𝑥^1/𝑝. (23) Thus, (23) is closer to Definition 3 than (22). Also, (23) seems to have stemmed from the famous technique due to Uchiyama [22]. We take advantage of equivalence of Theorem 4 in the proof of Theorem 6.

To establish Theorem 4, we will need an auxiliary vector-valued estimate of the Hardy-Littlewood maximal operator 𝑀. Define the Hardy-Littlewood maximal operator 𝑀 by 𝑀𝑓(𝑥)∶=sup_𝑟>01|𝑄(𝑥,𝑟)|_{𝑄(𝑥,𝑟)}𝑓𝑦𝑑𝑦. (24) The following proposition is proven in our earlier paper [12]. This is an extension of [23] to 𝐵̇(𝐿_𝑝,𝜆)(ℝ^𝑛).

Proposition 5. Let 1<𝑝<∞,1<𝑢≤∞,𝜎≥0, and 𝜆<0. Assume in addition that 𝜎+𝜆<0. Then we have ^∞_𝑗=1𝑀𝑓_𝑗^𝑢1/𝑢_{̇𝐵𝜎(𝐿𝑝,𝜆)}≤𝐶^∞_𝑗=1𝑓_𝑗^𝑢1/𝑢_{̇𝐵𝜎(𝐿𝑝,𝜆)} (25) for some 𝐶>0 independent of {𝑓_𝑗}^∞_𝑗=1⊂𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛), where we modify (25) obviously when 𝑢=∞.

Chiarenza and Frasca obtained the boundedness of Hardy-Littlewood maximal operators on global Morrey spaces in [24]. In [11] Burenkov and Guliyev considered local Morrey-type spaces, where they showed that maximal operators are bounded [25–27].

With Theorem 4 and Proposition 5 in mind, we investigate the boundedness property of 𝐼_𝛼 again as we announced in the beginning. More precisely, we shall provide an alternative proof of the following theorems, which were proven earlier in [10, 12].

Theorem 6 (see [10, 12]). Suppose that the parameters 𝛼,𝜎,𝑝,𝜆,𝑞, and 𝜇 satisfy 0<𝛼<𝑛,𝜎≥0,𝜆,𝜇<0,1<𝑝,𝑞<∞. (26) Assume in addition that 𝜇=𝜆+𝛼,𝑞=𝜆𝑝𝜇=𝜆𝑝,𝑛𝜆+𝛼𝜆+𝜎+𝛼≤0,𝜆+𝑝≥0. (27) Then 𝐼_𝛼 is a bounded operator from 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) to 𝐵̇_𝜎(𝐿_𝑞,𝜇)(ℝ^𝑛).

Theorem 7 (see [10, 12]). Suppose that the parameters 𝛼,𝜎,𝜆,𝑞, and 𝜇 satisfy (26) and 𝜆𝜇=𝜆+𝛼,𝑞=𝜇=𝜆,𝜆+𝛼𝜆+𝜎+𝛼<0,𝜆+𝑛≥0 (28) with 𝑝=1. Then 𝐼_𝛼 is a bounded linear operator from 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) to 𝐵̇_𝜎(𝑊𝐿_𝑞,𝜇)(ℝ^𝑛).

We can also consider Campanato spaces and Lipschitz spaces in this framework. First, let 𝒫_𝑑(ℝ^𝑛) be the set of all polynomials having a degree at most 𝑑∈{0,1,2,…}. For a cube 𝑄, a locally integrable function 𝑓 over 𝑄 and a nonnegative integer 𝑑, there exists a unique polynomial 𝑃∈𝒫_𝑑(ℝ^𝑛) such that, for all 𝑞∈𝒫_𝑑(ℝ^𝑛), _𝑄𝑓(𝑥)−𝑃(𝑥)𝑞(𝑥)𝑑𝑥=0. (29) Denote this unique polynomial 𝑃 by 𝑃^𝑑_𝑄𝑓. It follows immediately from the definition that 𝑃^𝑑_𝑄𝑔=𝑔 if 𝑔∈𝒫_𝑑(ℝ^𝑛). We can characterize Λ spaces and Λ̇ spaces (cf. [28]).

Definition 8. Let 1≤𝑝≤∞ and −𝑛/𝑝≤𝜆<𝑑. Then, for 𝑓∈𝐿^𝑝_loc(ℝ^𝑛), define that 𝑓_Λ^(𝑑)_𝑝,𝜆∶=sup_𝑟≥11𝑟^𝜆1𝑄_𝑟𝑄𝑟𝑓𝑦−𝑃^𝑑_𝑄𝑟𝑓𝑦^𝑝𝑑𝑦^1/𝑝,𝑓_̇Λ^(𝑑)_𝑝,𝜆∶=sup_𝑟>01𝑟^𝜆1𝑄_𝑟𝑄𝑟𝑓𝑦−𝑃^𝑑_𝑄𝑟𝑓𝑦^𝑝𝑑𝑦^1/𝑝, (30) where we use the obvious modification when 𝑝=∞. The spaces Λ^(𝑑)_𝑝,𝜆(ℝ^𝑛) and Λ̇^(𝑑)_𝑝,𝜆(ℝ^𝑛) are the sets (modulo 𝒫_𝑑(ℝ^𝑛)) of all 𝑓∈𝐿^𝑝_loc(ℝ^𝑛) for which the quantities ‖𝑓‖_Λ^(𝑑)_𝑝,𝜆 and ‖𝑓‖_̇Λ^(𝑑)_𝑝,𝜆 are finite, respectively.

Note that, in particular, Λ⁽⁰⁾_𝑝,𝜆ℝ^𝑛=CMO^𝑝,𝜆ℝ^𝑛,Λ̇⁽⁰⁾_𝑝,𝜆ℝ^𝑛=CBMO^𝑝,𝜆ℝ^𝑛 (31) with norm coincidence. For the definition of CMO^𝑝,𝜆(ℝ^𝑛) and CBMO^𝑝,𝜆(ℝ^𝑛), we refer to [12].

Definition 9. For 𝜎≥0, 1≤𝑝≤∞, 𝜆≥−𝑛/𝑝 and 𝑑∈{0,1,2,…} with 𝑑≥𝜆, let 𝐵̇_𝜎(ℒ^(𝑑)_𝑝,𝜆)(ℝ^𝑛) be the set of all 𝑓∈𝐿^𝑝_loc(ℝ^𝑛) such that ‖𝑓‖_{̇𝐵𝜎(ℒ}^(𝑑)_𝑝,𝜆)<∞, where 𝑓_{̇𝐵𝜎(ℒ}^(𝑑)_𝑝,𝜆)∶=sup_𝑟>01𝑟^𝜎×sup_{𝑄𝑄∈𝒬𝑟}1ℓ(𝑄)^𝜆×1|𝑄|_𝑄𝑓𝑦−𝑃^𝑑_𝑄𝑓𝑦^𝑝𝑑𝑦^1/𝑝, (32) where we use the obvious modification when 𝑝=∞.

Observe that the quotient space 𝐵̇_𝜎(ℒ^(𝑑)_𝑝,𝜆)(ℝ^𝑛)/𝒫_𝑑(ℝ^𝑛) is a Banach space equipped with the norm ‖𝑓‖_{̇𝐵𝜎(ℒ}^(𝑑)_𝑝,𝜆).

Proposition 10. Let 1≤𝑝≤∞, 𝜎≥0, 𝜆<0. Assume that −𝑛/𝑝≤𝜆 and that 𝜎+𝜆<0. Then 𝐵̇_𝜎ℒ⁽⁰⁾_𝑝,𝜆ℝ^𝑛/𝒫₀ℝ^𝑛≈𝐵̇_𝜎𝐿_𝑝,𝜆ℝ^𝑛, (33) with equivalent norms.

Now we define a function space by way of difference. For ℎ∈ℝ^𝑛, an integer 𝐿∈ℕ and a function 𝑓∶ℝ^𝑛→ℂ, we define Δ⁰_ℎ𝑓∶=𝑓,Δ^𝐿_ℎ𝑓∶=Δ_ℎ^𝐿−1𝑓(⋅+ℎ)−Δ_ℎ^𝐿−1𝑓 (34) inductively.

Definition 11. For 𝜎≥0, 𝜃>0 and 𝐿∈{0,1,2,…} with 𝐿>𝜃, let 𝐵̇_𝜎(Lip_𝜃^(𝐿))(ℝ^𝑛) be the set of all continuous functions 𝑓 such that ‖𝑓‖_{̇𝐵𝜎(Lip𝜃}^(𝐿)₎<∞, where 𝑓_{̇𝐵𝜎(Lip𝜃}^(𝐿)₎∶=sup_𝑟>01𝑟^𝜎×sup_{𝑄𝑄∈𝒬𝑟}sup_𝑥∈𝑄sup_{𝑦(𝑄)≤ℓ}1ℓ(𝑄)^𝜃Δ^𝐿_𝑦𝑓(𝑥). (35)
Then 𝐵̇_𝜎(Lip_𝜃^(𝐿))(ℝ^𝑛) is a Banach space equipped with the norm 𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝜃}^(𝐿)₎+𝑓(0), (36) and also the quotient space 𝐵̇_𝜎(Lip_𝜃^(𝐿))(ℝ^𝑛)/𝒫_𝐿−1(ℝ^𝑛) is a Banach space equipped with the norm ‖𝑓‖_{̇𝐵𝜎(Lip𝜃}^(𝐿)₎.

Definition 12. For 𝜎≥0 and 𝜃>0 let 𝐵̇_𝜎(Lip^𝐷_𝜃)(ℝ^𝑛) be the set of all 𝑓∈𝒮^′(ℝ^𝑛) such that ‖𝑓‖_{̇𝐵𝜎(Lip}^𝐷_𝜃)<∞, where 𝑓_{̇𝐵𝜎(Lip𝐷𝜃)}∶=sup_𝑟>01𝑟^𝜎sup_{𝑄𝑄∈𝒟𝑟}sup_𝑥∈𝑄2^𝐽𝑄𝜃𝜑_𝐽𝑄(𝐷)𝑓(𝑥). (37)
In the present paper, we aim to show the following equivalence as well. Here, for 𝑡∈ℝ, we let [𝑡] the largest integer 𝑚 such that 𝑚≤𝑡.

Theorem 13. Let 𝜎≥0 and 𝜃>0. Assume that the integer 𝑑 satisfies 𝑑>[𝜎+𝜃]. (1)If 𝑓∶ℝ^𝑛→ℂ is a continuous function such that the quantity ‖𝑓‖_{̇𝐵𝜎(𝐿𝑖𝑝𝜃}^(𝑑+1)₎ is finite, then 𝑓∈𝒮^′(ℝ^𝑛) and the inequality ‖𝑓‖_{̇𝐵𝜎(𝐿𝑖𝑝}^𝐷_𝜃)≲‖𝑓‖_{̇𝐵𝜎(𝐿𝑖𝑝𝜃}^(𝑑+1)₎ holds. (2)If 𝑓∈𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝜃)(ℝ^𝑛), then 𝑓 can be represented by a continuous function and the inequality ‖𝑓‖_{̇𝐵𝜎(𝐿𝑖𝑝𝜃}^(𝑑+1)₎≲‖𝑓‖_{̇𝐵𝜎(𝐿𝑖𝑝}^𝐷_𝜃) holds.

Remark 14. It is absolutely necessary to assume that 𝑓 is a continuous function in Theorem 13, when 𝑑≥1. We remark that there exists a discontinuous function 𝑓 such that Δ_ℎ^𝑑+1𝑓(𝑥)=0 for all 𝑥,ℎ∈ℝ^𝑛. See [29, Proposition A1] for such an example constructed algebraically.

Now we explain notations and we describe its organization of the present paper. We use the following notations for the inequalities. First, we use standard notation for inequalities. For example, in the present paper a chain of inequalities of the form 𝐴₁≲𝐴₂≤𝐴₃∼𝐴₄ (38) appears in (110) below. The inequality (38) means that there exist 𝐶₁,𝐶₂,𝐶₃>0 such that 𝐴₁≤𝐶₁𝐴₂,𝐴₂≤𝐴₃,𝐴₃≤𝐶₂𝐴₄,𝐴₄≤𝐶₃𝐴₃. (39) If the implicit constants in ≲ or ∼ do depend on some important parameters 𝛼,𝛽,…, then we write ≲_{𝛼,𝛽,…} or ∼_{𝛼,𝛽,…}. We shall prove Theorem 4 in Section 3. We prove Theorem 6 in Section 4. Theorem 7 is covered in Section 5. We prove Theorem 13 in Section 6. Finally in Section 7 we present another application of Theorem 4 by showing that the Fourier multiplier is bounded on 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛).

2. Preliminaries

In the present paper, we frequently use the following fundamental inequalities.

