Frequency-Uniform Decomposition, Function Spaces 𝑋^𝑠_𝑝,𝑞, and Applications to Nonlinear Evolution Equations

Abstract

By combining frequency-uniform decomposition with 𝐿^𝑝(ℓ^𝑞), we introduce a new class of function spaces (denoted by 𝑋^𝑠_𝑝,𝑞). Moreover, we study the Cauchy problem for the generalized NLS equations and Ginzburg-Landau equations in 𝐿^𝑟(0,𝑇;𝑋^𝑠_𝑝,1).

1. Introduction and Notation

In this paper, we introduce a new space (denoted by 𝑋^𝑠_𝑝,𝑞) by combining frequency-uniform decomposition with 𝐿^𝑝(ℓ^𝑞). In this new space, we discuss the semilinear estimates, Schwartz estimates, embedding properties between 𝑋^𝑠_𝑝,𝑞 and Triebel-Lizorkin space 𝐹^𝑠_𝑝,𝑞, and so forth. Moreover, we study the well-posedness of NLS equations and Ginzburg-Landau equations in 𝐿^𝑟(0,𝑇;𝑋^𝑠_𝑝,1).

In recent decades, the well-posedness of Schrödinger equations and Ginzburg-Landau equations is developed quickly. In this paper, we mainly study the local Cauchy problem of NLS equations and Ginzburg-Landau equations in this new space 𝑋^𝑠_𝑝,𝑞. Many people have studied these problems in Bessel potential and Besov spaces, for example, see [1–9], and so forth. We know that Besov spaces and Triebel spaces are generated by combining Littlewood-Paley decomposition with ℓ^𝑞(𝐿^𝑝) and 𝐿^𝑝(ℓ^𝑞) (one can refer to [10] for the properties of Besov and Triebel spaces.). Next, we give a brief introduction of these spaces (see [10, 11]).

To define the Littlewood-Paley decomposition, it is convenient to consider a radial function 𝜓∶ℝ^𝑛→[0,1] such that 𝜓(𝜉)=1,𝜉≤1;smooth𝜉𝜉,1<<2;0,≥2,(1) and to define the sequence {𝜑_𝑘}^∞₀ by 𝜑_𝑘(𝜉)=𝜑2^−𝑘𝜉−𝜑2^−𝑘+1𝜉𝜑,𝑘∈ℕ;₀(𝜉)=1−^∞_𝑘=1𝜑_𝑘(𝜉)=𝜓(𝜉).(2) Since supp𝜑⊂{𝜉∶2⁻¹≤|𝜉|≤2}, we can easily see that supp𝜑_𝑘⊂{𝜉∶2^𝑘−1≤|𝜉|≤2^𝑘+1}, 𝑘∈ℕ, and supp𝜑₀⊂{𝜉∶|𝜉|≤2}. Define Δ_𝑘=𝐹⁻¹𝜑_𝑘𝐹,𝑘∈ℕ∪{0},(3) where 𝐹 is the Fourier transform and 𝐹⁻¹ is the inverse Fourier transform. {Δ_𝑘}^∞₀ is said to be Littlewood-Paley decomposition operator. Formally, we see that ^∞_𝑘=0Δ_𝑘=𝐼,in𝒮^′.(4) We now define the Besov space 𝐵^𝑠_𝑝,𝑞(ℝ^𝑛) for −∞<𝑠<∞ and 0<𝑝,𝑞≤∞ by 𝐵^𝑠_𝑝,𝑞ℝ^𝑛=𝑓∈𝑆^′ℝ^𝑛∶𝑓_{𝐵𝑠𝑝,𝑞},𝑓<∞_{𝐵𝑠𝑝,𝑞}∶=^∞_𝑘=02^𝑘𝑠𝑞Δ_𝑘𝑓^𝑞_𝑝^1/𝑞,(5) and Triebel space 𝐹^𝑠_𝑝,𝑞(ℝ^𝑛) for −∞<𝑠<∞ and 0<𝑝<∞, 0<𝑞≤∞ by 𝐹^𝑠_𝑝,𝑞ℝ^𝑛=𝑓∈𝑆^′ℝ^𝑛∶𝑓_{𝐹𝑠𝑝,𝑞},𝑓<∞_{𝐹𝑠𝑝,𝑞}∶=^∞_𝑘=02^𝑘𝑠𝑞Δ_𝑘𝑓^𝑞1/𝑞_𝑝.(6)

Besov and Triebel spaces had been formulated during 1960s–1980s, which have been widely applied in recent years. Besov spaces 𝐵^𝑠_𝑝,𝑞 and Triebel spaces 𝐹^𝑠_𝑝,𝑞 are introduced by combining Littlewood-Paly decomposition operator with the function spaces ℓ^𝑞(𝐿^𝑝) and 𝐿^𝑝(ℓ^𝑞). Similar to Besov spaces and Triebel spaces, one can use frequency-uniform decomposition and ℓ^𝑞(𝐿^𝑝), 𝐿^𝑝(ℓ^𝑞) to generate a new class of function spaces. Actually, the spaces introduced by frequency-uniform decomposition and ℓ^𝑞(𝐿^𝑝) are modulation spaces. Many people have studied the well-posedness of evolution equations in these spaces (e.g., see [12–16], etc.). In this paper, we will consider the spaces generated by the frequency-uniform decomposition and 𝐿^𝑝(ℓ^𝑞). Firstly, we will recall the definition of the frequency-uniform decomposition and modulation spaces. In the 1930s, Wiener [17] first introduced the frequency-uniform decomposition. So, sometimes we call it Wiener decomposition of ℝ^𝑛 that is roughly denoted by □_𝑘∼𝐹⁻¹𝜒_𝑄𝑘𝐹,𝑘∈ℤ^𝑛,(7) where 𝜒_𝐸 denotes the characteristic function on the set 𝐸. Since 𝑄_𝑘 is a translation of 𝑄₀, □_𝑘 have the same localized structures in frequency space, which are said to be the frequency-uniform decomposition operators. Since 𝜒_𝑄𝑘 can not make differential operations, one needs to redefine it by smooth truncation function (see [12, 14, 18, 19]).

Now we give an exact definition on modulation spaces. We first denote |𝜉|_∞∶=max_{𝑖=1,...,𝑛}|𝜉_𝑖|. Let 𝜌∈𝑆(ℝ^𝑛)∶ℝ^𝑛→[0,1] be a smooth radial bump function adapted to {𝜉∶|𝜉_𝑖|_∞⩽1} with 𝜌(𝜉)=1 if |𝜉|_∞⩽1/2 and 𝜌(𝜉)=0 if |𝜉|_∞⩾1. Let 𝜌_𝑘 be a translation of 𝜌: 𝜌_𝑘(𝜉)=𝜌(𝜉−𝑘),𝑘∈ℤ^𝑛.(8) Since 𝜌_𝑘(𝜉)=1 in the unit closed cube 𝑄_𝑘∶{𝜉∈ℝ^𝑛∶|𝜉−𝑘|_∞⩽1/2} and {𝑄_𝑘}_𝑘∈ℤ^𝑛 is a covering of ℝ^𝑛, one has that for all 𝜉∈ℝ^𝑛, ∑_𝑘∈ℤ^𝑛𝜌_𝑘(𝜉)⩾1. Then, we write 𝜎_𝑘(𝜉)=𝜌_𝑘(𝜉)_{𝑘∈ℤ𝑛}𝜌_𝑘(𝜉)⁻¹,𝑘∈ℤ^𝑛.(9) It is easy to see that 𝜎_𝑘(𝜉)≥𝑐,𝜉∈𝑄_𝑘;supp𝜎_𝑘⊂𝜉∶𝜉−𝑘_∞≤1_{𝑘∈ℤ𝑛};𝜎_𝑘(𝜉)≡1.∀𝜉∈ℝ^𝑛;𝐷^𝛼𝜎_𝑘(𝜉)≤𝐶_|𝛼|,∀𝜉∈ℝ^𝑛,𝛼∈(𝑁∪{0})^𝑛.(∗∗∗) Hence, the set Υ_𝑛=𝜎_{𝑘𝑘∈ℤ𝑛}∶𝜎_{𝑘𝑘∈ℤ𝑛}satisfy(∗∗∗)(10) is nonvoid. Let {𝜎_𝑘}_𝑘∈ℤ^𝑛∈Υ_𝑛 be a function sequence. Denote □_𝑘∶=𝐹⁻¹𝜎_𝑘𝐹,𝑘∈ℤ^𝑛,(11) which is said to be the frequency-uniform decomposition operator. For any 𝑘∈ℤ^𝑛, we set |𝑘|=|𝑘₁|+⋯+|𝑘_𝑛|, ⟨𝑘⟩=1+|𝑘|. For any 𝑠∈ℝ, 0<𝑝,𝑞≤∞, we denote 𝑀^𝑠_𝑝,𝑞ℝ^𝑛=𝑓∈𝑆^′ℝ^𝑛∶𝑓_{𝑀𝑠𝑝,𝑞}=_{𝑘∈ℤ𝑛}⟨𝑘⟩^𝑠𝑞□_𝑘𝑓^𝑞_𝐿𝑝^1/𝑞<∞.(12)𝑀^𝑠_𝑝,𝑞∶=𝑀^𝑠_𝑝,𝑞(ℝ^𝑛) is said to be a modulation space, which was first introduced by Feichtinger [19] in the case 1⩽𝑝,𝑞⩽∞ (Appendix Theorem E shows the properties on 𝑀^𝑠_𝑝,𝑞(ℝ^𝑛)). For any 0⩽𝑠<∞, 0<𝑝,𝑞⩽∞, we denote 𝐸^𝑠_𝑝,𝑞ℝ^𝑛=𝑓∈𝑆^′ℝ^𝑛∶𝑓_{𝐸𝑠𝑝,𝑞}=_{𝑘∈ℤ𝑛}2^{|𝑘|𝑠𝑞}□_𝑘𝑓^𝑞_𝐿𝑝^1/𝑞<∞.(13)𝐸^𝑠_𝑝,𝑞∶=𝐸^𝑠_𝑝,𝑞(ℝ^𝑛) was first introduced by Baoxiang et al. [12].

