Convolution estimates for measures on some complex curves

Abstract

We consider the convolution operator for a measure supported on complex curves. The measure which we consider here is an analogue of the affine arclength measure for real curves. By modifying a combinatorial argument called the band structure argument, we prove the (nearly) optimal Lorentz space estimates. This includes the optimal strong type estimates as special cases. The complex curves we consider here are the ones considered for the Fourier restriction estimates for complex curves in [1].

Uniform Convolution and Fourier Restriction estimates for complex polynomial curves in $\mathbb{C}^3$

“…In [6], Chung and Ham established similar results, considering surfaces that could be represented as complex curves in C d . They established these bounds for curves of the form z, z 2 • • • z N in C d , and z, z 2 , φ(z) in C 3 , for analytic φ.…”

“…They do this by using a more powerful analogue to our Lemma 3.1, established in the complex Fourier Restriction paper [1], involving a lower bound in terms of the arithmetic means of {L Γ (z i )}, as opposed to the geometric mean. This method suffices for the complex curves in question in [6], but in general does not hold. In this paper, we do not utilise a lower bound in terms of the arithmetic mean, but instead adapt the geometric mean bound deriving from [8] into the complex case, and achieve the overall estimate using that.…”

“…Furthermore, if l = 1, the desired inequality is trivial. Applying this to T k with p = 6 7−k , which yields q = 3 for each k, we get:…”

A theory of complex oscillatory integrals: A case study

“…This is the real version of the example (6). Here the problem is that the function f and its derivative f ′ are conspiring to produce many zeros of f at places there the derivative f ′ is large.…”

“…The problem of fourier restriction to complex curves has been studied by a number of authors; see for example, [5], [6] and more recently [7] and [10] where positive results have been obtained. The question arises whether the results are sharp.…”