arXiv subjects

Karol Lesnik

Publications and source records attributed to Karol Lesnik.

8 recordsLinked to original sources

arXiv2024

Gagliardo-Nirenberg inequality via a new pointwise estimate

We prove a new type of pointwise estimate of the Kalamajska-Mazya-Shaposhnikova type, where sparse averaging operators replace the maximal operator. It allows us to extend the Gagliardo-Nirenberg interpolation inequality to all rearrangement invariant Banach function spaces without any assumptions on their upper Boyd index, i.e. omitting problems caused by unboundedness of maximal operator on spaces close to $L^1$. In particular, we remove unnecessary assumptions from the Gagliardo-Nirenberg inequality in the setting of Orlicz and Lorentz spaces. The applied method is new in this context and may be seen as a kind of sparse domination technique fitted to the context of rearrangement invariant Banach function spaces.

math.FA↗

arXiv2022

Optimal Gagliardo-Nirenberg interpolation inequality for rearrangement invariant spaces

We prove optimality of the Gagliardo-Nirenberg inequality $$ \|\nabla u\|_{X}\lesssim\|\nabla^2 u\|_Y^{1/2}\|u\|_Z^{1/2}, $$ where $Y, Z$ are rearrangement invariant Banach function spaces and $X=Y^{1/2}Z^{1/2}$ is the Calderón--Lozanovskii space. By optimality, we mean that for a certain pair of spaces on the right-hand side, one cannot reduce the space on the left-hand, remaining in the class of rearrangement invariant spaces. The optimality for the Lorentz and Orlicz spaces is given as a consequence, exceeding previous results. We also discuss pointwise inequalities, their importance and counterexample prohibiting an improvement.

math.FA↗

arXiv2019

Weakly compact sets and weakly compact pointwise multipliers in Banach function lattices

We prove that the class of Banach function lattices in which all relatively weakly compact sets are equi-integrable sets (i.e. spaces satisfying the Dunford-Pettis criterion) coincides with the class of 1-disjointly homogeneous Banach lattices. A new examples of such spaces are provided. Furthermore, it is shown that Dunford-Pettis criterion is equivalent to de la Vallee Poussin criterion in all rearrangement invariant spaces on the interval. Finally, the results are applied to characterize weakly compact pointwise multipliers between Banach function lattices.

math.FA↗

arXiv2018

Toeplitz and Hankel operators between distinct Hardy spaces

The paper gives the background for Toeplitz $T_a$ and Hankel $H_a$ operators acting between distinct Hardy type spaces over the unit circle $\mathbb{T}$. We characterize possible symbols of such operators and prove general versions of Brown-Halmos and Nehari theorems. The lower bound for measure of noncomactness of Toeplitz operator is also found. Our approach allows Hardy spaces associated with arbitrary rearrangement invariant spaces, but part of the results is new even for the classical case of $H^p$ spaces.

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arXiv2018

Symmetrization, factorization and arithmetic of quasi-Banach function spaces

We investigate relations between symmetrizations of quasi-Banach function spaces and constructions such as Calderon-Lozanovskii spaces, pointwise product spaces and pointwise multipliers. We show that under reasonable assumptions the symmetrization commutes with these operations. We determine also the spaces of pointwise multipliers between Lorentz spaces and Cesaro spaces. Developed methods may be regarded as an arithmetic of quasi-Banach function spaces and proofs of Theorems 3, 4 and 6 give a kind of tutorial for these methods. Finally, the above results will be used in proofs of some factorization results.

math.FA↗

arXiv2015

Interpolation of abstract Cesaro, Copson and Tandori spaces

We study real and complex interpolation of abstract Cesàro, Copson and Tandori spaces, including the description of Calderón-Lozanovski{\v ı} construction for those spaces. The results may be regarded as generalizations of interpolation for Cesàro spaces $Ces_p(I)$ in the case of real method, but they are new even for $Ces_p(I)$ in the case of complex method. Some results for more general interpolation functors are also presented. The investigations show an interesting phenomenon that there is a big difference between interpolation of Cesàro function spaces in the cases of finite and infinite interval.

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arXiv2015

Monotone substochastic operators and a new Calderon couple

An important result on submajorization, which goes back to Hardy, Littlewood and Pólya, states that $b\preceq a$ if and only if there is a doubly stochastic matrix $A$ such that $b=Aa$. We prove that under monotonicity assumptions on vectors $a$ and $b$ respective matrix $A$ may be chosen monotone. This result is then applied to show that $(\widetilde{L^p},L^{\infty})$ is a Calderón couple for $1\leq p<\infty $, where $\widetilde{L^{p}}$ is the Köthe dual of the Cesàro space $Ces_{p'}$ (or equivalently the down space $L^{p'}_{\downarrow}$). In particular, $(\widetilde{L^1},L^{\infty})$ is a Calderón couple and this complements the result of [MS06] where it was shown that $(L^{\infty}_{\downarrow},L^{1})$ is a Calderón couple.

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arXiv2012

Pointwise multipliers of Calderón-Lozanovskii spaces

Several results concerning multipliers of symmetric Banach function spaces are presented firstly. Then the results on multipliers of Calderón-Lozanovskii spaces are proved. We investigate assumptions on a Banach ideal space E and three Young functions φ_1, φ_2 and φ, generating the corresponding Calderón-Lozanovskii spaces E_{φ_1}, E_{φ_2}, E_φ so that the space of multipliers M(E_{φ_1}, E_φ) of all measurable x such that x,y \in E_φ for any y \in E_{φ_1} can be identified with E_{φ_2}. Sufficient conditions generalize earlier results by Ando, O'Neil, Zabreiko-Rutickii, Maligranda-Persson and Maligranda-Nakai. There are also necessary conditions on functions for the embedding M(E_{φ_1}, E_φ) \subset E_{φ_2} to be true, which already in the case when E = L^1, that is, for Orlicz spaces M(L^{φ_1}, L^φ) \subset L^{φ_2} give a solution of a problem raised in the book [Ma89]. Some properties of a generalized complementary operation on Young functions, defined by Ando, are investigated in order to show how to construct the function φ_2 such that M(E_{φ_1}, E_φ) = E_{φ_2}. There are also several examples of independent interest.

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