We now introduce a new class of function spaces (denoted by 𝑋^𝑠_𝑝,𝑞) by combining the frequency-uniform decomposition with 𝐿^𝑝(ℓ^𝑞). For any 0<𝑝<∞, 0<𝑞⩽∞, 0⩽𝑠<∞, we denote 𝑋^𝑠_𝑝,𝑞ℝ^𝑛∶=𝑓∈𝑆^′ℝ^𝑛∶𝑓_{𝑋𝑠𝑝,𝑞}=_{𝑘∈ℤ𝑛}2^{|𝑘|𝑠𝑞}□_𝑘𝑓^𝑞1/𝑞_𝐿𝑝<∞.(14) Moreover, we denote 𝐿^𝑟ℝ,𝑋^𝑠_𝑝,𝑞∶=𝑓∈𝒮^′∶_ℝ𝑓^𝑟_{𝑋𝑠𝑝,𝑞}𝑑𝑡^1/𝑟<∞.(15)

Main Results

Theorem 1. Assume 𝑓(𝑢)=𝑢|𝑢|^𝑘, 𝑘=2𝑚, 𝑚∈ℤ⁺, and 𝑟⩾𝑘+2; then for any initial data 𝑢₀∈𝑋^𝑠_1,1, 0⩽𝑠<∞, there exists 𝑇^∗∶=𝑇^∗(‖𝑢₀‖_𝑋𝑠1,1)>0 such that the initial value problem 𝑢(𝑡)=𝑆(𝑡)𝑢₀−𝑖^𝑡₀𝑆(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏,𝑁𝐿𝑆𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛(16) has a unique solution 𝑢∈𝐿^𝑟_loc0,𝑇^∗;𝑋^𝑠_1,1.(17) Moreover, if 𝑇^∗<∞, then ‖𝑢‖_𝐿^𝑟_(0,𝑇^∗_;𝑋^𝑠_1,1)=∞,(18) where 𝑆(𝑡)=𝐹⁻¹𝑒^{−𝑖𝑡|𝜉|2}𝐹.

Theorem 2. Assume 𝑓(𝑢)=(𝑏+𝑖𝛽)𝑢|𝑢|^𝑘+𝜇𝑢, 𝑘=2𝑚, 𝑚∈ℤ⁺, and 𝑟⩾𝑘+2. Assume also 𝑎,𝑏>0, 𝛼,𝛽,𝜇∈ℝ. Then, for any initial data 𝑢₀∈𝑋^𝑠_𝑝,1, 1⩽𝑝<∞, 0⩽𝑠<∞, there exists 𝑇^∗∶=𝑇^∗(‖𝑢₀‖_{𝑋𝑠𝑝,1})>0 such that the initial value problem (complex Ginzburg-Landau equation) 𝑢(𝑡)=𝑈(𝑡)𝑢₀−𝑖^𝑡₀𝑈(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏(19) has a unique solution 𝑢∈𝐿^𝑟_loc0,𝑇^∗;𝑋^𝑠_𝑝,1.(20) Moreover, if 𝑇^∗<∞, then ‖𝑢‖_𝐿^𝑟_(0,𝑇^∗_;𝑋^𝑠_𝑝,1)=∞,(21) where 𝑈(𝑡)=𝐹⁻¹𝑒^{−𝑡𝑃(𝜉)}𝐹, 𝑃(𝜉)=1+(𝑎+𝑖𝛼)|𝜉|².

This paper is organized as follows. In Section 2, we will state some properties on 𝑋^𝑠_𝑝,𝑞, which is useful to establish the following inclusions. Section 3 is devoted to considering the multiplication algebra of 𝑋^𝑠_𝑝,𝑞. Some dispersive smooth effects for the Schrödinger and Ginzburg-Landau semigroups will be given in Section 4. Theorems 1 and 2 will be proved in Section 5.

2. The Properties on 𝑋^𝑠_𝑝,𝑞

In order to study the Cauchy problem in 𝑋^𝑠_𝑝,𝑞, we first give some properties on 𝑋^𝑠_𝑝,𝑞 as follows.

Proposition 3. Letting 0⩽𝑠<∞, 0<𝑝<∞, 0<𝑞⩽∞, one has(1)if 𝑞₁≤𝑞₂, then 𝑋^𝑠_𝑝,𝑞1⊂𝑋^𝑠_𝑝,𝑞2;(22)(2)if 𝜀>0, then 𝑋^𝑠+𝜀_𝑝,𝑞1⊂𝑋^𝑠_𝑝,𝑞2;(23)(3)if 𝑝<∞, then 𝐸^𝑠_{𝑝,𝑝∧𝑞}⊂𝑋^𝑠_𝑝,𝑞⊂𝐸^𝑠_{𝑝,𝑝∨𝑞},,.𝑝∧𝑞=min𝑝,𝑞𝑝∨𝑞=max𝑝,𝑞(24)

Proof. Because ℓ^𝑝⊂ℓ^𝑝+𝑎, 𝑎≥0, we can get the results of (1) directly. For (2), we know that _{𝑘∈ℤ𝑛}2^{|𝑘|𝑠𝑞2}𝑎_𝑘^𝑞21/𝑞2=_{𝑘∈ℤ𝑛}2^{−|𝑘|𝜀𝑞2}2^{|𝑘|(𝑠+𝜀)𝑞2}𝑎_𝑘^𝑞21/𝑞2≲sup_𝑘∈ℤ^𝑛2^{|𝑘|(𝑠+𝜀)}𝑎_𝑘,(𝜀>0).(25)
Then, letting 𝑎_𝑘=□_𝑘𝑓, we get (2).
Finally, we prove (3). Let 𝑏_𝑘=2^|𝑘|𝑠□_𝑘𝑓. The proof can be classified into the following two kinds of cases.
Case  1 (𝑞≤𝑝). In this case, ℓ^𝑞⊂ℓ^𝑝. We have 𝑏_{𝑘ℓ𝑝(𝐿𝑝)}≤𝑏_{𝑘𝐿𝑝(ℓ𝑞)}≤𝑏_{𝑘ℓ𝑞(𝐿𝑝)}.(26) Actually, noticing that ℓ^𝑞⊂ℓ^𝑝, we have ‖𝑏_𝑘‖_{ℓ𝑝(𝐿𝑝)}=‖𝑏_𝑘‖_{𝐿𝑝(ℓ𝑝)}≤‖𝑏_𝑘‖_{𝐿𝑝(ℓ𝑞)}. So, the first part of the above inequality holds. On the other hand, by Minkowski’s inequality, we have 𝑏_{𝑘𝐿𝑝(ℓ𝑞)}=^∞_𝑘=0𝑏_𝑘^𝑞_𝐿^1/𝑞_𝑝/𝑞⩽^∞_𝑘=0𝑏_𝑘^𝑞_{𝐿𝑝/𝑞}^1/𝑞=𝑏_{𝑘ℓ𝑞(𝐿𝑝)}.(27) This proves the second part.
Case  2 (𝑝≤𝑞). By Minkowski’s inequality and ℓ^𝑝⊂ℓ^𝑞, we have 𝑏_{𝑘ℓ𝑞(𝐿𝑝)}=_ℝ𝑛𝑏_𝑘^𝑝𝑑𝑥_ℓ𝑞/𝑝^1/𝑝⩽_ℝ𝑛𝑏_𝑘^𝑝_ℓ𝑞/𝑝𝑑𝑥^1/𝑝=𝑏_{𝑘𝐿𝑝(ℓ𝑞)}⩽𝑏_{𝑘ℓ𝑝(𝐿𝑝)}.(28) This finishes the proof of this proposition.

Proposition 4 (completeness). For any 0⩽𝑠<∞, 0<𝑝<∞, 0<𝑞⩽∞, one has the following:(1)𝑋^𝑠_𝑝,𝑞 is a quasi-Banach space. Moreover, if 1⩽𝑝<∞, 1⩽𝑞⩽∞, then 𝑋^𝑠_𝑝,𝑞 is a Banach space;(2)𝑆(ℝ^𝑛)⊂𝑋⁰_𝑝,𝑞⊂𝑆^′(ℝ^𝑛);(3)if 0<𝑝,𝑞<∞, then 𝒮(ℝ^𝑛) is dense in 𝑋⁰_𝑝,𝑞.