Lemma 15 (see [30, page 466]). Let 𝜈,𝜇∈ℤ,𝑀,𝑁>0, and 𝐿∈ℕ∪{0} satisfy 𝜈≥𝜇,𝑁>𝑀+𝐿+𝑛. (40) Suppose that 𝜙∈𝐶^𝐿(ℝ^𝑛) such that ∇^𝐿𝜙(𝑥≤2)^{𝜇(𝑛+𝐿)}1+2^𝜇𝑥−𝑥_𝜙^𝑀 (41) for some 𝑥_𝜙∈ℝ^𝑛. Assume in addition that 𝜓 is a measurable function such that _ℝ𝑛𝑥^𝛽𝜓(𝑥)𝑑𝑥=0,𝛽≤𝐿−1 (42) and that 𝜓(𝑥≤2)^𝜈𝑛1+2^𝜈𝑥−𝑥_𝜓^𝑁 (43) for some 𝑥_𝜓∈ℝ^𝑛. Then we have _ℝ𝑛𝜙(𝑥)𝜓(𝑥)≲2𝑑𝑥^{𝜇𝑛−(𝜈−𝜇)𝐿}1+2^𝜇𝑥_𝜙−𝑥_𝜓^𝑀. (44)

To formulate the next lemma, we recall the definition of 𝐴_𝑝 with 1≤𝑝<∞. A measurable function 𝑤, which takes values in (0,∞) almost everywhere, is said to be an 𝐴_𝑝 weight or belongs to the class 𝐴_𝑝, if 𝐴_𝑝(𝑤)∶=sup_𝑄∈𝒬1|𝑄|_𝑄𝑤(𝑥)×1𝑑𝑥|𝑄|_𝑄𝑤(𝑥)^{−1/(𝑝−1)}𝑑𝑥^𝑝−1,𝐴<∞1<𝑝<∞₁(𝑤)∶=𝑤⁻¹𝑀𝑤_𝐿∞<∞𝑝=1. (45) For all 𝑝>1, it is easy to see that 𝐴₁⊂𝐴_𝑝 and that 𝐴_𝑝(𝑤)≤𝐴₁(𝑤).

Lemma 16 (see [31]). Let 1<𝑝<∞ and 𝜑 satisfy (17). Then, for 𝑤∈𝐴_𝑝, we have 𝑤^1/𝑝𝑓_𝐿𝑝∼_{𝑝,𝐴𝑝(𝑤)}𝑤^1/𝑝∞_𝑗=−∞𝜑_𝑗(𝐷)𝑓^21/2_𝐿𝑝 (46) for all 𝑓∈𝐿^𝑝_loc(𝑤)∩𝒮^′(ℝ^𝑛).

We also need a piece of information on dilation of the space 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛).

Lemma 17. Let 1≤𝑝<∞, 𝜆≥−𝑛/𝑝 and 𝜎≥0. Let 𝑡>0 and 𝑓∈𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛). Then 𝑓(𝑡⋅)_{̇𝐵𝜎(𝐿𝑝,𝜆)}=𝑡^𝜎+𝜆𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}. (47)

It is just a matter of handling the left-hand side carefully. But, for the sake of convenience, we supply the proof.

Proof. From the definition of ‖𝑓(𝑡⋅)‖_{̇𝐵𝜎(𝐿𝑝,𝜆)}, we deduce 𝑓(𝑡⋅)_{̇𝐵𝜎(𝐿𝑝,𝜆)}=sup_𝑟>01𝑟^𝜎sup_{()𝑄𝑥,𝑅∶𝑄𝑥,𝑅∈𝒟𝑟}1(2𝑅)^𝜆×1|𝑄(𝑥,𝑅)|_{𝑄(𝑥,𝑅)}𝑓𝑡𝑦^𝑝𝑑𝑦^1/𝑝=sup_𝑟>01𝑟^𝜎sup_{()𝑄𝑥,𝑅∶𝑄𝑥,𝑅∈𝒟𝑟}1(2𝑅)^𝜆×1|𝑄(𝑡𝑥,𝑡𝑅)|_{𝑄(𝑡𝑥,𝑡𝑅)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝=sup_𝑟>01𝑟^𝜎sup_{()𝑄𝑥,𝑅∶𝑄𝑡𝑥,𝑡𝑅∈𝒟𝑡𝑟}1(2𝑅)^𝜆×1|𝑄(𝑡𝑥,𝑡𝑅)|_{𝑄(𝑡𝑥,𝑡𝑅)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝=𝑡^𝜎+𝜆sup_𝑟>01(𝑡𝑟)^𝜎sup_{()𝑄𝑥,𝑅∶𝑄𝑡𝑥,𝑡𝑅∈𝒟𝑡𝑟}1(2𝑡𝑅)^𝜆×1|𝑄(𝑡𝑥,𝑡𝑅)|_{𝑄(𝑡𝑥,𝑡𝑅)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝=𝑡^𝜎+𝜆𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}, (48) proving the lemma.

In the course of the proof of Theorem 6, we need another piece of information on the space 𝐵_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛). Let 1≤𝑝≤∞, 𝜎≥0 and −𝑛/𝑝≤𝜆≤0. Define 𝐵_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) as the set of all 𝑓∈𝐿^𝑝_loc(ℝ^𝑛) such that ‖𝑓‖_{𝐵𝜎(𝐿𝑝,𝜆)}<∞, where 𝑓_{𝐵𝜎(𝐿𝑝,𝜆)}∶=sup_𝑟≥11𝑟^𝜎sup_{𝑄𝑄∈𝒬𝑟}1ℓ(𝑄)^𝜆1|𝑄|_𝑄𝑓𝑦^𝑝𝑑𝑦^1/𝑝. (49)

The next lemma concerns the norm of the translation operator.

Lemma 18. Let 1≤𝑝<∞, 𝜆≥−𝑛/𝑝 and 𝜎≥0. Then 𝑓⋅−𝑥_{0𝐵𝜎(𝐿𝑝,𝜆)}≲1+𝑥₀^𝜎𝑓_{𝐵𝜎(𝐿𝑝,𝜆)}. (50)

Proof. Let 𝑟≥1 and 𝑄=𝑄(𝑧,𝑠)∈𝒬(𝑄_𝑟) be fixed and consider 1𝐼∶=𝑟^𝜎1(2𝑠)^𝜆1|𝑄(𝑧,𝑠)|_{𝑄(𝑧,𝑠)}𝑓𝑦−𝑥₀^𝑝𝑑𝑦^1/𝑝. (51) Then we have 1𝐼=𝑟^𝜎1(2𝑠)^𝜆1𝑄𝑧−𝑥₀,𝑠_{𝑄(𝑧−𝑥0,𝑠)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝 (52) and 𝑄(𝑧−𝑥₀,𝑠)∈𝒬(𝑄_𝑟+|𝑥0|). Note that 𝑟^𝜎1+𝑥₀^𝜎≥𝑟+𝑥₀^𝜎 (53) since 𝑟≥1. This implies that I≤1+𝑥₀^𝜎1𝑟+𝑥₀^𝜎×1(2𝑠)^𝜆1𝑄𝑧−𝑥₀,𝑠_{𝑄(𝑧−𝑥0,𝑠)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝. (54) If we consider the supremum over 𝑟,𝑠>0 and 𝑧∈ℝ^𝑛 such that 𝑄(𝑧,𝑠)⊂𝑄_𝑟, then we obtain the desired result.

Lemma 19. Let 𝜑∈𝒮(ℝ^𝑛). Let 𝜎≥0, 1≤𝑝<∞ and 𝜆∈(−𝑛/𝑝,−𝜎). Then 𝜑(𝐷)𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}≲𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)} (55) for all 𝑓∈𝐵̇_𝜎(𝐿_𝑝,𝜆).

Proof. We have 𝑓⋅−𝑦_{𝐵𝜎(𝐿𝑝,𝜆)}≲1+𝑦^𝜎𝑓_{𝐵𝜎(𝐿𝑝,𝜆)}𝑦∈ℝ^𝑛 (56) from Lemma 18. Hence, by the triangle inequality, we obtain that 𝜑(𝐷)𝑓_{𝐵𝜎(𝐿𝑝,𝜆)}=𝑐_ℝ𝑛ℱ⁻¹𝜑𝑦𝑓⋅−𝑦𝑑𝑦_{𝐵𝜎(𝐿𝑝,𝜆)}≲_ℝ𝑛ℱ⁻¹𝜑𝑦⋅𝑓⋅−𝑦_{𝐵𝜎(𝐿𝑝,𝜆)}≲𝑑𝑦_ℝ𝑛ℱ⁻¹𝜑𝑦1+𝑦^𝜎⋅𝑓_{𝐵𝜎(𝐿𝑝,𝜆)}∼𝑓𝑑𝑦_{𝐵𝜎(𝐿𝑝,𝜆)}. (57) Meanwhile, since ∫_𝑄𝑟|𝑓(𝑥)|𝑑𝑥≤𝑟^{𝜎+𝜆+𝑛/𝑝}‖𝑓‖_{̇𝐵𝜎(𝐿𝑝,𝜆)}, for 𝑟∈(0,∞), we have, on 𝑄₁, 𝜑(𝐷)𝑓(𝑥)≲_ℝ𝑛𝑓𝑦1+𝑥−𝑦^{−𝜎−𝜆−(𝑛/𝑝)−1}≲𝑑𝑦_ℝ𝑛𝑓𝑦1+𝑦^{−𝜎−𝜆−(𝑛/𝑝)−1}≲𝑑𝑦_∞𝑘=02^{−(𝜎+𝜆+(𝑛/𝑝)+1)𝑘}_𝑄2𝑘𝑓𝑦≲𝑑𝑦_∞𝑘=02^{−(𝜎+𝜆+(𝑛/𝑝)+1)𝑘}2^{(𝜎+𝜆+(𝑛/𝑝))𝑘}𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}∼𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}. (58) Assuming that 𝜎+𝜆<0, we have 𝜑(𝐷)𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}≤𝜑(𝐷)𝑓_{𝐵𝜎(𝐿𝑝,𝜆)}+𝑓_{𝐿∞(𝑄1)}. (59) Putting (57)–(59) together, we obtain the desired result.

Lemma 20. Suppose that the parameters 𝜎,𝑝, and 𝜆 satisfy 𝜎≥0,𝜆<0,1≤𝑝<∞. (60) Let 𝜑∈𝒮(ℝ^𝑛). Then 𝜑_𝑘(𝐷)𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}=𝜑2^−𝑘𝐷𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}≲𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}. (61)

Proof. Note that 𝜑_𝑘(𝐷)𝑓(𝑥)=𝜑(𝐷)𝑓^†,𝑘(2^𝑘𝑥) for 𝑓^†,𝑘=𝑓(2^𝑘⋅)∈𝒮^′(ℝ^𝑛). Recall that 𝐵̇_𝜎(𝐿_𝑝,𝜆) has a scaling invariance, as we have verified in Lemma 17. So, to prove (61), we can assume that 𝑘=0. In this case, (61) is Lemma 19 itself.

We need the following sequence of functions.

Lemma 21. There exists a sequence of functions {𝑟_𝑗}^∞_𝑗=−∞⊂𝐿²([0,1]) such that, for any fixed constant 𝑝∈(0,∞), ¹₀^∞_𝑗=−∞𝑎_𝑗𝑟_𝑗(𝑡)^𝑝𝑑𝑡^1/𝑝∼^∞_𝑗=−∞𝑎_𝑗^21/2 (62) provided {𝑎_𝑗}^∞_𝑗=−∞∈ℓ²(ℤ).

In the present paper the sequence {𝑟_𝑗}^∞_𝑗=−∞ above is called a Rademacher sequence.

3. Littlewood-Paley Characterization of 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛)

3.1. Proof of Proposition 10

It follows from the definition of the norm ‖𝑓‖_{̇𝐵𝜎(𝐿𝑝,𝜆)} that 𝑓+𝐶_{̇𝐵𝜎(ℒ}⁽⁰⁾_𝑝,𝜆)≲𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)} (63) for any constant function 𝐶. Note that this implies that 𝐵̇_𝜎(𝐿_𝑝,𝜆) is a subset of 𝐵̇_𝜎(ℒ⁽⁰⁾_𝑝,𝜆).

Let 𝑓∈𝐵̇_𝜎(ℒ⁽⁰⁾_𝑝,𝜆)(ℝ^𝑛). Then, since 𝑝>1, we can use the Hölder inequality and we have 𝑃⁰_𝑄𝑠𝑓(0)−𝑃⁰_𝑄𝑟𝑓(0≤1)𝑄_𝑠𝑄𝑠𝑓(𝑥)−𝑃⁰_𝑄𝑟𝑓(0)𝑑𝑥≲𝑟^𝜎+𝜆𝑓_{̇𝐵𝜎(ℒ}⁽⁰⁾_{𝑝,𝜆)(ℝ𝑛)} (64) for 𝑟≤𝑠≤2𝑟. Assuming that 𝜎+𝜆<0, the sequence {𝑃⁰_𝑄𝑟𝑓(0)}_𝑟>0 is convergent by the Cauchy test. Thus, we can consider the mapping 𝜄∶𝑓∈𝐵̇_𝜎ℒ⁽⁰⁾_𝑝,𝜆ℝ^𝑛↦𝑓−lim_𝑅→∞𝑃⁰_𝑄𝑅𝑓(0). (65) Let us check that the range of 𝜄 is in 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛).

