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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.

In the first part of this paper are considered real polynomials$P$(ζ), ζ∈$R$^{n}, complete and nondegenerate in the sense that there is a set of (even) multi-indices α^{$j$},$j$=1,...,$N$, such that, for |ζ|>$K$, ζ real, $$cP(\xi ) \leqslant \sum {\xi ^{\alpha j} } \leqslant CP(\xi ).$$ (See V. P. Mihailov,$Soviet Math. Dokl. 164$(1965), MR 32: 6047).

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