Proof
 
Step  1. In this step, we prove that 𝑆(ℝ^𝑛)⊂𝑋⁰_𝑝,𝑞⊂𝑆^′(ℝ^𝑛). Actually, by 𝑆(ℝ^𝑛)⊂𝐸⁰_𝑝,𝑞⊂𝑆^′(ℝ^𝑛) [12] and 𝐸^𝑠_{𝑝,𝑝∧𝑞}⊂𝑋^𝑠_𝑝,𝑞⊂𝐸^𝑠_{𝑝,𝑝∨𝑞} (Proposition 3), we immediately obtain the result.
Step 2. 𝑋^𝑠_𝑝,𝑞 is a quasinormed space. We prove the completeness. Let {𝑓_ℓ}^∞_ℓ=1 be a Cauchy sequence in 𝑋^𝑠_𝑝,𝑞 (with respect to a fixed quasinorm in 𝑋^𝑠_𝑝,𝑞). Proposition 3 shows that {𝑓_ℓ}^∞_ℓ=1 is also a Cauchy sequence in 𝐸^𝑠_{𝑝,𝑝∨𝑞}. Because 𝐸^𝑠_{𝑝,𝑝∨𝑞} is a complete topological linear space, we can find a limit element 𝑓∈𝐸^𝑠_{𝑝,𝑝∨𝑞}. Then, 𝐹⁻¹𝜎_𝑘𝐹𝑓_ℓ converges to 𝐹⁻¹𝜎_𝑘𝐹𝑓 in 𝐸^𝑠_{𝑝,𝑝∨𝑞} if ℓ→∞. On the other hand, {𝐹⁻¹𝜎_𝑘𝐹𝑓_ℓ}^∞_ℓ=1 is a Cauchy sequence in 𝐿^𝑝(ℝ^𝑛) (𝑋^𝑠_𝑝,𝑞⊂𝐸^𝑠_{𝑝,𝑝∨𝑞}, 𝑠⩾0.). By Appendix Theorem B, it is also a Cauchy sequence in 𝐿^∞(ℝ^𝑛). This shows that the limiting element of {𝐹⁻¹𝜎_𝑘𝐹𝑓_ℓ}^∞_ℓ=1 in 𝐿^𝑝(ℝ^𝑛) (which is the same as in 𝐿^∞(ℝ^𝑛)) coincides with {𝐹⁻¹𝜎_𝑘𝐹𝑓}. Now it follows by standard arguments that 𝑓 belongs to 𝑋^𝑠_𝑝,𝑞 and that 𝑓_ℓ converges in 𝑋^𝑠_𝑝,𝑞 to 𝑓. Hence, 𝑋^𝑠_𝑝,𝑞 is complete.
Step 3. We prove that 𝒮(ℝ^𝑛) is dense in 𝑋⁰_𝑝,𝑞(ℝ^𝑛) if 0<𝑝<∞ and 0<𝑞<∞. Let 𝑓∈𝑋⁰_𝑝,𝑞; then we put 𝑓_𝑁(𝑥)=^𝑁_𝑘=0𝐹⁻¹𝜎_𝑘𝐹𝑓.(29) Of course, 𝑓_𝑁∈𝑋⁰_𝑝,𝑞(ℝ^𝑛). Consequently (by Appendix Theorem F), 𝑓−𝑓_{𝑁𝑋0𝑝,𝑞(ℝ𝑛)}⩽𝑐^∞1_{𝑘=𝑁𝑟=−1}𝐹⁻¹𝜎_𝑘𝜎_𝑘+𝑟𝐹𝑓^𝑞1/𝑞_{𝐿𝑝(ℝ𝑛)}⩽𝑐^∞_𝑘=𝑁𝐹⁻¹𝜎_𝑘𝐹𝑓^𝑞1/𝑞_{𝐿𝑝(ℝ𝑛)}.(30) Lebesgue’s bounded convergence theorem proves that the right-hand side of the above inequality tends to zero if 𝑁→∞. Hence, 𝑓_𝑁 approximates 𝑓 in 𝑋⁰_𝑝,𝑞(ℝ^𝑛). Next, we let 𝜑∈𝒮 with 𝜑(0)=1 and supp𝐹𝜑⊂{𝑦∶|𝑦|⩽1}. Let 𝑓_𝛿(𝑥)=𝜑(𝛿𝑥)𝑓(𝑥) with 0<𝛿<1. Then, (𝑓_𝑁)_𝛿∈𝒮(ℝ^𝑛) approximates 𝑓_𝑁 in 𝐿^Ω_𝑝∶={𝑓∈𝐿^𝑝∶supp𝑓⊂Ω} with Ω={𝑦∶|𝑦|⩽2^𝑁+2} if 𝛿→0. However this is also an approximation of 𝑓_𝑁 in 𝑋⁰_𝑝,𝑞(ℝ^𝑛). This proves that 𝒮(ℝ^𝑛) is dense in 𝑋⁰_𝑝,𝑞(ℝ^𝑛).

Proposition 5 (equivalent norm). Assume {𝜎_𝑘}_𝑘∈ℤ^𝑛, {𝜙_𝑘}_𝑘∈ℤ^𝑛∈Υ_𝑛, 0<𝑝<∞, 0<𝑞⩽∞. Then, {𝜎_𝑘}_𝑘∈ℤ^𝑛 and {𝜙_𝑘}_𝑘∈ℤ^𝑛 generate equivalent norms on 𝑋^𝑠_𝑝,𝑞.

Proof. Firstly, we have the following translation equality: 𝐹⁻¹𝑚𝐹𝑓(𝑥)=𝑒^𝑖𝑥𝑘𝐹⁻¹𝑚(⋅+𝑘)𝐹𝑒^{−𝑖𝑘𝑦}𝑓𝑦(𝑥).(31) For the sake of convenience, we denote □^𝜎_𝑘=𝐹⁻¹𝜎_𝑘𝐹,□^𝜙_𝑘=𝐹⁻¹𝜙_𝑘𝐹.(32) Then, we have □^𝜎_𝑘=_|ℓ|∞≤1□^𝜎_𝑘□^𝜙_𝑘+ℓ,□^𝜎_𝑘𝑓_{𝐿𝑝(ℓ𝑞)}≤_|ℓ|∞≤1□^𝜎_𝑘□^𝜙_𝑘+ℓ𝑓_{𝐿𝑝(ℓ𝑞)}.(33) By Appendix Theorem F, we have □^𝜎_𝑘□^𝜙_𝑘+ℓ𝑓_{𝐿𝑝(ℓ𝑞)}≲sup_𝑘𝜎_𝑘𝐻𝑠□^𝜙_𝑘+ℓ𝑓_{𝐿𝑝(ℓ𝑞)},1𝑠>𝑛−11∧𝑝2.(34) Then, by ‖𝜎_𝑘‖_𝐻𝑠<𝐶, we have □^𝜎_𝑘𝑓_{𝐿𝑝(ℓ𝑞)}≲_|ℓ|∞≤1□^𝜙_𝑘+ℓ𝑓_{𝐿𝑝(ℓ𝑞)}.(35) By the above discussion, we get ‖𝑓‖^{𝜎𝑘}_{𝑋0𝑝,𝑞}≲‖𝑓‖^{𝜙𝑘}_{𝑋0𝑝,𝑞}.

The inverse inequality can be proved similarly. Based on the above observations, one can also obtain that ‖𝑓‖^{𝜎𝑘}_{𝑋𝑠𝑝,𝑞}∼‖𝑓‖^{𝜙𝑘}_{𝑋𝑠𝑝,𝑞}.

Theorem 6. Assume 1⩽𝑝₂⩽𝑝₁<∞, 1⩽𝑞⩽∞, 0⩽𝑠<∞; then one has ‖𝑢‖_𝑋^𝑠_𝑝1,𝑞≲‖𝑢‖_𝑋^𝑠_𝑝2,𝑞.(36)

Proof. By the definition of 𝑋^𝑠_𝑝,𝑞, we have ‖𝑢‖_𝑋^𝑠_𝑝1,𝑞=_{𝑖∈ℤ𝑛}2^{|𝑖|𝑞𝑠}□_𝑖𝑢^𝑞1/𝑞_𝐿𝑝1=_{𝑖∈ℤ𝑛}2^{|𝑖|𝑞𝑠}□_{𝑖|ℓ|∞⩽1}□_𝑖+ℓ𝑢^𝑞1/𝑞_𝐿𝑝1=_{𝑖∈ℤ𝑛}2^{|𝑖|𝑞𝑠}𝜎^∨_𝑖∗_|ℓ|∞⩽1□_𝑖+ℓ𝑢^𝑞1/𝑞_𝐿𝑝1⩽_{𝑖∈ℤ𝑛}2^{|𝑖|𝑞𝑠}𝜎^∨_𝑖|∗|_|ℓ|∞⩽1□_𝑖+ℓ𝑢^𝑞1/𝑞_𝐿𝑝1=_{𝑖∈ℤ𝑛}2^{|𝑖|𝑞𝑠}𝜎^∨₀|∗|_|ℓ|∞⩽1□_𝑖+ℓ𝑢^𝑞1/𝑞_𝐿𝑝1≲_{𝑖∈ℤ𝑛}2^{|𝑖|𝑞𝑠}_|ℓ|∞⩽1□_𝑖+ℓ𝑢^𝑞1/𝑞_𝐿𝑝2≲_{𝑖∈ℤ𝑛}2^{|𝑖|𝑞𝑠}□_𝑖𝑢^𝑞1/𝑞_𝐿𝑝2.(37)
In this proof, we used ‖|𝜎^∨₀|∗𝑓‖_𝐿𝑝1⩽‖𝑓‖_𝐿𝑝2 (𝑝₁⩾𝑝₂, Young’s inequality) and Appendix Theorem G.