Let 𝑟>0 and 𝑄⊂𝑄_𝑟 be fixed. Let 𝑁 be the largest integer such that 2^𝑁𝑄⊂𝑄_2𝑟. Then observe that 2^𝑁+2𝑄⊃𝑄_𝑟. We decompose 𝑓−lim_𝑅→∞𝑃⁰_𝑄𝑅𝑓(0)=𝑓−𝑃⁰_𝑄𝑓(0)+𝑃⁰_2𝑁𝑄𝑓(0)−𝑃⁰_𝑄𝑟𝑓(0)+^∞_𝑖=1𝑃⁰_{2𝑖−1𝑄𝑟}𝑓(0)−𝑃⁰_{2𝑖𝑄𝑟}𝑓(0)−^𝑁_𝑙=1𝑃⁰_2𝑙𝑄𝑓(0)−𝑃⁰_{2𝑙−1𝑄}𝑓(0). (66) We consider the 𝐿^𝑝-norm over 𝑄 and multiply 1/(𝑟^𝜎ℓ(𝑄)^{𝜆+𝑛/𝑝}).

As for the first term, directly from the definition of ‖𝑓‖_ℒ⁽⁰⁾_𝑝,𝜆, we obtain that 1𝑟^𝜎ℓ(𝑄)^{𝜆+𝑛/𝑝𝑄}𝑓(𝑥)−𝑃⁰_𝑄𝑓(0)𝑑𝑥≤𝑓_ℒ⁽⁰⁾_𝑝,𝜆. (67) Also, a geometric observation shows that the second term can be estimated similarly. Since 𝜆<0, we obtain that 1𝑟^𝜎ℓ(𝑄)^𝜆𝑃⁰_2𝑁𝑄𝑓(0)−𝑃⁰_𝑄𝑟𝑓(0≤1)𝑟^𝜎2^𝑁ℓ(𝑄)^𝜆𝑃⁰_2𝑁𝑄𝑓(0)−𝑃⁰_𝑄𝑟𝑓(0≲𝑓_{ℒ(0)𝑝,𝜆}). (68) By using (64) and 𝜆<0, we can handle the third term: 1𝑟^𝜎ℓ(𝑄)^𝜆∞_𝑖=1𝑃⁰_{2𝑖−1𝑄𝑟}𝑓(0)−𝑃⁰_{2𝑖𝑄𝑟}𝑓(0)≤1𝑟^𝜎ℓ(𝑄)^𝜆∞_𝑖=1𝑃⁰_{2𝑖−1𝑄𝑟}𝑓(0)−𝑃⁰_{2𝑖𝑄𝑟}𝑓(0≤1)𝑟^𝜎2^𝑁ℓ(𝑄)^𝜆∞_𝑖=1𝑃⁰_{2𝑖−1𝑄𝑟}𝑓(0)−𝑃⁰_{2𝑖𝑄𝑟}𝑓(0). (69) In view of the way in which we chose 𝑁, we obtain that 1𝑟^𝜎ℓ(𝑄)^𝜆∞_𝑖=1𝑃⁰_{2𝑖−1𝑄𝑟}𝑓(0)−𝑃⁰_{2𝑖𝑄𝑟}𝑓(0)≲1𝑟^∞𝜎+𝜆_𝑖=1𝑃⁰_{2𝑖−1𝑄𝑟}𝑓(0)−𝑃⁰_{2𝑖𝑄𝑟}𝑓(0≲𝑓_{ℒ(0)𝑝,𝜆}). (70) In summary, 1𝑟^𝜎ℓ(𝑄)^𝜆∞_𝑖=1𝑃⁰_{2𝑖−1𝑄𝑟}𝑓(0)−𝑃⁰_{2𝑖𝑄𝑟}𝑓(0)≲𝑓_ℒ⁽⁰⁾_𝑝,𝜆. (71) It remains to estimate the fourth term. We employ the following estimate: 𝑃⁰_2𝑙𝑄𝑓(0)−𝑃⁰_{2𝑙−1𝑄}𝑓(0)≲𝑟^𝜎ℓ2^𝑙𝑄^𝜆𝑓_ℒ⁽⁰⁾_𝑝,𝜆. (72) Since 𝜆<0, (72) is summable and we obtain that 1𝑟^𝜎ℓ(𝑄)^𝜆𝑁_𝑙=1𝑃⁰_2𝑙𝑄𝑓(0)−𝑃⁰_{2𝑙−1𝑄}𝑓(0)≲𝑓_ℒ⁽⁰⁾_𝑝,𝜆, (73) where the implicit constant in (73) is independent of 𝑁.

It follows from (67), (68), (71), and (73) that 𝜄 sends 𝐵̇_𝜎(ℒ⁽⁰⁾_𝑝,𝜆)(ℝ^𝑛) to 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) boundedly; 𝜄𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}≲𝑓_{̇𝐵𝜎(ℒ}⁽⁰⁾_𝑝,𝜆) (74) for all 𝑓∈𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛).

Meanwhile, it follows from the norm ‖⋅‖_{̇𝐵𝜎(𝐿𝑝,𝜆)} that lim_𝑅→∞𝑃⁰_𝑄𝑅𝑓(0)=0 (75) for all 𝑓∈𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛). In view of (63), 𝜄 is a surjection: 𝜄𝐵̇_𝜎ℒ⁽⁰⁾_𝑝,𝜆ℝ^𝑛=𝐵̇_𝜎𝐿_𝑝,𝜆ℝ^𝑛. (76)

Finally observe that, for 𝑓∈𝐵̇_𝜎(ℒ⁽⁰⁾_𝑝,𝜆)(ℝ^𝑛), 𝜄(𝑓)=0 if and only if 𝑓 is a constant function, that is, 𝑓∈𝒫₀(ℝ^𝑛). Namely, Ker(𝜄)=𝒫₀ℝ^𝑛. (77) Thus, from (75), (76) and (77), we conclude (33).

3.2. Proof of Theorem 4 Part (a)

For 1<𝑝<∞ and 𝜎∈ℝ, let us define the function space 𝑢-𝐿^𝑝_loc(⟨⋅⟩^𝜎)(ℝ^𝑛) of uniformly 𝐿^𝑝_loc(⟨⋅⟩^𝜎)(ℝ^𝑛) functions by 𝑢-𝐿^𝑝_loc⟨⋅⟩^𝜎ℝ^𝑛=𝑓∈𝐿^𝑝_locℝ^𝑛∶𝑓_𝑢-𝐿^𝑝_{loc(⟨⋅⟩}^𝜎₎<∞, (78) where, for 𝑚∈ℤ^𝑛, we write ⟨𝑚⟩∶=1+|𝑚|², and the norm is given by 𝑓_𝑢-𝐿^𝑝_{loc(⟨⋅⟩}^𝜎₎=sup_{𝑚∈ℤ𝑛}⟨𝑚⟩^𝜎𝑝_{𝑚+[0,1]𝑛}𝑓𝑦^𝑝𝑑𝑦^1/𝑝. (79) Then from the definition of the norms (10) and (78), the following chain of continuous inclusion holds. For −𝑛/𝑝≤𝜆, 𝐵̇_𝜎𝐿_𝑝,𝜆ℝ^𝑛↪𝑢-𝐿^𝑝_loc⟨⋅⟩^−𝜎ℝ^𝑛↪𝒮^′ℝ^𝑛. (80) Thus, (a) follows.

Remark 22. The space 𝑢-𝐿^𝑝_loc(⟨⋅⟩^−𝜎)(ℝ^𝑛) is a special case of amalgams investigated in [32].

3.3. Proof of Theorem 4 Part (b)

Let 𝜑∈𝒮(ℝ^𝑛) be a fixed function satisfying (17). It follows from (17) that there exist 𝐽₋ and 𝐽₊ such that ^𝐽+_𝑗=𝐽−𝜑(𝜉)𝜑2^−𝑗𝜉=𝜑(𝜉). (81) Let 𝑓∈𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆)(ℝ^𝑛) and fix a dyadic cube 𝑄_∗ such that 0∈𝑄_∗. We are going to show that 𝑔₁∶=lim^𝐽_𝐿→∞^𝑄∗_𝑗=−𝐿𝜑_𝑗(𝐷)𝑓∈𝐿^∞𝑄_∗∩𝒮^′ℝ^𝑛 (82) converges in the topology of 𝐿^∞(𝑄_∗)∩𝒮^′(ℝ^𝑛), and that 𝑔₂∶=lim^𝐿_{𝐿→∞𝑗=𝐽𝑄∗}𝜑_𝑗(𝐷)𝑓∈𝐿^𝑝𝑄_∗∩𝒮^′ℝ^𝑛 (83) converges in the topology of 𝐿^𝑝(𝑄_∗)∩𝒮^′(ℝ^𝑛).

The presice meaning of (82) and (83) is that as follows: (82) means 𝑔₁−^𝐽𝑄∗_𝑗=−𝐿𝜑_𝑗(𝐷)𝑓_{𝐿∞(𝑄∗)}→0,_{𝐽𝑄∗𝑗=−𝐿}𝜑_𝑗(𝐷)𝑓,𝜏→𝑔₁,𝜏𝜏∈𝒮ℝ^𝑛 (84) as 𝐿→∞ and (83) means that 𝑔₂−^𝐿−1_{𝑗=𝐽𝑄∗}𝜑_𝑗(𝐷)𝑓_{𝐿𝑝(𝑄∗)}→0_{𝐿−1𝑗=𝐽𝑄∗}𝜑_𝑗(𝐷)𝑓,𝜏→𝑔₂,𝜏𝜏∈𝒮ℝ^𝑛 (85) as 𝐿→∞. Once (82) and (83) are proven, we will have 𝑔∶=^∞_𝑗=−∞𝜑_𝑗(𝐷)𝑓∈𝐿^𝑝𝑄_∗∩𝒮^′ℝ^𝑛 (86) converges in the topology of 𝐿^𝑝(𝑄_∗)∩𝒮^′(ℝ^𝑛) and that 𝑓−𝑔 is a polynomial. Hence it follows that 𝑓∈𝐿^𝑝_loc(ℝ^𝑛). Remark also that the convergence in 𝒮^′(ℝ^𝑛) of the sum defining 𝑔₂ is a generality. So let us prove (82) and (83).

