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Nonconvolution transforms with oscillating kernels that map $Ḃ_{1}^{0, 1}$ into itself

G. Sampson — 1993

Studia Mathematica

We consider operators of the form $(Ω f) (y) = ʃ_{- \infty}^{\infty} Ω (y, u) f (u) d u$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h \in L^{\infty}$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space $Ḃ_{1}^{0, 1}$ (= B) into itself. In particular, all operators with $h (y) = e^{{i | y |}^{a}}$ , a > 0, a ≠ 1, map B into itself.

$L^{2}$ Estimates for Oscillatory Integrals

G. Sampson — 1998

Colloquium Mathematicae

A note on weak estimates for oscillating kernels

G. Sampson — 1981

Studia Mathematica

$L^{p}$ type mapping estimates for oscillatory integrals in higher dimensions

G. Sampson — 2006

Studia Mathematica

We show in two dimensions that if $K f = \int_{ℝ ₊ ²} k (x, y) f (y) d y$ , $k (x, y) = (e^{i x^{a} \cdot y^{b}}) {/ (| x - y |}^{η})$ , p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), $v_{p} (y) = y^{(p / p^{'}) (1 ̅ - b / a)}$ , then ${| | K f | |}_{p} {\leq C | | f | |}_{p, v_{p}}$ if η + α₁ + α₂ < 2, $α_{j} = 1 - b_{j} / a_{j}$ , j = 1,2. Our methods apply in all dimensions and also for more general kernels.