Theorem 7. Assume 𝑓∈𝐿²; then one has 𝑓_𝐿2∼𝑓_𝑋02,2.(38)

Proof. We first prove ‖𝑓‖_𝐿2≲‖𝑓‖_𝑋02,2. Consider the following: 𝑓_𝐿2=_{ℝ𝑛𝑖∈ℤ𝑛}□_𝑖𝑓²𝑑𝑥^1/2=_{ℝ𝑛𝑖∈ℤ𝑛}𝜎_𝑖𝐹𝑓²𝑑𝑥^1/2≲_{ℝ𝑛𝑖∈ℤ𝑛}𝜎_𝑖𝐹𝑓²𝑑𝑥^1/2=_{ℝ𝑛𝑖∈ℤ𝑛}𝐹⁻¹𝜎_𝑖𝐹𝑓²𝑑𝑥^1/2=_{ℝ𝑛𝑖∈ℤ𝑛}𝐹⁻¹𝜎_𝑖𝐹𝑓^21/22𝑑𝑥^1/2=𝑓_𝑋02,2.(39) The inverse inequality can be proved similarly (∑_𝑖∈ℤ^𝑛|𝜎_𝑖𝐹𝑓|²≲|∑_𝑖∈ℤ^𝑛𝜎_𝑖𝐹𝑓|²). Moreover, by the similar discussion as mentioned previously, we can also obtain that ‖(∑_𝑘∈ℤ^𝑛|𝐹⁻¹(1+|𝜉|²)^𝑠/2𝜎_𝑘𝐹𝑓|²)^1/2‖_𝐿2∼‖𝑓‖_𝐻𝑠2.

Theorem 8. Let 0<𝑠<∞, 0<𝑝<∞, 0<𝑞⩽∞. Then, one has 𝑋^𝑠_𝑝,𝑞ℝ^𝑛⊂𝐶^∞ℝ^𝑛.(40)

Proof. One can obtain the result directly by Appendix Theorem D and Proposition 3.

Theorem 9. Let 0<𝑝<∞, 0<𝑞⩽1. One has 𝑋⁰_𝑝,𝑞ℝ^𝑛⊂𝐹⁰_𝑝,𝑞ℝ^𝑛.(41)

Proof. Let 𝑎_𝑘=max(0,2^𝑘−1−√𝑛), 𝑏_𝑘=2^𝑘+1+√𝑛. We can easily find that if |𝑖|∈[𝑎_𝑘,𝑏_𝑘], then Δ_𝑘□_𝑖𝑓=0. For 𝑝>0, 𝑞⩽1, by Appendix Theorem F, we have Δ_𝑘𝑓_{𝐿𝑝(ℓ𝑞)}=_𝑘Δ_𝑘𝑓^𝑞1/𝑞_𝐿𝑝⩽_{𝑘𝑖∈ℤ𝑛,|𝑖|∈[𝑎𝑘,𝑏𝑘]}Δ_𝑘□_𝑖□_𝑖𝑓^𝑞1/𝑞_𝐿𝑝≲_{𝑘𝑖∈ℤ𝑛,|𝑖|∈[𝑎𝑘,𝑏𝑘]}𝐹⁻¹𝜑_𝑘𝜎_𝑖𝐹□_𝑖𝑓^𝑞1/𝑞_𝐿𝑝≲_{𝑖∈ℤ𝑛|𝑖|𝑘,∈[𝑎𝑘,𝑏𝑘]}𝐹⁻¹𝜑_𝑘𝜎_𝑖𝐹□_𝑖𝑓^𝑞1/𝑞_𝐿𝑝≲_{𝑖∈ℤ𝑛}□_𝑖𝑓^𝑞1/𝑞_𝐿𝑝.(42) This proves the theorem.

Theorem 10. Let 𝐹^𝑠_𝑝,𝑞(ℝ^𝑛) be Triebel space; then one has 𝐹^𝑠_2,𝑞ℝ^𝑛⊂𝑋⁰_2,𝑞ℝ^𝑛1,𝑠>𝑛𝑞−12,𝑋0<𝑞<2,_02,𝑞ℝ^𝑛⊂𝐹^𝑠_2,𝑞ℝ^𝑛1,𝑠<𝑛𝑞−12,2<𝑞⩽∞.(43)

Proof. By Appendix Theorem A, we know that 𝐻^𝑠⊂𝐸⁰_2,𝑞1;𝑠>𝑛𝑞−12,0<𝑞<2.(44) Then, by the embedding between 𝐸^𝑠_𝑝,𝑞 and 𝑋^𝑠_𝑝,𝑞 (Proposition 3), we have (for any given 𝜀>0) 𝐹^{𝑛(1/𝑞−1/2)+𝜀}_2,𝑞⊂𝐹^{𝑛(1/𝑞−1/2)+𝜀}_2,2=𝐻^{𝑛(1/𝑞−1/2)+𝜀}⊂𝐸⁰_2,𝑞⊂𝑋⁰_2,𝑞.(45) This proves the first result (𝐹^𝑠_2,2∼𝐻^𝑠, [11]).
On the other hand, by Appendix Theorem A, we have 𝐸⁰_2,𝑞⊂𝐻^𝑠1;𝑠<𝑛𝑞−12,2<𝑞⩽∞.(46)
Then, by the embedding between 𝐸^𝑠_𝑝,𝑞 and 𝑋^𝑠_𝑝,𝑞 (Proposition 3), we have 𝑋⁰_2,𝑞⊂𝐸⁰_2,𝑞⊂𝐻^{𝑛(1/𝑞−1/2)−𝜀}⊂𝐹^{𝑛(1/𝑞−1/2)−𝜀}_2,𝑞.(47)
This implies the theorem.

Theorem 11. Let 𝑠>𝑛(1/𝑞−1/2), 0<𝑞<2. Then, one has 𝑓_𝑋02,𝑞⩽𝐶_𝑠1+ln1+𝑓_𝐻𝑠^{(1/𝑞−1/2)}𝑓_{𝐻𝑛(1/𝑞−1/2)}.(48)

Proof. By the definition of 𝑋^𝑠_𝑝,𝑞, we have 𝑓_𝑋02,𝑞=_{𝑘∈ℤ𝑛}□_𝑘𝑓^𝑞1/𝑞_𝐿2≲_{|𝑘|⩽𝑇}□_𝑘𝑓^𝑞1/𝑞_𝐿2+_|𝑘|>𝑇□_𝑘𝑓^𝑞1𝑞_𝐿2=𝐴+𝐵.(49) Then, by Appendix Theorem F, Theorem 7 and Hölder’s inequality, the term 𝐴 can be dominated by 𝐴=_{|𝑘|⩽𝑇}𝐹⁻¹(1+|𝑘|)²1+𝜉^{2𝑛(1/𝑞−1/2)/2}×1+|𝜉|²(1+|𝑘|)^{2𝑛(1/𝑞−1/2)/2}𝜎_𝑘𝐹𝑓^𝑞1/𝑞_𝐿2≲_{|𝑘|⩽𝑇}𝐹⁻¹1+𝜉²(1+|𝑘|)^{2𝑛(1/𝑞−1/2)/2}𝜎_𝑘𝐹𝑓^𝑞1/𝑞_𝐿2≲_{|𝑘|⩽𝑇}(1+|𝑘|)^{−𝑛(1/𝑞−1/2)}×_{|𝑘|⩽𝑇}𝐹⁻¹1+|𝜉|^{2𝑛(1/𝑞−1/2)/2}𝜎_𝑘𝐹𝑓^21/2_𝐿2≲_{0⩽𝑖⩽𝑇}(1+𝑖)^{−1(1/𝑞−1/2)}𝑓_{𝐻𝑛(1/𝑞−1/2)}≲[ln(1+𝑇)]^{(1/𝑞−1/2)}𝑓_{𝐻(𝑛(1/𝑞−1/2))}.(50) On the other hand, by the similar discussion as the above, we have 𝐵≲_|𝑘|>𝑇(1+|𝑘|)^{−2𝑠𝑞/(2−𝑞)(1/𝑞−1/2)}×_|𝑘|>𝑇𝐹⁻¹1+𝜉^2𝑠/2𝜎_𝑘𝐹𝑓^21/2_𝐿2≲_{𝑖>𝑇|𝑘|=𝑖}(1+𝑖)^{−2𝑠𝑞/(2−𝑞)(1/𝑞−1/2)}𝑓_𝐻𝑠≲_𝑖>𝑇(1+𝑖)^{−2𝑠𝑞/(2−𝑞)+𝑛−1(1/𝑞−1/2)}𝑓_𝐻𝑠⩽𝐶_𝑠𝑇^{−𝑠+𝑛(1/𝑞−1/2)}𝑓_𝐻𝑠.(51) Let 𝑇=max1,𝑓_𝐻^{1/(𝑠−𝑛(1/𝑞−1/2))}_𝑠.(52) From the above discussion, we get the result.

Remark 12. By Theorem 9, we have 𝑋⁰_𝑝,1ℝ^𝑛⊂𝐹⁰_𝑝,1ℝ^𝑛.(53) Then, by Theorem 11, we have 𝐻^𝑠ℝ^𝑛⊂𝑋⁰_2,1ℝ^𝑛⊂𝐹⁰_2,1ℝ^𝑛𝑛,𝑠>2.(54) So, 𝑋⁰_2,1 is an intermediate space between 𝐻^𝑠(𝑠>𝑛/2) and 𝐹⁰_2,1.

3. Multilinear Estimate in 𝑋^𝑠_𝑝,𝑞

It is well known that 𝐵^𝑠_𝑝,𝑞 is a multiplication algebra if 𝑠>𝑛/𝑝, see [2]. The regularity indices, for which 𝑋^𝑠_𝑝,𝑞 constitutes a multiplication algebra, are quite different from those of Besov space.