Let us begin with proving (82). To this end we take a dyadic cube 𝑄 containing 𝑄_∗. Since we are assuming (81), we deduce that 𝜑_𝐽𝑄(𝐷)𝑓(𝑥≲_{𝐽+𝑙=𝐽−})ℱ⁻¹𝜑_{𝑙+𝐽𝑄}∗𝜑_𝐽𝑄(𝐷)𝑓(𝑥≤_{𝐽+𝑙=𝐽−ℝ𝑛})ℱ⁻¹𝜑_{𝑙+𝐽𝑄}𝑥−𝑦𝜑_𝐽𝑄(𝐷)𝑓𝑦𝑑𝑦𝑥∈ℝ^𝑛. (87) Since ℱ⁻¹𝜑∈𝒮(ℝ^𝑛), we have 𝜑_𝐽𝑄(𝐷)𝑓(𝑥≲1)|𝑄|_ℝ^𝑛𝜑_𝐽𝑄(𝐷)𝑓𝑦1+ℓ(𝑄)⁻¹𝑦^{2𝑛+2𝜎+2}𝑑𝑦(𝑥∈𝑄) (88) from (19) and (20). By the Hölder inequality and the fact that 𝜎≥0, we have 𝜑_𝐽𝑄(𝐷)𝑓(𝑥≲1)|𝑄|_ℝ^𝑛𝑑𝑦1+ℓ(𝑄)⁻¹𝑦^{𝑝′(𝑛+𝜎+1)1/𝑝′}×_ℝ𝑛𝜑_𝐽𝑄(𝐷)𝑓𝑦^𝑝𝑑𝑦1+ℓ(𝑄)⁻¹𝑦^{𝑝(𝑛+𝜎+1)1/𝑝}∼1|𝑄|_ℝ^𝑛𝜑_𝐽𝑄(𝐷)𝑓𝑦^𝑝1+ℓ(𝑄)⁻¹𝑦^{𝑝(𝑛+𝜎+1)}𝑑𝑦^1/𝑝. (89) for all 𝑥∈𝑄. By decomposing the last integral dyadically, we obtain that 𝜑_𝐽𝑄(𝐷)𝑓(𝑥≲_∞𝑙=0)2^{−𝑙𝑝(𝑛+𝜎+1)}1|𝑄|₂^𝑙_𝑄𝜑_𝐽𝑄(𝐷)𝑓𝑦^𝑝𝑑𝑦^1/𝑝. (90) Observe from Definition 3 that 𝜑_𝐽𝑄(𝐷)𝑓_{𝐿𝑝(2𝑙𝑄)}≲2^𝑙𝑄^{((𝜎+𝜆)/𝑛)+(1/𝑝)}𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)} (91) for all 𝑙≥0. Thus, we obtain 𝜑_𝐽𝑄(𝐷)𝑓(𝑥)=|𝑄|^{∞(𝜆+𝜎)/𝑛}_𝑙=02^{−𝑙(𝑛+𝜎+1)+𝑙(𝜎+𝜆+(𝑛/𝑝))}𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}≲|𝑄|^{(𝜎+𝜆)/𝑛}𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}(𝑥∈𝑄). (92) Hence, from (92) and the fact that 𝜆+𝜎<0, we have ^{𝐽𝑄∗−1}_𝑗=−∞𝜑_𝑗(𝐷)𝑓_{𝐿∞(𝑄∗)}≤^{𝐽𝑄∗−1}_𝑗=−∞sup_{𝑥∈𝑄∗}𝜑_𝑗(𝐷)𝑓(𝑥≲𝑄_{∗(𝜎+𝜆)/𝑛})𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}, (93) which proves the embedding into 𝐿^∞(𝑄_∗). To prove that the sum defining 𝑔₁ converges in the topology of 𝒮^′(ℝ^𝑛), we first fix a cube 𝑅 containing 𝑄_∗ and observe ^𝐽𝑄∗_{𝑗=𝐽𝑅}𝜑_𝑗(𝐷)𝑓_{𝐿𝑝(𝑅)}≲|𝑅|^{((𝜎+𝜆)/𝑛)+(1/𝑝)}ℓ(𝑅)×log2+ℓ𝑄_∗𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)} (94) from (91) and the fact that 𝜎+𝜆+𝑛/𝑝≥0. Meanwhile, again by (92), we have 𝜑_𝐽𝑄(𝐷)𝑓_{𝐿∞(𝑄)}≲|𝑄|^{(𝜎+𝜆)/𝑛}𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)} (95) for all cubes 𝑄 containing 𝑅. As a result, we obtain that ^𝐽𝑅_𝑗=−∞𝜑_𝑗(𝐷)𝑓_{𝐿𝑝(𝑅)}≲|𝑅|^{((𝜎+𝜆)/𝑛)+(1/𝑝)}𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)} (96) from the fact that 𝜎+𝜆<0. It follows from (94) and (96) that the sum defining 𝑔₁ converges in 𝒮^′(ℝ^𝑛).

Now let us prove (83). First, choose 𝜃∈(1+(𝜆+𝜎)𝑝/𝑛,1). By virtue of Lemma 16 and the fact that 𝐴_𝑝((𝑀𝜒_𝑄∗)^𝜃) has a bound independent of 𝑄_∗∈𝒟 [33], we obtain that ^∞_{𝑗=𝐽𝑄∗}𝜑_𝑗(𝐷)𝑓_{𝐿𝑝(𝑄∗)}≤𝑀𝜒_𝑄∗^∞𝜃/𝑝_{𝑗=𝐽𝑄∗}𝜑_𝑗(𝐷)𝑓_𝐿𝑝≲𝑀𝜒_𝑄∗^𝜃/𝑝∞_{𝑗=𝐽𝑄∗−𝑗0}𝜑_𝑗(𝐷)𝑓^21/2_𝐿𝑝. (97) Here for the last inequality, we employed a geometric observation of the support of 𝜑. Therefore, 𝑗₀ of the last inequality depends only on 𝜑.

Since 𝑀𝜒_𝑄∗(𝑥)∼(1+ℓ(𝑄_∗)⁻¹|𝑥−𝑐(𝑄_∗)|)^−𝑛, we have ^∞_{𝑗=𝐽𝑄∗}𝜑_𝑗(𝐷)𝑓_{𝐿𝑝(𝑄∗)}≲^∞_{𝑙=−𝑗0}2^{−𝑙𝑛𝜃/𝑝∞}_{𝑗=𝐽𝑄∗−𝑗0}𝜑_𝑗(𝐷)𝑓^21/2_{𝐿𝑝(2𝑙𝑄∗)}≲^∞_{𝑙=−𝑗0}2^{−𝑙𝑛𝜃/𝑝}ℓ2^𝑙𝑄_∗^{𝜎+𝜆+(𝑛/𝑝)}𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (98) Recall that we are assuming that 𝜃∈(1+(𝜆+𝜎)𝑝/𝑛,1). So (98) is summable and we have ^∞_{𝑗=𝐽𝑄∗}𝜑_𝑗(𝐷)𝑓_{𝐿𝑝(𝑄∗)}≲ℓ𝑄_∗^{𝜎+𝜆+(𝑛/𝑝)}𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}, (99) showing (83).

With (82) and (83), the proof of (b) is now complete.

Remark 23. We did not use the structure of 𝜑_𝑗(𝐷)𝑓 in the proof. Therefore, we can deduce the following variant.

Proposition 24. Suppose that we are given a collection {𝐹_𝑗}^∞_𝑗=−∞⊂𝒮^′(ℝ^𝑛) such that supp(ℱ𝐹_𝑗)⊂supp(𝜑_𝑗) and that sup_𝑟>01𝑟^𝜎sup_{𝑄𝑄∈𝒟𝑟}1ℓ(𝑄)^𝜆1|𝑄|_𝑄^∞_{𝑗=𝐽𝑄}𝐹_𝑗𝑦^2𝑝/2𝑑𝑦^1/𝑝<∞. (100) Then 𝐺∶=∑^∞_𝑗=−∞𝐹_𝑗 converges in the topology of 𝒮^′(ℝ^𝑛).

3.4. Proof of Theorem 4 Part (c)

Keeping what we proved in Parts (a) and (b) in mind, we shall now describe the isomorphism between 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) and 𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆)(ℝ^𝑛). If 𝑓∈𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛), then we claim that 𝑓∈𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆)(ℝ^𝑛) and that 𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}≲𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}. (101) The proof of (101) is comparably easy.

Let us start to prove (101). Let 𝑟>0 and 𝑄∈𝒟(𝑄_𝑟) be fixed. Also, fix a Rademacher sequence {𝑟_𝑗}^∞_𝑗=−∞⊂𝐿²([0,1]). Define 𝐾_𝑄(𝑥;𝑡)∶=^∞_{𝑗=𝐽𝑄}2^𝑗𝑛𝑟_𝑗(𝑡)ℱ⁻¹𝜑2^𝑗𝑥𝑥∈ℝ^𝑛,𝑡>0. (102) Then we have ^∞_{𝑗=𝐽𝑄}𝑟_𝑗(𝑡)𝜑_𝑗(𝐷)𝑓=𝐾_𝑄()1⋅;𝑡∗𝑓,𝑟^𝜎_𝑄^∞_{𝑗=𝐽𝑄}𝜑_𝑗(𝐷)𝑓𝑦^2𝑝/2𝑑𝑦|𝑄|^{(𝜆𝑝/𝑛)+11/𝑝}∼1𝑟^𝜎_𝑄×[0,1]𝐾_𝑄(⋅;𝑡)∗𝑓𝑦^𝑝𝑑𝑦𝑑𝑡|𝑄|^{(𝜆𝑝/𝑛)+11/𝑝} (103) by the property of the Rademacher sequence. Let us set 𝑓₁∶=𝑓𝜒_2𝑄,𝑓₂∶=𝑓−𝑓₁. (104) Then we have 𝐾_𝑄(𝑥;𝑡)≲|𝑥|^−𝑛,𝜕_𝑘𝐾_𝑄(𝑥;𝑡)≲|𝑥|^−𝑛−1,ℱ𝐾_𝑄(⋅;𝑡)(𝜉)≲1, (105) since 𝑟_𝑗(𝑡)∈{0,1,−1} with constant independent of 𝑄.

By the Calderón-Zygmund theory for 𝐿^𝑝(ℝ^𝑛), we have 1𝑟^𝜎1_0𝑄𝐾_𝑄(⋅;𝑡)∗𝑓₁𝑦^𝑝𝑑𝑦𝑑𝑡|𝑄|^{(𝜆𝑝/𝑛)+11/𝑝}≲1𝑟^𝜎_2𝑄𝑓𝑦^𝑝𝑑𝑦|𝑄|^{(𝜆𝑝/𝑛)+11/𝑝}. (106) As for 𝑓₂, we invoke the following pointwise estimate obtained from (105): 𝐾_𝑄(⋅;𝑡)∗𝑓₂𝑦≲_{ℝ𝑛⧵2𝑄}𝑓(𝑧)|𝑧−𝑐(𝑄)|^𝑛𝑑𝑧𝑦∈𝑄. (107) We write I_𝑘∶=_{2𝑘+1𝑄⧵2𝑘𝑄}𝑓(𝑧)|𝑧−𝑐(𝑄)|^𝑛∼1𝑑𝑧2^𝑘𝑄^{2𝑘+1𝑄⧵2𝑘𝑄}𝑓(𝑧)𝑑𝑧,𝑘=1,2,…. (108) Notice that I_𝑘≲2^𝑘+1𝑟^𝜎2^𝑘+1ℓ(𝑄)^𝜆𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}, (109) since 2^𝑘+1𝑄⊂𝑄₂^𝑘+1_𝑟. Assuming that 𝜎+𝜆<0, we have ^∞_𝑘=1I_𝑘𝑟^𝜎|𝑄|^𝜆/𝑛≲^∞_𝑘=12^𝑘+1𝑟^𝜎2^𝑘+1ℓ(𝑄)^𝜆𝑟^𝜎|𝑄|^𝜆/𝑛𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}≲^∞_𝑘=12^{(𝜎+𝜆)𝑘}𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}∼𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}. (110) Consequently, from (106) and (110), we obtain (101).

The heart of the matter is how to construct the inverse mapping from 𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆)(ℝ^𝑛) to 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛). Let 𝑓∈𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆)(ℝ^𝑛). Then we have established in (b) that 𝑓∶=^∞_𝑗=−∞𝜑_𝑗(𝐷)𝑓 (111) exists in the topology of 𝐿^𝑝_loc(ℝ^𝑛)∩𝒮^′(ℝ^𝑛) in view of (92) and (99). We claim that 𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}≲𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (112) Here and below in proving (112), we assume that 𝑓=𝑓 by replacing 𝑓 with 𝑓.

Let us prove (112). Suppose that we are given a dyadic cube 𝑄 and 𝑟>0 such that 𝑄∈𝒟(𝑄_𝑟). We estimate 1𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)𝑄}𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝. (113) First, we expand 𝑓 by using (111). If we denote 𝜑_𝑗(𝜉)∶=^𝐽+_𝑙=𝐽−𝜑2^{−𝑗−𝑙}𝜉,𝜉∈ℝ^𝑛 (114) for 𝑗∈ℤ, then we have 𝑓(𝑥)=^∞_𝑗=−∞𝜑_𝑗(𝐷)𝜑_𝑗(𝐷)𝑓(𝑥), (115) where the convergence, as we have established in Part (b), takes places in 𝐿^𝑝_loc(ℝ^𝑛)∩𝒮^′(ℝ^𝑛).

We set 𝐽₁1∶=𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)𝑄∞}_{𝑗=𝐽𝑄}𝜑_𝑗(𝐷)𝜑_𝑗(𝐷)𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝,𝐽_2,𝑘1∶=𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)𝑄}𝜑_𝑘(𝐷)𝜑_𝑘(𝐷)𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝 (116) for 𝑘∈ℤ.

We decompose (83) by using 𝐽₁ and 𝐽_2,𝑘. 1𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)𝑄}𝑓(𝑥)^𝑝𝑑𝑥^1/𝑝≤𝐽₁+^𝐽𝑄−1_𝑘=−∞𝐽_2,𝑘. (117) by using (115).

