On Laplace–Carleson embeddings, and $L^p$-mapping properties of the Fourier transform

Abstract

We investigate so-called Laplace-Carleson embeddings for large exponents. In particular, we extend some results by Jacob, Partington, and Pott. We also discuss some related results for Sobolev-and Besov spaces, and mapping properties of the Fourier transform. These variants of the Hausdorff-Young theorem appear difficult to find in the literature. We conclude the paper with an example related to an open problem.

Laplace-Carleson embeddings and infinity-norm admissibility

“…. , N − 1, we obtain the required norm bound of the first term in (12), with a constant only depending on N (therefore on ǫ, and hence only on c, C). To control the second term in (12), just consider f = e itc I χ (0,1) , where c I is the midpoint of the interval I.…”

“…For 1 ≤ p ≤ 2, and p ′ ≤ q < ∞ (this corresponds to the region I in Figure 1), L : L p → L q (C + , dµ) if and only if (6) holds, see [8,Theorem 3.2]. In [12,Theorem 1.1], this result was extended to 2 < p ≤ q < ∞ (region II in Figure 1). For 2 ≤ q < p < ∞ (region III in Figure 1), ( 6) is sufficient if µ has support on a vertical strip, but not if µ has support on a sector, see [8, Theorem 3.6 and Theorem 3.5].…”

Nuclear Fourier Transforms

“…The only paper known to us which fits in the above scheme is [23]. It is one of the main aims of this paper to justify the continuous mapping…”