Theorem 13. Let 0⩽𝑠<∞, 0<𝑝⩽𝑝₁, 𝑝₂<∞, 0<𝑞⩽∞. If 1/𝑝=1/𝑝₁+1/𝑝₂, then one has 𝑓𝑔_{𝑋𝑠𝑝,𝑞}≲𝐶2^𝐶𝑠𝑓_{𝑋𝑠𝑝1,𝑞∧1}𝑔_{𝑋𝑠𝑝2,𝑞∧1},(55) where 𝐶 is independent of 𝑠,𝑞.

Proof. Recall that we denote 𝑘∈ℤ^𝑛 and |𝑘|=|𝑘₁|+⋯+|𝑘_𝑛|. We have 𝑓𝑔_{𝑋𝑠𝑝,𝑞}=_{𝑘∈ℤ𝑛}2^{|𝑘|𝑠𝑞}□_𝑘𝑓𝑔^𝑞1/𝑞_𝐿𝑝.(56) By ∑_𝑘∈ℤ^𝑛𝜎_𝑘=1, we can obtain the decomposition of 𝑓𝑔 as follows: 𝑓𝑔=_{𝑖,𝑗∈ℤ𝑛}□_𝑖𝑓□_𝑗𝑔.(57) Then, we have □_𝑘𝑓𝑔=_{𝑖,𝑗∈ℤ𝑛}□_𝑘□_𝑖𝑓□_𝑗𝑔.(58) It is easy to see that 𝐹(□_𝑖𝑓□_𝑗𝑔)=(𝜎_𝑖𝑓)∗(𝜎_𝑗𝑔) and supp𝜎_𝑖𝑓∗𝜎_𝑗𝑔⊂𝜉∶𝜉−𝑖−𝑗⩽2√2𝑛.(59) Then, we have □_𝑘□_𝑖𝑓□_𝑗𝑔=□_{𝑘𝑖∈ℤ𝑛}□_𝑖𝑓□_{𝑗(𝑖,𝑘)}𝑔,𝑘−𝑖−𝑗(𝑖,𝑘)⩽𝐶(𝑛).(60)
Case  1  (𝑞⩽1). Based on the previous observations and Appendix Theorem F, we have 𝑓𝑔_{𝑋𝑠𝑝,𝑞}=_{𝑘∈ℤ𝑛}2^{|𝑘|𝑠𝑞}□_𝑘𝑓𝑔^𝑞1/𝑞_𝐿𝑝⩽_{𝑘∈ℤ𝑛}2^{|𝑘|𝑠𝑞}□_{𝑘𝑖∈ℤ𝑛}□_𝑖𝑓□_{𝑗(𝑖,𝑘)}𝑔^𝑞1/𝑞_𝐿𝑝⩽_{𝑘∈ℤ𝑛𝑖∈ℤ𝑛}2^{|𝑘|𝑠𝑞}□_𝑖𝑓□_{𝑗(𝑖,𝑘)}𝑔^𝑞1/𝑞_𝐿𝑝⩽_{𝑘∈ℤ𝑛𝑖∈ℤ𝑛}2^{|𝑘−𝑖−𝑗(𝑖,𝑘)|𝑠𝑞}2^|𝑖|𝑠□_𝑖𝑓2^{|𝑗(𝑖,𝑘)|𝑠}□_{𝑗(𝑖,𝑘)}𝑔^𝑞1/𝑞_𝐿𝑝≲^𝐶(𝑛)_{𝑡=0𝑘∈ℤ𝑛𝑖∈ℤ𝑛,𝑗=𝑘−𝑖±𝑡}2^{|𝑘−𝑖−𝑗|𝑠𝑞}×2^|𝑖|𝑠□_𝑖𝑓2^|𝑗|𝑠□_𝑗𝑔^𝑞1/𝑞_𝐿𝑝⩽𝐶2^𝐶𝑠_{𝑖∈ℤ𝑛}2^{|𝑖|𝑠𝑞}□_𝑖𝑓^𝑞1/𝑞_{𝑗∈ℤ𝑛}2^{|𝑗|𝑠𝑞}□_𝑗𝑔^𝑞1/𝑞_𝐿𝑝⩽𝐶2^𝐶𝑠𝑓_{𝑋𝑠𝑝1,𝑞}𝑔_{𝑋𝑠𝑝2,𝑞},(61) where we used |𝑘−𝑖−𝑗(𝑖,𝑘)|<𝐶(𝑛) and the following Young’s inequality: 𝑎_𝑖∗𝑏_𝑗ℓ𝑝⩽𝑎_𝑖ℓ𝑝1𝑏_𝑗ℓ𝑝21,1+𝑝=1𝑝₁+1𝑝₂.(62)
Case  2 (𝑞>1). By the definition of 𝑋^𝑠_𝑝,𝑞 and Appendix Theorem F, we have 𝑓𝑔_{𝑋𝑠𝑝,𝑞}=_{𝑘∈ℤ𝑛}2^{|𝑘|𝑠𝑞}□_𝑘𝑓𝑔^𝑞1/𝑞_𝐿𝑝⩽_{𝑘∈ℤ𝑛}2^|𝑘|𝑠□_𝑘𝑓𝑔_𝐿𝑝,ℓ¹⊂ℓ^𝑞,if⩽𝑞⩾1_{𝑘∈ℤ𝑛}2^|𝑘|𝑠□_{𝑘𝑖∈ℤ𝑛}□_𝑖𝑓□_{𝑗(𝑖,𝑘)}𝑔_𝐿𝑝⩽_{𝑘∈ℤ𝑛}2^|𝑘|𝑠_{𝑖∈ℤ𝑛}□_𝑖𝑓□_{𝑗(𝑖,𝑘)}𝑔_𝐿𝑝≲^𝐶(𝑛)_{𝑡=0𝑘∈ℤ𝑛}2^|𝑘|𝑠_{𝑖∈ℤ𝑛,𝑗=𝑘−𝑖±𝑡}□_𝑖𝑓□_𝑗𝑔_𝐿𝑝⩽^𝐶(𝑛)_{𝑡=0𝑘∈ℤ𝑛}2^{|𝑘−𝑖−𝑗|𝑠}_{𝑖∈ℤ𝑛,𝑗=𝑘−𝑖±𝑡}2^|𝑖|𝑠□_𝑖𝑓2^|𝑗|𝑠□_𝑗𝑔_𝐿𝑝⩽𝐶2^𝐶𝑠_{𝑖∈ℤ𝑛}2^𝑖𝑠□_𝑖𝑓_{𝑗∈ℤ𝑛}2^𝑗𝑠□_𝑗𝑔_𝐿𝑝⩽𝐶2^𝐶𝑠𝑓_{𝑋𝑠𝑝1,1}𝑔_{𝑋𝑠𝑝2,1}.(63) By the above discussion, we can obtain the claimed results.

Corollary 14. Let 0⩽𝑠<∞, 0<𝑝⩽𝑝_𝑖<∞, 𝑖=1,…,𝑁, 0<𝑞⩽∞. If 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑁, then one has ^𝑁_𝑖=1𝑢_{𝑖𝑋𝑠𝑝,𝑞}≲𝐶^𝑁2^{𝑁𝐶𝑁𝑠}_𝑖=1𝑢_{𝑖𝑋𝑠𝑝𝑖,𝑞∧1},(64) where 𝐶 is independent of 𝑠,𝑞.

Proof. Let 1/𝑝=1/𝑝₁+1/𝑝^∗, where 1/𝑝^∗=1/𝑝₂+⋯+1/𝑝_𝑁. Then, we have ^𝑁_𝑖=1𝑢_{𝑖𝑋𝑠𝑝,𝑞}⩽𝐶2^𝐶𝑠𝑢_{1𝑋𝑠𝑝1,𝑞∧1}^𝑁_𝑖=2𝑢_{𝑖𝑋𝑠𝑝∗,𝑞∧1}.(65) By induction, we can easily see that the claimed result holds.

4. Smooth Effects of the Schrödinger and Ginzburg-Landau Semigroups

In this section, we will discuss a kind of Strichartz’s estimates. This kind of estimates is first introduced by Strichartz in [20], then developed by [18, 21, 22], and so forth.

Put 𝑆(𝑡)=𝐹⁻¹𝑒^{−𝑖𝑡|𝜉|2}𝑈(𝑡)𝐹,=𝐹⁻¹𝑒^{−𝑡𝑃(𝜉)}𝐹,𝑃(𝜉)=1+(𝑎+𝑖𝛼)𝜉²,𝑎>0,𝛼∈ℝ.(66) Our aim is to derive the estimates of 𝑆(𝑡), 𝑈(𝑡) in the spaces 𝑋^𝑠_𝑝,𝑞.

Theorem 15. Assume 2⩽𝑝<∞, 𝑝^′⩽𝑞⩽𝑝, 1/𝑝+1/𝑝^′=1, 0⩽𝑠<∞; then one has 𝑆(𝑡)𝑓_{𝑋𝑠𝑝,𝑞}≲(1+|𝑡|)^{−𝑛(1/2−1/𝑝)}𝑓_{𝑋𝑠𝑝′,𝑞}.(67)

Proof
 
Step  1. By Appendix Theorem C, for 2⩽𝑝<∞, we have □_𝑘𝑆(𝑡)𝑓_𝐿𝑝≲(1+|𝑡|)^{−𝑛(1/2−1/𝑝)}□_𝑘𝑓_𝐿𝑝′.(68)
Thus, for 0⩽𝑠<∞, we have 𝑆(𝑡)𝑓_{𝐸𝑠𝑝,𝑞}≲(1+|𝑡|)^{−𝑛(1/2−1/𝑝)}𝑓_{𝐸𝑠𝑝′,𝑞}.(69)
Step  2. By Proposition 3, for 𝑝^′⩽𝑞⩽𝑝, we have 𝑆(𝑡)𝑓_{𝑋𝑠𝑝,𝑞}⩽𝑆(𝑡)𝑓_{𝐸𝑠𝑝,𝑞}⩽(1+𝑡)^{−𝑛(1/2−1/𝑝)}𝑓_{𝐸𝑠𝑝′,𝑞}⩽(1+𝑡)^{−𝑛(1/2−1/𝑝)}𝑓_{𝑋𝑠𝑝′,𝑞}.(70) This proves the theorem.