We estimate 𝐽₁. Let 𝜃 be an auxiliary parameter again, which is taken so that ()𝑝𝜃∈1+𝜆+𝜎𝑛,1. (118) By using the maximal operator 𝑀, we have 𝐽₁≤1𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)}×_ℝ𝑛^∞_{𝑗=𝐽𝑄}𝜑_𝑗(𝐷)𝜑_𝑗(𝐷)𝑓(𝑥)^𝑝𝑀𝜒_𝑄(𝑥)^𝜃𝑑𝑥^1/𝑝. (119) By Lemma 16, we have 𝐽₁≲1𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)}×_ℝ𝑛^∞_{𝑗=𝐽𝑄}𝜑_𝑗(𝐷)𝑓(𝑥)^2𝑝/2𝑀𝜒_𝑄(𝑥)^𝜃𝑑𝑥^1/𝑝. (120) We write 𝑎_𝑘1∶=𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)}×_2𝑘𝑄^∞_{𝑗=𝐽𝑄}|𝜑_𝑗(𝐷)𝑓(𝑥)|^2𝑝/2𝑑𝑥^1/𝑝. (121) We decompose again the right-hand side of (120) dyadically. 𝐽₁≲^∞_𝑘=12^{−𝑘𝑛𝜃/𝑝}𝑎_𝑘. (122) For each 𝑘, we have 2^𝑘𝑄⊂𝑄₂^𝑘_𝑟, since 𝑄⊂𝑄_𝑟. Thus, 𝑎_𝑘=2^{𝑘𝜎+𝑘(𝜆+(𝑛/𝑝))}2^𝑘𝑟^𝜎2^𝑘𝑄^{(𝜆/𝑛)+(1/𝑝)}×_2𝑘𝑄^∞_{𝑗=𝐽𝑄}|𝜑_𝑗(𝐷)𝑓(𝑥)|^2𝑝/2𝑑𝑥^1/𝑝≲2^{𝑘(𝜎+𝜆+(𝑛/𝑝))}𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (123) Consequently, from (118), (122), and (123), we obtain that 𝐽₁≲^∞_𝑘=𝐾2^{−(𝑘𝑛𝜃/𝑝)}𝑎_𝑘≲^∞_𝑘=12^{𝑘(𝜎+𝜆+(𝑛(1−𝜃)/𝑝))}𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}≲𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (124) Thus, the estimate for 𝐽₁ is now valid.

As for 𝐽_2,𝑘 we choose 𝐿^∗≫1 first. Since 𝜑∈𝒮, we have 𝐽_2,𝑘=(2𝜋)^−𝑛/2⋅1𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)}×_𝑄ℝ𝑛ℱ⁻¹𝜑_𝑘𝑥−𝑦𝜑_𝑘(𝐷)𝑓𝑦𝑑𝑦^𝑝𝑑𝑥^1/𝑝≲1𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)}×_𝑄ℝ𝑛2^𝑘𝑛𝜑_𝑘(𝐷)𝑓𝑦2^𝑘𝑥−𝑦^2𝐿∗𝑑𝑦^𝑝𝑑𝑥^1/𝑝. (125) By the Hölder inequality, we have 𝐽_2,𝑘≲1𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)}×_𝑄ℝ𝑛2^𝑘𝑛2^𝑘𝑥−𝑦^{𝐿∗𝑝′}𝑑𝑦^𝑝/𝑝′×_ℝ𝑛2^𝑘𝑛𝜑_𝑘(𝐷)𝑓𝑦^𝑝2^𝑘𝑥−𝑦^𝐿∗𝑝𝑑𝑦𝑑𝑥^1/𝑝∼1𝑟^𝜎|𝑄|^𝜆/𝑛×_ℝ𝑛1|𝑄|_𝑄2^𝑘𝑛2^𝑘𝑥−𝑦^𝐿∗𝑝×𝜑𝑑𝑥_𝑘(𝐷)𝑓𝑦^𝑝𝑑𝑦^1/𝑝. (126) We have, for 𝑘≤𝐽_𝑄 and 𝑦∈ℝ^𝑛, 1|𝑄|_𝑄2^𝑘𝑛2^𝑘𝑥−𝑦^𝐿∗𝑝2𝑑𝑥≲^𝑘𝑛2^𝑘𝑦−𝑐(𝑄)^𝐿∗𝑝. (127) Consequently, from (126) and (127), we have 𝐽_2,𝑘≲1𝑟^𝜎|𝑄|^{𝜆/𝑛ℝ𝑛}2^𝑘𝑛𝜑_𝑘(𝐷)𝑓𝑦^𝑝2^𝑘𝑦−𝑐(𝑄)^𝐿∗𝑝𝑑𝑦^1/𝑝. (128) We decompose the integral (128) dyadically: 𝐽_2,𝑘≲1𝑟^𝜎|𝑄|^𝜆/𝑛×^∞_𝑙=12^{(𝑘𝑛/𝑝)−𝑙𝐿∗}_{𝑄(𝑐(𝑄),2−𝑘+𝑙)}𝜑_𝑘(𝐷)𝑓𝑦^𝑝𝑑𝑦^1/𝑝. (129) Notice that 𝑄(𝑐(𝑄),2^−𝑘+𝑙)⊂𝑄_𝑟+2^−𝑘+𝑙, since 𝑐(𝑄)∈𝑄⊂𝑄_𝑟. Consequently, assuming that 𝐿^∗≫1, we have 𝐽_2,𝑘≲1𝑟^𝜎|𝑄|^∞𝜆/𝑛_𝑙=12^{(𝑙𝑛/𝑝)−𝑙𝐿∗}𝑟+2^{−𝑘+𝑙𝜎}×𝑄𝑐(𝑄),2^{−𝑘+𝑙𝜆/𝑛}𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}=1𝑟^𝜎|𝑄|^𝜆/𝑛𝑟+2^−𝑘𝜎2^−𝑘𝜆𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (130) We are going to add (130) over 𝑘≤𝐽_𝑄. To this end, observe that ^𝐽𝑄−1_𝑘=−∞𝑟+2^−𝑘𝜎2^−𝑘𝜆=1log2^𝐽𝑄−1_𝑘=−∞^2−𝑘+1₂^−𝑘𝑟+2^−𝑘𝜎2^−𝑘𝜆𝑑𝑡𝑡∼^𝐽𝑄−1_𝑘=−∞^2−𝑘+1_2−𝑘(𝑟+𝑡)^𝜎𝑡^𝜆𝑑𝑡𝑡=^∞_(𝑄)2ℓ(𝑟+𝑡)^𝜎𝑡^𝜆𝑑𝑡𝑡. (131) If we change variables, then we have ^𝐽𝑄−1_𝑘=−∞𝑟+2^−𝑘𝜎2^−𝑘𝜆∼^∞_(𝑄)2ℓ(𝑟+𝑡)^𝜎𝑡^𝜆𝑑𝑡𝑡=𝑟^𝜎+𝜆∞_{(𝑄)2ℓ/𝑟}(1+𝑣)^𝜎𝑣^𝜆𝑑𝑣𝑣. (132) Now that ℓ(𝑄)≤𝑟 and 𝜎+𝜆<0, we obtain ^𝐽𝑄−1_𝑘=−∞𝑟+2^−𝑘𝜎2^−𝑘𝜆∼𝑟^𝜎+𝜆2_{(𝑄)2ℓ/𝑟}𝑣^𝜆𝑑𝑣𝑣+^∞₂(1+𝑣)^𝜎𝑣^𝜆𝑑𝑣𝑣∼𝑟^𝜎ℓ(𝑄)^𝜆. (133) As a result, from (133) and the fact that ℓ(𝑄)≤𝑟, the estimate (130) is summable over 𝑘≤𝐽_𝑄−1, and we obtain that ^𝐽𝑄−1_𝑘=−∞𝐽_2,𝑘≲^𝐽𝑄−1_𝑘=−∞𝑟+2^−𝑘𝜎2^−𝑘𝜆𝑟^𝜎|𝑄|^𝜆/𝑛∼(𝑟+ℓ(𝑄))^𝜎ℓ(𝑄)^𝜆𝑟^𝜎|𝑄|^𝜆/𝑛𝑓_{𝐿𝐷𝑝,𝜆}∼𝑓_{𝐿𝐷𝑝,𝜆}. (134) Consequently, (83) is proved, and we conclude the proof of Theorem 4(c).

3.5. Proof of Theorem 4 Part (d)

An important corollary of Proposition 5 is that we can relax the condition (17) on 𝜑, which is stronger than (d) in Theorem 4.

Corollary 25. Let 𝜑^∗∈𝐶^∞_{𝑐𝑜𝑚𝑝}(ℝ^𝑛⧵{0}) be such that ^∞_𝑗=−∞𝜑^∗2^−𝑗𝜉≳1𝜉∈ℝ^𝑛⧵{0}. (135) Define 𝑓^∗_{̇𝐵𝜎𝐿𝐷𝑝,𝜆}∶=sup_𝑟>01𝑟^𝜎sup_{𝑄𝑄∈𝒟𝑟}1ℓ(𝑄)^𝜆×1|𝑄|_𝑄^∞_{𝑗=𝐽𝑄}𝜑^∗_𝑗(𝐷)𝑓𝑦^2𝑝/2𝑑𝑦^1/𝑝. (136) Then the norm equivalence 𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}∼𝑓^∗_{̇𝐵𝜎𝐿𝐷𝑝,𝜆} (137) holds.

Proof. Choose integers 𝐽₋ and 𝐽₊ so that Θ(𝜉)∶=^𝐽+_𝑘=𝐽−𝜑^∗2^−𝑘𝜉>0,𝜉∈supp𝜑. (138) Define 𝜑^∗∗(𝜉)𝜑(𝜉)∶=Θ(𝜉)𝜑,0onsuppoutsidesupp𝜑. (139) Then 𝜑^∗∗∈𝐶^∞_comp(ℝ^𝑛⧵{0}) by virtue of (138). Observe also that 𝜑^∗∗Θ=𝜑 (140) in view of the size of supports. Hence, we have 𝜑_𝑗(𝐷)𝑓(𝑥)≲^𝐽+_𝑘=𝐽−𝜑^∗_𝑗+𝑘(𝐷)𝜑_𝑗^∗∗(𝐷)𝑓(𝑥≲_{𝐽+𝑘=𝐽−})𝑀𝜑^∗_𝑗+𝑘(𝐷)𝑓(𝑥). (141) Since the numbers 𝐽_± appearing in (138) are finite, we have 𝑓^∗_{̇𝐵𝜎𝐿𝐷𝑝,𝜆}≲𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)} (142) from (141) and Proposition 5. The proof of 𝑓^∗_{̇𝐵𝜎𝐿𝐷𝑝,𝜆}≳𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (143) is analogous; just swap the role of 𝜑 and 𝜑^∗. With (142) and (143), we see that the norms in (137) are equivalent.

4. 𝐵̇_𝜎(𝐿_𝑝,𝜆)-𝐵̇_𝜎(𝐿_𝑞,𝜇) Boundedness of 𝐼_𝛼, Proof of Theorem 6

To prove Theorem 6, we need the lemmas. Note that Lemma 26 can be seen as the Plancherel-Polya-Nikolskij inequality for 𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆).

Lemma 26. Suppose that the parameters 𝜎,𝑝, and 𝜆 satisfy 𝜎≥0,𝜆<0,1≤𝑝<∞. (144) Let 𝜑 be chosen so that it satisfies (17). Then we have 1𝑟^𝜎|𝑄|^𝜆/𝑛𝜑_𝐽𝑄(𝐷)𝑓_{𝐿∞(𝑄)}≲𝜑_𝐽𝑄(𝐷)𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}≲𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)} (145) for all 𝑟>0 and 𝑄∈𝒟(𝑄_𝑟).

Proof. The right-hand inequality is a consequence of Lemma 20.
Let 𝑥∈𝑄 be fixed. We choose 𝐿∈ℕ sufficiently large. Also, take a compactly supported function 𝜑 so that 𝜑 equals 1 on supp(𝜑) as we did in (138). Then we have 1𝑟^𝜎|𝑄|^𝜆/𝑛𝜑_𝐽𝑄(𝐷)𝜑_𝐽𝑄(𝐷)𝑓(𝑥=())2𝜋^−𝑛/2⋅1𝑟^𝜎|𝑄|^{(𝜆/𝑛)+1}×_ℝ𝑛ℱ⁻¹𝜑𝑥−𝑦ℓ(𝑄)𝜑_𝐽𝑄(𝐷)𝑓𝑦≲1𝑑𝑦𝑟^𝜎|𝑄|^{(𝜆/𝑛)+1}×_ℝ𝑛𝑐(𝑄)1+−𝑦ℓ(𝑄)^−𝐿𝜑_𝐽𝑄(𝐷)𝑓𝑦𝑑𝑦. (146) We decompose ℝ^𝑛 dyadically. Then we have 1𝑟^𝜎|𝑄|^𝜆/𝑛𝜑_𝐽𝑄(𝐷)𝜑_𝐽𝑄(𝐷)𝑓(𝑥≲_∞𝑙=1)2^−𝑙𝐿𝑟^𝜎|𝑄|^{(𝜆/𝑛)+12𝑙𝑄}𝜑_𝐽𝑄(𝐷)𝑓𝑦≲𝑑𝑦_∞𝑙=12^{−𝑙(𝐿−𝜎−𝑛−𝜆)}2^𝑙𝑟^𝜎2^𝑙𝑄^{(𝜆/𝑛)+12𝑙𝑄}𝜑_𝐽𝑄(𝐷)𝑓𝑦≲𝜑𝑑𝑦_𝐽𝑄(𝐷)𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (147) Thus, the proof is now complete.

