arXiv subjects

Petr Gurka

Publications and source records attributed to Petr Gurka.

3 recordsLinked to original sources

arXiv2025

Quantitative Non-Compactness Properties of the Fourier Transform on Optimal Spaces

We establish that the Fourier transform $\mathcal{F}: L^p(\mathbb{R}^d)\to L^{p',p}(\mathbb{R}^d)$, for $d\in\mathbb{N}$ and $1<p<2$, is not strictly singular, thereby confirming the optimality of the source and target spaces. A~similar result is obtained for Fourier series on $L^p(\mathbb{T}^n)$, with sequence Lorentz spaces as the target. These findings complement known results, which state that $\mathcal{F}: L^p(\mathbb{R}^d)\to L^{p'}(\mathbb{R}^d)$ is finitely strictly singular and then also strictly singular, and provide further insight into the degrees of non-compactness of~$\mathcal{F}$.

math.FA↗

arXiv2024

Existence of an extremal function of Sobolev critical embedding with an $\alpha$-homogeneous weight

In our previous publication [{\em Calc. Var. Partial Differential Equations}, 60(1):Paper No. 16, 27, 2021], we delved into examining a critical Sobolev-type embedding of a Sobolev weighted space into an exponential weighted Orlicz space. We specifically determined the optimal Moser-type constant for this embedding, utilizing the monomial weight introduced by Cabr\'e and Ros-Oton [{\em J. Differential Equations}, 255(11):4312--4336, 2013]. Towards the conclusion of that paper, we pledged to explore the existence of an extremal function within this framework. In this current work, we not only provide a positive affirmation to this inquiry but extend it to a broader range of weights known as \emph{$\alpha$-homogeneous weights}.

math.AP↗

arXiv2023

Quantitative analysis of optimal Sobolev-Lorentz embeddings with $\alpha$-homogeneous weights

Optimal weighted Sobolev-Lorentz embeddings with homogeneous weights in open convex cones are established, with the exact value of the optimal constant. These embeddings are non-compact, and this paper investigates the structure of their non-compactness quantitatively. Opposite to the previous results in this direction, the non-compactness in this case does not occur uniformly over all sub-domains of the underlying domain, and the problem is not translation invariant, and so these properties cannot be exploited here. Nevertheless, by developing a new approach based on a delicate interplay between the size of suitable extremal functions and the size of their supports, the exact values of the (ball) measure of non-compactness and of all injective strict s-numbers (in particular, of the Bernstein numbers) are obtained. Moreover, it is also shown that the embedding is not strictly singular.

math.FA↗