Theorem 16. Assume 𝑟⩾1, 0⩽𝑠<∞, 𝑝^′⩽𝑞⩽𝑝, 2⩽𝑝<∞, 𝐴𝑓=∫^𝑡₀𝑆(𝑡−𝜏)𝑓(𝜏)𝑑𝜏; then one has 𝐴𝑓_{𝐿𝑟(−𝑇,𝑇;𝑋𝑠𝑝,𝑞)}≲𝑇^2/𝑟𝑓_{𝐿𝑟′(−𝑇,𝑇;𝑋𝑠𝑝′,𝑞)}.(71)

Proof. By Theorem 15, we have 𝐴𝑓_{𝐿𝑟(−𝑇,𝑇;𝑋𝑠𝑝,𝑞)}⩽^𝑡₀𝑆(𝑡−𝜏)𝑓(𝜏)_{𝑋𝑠𝑝,𝑞}𝑑𝜏_{𝐿𝑟(−𝑇,𝑇)}⩽^𝑡₀(1+|𝑡−𝜏|)^{−𝑛(1/2−1/𝑝)}𝑓(𝜏)_{𝑋𝑠𝑝′,𝑞}𝑑𝜏_{𝐿𝑟(−𝑇,𝑇)}≲^∞₀𝜒_{𝜏∈[0,𝑡]}𝑓(𝜏)_{𝑋𝑠𝑝′,𝑞}𝑑𝜏_{𝐿𝑟(−𝑇,𝑇)}≲𝑇^1/𝑟𝑇_−𝑇𝑓(𝜏)^𝑟_{𝐿𝑟′(−𝑇,𝑇;𝑋𝑠𝑝′,𝑞)}𝑑𝑡^1/𝑟≲𝑇^2/𝑟𝑓_{𝐿𝑟′(−𝑇,𝑇;𝑋𝑠𝑝′,𝑞)}.(72) This proves the theorem.

Theorem 17. Letting 0⩽𝑠<∞, one has 𝑆(𝑡)𝑓_𝑋𝑠1,1⩽(1+|𝑡|)^𝑛/2𝑓_𝑋𝑠1,1,(73) which holds for all 𝑓∈𝑋^𝑠_1,1.

Proof. By [18] (Appendix Theorem H), we have □_𝑘𝑆(𝑡)𝑓_𝐿𝑝≲(1+|𝑡|)^{𝑛|1/2−1/𝑝|}□_𝑘𝑓_𝐿𝑝,1⩽𝑝⩽∞.(74) Then, we have 𝑆(𝑡)𝑓_𝑋𝑠1,1=_{𝑘∈ℤ𝑛}2^|𝑘|𝑠□_𝑘𝑆(𝑡)𝑓_𝐿1=_{𝑘∈ℤ𝑛}2^|𝑘|𝑠□_𝑘𝑆(𝑡)𝑓_𝐿1≲(1+|𝑡|)^𝑛/2_{𝑘∈ℤ𝑛}2^|𝑘|𝑠□_𝑘𝑓_𝐿1≲(1+|𝑡|)^𝑛/2𝑓_𝑋𝑠1,1.(75) This proves the theorem.

Theorem 18. Assume 𝑟⩾1,0⩽𝑠<∞, 𝐴𝑓=∫^𝑡₀𝑆(𝑡−𝜏)𝑓(𝜏)𝑑𝜏; then one has 𝐴𝑓_{𝐿𝑟(−𝑇,𝑇;𝑋𝑠1,1)}≲(1+𝑇)^𝑛/2𝑇^2/𝑟𝑓_{𝐿𝑟′(−𝑇,𝑇;𝑋𝑠1,1)}.(76)

Proof. By Theorem 17, we have 𝐴𝑓_{𝐿𝑟(−𝑇,𝑇;𝑋𝑠1,1)}⩽^𝑡₀𝑆(𝑡−𝜏)𝑓(𝜏)_𝑋𝑠1,1𝑑𝜏_{𝐿𝑟(−𝑇,𝑇)}⩽^𝑡₀(1+|𝑡−𝜏|)^𝑛/2𝑓(𝜏)_𝑋𝑠1,1𝑑𝜏_{𝐿𝑟(−𝑇,𝑇)}≲(1+𝑇)^𝑛/2∞₀𝜒_{𝜏∈[0,𝑡]}𝑓(𝜏)_𝑋𝑠1,1𝑑𝜏_{𝐿𝑟(−𝑇,𝑇)}≲(1+𝑇)^𝑛/2𝑇^1/𝑟𝑇_−𝑇𝑓(𝜏)^𝑟_{𝐿𝑟′(−𝑇,𝑇;𝑋𝑠1,1)}𝑑𝑡^1/𝑟≲(1+𝑇)^𝑛/2𝑇^2/𝑟𝑓_{𝐿𝑟′(−𝑇,𝑇;𝑋𝑠1,1)}.(77) This proves the theorem.

Theorem 19. Assume 𝑟⩾𝑘+2, 0⩽𝑠<∞. Assume also 𝑓(𝑢)=𝑢|𝑢|^𝑘, 𝑘=2𝑚, 𝑚∈ℤ⁺, 𝐴(𝑓)=∫^𝑡₀𝑆(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏. Then, one has 𝐴𝑓_{𝐿𝑟(−𝑇,𝑇;𝑋𝑠1,1)}≲(1+𝑇)^𝑛/2𝑇^{1−𝑘/𝑟}‖𝑢‖_𝐿^𝑘+1𝑟_{(−𝑇,𝑇;𝑋}^𝑠_1,1).(78)

Proof. By the definition, Theorems 18 and 13, we have 𝐴𝑓_{𝐿𝑟(−𝑇,𝑇;𝑋𝑠1,1)}=^𝑡₀𝑆(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏_{𝐿𝑟(−𝑇,𝑇;𝑋𝑠1,1)}≲(1+𝑇)^𝑛/2𝑇^2/𝑟𝑓_{𝐿𝑟′(−𝑇,𝑇;𝑋𝑠1,1)}≲(1+𝑇)^𝑛/2𝑇^2/𝑟𝑇_−𝑇𝑓(𝑢(𝜏))^𝑟′_𝑋𝑠1,1𝑑𝜏^1/𝑟′≲(1+𝑇)^𝑛/2𝑇^2/𝑟𝑇_−𝑇‖𝑢(𝜏)‖^{(𝑘+1)𝑟′}_{𝑋𝑠𝑘+1,1}𝑑𝜏^1/𝑟′≲(1+𝑇)^𝑛/2𝑇^2/𝑟𝑇_−𝑇‖𝑢(𝜏)‖^𝑟_𝑋^𝑠_1,1𝑑𝜏^{(𝑘+1)/𝑟}𝑇^{(𝑟−𝑘−2)/𝑟}≲(1+𝑇)^𝑛/2𝑇^{1−𝑘/𝑟}‖𝑢‖_𝐿^𝑘+1𝑟_{(−𝑇,𝑇;𝑋}^𝑠_1,1).(79) This proves the theorem.

Theorem 20. Let 0<𝑝<∞, 0<𝑞⩽∞, 0⩽𝑠<∞; then 𝑈(𝑡)𝑓_{𝑋𝑠𝑝,𝑞}⩽𝐶𝑓_{𝑋𝑠𝑝,𝑞},(80) which holds for all 𝑓∈𝑋^𝑠_𝑝,𝑞.

Proof
 
Step  1 (see [12]). Let 𝐿>𝑛(𝑛/min(1,𝑝,𝑞)−1/2) and 𝐿∈ℤ⁺; then we have ‖𝜎_𝑘𝑒^{−𝑡𝑃(𝜉)}‖_𝐻𝐿≲1. Actually, in view of the Leibnitz rule, we have 𝐷^𝐿𝜎_𝑘𝑒^{−𝑡𝑃(𝜉)}⩽𝐶_{𝐿1+𝐿2=𝐿}𝐷^𝐿1𝜎_𝑘𝐷^𝐿2𝑒^{−𝑡𝑃(𝜉)}⩽𝐶_{0⩽𝑠⩽𝐿}𝐷^𝑠𝑒^{−𝑡𝑃(𝜉+𝑘)}𝜒_{|𝜉|∞⩽1}.(81)
Let 𝛽=(𝛽₁,...,𝛽_𝑞), and let Λ^𝑠_𝑞={𝛽∶𝛽₁+⋯+𝛽_𝑞=𝑠,𝛽₁,...,𝛽_𝑞⩾1}. It is easy to see that 𝐷^𝑠𝑒^𝑓(𝜂)=𝑒^{𝑓(𝜂)𝑠}_{𝑞=1Λ𝑠𝑞}𝐶_𝛽^𝑞_𝑖=1𝐷^𝛽𝑖𝑓(𝜉).(82)
Let 𝑓(𝜉)=−𝑡𝑃(𝜉); one easily sees that 𝐷^𝛽𝑖𝑓(𝜉)⩽𝐶𝑡1+𝜉+𝑘²,𝑘∈ℤ.(83)
Hence, we have 𝐷^𝑠𝑒^𝑓(𝜂)⩽𝑒^{−𝑐𝑡(1+|𝜉+𝑘|2)𝑠}_𝑞=1𝑡1+𝜉+𝑘^2𝑞⩽𝑒^{−𝑐𝑡(1+|𝜉+𝑘|2)}.(84)
Combining the previous discussion, we have 𝜎_𝑘𝑒^{−𝑡𝑃(𝜉)}_𝐻𝐿≲𝐶𝑒^{−𝑐𝑡(1+|𝑘|2)}.(85)
Step  2. By Step 1, we have sup_𝑘‖𝜎_𝑘𝑒^{−𝑡𝑃(𝜉)}‖_𝐻𝐿≲1. Then by Appendix Theorem F, we can obtain that 𝑈(𝑡)𝑓_{𝑋𝑠𝑝,𝑞}⩽𝐶sup_𝑘𝜎_𝑘𝑒^{−𝑡𝑃(𝜉)}_𝐻𝑠𝑓_{𝑋𝑠𝑝,𝑞}≲𝑓_{𝑋𝑠𝑝,𝑞}.(86) This proves the theorem.