In the course of the proof, we obtained the following chain of inequalities.

Corollary 27. Suppose that the parameters 𝜎,𝑝, and 𝜆 satisfy 𝜎≥0,𝜆<0,1≤𝑝<∞. (148) Let 𝜑 be chosen so that it satisfies (17). Then one has 𝜑_𝑗(𝐷)𝑓_{𝐿∞(𝑄𝑟)}≲2^−𝑗𝜆𝑟^𝜎𝜑_𝑗(𝐷)𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}∼2^−𝑗𝜆𝑟^𝜎𝜑_𝑗(𝐷)𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}≲2^−𝑗𝜆𝑟^𝜎𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)} (149) for 𝑓∈𝐵̇_𝜎(𝐿^𝐷_𝑝,𝜆)(ℝ^𝑛), 𝑗∈ℤ and 𝑟>0 such that 2^−𝑗<𝑟.

The following estimate is somehow well known. Here we remark that the following form was recorded in [34]. Here we change variables 𝑗↦−𝑗 to transform the result into the one needed in the present paper.

Lemma 28. Let 𝐴,𝐵,𝜅₁,𝜅₂>0. Then min2^∞_𝑗=−∞^𝑗𝜅1𝐴,2^−𝑗𝜅2𝐵≲𝐴^{𝜅2/(𝜅1+𝜅2)}𝐵^{𝜅1/(𝜅1+𝜅2)}. (150)

Now we prove Theorem 6. Given 𝑓∈𝐵̇_𝜎(𝐿_𝑝,𝜆), we shall define 𝐼_𝛼𝑓(𝑥)=^∞_𝑗=−∞𝐼_𝛼𝜑_𝑗(𝐷)𝑓(𝑥). (151) Note that, by using the Fourier transform, 𝐼_𝛼[𝜑_𝑗(𝐷)𝑓] can be defined. Once we show that {𝐼_𝛼[𝜑_𝑗(𝐷)𝑓]}^∞_𝑗=1 satisfies the condition in Proposition 24, then by Proposition 24  𝐼_𝛼 can be defined. Since this essentially amounts to showing 𝐼_𝛼𝑓_{̇𝐵𝜎(𝐿𝐷𝑞,𝜇)}<∞ (152) by assuming that we can define 𝐼_𝛼𝑓, we omit the detail.

Since 1<𝑞<∞, we have 𝐼_𝛼𝑓_{̇𝐵𝜎(𝐿𝑞,𝜇)}∼𝐼_𝛼𝑓_{̇𝐵𝜎(𝐿𝐷𝑞,𝜇)} (153) by virtue of Theorem 4. Likewise since 1<𝑝<∞, we have 𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}∼𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (154) Let 𝑄 be a fixed dyadic cube. Define 𝜑^∗(𝜉)∶=|𝜉|^𝛼𝜑(𝜉). Then we have (135) and 𝜑^∗_𝑗(𝜉)∶=2^−𝑗𝜉^𝛼𝜑_𝑗(𝜉) (155) for 𝜉∈ℝ^𝑛 and 𝑗∈ℤ. Thus, we have 1𝑟^𝜎|𝑄|^𝜇/𝑛∞_{𝑗=𝐽𝑄}𝜑^∗_𝑗(𝐷)𝐼_𝛼𝑓(𝑥)^21/21=𝑐𝑟^𝜎|𝑄|^𝜇/𝑛∞_{𝑗=𝐽𝑄}2^−2𝑗𝛼𝜑_𝑗(𝐷)𝑓(𝑥)^21/2 (156) since ℱ𝐼_𝛼𝑓(𝜉)=𝑐|𝜉|^−𝛼ℱ𝑓(𝜉). If we use the embedding ℓ¹↪ℓ², then we have 1𝑟^𝜎|𝑄|^𝜇/𝑛∞_{𝑗=𝐽𝑄}2^−2𝑗𝛼𝜑_𝑗(𝐷)𝑓(𝑥)^21/2≲1𝑟^𝜎|𝑄|^∞𝜇/𝑛_{𝑗=𝐽𝑄}2^−𝑗𝛼𝜑_𝑗(𝐷)𝑓(𝑥). (157) By Corollary 27 and 𝜇=𝛼+𝜆, we have 1𝑟^𝜎|𝑄|^∞𝜇/𝑛_{𝑗=𝐽𝑄}2^−𝑗𝛼𝜑_𝑗(𝐷)𝑓(𝑥≲1)|𝑄|^𝜇/𝑛×^∞_{𝑗=𝐽𝑄}𝜑min^𝑗(𝐷)𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}2^𝑗𝜇,12^𝑗𝛼𝑟^𝜎∞_{𝜈=𝐽𝑄}𝜑_𝜈(𝐷)𝑓(𝑥)^21/2≲1|𝑄|^𝜇/𝑛×^∞_𝑗=−∞𝑓min^{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}2^𝑗𝜇,12^𝑗𝛼𝑟^𝜎∞_{𝜈=𝐽𝑄}𝜑_𝜈(𝐷)𝑓(𝑥)^21/2. (158) Now we invoke Lemma 28 with 𝐴=𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}1,𝐵=𝑟^𝜎∞_{𝜈=𝐽𝑄}𝜑_𝜈(𝐷)𝑓(𝑥)^21/2,𝜅₁=−𝜇,𝜅₂=𝛼. (159) Since 𝜇=𝛼+𝜆, we have 𝜅₂/(𝜅₁+𝜅₂)=−𝛼/𝜆,𝜅₁/(𝜅₁+𝜅₂)=1+(𝛼/𝜆). Thus, from Lemma 28, we obtain that 1𝑟^𝜎|𝑄|^𝜇/𝑛∞_{𝑗=𝐽𝑄}2^−𝑗𝛼𝜑_𝑗(𝐷)𝑓(𝑥)≲𝐴^{𝜅2/(𝜅1+𝜅2)}𝐵^{𝜅2/(𝜅1+𝜅2)}=1|𝑄|^𝜇/𝑛𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}^−𝛼/𝜆×1𝑟^𝜎∞_{𝜈=𝐽𝑄}𝜑_𝜈(𝐷)𝑓(𝑥)^{21/21+(𝛼/𝜆)}. (160) We consider the 𝐿^𝑞(𝑄)-norm of the both sides. Then by the relation 1+𝛼/𝜆=𝜇/𝜆=𝑝/𝑞, we deduce 1𝑟^𝜎|𝑄|^{(𝜇/𝑛)+(1/𝑞)𝑄∞}_{𝑗=𝐽𝑄}2^−𝑗𝛼𝜑_𝑗(𝐷)𝑓𝑦^𝑞𝑑𝑦^1/𝑞≲𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}^−𝛼/𝜆×1𝑟^𝜎𝑝|𝑄|^{(𝜆𝑝/𝑛)+1𝑄∞}_{𝜈=𝐽𝑄}𝜑_𝜈(𝐷)𝑓𝑦^2𝑝/2𝑑𝑦^1/𝑞=𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}^−𝛼/𝜆×1𝑟^𝜎𝑝|𝑄|^{(𝜆𝑝/𝑛)+1𝑄∞}_{𝜈=𝐽𝑄}𝜑_𝜈(𝐷)𝑓𝑦^2𝑝/2𝑑𝑦^{(𝜆+𝛼)/(𝜆𝑝)}. (161) In view of the definition of ‖𝑓‖_{̇𝐵𝜎(𝐿}^𝐷_𝑝,𝜆), we have 1𝑟^𝜎|𝑄|^{(𝜇/𝑛)+(1/𝑞)𝑄∞}_{𝑗=𝐽𝑄}2^−𝑗𝛼𝜑_𝑗(𝐷)𝑓𝑦^𝑞𝑑𝑦^1/𝑞≲𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (162) Putting (156), (157) and (162) together, we obtain 1𝑟^𝜎|𝑄|^𝜇/𝑛1|𝑄|_𝑄^∞_{𝑗=𝐽𝑄}𝜑^∗_𝑗(𝐷)𝐼_𝛼𝑓(𝑥)^2𝑞/2𝑑𝑥^1/𝑞≲𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (163) Consequently, from (153), Corollary 25 and the definition of the norm 𝐵̇_𝜎(𝐿^𝐷_𝑞,𝜇)(ℝ^𝑛), we obtain 𝐼_𝛼𝑓_{̇𝐵𝜎(𝐿𝑞,𝜇)}≲𝐼_𝛼𝑓_{̇𝐵𝜎(𝐿𝐷𝑞,𝜇)}≲𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (164) By combining (164) with (154), we conclude that 𝐼_𝛼 is a bounded operator from 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛) to 𝐵̇_𝜎(𝐿_𝑞,𝜇)(ℝ^𝑛).

Remark 29. Suppose that the parameters 𝛼,𝜎,𝑝,𝑞,𝜇, and 𝜆 satisfy 0<𝛼<𝑛,𝜎≥0,𝜆,𝜇<0,1≤𝑝<∞,1<𝑞<∞ (165) and (27). If we reexamine (157), we see that 𝐼_𝛼𝑓_{̇𝐵𝜎(𝐿𝑞,𝜇)}∼sup_{𝑄𝑟>0,𝑄∈𝒟𝑟}1𝑟^𝜎|𝑄|^{(𝜇/𝑛)+(1/𝑞)}×_𝑄^∞_{𝑗=𝐽𝑄}2^−𝑗𝛼𝜑_𝑗(𝐷)𝑓𝑦^𝑞𝑑𝑦^1/𝑞≲^∞_𝜈=−∞𝜑_𝜈(𝐷)𝑓^21/2_{̇𝐵𝜎(𝐿𝑝,𝜆)}≲^∞_𝜈=−∞𝜑_𝜈(𝐷)𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}. (166)

5. 𝐵̇_𝜎(𝐿_𝑝,𝜆)-𝐵̇_𝜎(𝑊𝐿_𝑞,𝜇) Boundedness of 𝐼_𝛼, Proof of Theorem 7

The definition of 𝐼_𝛼𝑓 can be justified in the same was as Theorem 6. So we assume that 𝐼_𝛼𝑓 is already defined.

Let 𝑟>0 and a cube 𝑄 be fixed so that 𝑄⊂𝑄_𝑟. Set 𝑞₀∶=𝜆𝑝/(𝜆+𝛼−𝜀) and 𝑞₁∶=𝜆𝑝/(𝜆+𝛼+𝜀) for some small 𝜀>0. More precisely, we choose 𝜀>0 so that 𝜇+𝜀<0,𝑞₀,𝑞₁∈(1,∞),𝜆+𝜎+𝛼+𝜀<0. (167) By interpolation described in [35] we have 𝐼_𝛼𝑓_{𝑊𝐿𝑞(𝑄)}∼𝐼_𝛼𝑓_{[𝐿𝑞0(𝑄),𝐿𝑞1(𝑄)]1/2,∞}. (168) We calculate the 𝐾-functional. By the definition of the 𝐾-functional [35, Section 3.1] and by Remark 29, we have 𝐾2^2𝑗𝜀|𝑄|^{−2𝜀/(𝜆𝑝)}ℓ(𝑄)^−2𝜀,𝐼_𝛼𝑓;𝐿^𝑞0(𝑄),𝐿^𝑞1(𝑄)=inf𝑔_{0𝐿𝑞0(𝑄)}+2^2𝑗𝜀|𝑄|^{−2𝜀/(𝜆𝑝)}𝑟^−2𝜀𝑔_{1𝐿𝑞1(𝑄)}∶𝑔₀∈𝐿^𝑞0(𝑄),𝑔₁∈𝐿^𝑞1(𝑄),𝐼_𝛼𝑓=𝑔₀+𝑔₁. (169) Note that (1/𝑞)−(1/𝑞₀)=𝜀/(𝜆𝑝),(1/𝑞)−(1/𝑞₁)=−𝜀/(𝜆𝑝). Hence we have 1𝑟^𝜎ℓ(𝑄)^{(𝑛/𝑞)+𝜇}|𝑄|^{𝜀/(𝜆𝑝)}ℓ(𝑄)^𝜀×𝐾2^2𝑗𝜀|𝑄|^{−2𝜀/(𝜆𝑝)}ℓ(𝑄)^−2𝜀,𝐼_𝛼𝑓;𝐿^𝑞0(𝑄),𝐿^𝑞1(𝑄)≤inf𝑔_{0̇𝐵𝜎(𝐿𝑞0,𝜇−𝜀)}+2^2𝑗𝜀𝑔_{1̇𝐵𝜎(𝐿𝑞1,𝜇+𝜀)}∶𝑔₀∈𝐿^𝑞0(𝑄),𝑔₁∈𝐿^𝑞1(𝑄),𝐼_𝛼𝑓=𝑔₀+𝑔₁. (170)

