Strongly singular integrals along curves on α-modulation spaces

In this paper, we study the strongly singular integrals Tn,β,γf(x)=p.v.∫−11f(x−Γθ(t))e−2πi|t|−βt|t|γdt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{n, \beta, \gamma}f(x)=\mathrm{p.v.} \int_{-1}^{1}f\bigl(x-\Gamma_{\theta}(t) \bigr)\frac {e^{-2\pi i \vert t \vert ^{-\beta}}}{t \vert t \vert ^{\gamma}}\,dt $$\end{document} along homogeneous curves Γθ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Gamma_{\theta}(t)$\end{document}. We prove that Tn,β,γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{n, \beta, \gamma}$\end{document} is bounded on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces.

Later, Cheng-Zhang [] and Cheng [] extended the results to the modulation space. They showed that the strongly singular integral T n,β,γ is bounded on the modulation spaces M s p,q for all p > . It is worth to point out that the modulation space is a better substitution to study the strongly singular integrals because there is no restriction on the index p.
Here we will consider the strongly singular integrals along homogeneous curves T n,β,γ on the α-modulation spaces. The α-modulation spaces M s,α p,q were first introduced by Gröbner in []. They contain the inhomogeneous Besov spaces B s p,q in the limit case α =  and the classical modulation spaces M s p,q in the case α = , respectively. It is proposed as an intermediate function space; see [, ] for more details.
In recent years, there were numerous papers on these spaces and its applications, such as [-] and the references therein. Motivated by the work of Cheng-Zhang [] on the modulation spaces, one naturally expects that the strongly singular integral operators T n,β,γ have the boundedness property on the α-modulation spaces for all  ≤ α ≤ . In this paper, we will affirm this. This paper is organized as follows. In Section , we will recall the definition of the αmodulation spaces and the Besov spaces. Some lemmas will also be presented in this section. In Section , we will give the main results and prove the theorems. In addition, we will consider the strongly singular integrals along a well-curved (t) in R n . Throughout this paper, we use the notation A B meaning that there is a positive constant C independent of all essential variables such that A ≤ CB. We denote A ∼ B to stand for A B and B A.

Preliminaries and lemmas
Before giving the definition of the α-modulation spaces, we introduce some notations frequently used in this paper. Let S = S(R n ) be the subspace of C ∞ (R n ) of Schwartz rapidly decreasing functions and S = S (R n ) be the space of all tempered distribution on R n . For k = (k  , k  , . . . , k n ) ∈ Z n , we denote We define the ball The Fourier transform F(f ) and the inverse Fourier transform F - (f ) are defined by To define the α-modulation spaces, we introduce the α-decomposition. Let ρ be a nonnegative smooth radial bump function supported in B(, ), satisfying ρ(ξ ) =  for |ξ | <  and ρ(ξ ) =  for |ξ | ≥ . For any k = (k  , k  , . . . , k n ) ∈ Z n , we set It is easy to check that {η α k } k∈Z n satisfy and Corresponding to the above sequence {η α k } k∈Z n , we can construct an operator sequence For  ≤ α < ,  < p, q ≤ ∞, s ∈ R, using this decomposition, we define the α-modulation spaces as We have the usual modification when p, q = ∞. We denote M s, p,q = M s p,q . It is the classical modulation space. Its related decomposition is called uniform decomposition; see [, ] and [] for details. In order to define the Besov spaces, we introduce the dyadic decomposition. Let ψ be a smooth bump function supported in the ball {ξ : |ξ | ≤   }. We may Define the Littlewood-Paley (or dyadic) decomposition operators as With the usual modification when p, q = ∞. Obviously, the α-decomposition is bigger than the uniform decomposition and thinner than the dyadic decomposition. This decomposition on frequency extends the dyadic and the uniform decomposition.
In order to prove the theorems, we also need some lemmas.

Lemma . Van der Corput lemma ([], p.). Let ϕ and φ be real valued smooth functions on the interval (a, b) and k
Denote (k) = {j ∈ Z n : η α k · η α j = }. Then by the support condition (.), we have Recall that the choice of η α j satisfies j∈Z n η α j ≡ . By Minkowski's inequality, we obtain Thus, the first part of the lemma holds. The second part of this lemma is similar to the proof of the first part. Here we omit the details.

Main results and proofs
Proof By checking the following proof, we can only consider the operator Then we can decompose T β,γ as Using the Fourier transformation, the operator T j can be written as Let α k,j be the kernel of α k T j , so we have By the Young inequality, we obtain B(, r). By a simple substitution and the Fubini theorem, we get First, let us estimate I j . We divide it into two cases. Case . If k  ≥ , then By the Van der Corput lemma, we have Then, by a substitution and the Fubini theorem, we obtain By the Van der Corput lemma, we have Thus, similar to (.), we get Combining (.) and (.), we have Therefore, noticing β > γ and combining with (.), we get So, by Lemma .(), we have We finished the proof of Theorem ..
Proof As the proof of Theorem ., using the Fourier transformation, the operator T β,γ can be written as Let j,β,γ be the kernel of j T β,γ , so we have Using the Young inequality, we obtain First, let us estimate j,β,γ . By a substitution and integrating by parts, we have Then, using the Fubini theorem, we get Thus, using a substitution and the Minkowski inequality, it follows that Now, we estimate F - (η α k (·)m n,j (·)) L  . By scaling, we can assume θ = (, , . . . , ) in the definition of θ (t). Letη α . By a simple substitution and the Fubini theorem, we obtain Denote φ(t) = -π[t -β + n l= k l k α -α t p l ]. By a normal computation, we get  sitions(see Section  for details), we will lose the regularity of the space by our method.
Maybe we need some new ideas to overcome this limitation.

Remark . In []
, the authors mention the fact that the α-modulation space cannot be obtained by interpolation between modulation spaces α =  and Besov spaces α = . Thus, it shows that our proofs for  < α <  in Theorems . and . are meaningful.

Conclusions
In this paper, using the equivalent discrete definition of α-modulation spaces, combining the Fourier transform and Van der Corput lemma, we obtained the strongly singular integrals along homogeneous curves are bounded on the α-modulation spaces for all  ≤ α ≤ . Our results extend the main results in []. Our method is also different from [].