Abstract
Uniform endpoint Lorentz norm improving estimates for convolution operators with affine arclength measure supported on simple plane curves are established. The estimates hold for a wide class of simple curves, and the condition is stated in terms of averages of the square of the affine arclength weight, extending previously known results.
Highlights
Let φ : (a, b) → R be a C function such that φ (t) ≥ for all t ∈ (a, b)
The measure ω(t) dt supported on the curve (t, φ(t)) is known as the affine arclength measure, which is based on the affine arclength parameter as in [ ], and was introduced by Drury and Marshall [ ] in dealing with the Fourier restriction problem related to curves, and later by Drury [ ] in studying convolution operators with measures supported on curves
One big benefit of using the affine arclength measure in place of the Euclidean arclength measure + φ (t) dt has been its effect of mitigating degeneracies and it is believed that various uniform sharp estimates hold for a wide class of curves
Summary
Let φ : (a, b) → R be a C function such that φ (t) ≥ for all t ∈ (a, b). In this paper, we consider the convolution operator T given by bT f (x , x ) = f x – t, x – φ(t) ω(t) dt ( . )a for f ∈ C ∞(R ). We consider the convolution operator T given by b Many conditions to guarantee optimal uniform L / -L estimates have been known so far. (Choi [ ]) Let J be an open interval in R, and φ : J → R be a C function such that φ ≥ .
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