Let us define 𝑔₀∶=𝐼_𝛼−𝜀^𝑗_𝑘=−∞2^𝑘𝜀𝜑_𝑘(𝐷)𝑓,𝑔₁∶=𝐼_𝛼+𝜀^∞_𝑘=𝑗+12^−𝑘𝜀𝜑_𝑘(𝐷)𝑓. (171) By Lemma 20 and Remark 29, we have 𝑔_{0̇𝐵𝜎(𝐿𝑞0,𝜇−𝜀)}≲^𝑗_𝑘=−∞2^𝑘𝜀𝜑_𝑘(𝐷)𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)},𝑔_{1̇𝐵𝜎(𝐿𝑞1,𝜇+𝜀)}≲^∞_𝑘=𝑗+12^−𝑘𝜀𝜑_𝑘(𝐷)𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)},𝜑_𝑘(𝐷)𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}≲𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}. (172)(173)(174) From (172) and (173), we deduce that 1𝑟^𝜎ℓ(𝑄)^{(𝑛/𝑞)+𝜇}|𝑄|^{𝜀/(𝜆𝑝)}ℓ(𝑄)^𝜀×𝐾2^2𝑗𝜀|𝑄|^{−2𝜀/(𝜆𝑝)}ℓ(𝑄)^−2𝜀,𝐼_𝛼𝑓;𝐿^𝑞0(𝑄),𝐿^𝑞1(𝑄)≲^∞_𝑘=−∞2^{min(𝑘,2𝑗−𝑘)𝜀}𝜑_𝑘(𝐷)𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}𝑗∈ℤ. (175) If we insert this estimate and (174), then we have 1𝑟^𝜎ℓ(𝑄)^{(𝑛/𝑞)+𝜇}𝐼_𝛼𝑓_{[𝐿𝑞0(𝑄),𝐿𝑞1(𝑄)]1/2,∞}∼1𝑟^𝜎ℓ(𝑄)^{(𝑛/𝑞)+𝜇}|𝑄|^{𝜀/(𝜆𝑝)}ℓ(𝑄)^𝜀×𝐾2^2𝑗𝜀|𝑄|^{−2𝜀/(𝜆𝑝)}ℓ(𝑄)^−2𝜀,𝐼_𝛼𝑓;𝐿^𝑞0(𝑄),𝐿^𝑞1(𝑄)≲sup_𝑗∈ℤ2^{∞−𝑗𝜀}_𝑘=−∞2^{min(𝑘,2𝑗−𝑘)𝜀}𝜑_𝑘(𝐷)𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}≲𝑓_{̇𝐵𝜎(𝐿𝑝,𝜆)}. (176) The proof is therefore complete.

6. Structure of 𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝜃), Proof of Theorem 13

We prove Theorem 13. Let us recall (13), which we use now. To prove Theorem 13, we need the following lemma.

Lemma 30. For all multi-indices 𝛽, 𝑟>0, 𝑓∈𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝜃)(ℝ^𝑛), we have sup_𝑟>01𝑟^𝜎sup_{𝑄𝑥∈𝑄∈𝒬𝑟}1ℓ(𝑄)^𝜃2^{−𝐽𝑄|𝛽|}𝜕^𝛽𝜑_𝐽𝑄(𝐷)𝑓(𝑥≲𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}). (177)

Proof. To prove this, we fix 𝑥∈ℝ^𝑛,𝑟>0 and 𝑄∈𝒬 so that 𝑥∈𝑄⊂𝑄_𝑟. (178) Also, we take 𝜏∈𝐶^∞_comp(ℝ^𝑛⧵{0}) so that 𝜏≡1 on supp(𝜑). We set 𝜏_𝐽𝑄(𝜉)∶=𝜏(2^−𝐽𝑄𝜉) for 𝜉∈ℝ^𝑛 as usual. Then we have 2^{−𝐽𝑄|𝛽|}𝑟^𝜎|𝑄|^𝜃/𝑛𝜕^𝛽𝜑_𝐽𝑄(𝐷)𝑓(𝑥∼2_{−𝐽𝑄|𝛽|})𝑟^𝜎|𝑄|^{𝜃/𝑛ℝ𝑛}ℱ⁻¹𝜉^𝛽𝜏_𝐽𝑄𝑦𝜑_𝐽𝑄(𝐷)𝑓≲1𝑥−𝑦𝑑𝑦𝑟^𝜎|𝑄|^{𝜃/𝑛ℝ𝑛}2^𝐽𝑄𝑛𝜑_𝐽𝑄(𝐷)𝑓𝑥−𝑦1+2^𝐽𝑄𝑦^𝑛+𝜎+1≲𝑓𝑑𝑦_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}1𝑟^𝜎_ℝ^𝑛2^𝐽𝑄𝑛𝑟+𝑦^𝜎1+2^𝐽𝑄𝑦^𝑛+𝜎+1=𝑓𝑑𝑦_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)ℝ𝑛}2^𝐽𝑄𝑛1+𝑟⁻¹𝑦^𝜎1+2^𝐽𝑄𝑦^𝑛+𝜎+1𝑑𝑦. (179) Now that 2𝑟>2^−𝐽𝑄, we have 2^{−𝐽𝑄|𝛽|}𝑟^𝜎|𝑄|^𝜃/𝑛𝜕^𝛽𝜑_𝐽𝑄(𝐷)𝑓(𝑥≲𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)ℝ𝑛})2^𝐽𝑄𝑛1+2^𝐽𝑄𝑦^𝜎1+2^𝐽𝑄𝑦^𝑛+𝜎+1∼𝑓𝑑𝑦_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}. (180) This proves the lemma.

Proposition 31 (see [36]). Let 𝑑∈ℕ∪{0}. Suppose that we are given a sequence {𝑓_𝑗}^∞_𝑗=1 of 𝒮^′(ℝ^𝑛) such that {𝜕^𝛽𝑓_𝑗}^∞_𝑗=1 is convergent for each 𝛽 with |𝛽|=𝑑+1. Then there exists a sequence of polynomials {𝑃_𝑗}^∞_𝑗=1⊂𝒫_𝑑 such that {𝑓_𝑗+𝑃_𝑗}^∞_𝑗=1 is convergent in 𝒮^′(ℝ^𝑛).

The next proposition reveals the fundamental structure of 𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝜃)(ℝ^𝑛).

Proposition 32. Assume that the parameters 𝜎,𝜃∈ℝ and 𝑑∈ℤ satisfy 𝜎≥0,𝜃>0,𝑑>[𝜎+𝜃]. (181) Let 𝑓∈𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝜃)(ℝ^𝑛).   (1)There exists a sequence of polynomials {𝑃_𝑗}^∞_𝑗=1 of degree at most 𝑑 such that lim_𝑗→∞𝑃_𝑗+^∞_𝑘=−𝑗𝜑_𝑘(𝐷)𝑓 (182) converges in 𝒮^′(ℝ^𝑛).  (2) If we denote by 𝑔 the limit (182), then 𝑔_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}=𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}<∞. (183)  (3) The limit 𝑔 is determined modulo a difference of polynomials of order 𝑑 no matter how we choose {𝑃_𝑗}^∞_𝑗=1.   (4) The distribution 𝑓−𝑔 is a polynomial.   (5) The distribution 𝑓 is a continuous function.

Proof. (1) Let 𝑗≤0 and 𝑥∈ℝ^𝑛. Choose 𝑚∈ℤ so that 2^𝑚≤|𝑥|+2^−𝑗≤2^𝑚+1. Then we have 𝑥∈𝑄⊂𝑄_{4|𝑥|+4𝑟} for some cube 𝑄 with ℓ(𝑄)=2^𝑚+1. By Lemma 30, we have 2^{−𝑗|𝛽|}|𝑥|+2^−𝑗𝜎2^−𝑗𝜃𝜕^𝛽𝜑_𝑗(𝐷)𝑓(𝑥)≲𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}𝑥∈ℝ^𝑛. (184) Consequently, then we have 𝜕^𝛽𝜑_𝑗(𝐷)𝑓(𝑥)≲2^{𝑗(|𝛽|−𝜃−𝜎)}(1+|𝑥|)^𝜎𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}. (185) The estimate (185) shows that ∑¹_𝑗=−∞𝜕^𝛽𝜑_𝑗(𝐷)𝑓(𝑥) converges in the topology of 𝒮^′(ℝ^𝑛), provided that |𝛽|=𝑑+1. So we are in the position of resorting to Proposition 31.  (2) This is clear since 𝜑_𝑗(𝐷)𝑓=𝜑_𝑗(𝐷)𝑔.   (3) Suppose that {𝑃_𝑗}^∞_𝑗=1 is a sequence of polynomials of degree 𝑑 such that ℎ=lim_𝑗→∞𝑃_𝑗+^∞_𝑘=−𝑗𝜑_𝑘(𝐷)𝑓 (186) converges in the topology of 𝒮^′(ℝ^𝑛). Then we claim 𝜕^𝛽ℎ=𝜕^𝛽𝑔 for all 𝛽 with length 𝑑+1. Indeed, 𝜕^𝛽ℎ=lim_𝑗→∞𝜕^𝛽𝑃_𝑗+^∞_𝑘=−𝑗𝜑_𝑘(𝐷)𝑓=lim_𝑗→∞𝜕^𝛽∞_𝑘=−𝑗𝜑_𝑘(𝐷)𝑓=lim_𝑗→∞𝜕^𝛽𝑃_𝑗+^∞_𝑘=−𝑗𝜑_𝑘(𝐷)𝑓=𝜕^𝛽𝑔. (187) Consequently, 𝑔 and ℎ coincide modulo a polynomial of order 𝑑.   (4) Just observe that 𝑓−𝑔 has frequency support in the origin.  (5) In view of (4), we have only to show that 𝑔 itself is a continuous function. As we can see from (185), if |𝛽|=𝑑+1, then the limit 𝑔_𝛽(𝑥)=⁰_𝑗=−∞𝜕^𝛽𝜑_𝑗(𝐷)𝑓(𝑥) (188) defines a 𝐶^∞(ℝ^𝑛) function. If 𝛽,𝛽^′,𝛾,𝛾^′ are multi-indices such that |𝛽|=|𝛾|=𝑑+1 and that 𝛽+𝛽^′=𝛾+𝛾^′, then 𝑔_𝛽+𝛽′(𝑥)=𝑔_𝛾+𝛾′(𝑥). (189) Consequently, there exists a 𝐶^∞(ℝ^𝑛) function 𝐺 of polynomial growth such that 𝜕^𝛽𝐺=𝑔_𝛽. Observe also that when 𝑗≥0, 𝜑_𝑗(𝐷)𝑓(𝑥)≲|𝑥|+2^−𝑗𝜎2^−𝑗𝜃𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}≤(|𝑥|+1)^𝜎2^−𝑗𝜃𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}. (190) Consequently, since 𝜃>0, by virtue of the Weierstrass test, we see the series 1(1+|𝑥|)^𝜎∞_𝑗=0𝜑_𝑗(𝐷)𝑓(𝑥) (191) converges uniformly to a continuous function 𝐽(𝑥). Thus, 𝑔(𝑥)∶=𝐺(𝑥)+(1+|𝑥|)^𝜎𝐽(𝑥) is a desired function.

 Here and below we assume that 𝑓 itself satisfies 𝑓=lim_𝑗→∞𝑃_𝑗+^∞_𝑘=−𝑗𝜑_𝑘(𝐷)𝑓 (192) and that 𝑓 is a continuous function.

Lemma 33. Let 𝑓∈𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝜃)(ℝ^𝑛) satisfy (192). Then we have ^𝑑+1_𝑚′=0(−1)^𝑚′𝑚𝑑+1_′𝑓𝑥−𝑚^′𝑦=^∞_𝑗=−∞^𝑑+1_𝑚′=0(−1)^𝑚′𝑚𝑑+1_′𝜑_𝑗(𝐷)𝑓𝑥−𝑚^′𝑦 (193) in the topology of 𝒮^′(ℝ^𝑛).

Remark that the left-hand side equals (−1)^𝑑+1Δ^𝑑+1_−𝑦𝑓(𝑥).

Proof. By (192), we have ^𝑑+1_𝑚′=0(−1)^𝑚′𝑚𝑑+1_′𝑓𝑥−𝑚^′𝑦=lim^∞_{𝐽→∞𝑗=−𝐽}^𝑑+1_𝑚′=0(−1)^𝑚′𝑚𝑑+1_′×𝑃_𝑗𝑥−𝑚^′𝑦+𝜑_𝑗(𝐷)𝑓𝑥−𝑚^′𝑦=lim^∞_{𝐽→∞𝑗=−𝐽}^𝑑+1_𝑚′=0(−1)^𝑚′𝑚𝑑+1_′𝜑_𝑗(𝐷)𝑓𝑥−𝑚^′𝑦=^∞_𝑗=−∞^𝑑+1_𝑚′=0(−1)^𝑚′𝑚𝑑+1_′𝜑_𝑗(𝐷)𝑓𝑥−𝑚^′𝑦. (194) The proof is therefore completed.