Theorem 21. Assume 1⩽𝑝<∞, 1⩽𝑞⩽∞, 𝑟⩾1,0⩽𝑠<∞, 𝐴𝑓=∫^𝑡₀𝑈(𝑡−𝜏)𝑓(𝜏)𝑑𝜏; then one has 𝐴𝑓_{𝐿𝑟(−𝑇,𝑇;𝑋𝑠𝑝,𝑞)}≲𝑇^2/𝑟𝑓_{𝐿𝑟′(−𝑇,𝑇;𝑋𝑠𝑝,𝑞)}.(87)

Proof. By Theorem 20, we have 𝐴𝑓_{𝐿𝑟(−𝑇,𝑇;𝑋𝑠𝑝,𝑞)}⩽^𝑡₀𝑈(𝑡−𝜏)𝑓(𝜏)_{𝑋𝑠𝑝,𝑞}𝑑𝜏_{𝐿𝑟(−𝑇,𝑇)}⩽^𝑡₀𝑓(𝜏)_{𝑋𝑠𝑝,𝑞}𝑑𝜏_{𝐿𝑟(−𝑇,𝑇)}≲^∞₀𝜒_{𝜏∈[0,𝑡]}𝑓(𝜏)_{𝑋𝑠𝑝,𝑞𝐿𝑟(−𝑇,𝑇)}≲𝑇^2/𝑟𝑓_{𝐿𝑟′(−𝑇,𝑇;𝑋𝑠𝑝,𝑞)}.(88) This proves the theorem.

Theorem 22. Assume 𝑏>0, 𝛽,𝜇∈ℝ, 𝑓(𝑢)=(𝑏+𝑖𝛽)𝑢|𝑢|^𝑘+𝜇𝑢,𝐴(𝑓)=∫^𝑡₀𝑈(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏. Assume also 𝑘=2𝑚, 𝑟⩾𝑘+2, 𝑚∈ℤ⁺, 1⩽𝑝<∞, 1⩽𝑞⩽∞. Then, one has 𝐴𝑓_{𝐿𝑟(0,𝑇;𝑋𝑠𝑝,𝑞)}≲𝑇^{1−𝑘/𝑟}‖𝑢‖_𝐿^𝑘+1_{𝑟0,𝑇;𝑋𝑠𝑝,1}+𝑇‖𝑢‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_𝑝,𝑞).(89)

Proof. By Theorems 20 and 21 and the multilinear estimate, we have 𝐴𝑓_{𝐿𝑟(−𝑇,𝑇;𝑋𝑠𝑝,𝑞)}≲𝑇^2/𝑟𝑓_{𝐿𝑟′(−𝑇,𝑇;𝑋𝑠𝑝,𝑞)}=𝑇^2/𝑟𝑇_−𝑇𝑓(𝑢(𝜏))^𝑟′_{𝑋𝑠𝑝,𝑞}𝑑𝜏^1/𝑟′≲𝑇^2/𝑟𝑇_−𝑇𝑢|𝑢|^𝑘𝑟′_{𝑋𝑠𝑝,𝑞}𝑑𝜏^1/𝑟′+𝑇^2/𝑟𝑇_−𝑇‖𝑢(𝜏)‖^𝑟′_𝑋^𝑠_𝑝,𝑞𝑑𝜏^1/𝑟′≲𝑇^2/𝑟𝑇_−𝑇‖𝑢(𝜏)‖^{(𝑘+1)𝑟′}_{𝑋𝑠()𝑘+1𝑝,𝑞∧1}𝑑𝜏^1/𝑟′+𝑇^2/𝑟𝑇_−𝑇‖𝑢(𝜏)‖^𝑟′_𝑋^𝑠_𝑝,𝑞𝑑𝜏^1/𝑟′≲𝑇^2/𝑟𝑇_−𝑇‖𝑢(𝜏)‖^𝑟_𝑋^𝑠_𝑝,1𝑑𝜏^{(𝑘+1)/𝑟}𝑇^{(𝑟−𝑘−2)/𝑟}+𝑇‖𝑢‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_𝑝,𝑞)≲𝑇^{1−𝑘/𝑟}‖𝑢‖_𝐿^𝑘+1𝑟_(0,𝑇;𝑋^𝑠_𝑝,1)+𝑇‖𝑢‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_𝑝,𝑞).(90) This proves the theorem.

5. Well-Posedness of Nonlinear Schrödinger Equation and Ginzburg-Landau Equation

In this section, we first study the well-posedness of the following Schrödinger equation: 𝑖𝑢_𝑡+Δ𝑢=𝑓(𝑢);𝑢(0,𝑥)=𝑢₀(𝑥).(91) The solution, 𝑢(𝑥,𝑡), of the above Cauchy problem is given by: 𝑢(𝑡)=𝑆(𝑡)𝑢₀−𝑖^𝑡₀𝑆(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏,(92) where 𝑆(𝑡)=𝐹⁻¹𝑒^{𝑖𝑡|𝜉|2}𝐹. Then, we study the well-posedness of the following complex Ginzburg-Landau equation: 𝑢_𝑡−(𝑎+𝑖𝛼)Δ𝑢+𝑏+𝑖𝛽|𝑢|^𝑘𝑢+𝜇+1𝑢=0,𝑢(0,𝑥)=𝑢₀(𝑥).(93) The solution, 𝑢(𝑥,𝑡), of the above Cauchy problem is given by 𝑢(𝑡)=𝑈(𝑡)𝑢₀−𝑖^𝑡₀𝑈(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏,(94) where 𝑎,𝑏>0, 𝛼,𝛽,𝜇∈ℝ, 𝑈(𝑡)=𝐹⁻¹𝑒^{−𝑡𝑃(𝜉)}𝐹, and 𝑃(𝜉)=1+(𝑎+𝑖𝛼)|𝜉|².

6. Proof of Theorem 1

Proof. Fix 𝑇>0, 𝛿>0 to be chosen later. Let 𝒟=𝑢∈𝐿^𝑟0,𝑇;𝑋^𝑠_1,1∶‖𝑢‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_1,1)<𝛿(95)
be equipped with the distance 𝑑(𝑢,𝜐)=‖𝑢−𝜐‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_1,1).(96)
It is easy to know that (𝒟,𝑑) is a complete metric space. Now we consider the following map: ℑ∶𝑢(𝑡)⟶𝑆(𝑡)𝑢₀−𝑖^𝑡₀𝑆(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏.(97)
We will prove that there exists 𝑇,𝛿>0, such that ℑ∶(𝒟,𝑑)→(𝒟,𝑑) is a strict contraction map.
By the nonlinear term estimate and the semigroup estimate, |𝑢|^𝑘𝑢_𝑋𝑠1,1≲‖𝑢‖_𝑋^𝑘+1𝑠_(𝑘+1),1,𝑆(𝑡)𝑢_0𝑋𝑠1,1≲(1+|𝑡|)^𝑛/2𝑢_0𝑋𝑠1,1,𝐴𝑓_{𝐿𝑟(0,𝑇;𝑋𝑠1,1)}≲(1+𝑇)^𝑛/2𝑇^{1−𝑘/𝑟}‖𝑢‖_𝐿^𝑘+1_{𝑟0,𝑇;𝑋𝑠1,1},(98) we have ‖ℑ𝑢‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_1,1)≲(1+𝑇)^𝑛/2𝑇^1/𝑟𝑢_0𝑋𝑠1,1+(1+𝑇)^𝑛/2𝑇^{1−𝑘/𝑟}‖𝑢‖_𝐿^𝑘+1_{𝑟0,𝑇;𝑋𝑠1,1}.(99) Then, we let 𝛿=2𝐶𝑢_0𝑋𝑠1,1,𝑇<1.(100) Then, we choose 𝑇 such that 2𝐶𝑇^{(1−𝑘/𝑟)}𝛿^𝑘⩽12.(101) Then, we have ‖ℑ𝑢‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_1,1)<𝛿.(102) Moreover, we have that ‖ℑ𝑢−ℑ𝜐‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_1,1)⩽12‖𝑢−𝜐‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_1,1).(103) In this way, we prove that ℑ∶(𝒟,𝑑)→(𝒟,𝑑) is a strict contraction map. By Banach’s fixed-point theorem, there exists a unique solution 𝑢∈𝒟 that satisfies the conditions in the theorem. Using standard argument, we can extend the solution, considering the map ℑ∶𝑢(𝑡)⟶𝑆(𝑡−𝑇)𝑢_𝑇−𝑖^𝑡_𝑇𝑆(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏,(104) where 𝑢(𝑇) is the solution that we have chosen. We claim 𝑢(𝑇)∈𝑋^𝑠_1,1; one can refer to [23] for the details. In this way, we can find a maximum 𝑇^∗>0 which satisfies the conditions in the theorem.

