Abstract
In this paper, we study the strongly singular integrals \t\t\tTn,β,γf(x)=p.v.∫−11f(x−Γθ(t))e−2πi|t|−βt|t|γdt\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\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 Gamma_{theta}(t). We prove that T_{n, beta, gamma} is bounded on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces.
Highlights
The two dimension strongly singular integrals along curves Tβ,γ are defined by e– π i|t|–βTβ,γ (f )(x, y) = p.v. f x – t, y – (t) t|t|γ dt, where x, y ∈ R, β, γ >
In this paper, we study the strongly singular integrals e–2π i|t|–β
We prove that Tn,β,γ is bounded on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces
Summary
Chandarana [ ] extended the result to the general curves (t, |t|m) or (t, sgn(t)|t|m) with m ≥ and showed that Tβ,γ is bounded on Lp(R ) for γ (β + ). Cheng-Zhang [ ] and Cheng [ ] extended the results to the modulation space They showed that the strongly singular integral Tn,β,γ is bounded on the modulation spaces Mps ,q for all p >. We will consider the strongly singular integrals along homogeneous curves Tn,β,γ on the α-modulation spaces. We will consider the strongly singular integrals along a well-curved (t) in Rn. 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.
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