Proof of Theorem 13. Choose 𝜏∈𝒮(ℝ^𝑛) such that 𝜒_𝑄1/2≤𝜏≤𝜒_𝑄1. Let us define 𝑇_𝑗(𝑥)∶=^𝑑+1_𝑚′=1^𝑑+1_𝑚=1(−1)^{𝑑+𝑚+𝑚′}()!𝑚𝑑+1𝑑+1_′𝑚×𝑚𝑑+1_𝑑+12^−𝑗𝑚𝑚^′𝑛ℱ⁻¹𝜏𝑥2^−𝑗𝑚𝑚^′. (195) Then ℱ𝑇_𝑗 equals a nonzero constant near a neighborhood of the origin and has compact support. Let us set 𝜑≡ℱ𝑇₁−ℱ𝑇₀. From the above observation, 𝜑 can be used for the definition of 𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝜃)(ℝ^𝑛).
Suppose that 𝑓 is a continuous function such that the quantity ‖𝑓‖_{̇𝐵𝜎(𝐿𝑖𝑝𝜃)} is finite. Let 𝑄 be a cube such that 𝑄⊂𝑄_𝑟 and fix 𝑥∈𝑄. Then we have 𝑇_𝑗∗𝑓(𝑥)=^𝑑+1_𝑚′=1^𝑑+1_𝑚=1(−1)^{𝑑+𝑚+𝑚′}()!𝑚𝑑+1𝑑+1_′𝑚×𝑚𝑑+1_𝑑+12^−𝑗𝑚𝑚^′𝑛ℱ⁻¹𝜏⋅2^−𝑗𝑚𝑚^′∗𝑓(𝑥) (196) and hence 𝜑_𝑗(𝐷)𝑓(𝑥)=^𝑑+1_𝑚=0𝑚𝑑+1𝑚^𝑑+1(−1)^𝑑+𝑚()!×𝑑+1_ℝ𝑛ℱ⁻¹𝜏𝑦^𝑑+1_𝑚′=0(−1)^𝑚′𝑚𝑑+1_′×𝑓𝑥−2^−𝑗−1𝑚𝑚^′𝑦−𝑓𝑥−2^−𝑗𝑚𝑚^′𝑦𝑑𝑦. (197) Observe that 𝑥∈𝑄𝑥,2^−𝑗𝑦⊂𝑄_|𝑥|+2^−𝑗_|𝑦|,2^−𝑗𝑦∈𝑄₂^−𝑗_|𝑦|, (198) and hence, for 𝑚=0,1,2,…,𝑑+1, ^𝑑+1_𝑚′=0(−1)^𝑚′𝑚𝑑+1_′𝑓𝑥−2^−𝑗𝑚𝑚^′𝑦=Δ₂^𝑑+1_{−𝑗𝑚𝑦}𝑓(𝑥≤|𝑥|)+2^−𝑗𝑦^𝜎2^−𝑗𝑦^𝜃𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝜃}^(𝑑+1)₎≤|𝑥|+|𝑥|⋅𝑦+2^−𝑗𝑦+2^−𝑗𝜎×2^−𝑗𝑦^𝜃𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝜃}^(𝑑+1)₎=|𝑥|+2^−𝑗𝜎1+𝑦^𝜎2^−𝑗𝑦^𝜃𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝜃}^(𝑑+1)₎. (199) From this estimate and the fact that 𝜃>0 and 𝜎≥0, we obtain that 𝜑_𝑗(𝐷)𝑓(𝑥)≲|𝑥|+2^−𝑗𝜎2^−𝑗𝜃𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝜃}^(𝑑+1)₎. (200) The estimate (200) shows that 𝑓∈𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝜃)(ℝ^𝑛).
Conversely, suppose that 𝑓∈𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝜃)(ℝ^𝑛). We need to prove that 𝑓∈𝐵̇_𝜎(𝐿𝑖𝑝_𝜃^(𝑑+1))(ℝ^𝑛), that is, we need to find a constant 𝐶>0 such that 1𝑟^𝜎ℓ(𝑄)^𝜃Δ^𝑑+1_−𝑦𝑓(𝑥)≤𝐶, (201) for any 𝑟>0, any cube 𝑄 and any pair of points 𝑥,𝑦 satisfying 𝑥∈𝑄∈𝒬(𝑄_𝑟) and |𝑦|≤ℓ(𝑄). Then, from 𝑓∈𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝜃)(ℝ^𝑛), we have ^𝑑+1_𝑚′=0(−1)^𝑚′𝑚𝑑+1_′𝑓𝑥−𝑚^′𝑦=^∞_𝑗=−∞^𝑑+1_𝑚′=0(−1)^𝑚′𝑚𝑑+1_′𝜑_𝑗(𝐷)𝑓𝑥−𝑚^′𝑦 (202) by Lemma 33. Then by the mean value theorem, we have Δ^𝑑+1_−𝑦𝑓(𝑥)=^𝑑+1_𝑚′=0(−1)^𝑚′𝑚𝑑+1_′𝜑_𝑗(𝐷)𝑓𝑥−𝑚^′𝑦≲𝑦^𝑑+1sup_{|𝑧|≤(𝑦𝑑+1)}∇^𝑑+1𝜑_𝑗(𝐷)𝑓(𝑥+𝑧) (203) and, by using the triangle inequality applied directly to the summation defining Δ^𝑑+1_−𝑦𝑓(𝑥), we have Δ^𝑑+1_−𝑦𝑓(𝑥)≲sup_{|𝑧|≤(𝑦𝑑+1)}𝜑_𝑗(𝐷)𝑓(𝑥+𝑧). (204) Consequently, we have Δ^𝑑+1_−𝑦𝑓(𝑥≲_{∞𝑗=−∞})min𝑦^𝑑+1sup_{|𝑧|≤(𝑦𝑑+1)}∇^𝑑+1𝜑_𝑗(𝐷)𝑓(𝑥+𝑧),sup_{|𝑧|≤(𝑦𝑑+1)}𝜑_𝑗(𝐷)𝑓(𝑥+𝑧). (205) It follows directly from the definition of the norm that sup_{|𝑧|≤(𝑦𝑑+1)}𝜑_𝑗(𝐷)𝑓(𝑥+𝑧)≲2^−𝑗𝜃2^−𝑗+|𝑥|+𝑦^𝜎𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}∼2^−𝑗𝜃|𝑥|+𝑦^𝜎+2^{−𝑗(𝜃+𝜎)}𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)} (206) and from Lemma 30 and (181) that 𝑦^𝑑+1sup_{|𝑧|≤(𝑦𝑑+1)}∇^𝑑+1𝜑_𝑗(𝐷)𝑓(≲2𝑥+𝑧)_{𝑗(𝑑+1−𝜃)}|𝑥|+𝑦^𝜎𝑦^𝑑+1+2^{𝑗(𝑑+1−𝜃−𝜎)}𝑦^𝑑+1×𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}. (207) Consequently, by Lemma 28, we have Δ^𝑑+1_−𝑦𝑓(𝑥)≲^∞_𝑗=−∞2^−𝑗𝜃|𝑥|+𝑦^𝜎+2^{−𝑗(𝜃+𝜎)}×min1,2^𝑗(𝑑+1)𝑦^𝑑+1𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}≲𝑦^𝜃|𝑥|+𝑦^𝜎𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}. (208) Now that |𝑦|≤ℓ(𝑄) and 𝑥∈𝑄⊂𝑄_𝑟, we deduce that Δ^𝑑+1_−𝑦𝑓(𝑥)≤ℓ(𝑄)^𝜃𝑟^𝜎𝑓_{̇𝐵𝜎(𝐿𝑖𝑝𝐷𝜃)}. (209) This concludes the proof of Theorem 13.

Going through an argument similar to Theorem 6, we obtain the following.

Theorem 34. Let 𝛼>0, 𝜎≥0 and 𝜃>0. Then the fractional integral operator 𝐼_𝛼 is bounded from 𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝜃)(ℝ^𝑛) to 𝐵̇_𝜎(𝐿𝑖𝑝^𝐷_𝛼+𝜃)(ℝ^𝑛).

As a corollary of Theorems 13 and 34, we have the following conclusion.

Corollary 35. Let 𝛼>0, 𝜎≥0 and 𝜃>0. Assume in addition that the integer 𝑑 satisfies 𝑑>[𝜎+𝜃+𝛼]. Then the fractional integral operator 𝐼_𝛼 is bounded from 𝐵̇_𝜎(𝐿𝑖𝑝_𝜃^(𝑑+1))(ℝ^𝑛) to 𝐵̇_𝜎(𝐿𝑖𝑝^(𝑑+1)_𝛼+𝜃)(ℝ^𝑛).

7. Application of Theorem 4 to the Fourier Multipliers

As a corollary of Theorem 4, the boundedness of singular integral operators follows.

Corollary 36. Let 𝑚∈𝐶^∞(ℝ^𝑛⧵{0}) be such that 𝑝_𝛽(𝑚)∶=sup_{𝜉∈ℝ𝑛⧵{0}}𝜉^|𝛽|𝜕^𝛽𝑚(𝜉)<∞ (210) for all multi-indices 𝛽. Then the mapping 𝑓∈𝐿²(ℝ^𝑛)↦ℱ⁻¹[𝑚⋅ℱ𝑓]∈𝐿²(ℝ^𝑛) extends naturally to a 𝐵̇_𝜎(𝐿_𝑝,𝜆)(ℝ^𝑛)-bounded operator.

Proof. To check this, by virtue of Theorem 4, we need only to verify 1𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)∞}_{𝑗=𝐽𝑄}𝜑_𝑗(𝐷)ℱ⁻¹𝑚⋅ℱ𝑓^21/2_{𝐿𝑝(𝑄)}≲𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)} (211) for all 𝑟>0 and 𝑄∈𝒟(𝑄_𝑟), where 𝜑∈𝐶^∞_comp(ℝ^𝑛⧵{0}) satisfies (17). As usual we take 𝐽₋ and 𝐽₊ so that it satisfies (138). Write 𝑚_𝑗≡^𝐽+_𝑙=𝐽−𝜑_𝑗+𝑙𝑚𝑗∈ℤ. (212) Then it is easy to verify that 𝑝_𝛽𝑚_𝑗=sup_{𝜉∈ℝ𝑛⧵{0}}𝜉^|𝛽|𝜕^𝛽𝑚_𝑗(𝜉)≲1 (213) and that 𝜑(2^−𝑗⋅)𝑚_𝑗=𝜑(2^−𝑗⋅)𝑚. Consequently, we have ℱ⁻¹𝑚_𝑗(𝑥)≲2^𝑗𝑛1+2^𝑗𝑥^−𝑛−1𝑥∈ℝ^𝑛. (214) Recall that we defined 𝜑_𝑗(𝐷)[ℱ⁻¹[𝑚⋅ℱ𝑓]]≡ℱ⁻¹[𝜑_𝑗⋅𝑚⋅ℱ𝑓]. Hence, a direct calculation yields 𝜑_𝑗(𝐷)ℱ⁻¹𝑚⋅ℱ𝑓=(2𝜋)^𝑛/2ℱ⁻¹𝑚_𝑗∗ℱ⁻¹𝜑_𝑗=()⋅ℱ𝑓2𝜋^𝑛/2ℱ⁻¹𝑚_𝑗∗𝜑_𝑗(𝐷)𝑓. (215) If we insert (214) to (215), then we obtain that 𝜑_𝑗(𝐷)ℱ⁻¹𝑚⋅ℱ𝑓(𝑥)≲𝑀𝜑_𝑗(𝐷)𝑓(𝑥). (216) If we combine Proposition 5 with (216), then we have ^∞_{𝑗=𝐽𝑄}𝜑_𝑗(𝐷)ℱ⁻¹𝑚⋅ℱ𝑓^21/2_{𝐿𝑝(𝑄)}≲^∞_{𝑗=𝐽𝑄}𝑀𝜑_𝑗(𝐷)𝑓^21/2_{𝐿𝑝(𝑄)}≲𝑟^𝜎|𝑄|^{(𝜆/𝑛)+(1/𝑝)}𝑓_{̇𝐵𝜎(𝐿𝐷𝑝,𝜆)}. (217) Thus, (211) is established.

Acknowledgments

The first author is partially supported by Grant-in-Aid for Scientific Research (C), no. 21540199 and no. 24540194, Japan Society for the Promotion of Science. The second author was partially supported by Grant for 2011 Overseas Researcher of Nihon University, Japan. The third author is partially supported by Grant-in-Aid for Scientific Research (C), no. 20540167 and no. 24540159, Japan Society for the Promotion of Science. The fourth author is supported financially by Grant-in-Aid for Young Scientists (B), no. 21740104 and no. 24740085, Japan Society for the Promotion of Science. The authors are thankful to the anonymous referee for his/her pointing out some improvements about the presentation of mathematics.