7. Proof of Theorem 2

Proof. Fix 𝑇>0, 𝛿>0 to be chosen later. Let 𝒟=𝑢∈𝐿^𝑟0,𝑇;𝑋^𝑠_𝑝,1∶‖𝑢‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_𝑝,1)<𝛿(105)
be equipped with the distance 𝑑(𝑢,𝜐)=‖𝑢−𝜐‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_𝑝,1).(106)
It is easy to know that (𝒟,𝑑) is a complete metric space. Now, we consider the following map: ℑ∶𝑢(𝑡)⟶𝑈(𝑡)𝑢₀−𝑖^𝑡₀𝑈(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏.(107)
We shall prove that there exists 𝑇,𝛿>0, such that ℑ∶(𝒟,𝑑)→(𝒟,𝑑) is a strict contraction map.
By the nonlinear term estimate and the semigroup estimate, |𝑢|^𝑘𝑢_{𝑋𝑠𝑝,1}≲‖𝑢‖_𝑋^𝑘+1𝑠_{𝑝(𝑘+1),1},𝑈(𝑡)𝑢_{0𝑋𝑠𝑝,1}≲𝑢_{0𝑋𝑠𝑝,1},𝐴𝑓_{𝐿𝑟(0,𝑇;𝑋𝑠𝑝,1)}≲𝑇^{1−𝑘/𝑟}‖𝑢‖_𝐿^𝑘+1𝑟_(0,𝑇;𝑋^𝑠_𝑝,1)+𝑇‖𝑢‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_𝑝,1),(108)
we have ‖ℑ𝑢‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_𝑝,1)≲𝑇^1/𝑟𝑢_{0𝑋𝑠𝑝,1}+𝑇^{1−𝑘/𝑟}‖𝑢‖_𝐿^𝑘+1𝑟_(0,𝑇;𝑋^𝑠_𝑝,1)+𝑇‖𝑢‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_𝑝,1).(109)
Then, we let 𝛿=3𝐶𝑢_{0𝑋𝑠𝑝,1}1,𝑇<3.(110)
Then, we choose 𝑇 such that 𝐶𝑇^{(1−𝑘/𝑟)}𝛿^𝑘⩽13.(111)
Then, we have ‖ℑ𝑢‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_𝑝,1)<𝛿.(112)
Moreover, we have that ‖ℑ𝑢−ℑ𝜐‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_𝑝,1)⩽23‖𝑢−𝜐‖_𝐿^𝑟_(0,𝑇;𝑋^𝑠_𝑝,1).(113)
In this way, we prove that ℑ∶(𝒟,𝑑)→(𝒟,𝑑) is a strict contraction map. By Banach’s fixed-point theorem, there exists a unique solution 𝑢∈𝒟 that satisfies the conditions in the theorem. Using standard argument, we can extend the solution considering the map ℑ∶𝑢(𝑡)⟶𝑈(𝑡−𝑇)𝑢_𝑇−𝑖^𝑡_𝑇𝑈(𝑡−𝜏)𝑓(𝑢(𝜏))𝑑𝜏,(114) where 𝑢(𝑇) is the solution that we have chosen. We claim 𝑢(𝑇)∈𝑋^𝑠_𝑝,1; one can refer to [23] for the details. In this way, we can find a maximum 𝑇^∗>0 which satisfies the conditions in the theorem.

Appendix

Theorem A. We have 𝐻^𝑠ℝ^𝑛⊂𝐸⁰_2,𝑞ℝ^𝑛1,𝑠>𝑛𝑞−12𝐿,0<𝑞<2,₂ℝ^𝑛=𝐸⁰_2,2ℝ^𝑛,𝐸⁰_2,𝑞ℝ^𝑛⊂𝐻^𝑠ℝ^𝑛1,𝑠<𝑛𝑞−12,2<𝑞⩽∞.(A.1)

Proof. One can refer to [12].

Theorem B. Assume 0<𝑝⩽𝑞⩽∞ Let Ω⊂ℝ^𝑛 be compact set, diamΩ<2𝑅. Then, there exists 𝐶(𝑝,𝑞,𝑅)>0, such that 𝑓_𝐿𝑞⩽𝐶𝑓_𝐿𝑝,∀𝑓∈𝐿^𝑝_Ω,(A.2) where 𝐿^𝑝_Ω={𝑓∈𝐿^𝑝∶supp𝑓⊂Ω}.

Proof. One can refer to [15, 18].

Theorem C. Assume 𝑠∈ℝ, 2⩽𝑝<∞, 1/𝑝+1/𝑝^′=1, Then we have □_𝑘𝑆(𝑡)𝑓_𝐿𝑝≲(1+|𝑡|)^{−𝑛(1/2−1/𝑝)}□_𝑘𝑓_𝐿𝑝′‖𝑆(𝑡)𝑓‖_𝑀^𝑠_𝑝,𝑞≲(1+|𝑡|)^{−𝑛(1/2−1/𝑝)}𝑓_{𝑀𝑠𝑝′,𝑞}.(A.3)

Proof. One can refer to [18].

Theorem D (infinite smoothness). Let 0<𝜆<∞, 0<𝑝,𝑞⩽∞. Then, one has 𝐸^𝜆_𝑝,𝑞ℝ^𝑛⊂𝐵^{𝑠𝑘}_𝑝,𝑞ℝ^𝑛⊂𝐶^∞ℝ^𝑛,(A.4)
if 𝑠_𝑘⩽𝜆2^𝑘/3𝑘, 𝑘∈𝑁.

Proof. One can refer to [12]. 

Theorem E (embedding). Assume 𝑠₁,𝑠₂∈ℝ, 0<𝑝₁,𝑝₂, 𝑞₁,𝑞₂⩽∞.(a)If 𝑠₂⩽𝑠₁, 𝑝₁⩽𝑝₂,𝑞₁⩽𝑞₂, then 𝑀^𝑠1_𝑝1,𝑞1⊂𝑀^𝑠2_𝑝2,𝑞2.(b) If 𝑞₂<𝑞₁, 𝑠₁−𝑠₂>𝑛/𝑞₂−𝑛/𝑞₁, then 𝑀^𝑠1_𝑝,𝑞1⊂𝑀^𝑠2_𝑝,𝑞2.(c) If 0⩽𝑠<∞, 0<𝑝₁⩽𝑝₂⩽∞, 0<𝑞₁⩽𝑞₂⩽∞, then 𝐸^𝑠_𝑝1,𝑞1⊂𝐸^𝑠_𝑝2,𝑞2.

Proof. One can refer to [12, 18, 24].

Theorem F. Assume 0<𝑝<∞, 0<𝑞⩽∞; Ω={Ω_𝑘}^∞_𝑘=0 is a sequence with compact support in ℝ^𝑛; let 𝑑_𝑘>0 be the diameter of Ω_𝑘. If 𝑥>𝑛(𝑛/min(1,𝑝,𝑞)−1/2), then there exists a constant 𝐶 such that 𝐹⁻¹𝑀_𝑘𝐹𝑓_{𝑘𝐿𝑝(ℓ𝑞)}⩽𝐶sup_𝑖𝑀_𝑖𝑑_𝑖⋅_𝐻𝑥𝑓_{𝑘𝐿𝑝(ℓ𝑞)},(A.5)
where {𝑓_𝑘}^∞_𝑘=0∈𝐿^𝑝_Ω(ℓ^𝑞),{𝑀_𝑘(𝑥)}^∞_𝑘=0⊂𝐻^𝑥.

Proof. One can refer to [11]. Note that 𝐿^𝑝_Ω(ℓ^𝑞)={𝑓∣𝑓={𝑓_𝑘}^∞_𝑘=0⊂𝑆^′, supp𝐹𝑓_𝑘⊂Ω_𝑘 if 𝑘=0,1,2... and ‖𝑓_𝑘‖_{𝐿𝑝(ℓ𝑞)}<∞}.

Theorem G. Let 0<𝑝,𝑞⩽∞ and (𝑋,𝜇),(𝑌,𝜐) be two measure space. Let 𝑇 be a positive linear operator mapping 𝐿^𝑝(𝑋) into 𝐿^𝑞(𝑌) (resp., into 𝐿^𝑞,∞(𝑌)) with norm 𝐴. Let 𝐵 be a Banach space. Then, 𝑇 has a 𝐵-valued extension 𝑇 that maps 𝐿^𝑝(𝑋,𝐵) into 𝐿^𝑞(𝑌,𝐵) (resp., into 𝐿^𝑞,∞(𝑌,𝐵)) with the same norm.

Proof. One can refer to [25].

Theorem H. Let 𝑠∈ℝ, 1⩽𝑝⩽∞, 0<𝑞<∞; then one has □_𝑘𝑆(𝑡)𝑓_𝐿𝑝≲(1+|𝑡|)^{𝑛|1/2−1/𝑝|}□_𝑘𝑓_𝐿𝑝𝑆(𝑡)𝑓_{𝑀𝑠𝑝,𝑞}≲(1+|𝑡|)^{𝑛|1/2−1/𝑝|}𝑓_{𝑀𝑠𝑝,𝑞}.(A.6)

Proof. One can refer to [18]. 

Acknowledgment

This paper is supported by NSF of China (Grant no. 11271330).