# Strongly singular integrals along curves on α-modulation spaces

Strongly singular integrals along curves on α-modulation spaces In this paper, we study the strongly singular integrals T n , β , γ f ( x ) = p . v . ∫ − 1 1 f ( x − Γ θ ( t ) ) e − 2 π i | t | − β t | t | γ d t $$T_{n, \beta, \gamma}f(x)=\mathrm{p.v.} \int_{-1}^{1}f\bigl(x-\Gamma_{\theta}(t) \bigr)\frac {e^{-2\pi i \vert t \vert ^{-\beta}}}{t \vert t \vert ^{\gamma}}\,dt$$ along homogeneous curves Γ θ ( t ) $\Gamma_{\theta}(t)$ . We prove that T n , β , γ $T_{n, \beta, \gamma}$ is bounded on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Inequalities and Applications Springer Journals

# Strongly singular integrals along curves on α-modulation spaces

, Volume 2017 (1) – Aug 9, 2017
13 pages

/lp/springer_journal/strongly-singular-integrals-along-curves-on-modulation-spaces-meeEPodUgr
Publisher
Springer Journals
Subject
Mathematics; Analysis; Applications of Mathematics; Mathematics, general
eISSN
1029-242X
D.O.I.
10.1186/s13660-017-1458-0
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we study the strongly singular integrals T n , β , γ f ( x ) = p . v . ∫ − 1 1 f ( x − Γ θ ( t ) ) e − 2 π i | t | − β t | t | γ d t $$T_{n, \beta, \gamma}f(x)=\mathrm{p.v.} \int_{-1}^{1}f\bigl(x-\Gamma_{\theta}(t) \bigr)\frac {e^{-2\pi i \vert t \vert ^{-\beta}}}{t \vert t \vert ^{\gamma}}\,dt$$ along homogeneous curves Γ θ ( t ) $\Gamma_{\theta}(t)$ . We prove that T n , β , γ $T_{n, \beta, \gamma}$ is bounded on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces.

### Journal

Journal of Inequalities and ApplicationsSpringer Journals

Published: Aug 9, 2017

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