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Krystian Kazaniecki

Publications and source records attributed to Krystian Kazaniecki.

12 recordsLinked to original sources
arXiv2025

Factoring the Sobolev embedding operator

The paper studies the factorization and summing properties of the Sobolev embedding operator. We propose two different approaches. One shows that the Sobolev embedding operator $S:W^{1,1}(\mathbb{T}^2)\hookrightarrow L_2(\mathbb{T}^2)$ factorises through the identical embedding $\ell_Φ\hookrightarrow\ell_2$ for some Young function with Matuszewska-Orlicz index 1. Proof of this fact is based on two results of independent interest. First, a necessary and sufficient conditions on a Young function $Φ$ and weight $Ψ$ for boundedness of the embedding of the Sobolev space $W^{1,1}(\mathbb{T}^2)$ into Besov-Orlicz space $B^Ψ_{Φ,1}(\mathbb{T}^2)$. Second, a generalization of the Marcinkiewicz sampling theorem to the context of Orlicz spaces. Another approach is based on the extrapolation of $(p,1)$-summing norm.

math.FA↗
arXiv2024

Schur property for jump parts of gradient measures

We consider weakly null sequences in the Banach space of functions of bounded variation $\mathrm{BV}(\mathbb{R}^d)$. We prove that for any such sequence $\{f_n\}$ the jump parts of the gradients of functions $f_n$ tend to $0$ strongly as measures. It implies that Dunford--Pettis property for the space $\mathrm{SBV}$ is equivalent to the Dunford--Pettis property for the Sobolev space $W^{1,1}.$

math.FA↗
arXiv2024

Trace operator on von Koch's snowflake

We study properties of the boundary trace operator on the Sobolev space $W^1_1(Ω)$. Using the density result by Koskela and Zhang, we define a surjective operator \mbox{$Tr: W^1_1(Ω_K)\rightarrow X(Ω_K)$}, where $Ω_K$ is von Koch's snowflake and $X(Ω_K)$ is a trace space with the quotient norm. Since $Ω_K$ is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý that there exists a right inverse to $Tr$, i.e. a linear operator $S: X(Ω_K) \rightarrow W^1_1(Ω_K)$ such that $Tr \circ S= Id_{X(Ω_K)}$. In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch's snowflake. Moreover we identify the isomorphism class of the trace space as $\ell_1$. As an additional consequence of our approach we obtain a simple proof of the Peetre's theorem about non-existence of the right inverse for domain $Ω$ with regular boundary, which explains Banach space geometry cause for this phenomenon.

math.FA↗
arXiv2024

Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm

In the present paper we identify those filtered probability spaces $(Ω,\, \mathcal{F},\, \left(\mathcal{F}_n\right),\, \mathbb{P})$ that determine already the martingale type of a Banach space $X$. We isolate intrinsic conditions on the filtration $(\mathcal{F}_n)$ of purely atomic $σ$-algebras which determine that the upper $\ell^p$ estimates \[ \|f\|_{L^p(Ω,\, X)}^p\leq C^p\left( \|\mathbb{E} f|\mathcal{F}_0\|^p_{L^p(Ω,\, X)}+\sum_{n=1}^{\infty} \|Δ_n f\|^p_{L^p(Ω,\, X)}\right),\qquad f\in L^p(Ω,X)\] imply that the Banach space $X$ is of martingale type $p$. Our paper complements \mbox{G. Pisier's} investigation \cite{Pisier1975} and continues the work by S. Geiss and second named author in \cite{Geiss2008}.

math.FA↗
arXiv2023

On Bernstein type quantitative estimates for Ornstein non-inequalities

For the sequence of multi-indexes $\{α_i\}_{i=1}^{m}$ and $β$ we study the inequality \[ \|D^β f\|_{L_1(\mathbb{T}^d)}\leq K_N \sum_{j= 1}^{m} \|D^{α_j}f\|_{L_1(\mathbb{T}^d)}, \] where $f$ is a trigonometric polynomial of degree at most $N$ on $d$-dimensional torus. Assuming some natural geometric property of the set $\{α_j\}\cup\{β\}$ we show that \[ K_{N}\geq C \left(\ln N\right)^ϕ, \] where $ϕ<1$ depends only on the set $\{α_j\}\cup\{β\}$.

math.FA↗
arXiv2015

A conditional regularity result for p-harmonic flows

We prove an $\varepsilon$-regularity result for a wide class of parabolic systems $$ u_t-\text{div}\big(|\nabla u|^{p-2}\nabla u) = B(u, \nabla u) $$ with the right hand side $B$ growing like $|\nabla u|^p$. It is assumed that the solution $u(t,\cdot)$ is uniformly small in the space of functions of bounded mean oscillation. The crucial tool is provided by a sharp nonlinear version of the Gagliardo-Nirenberg inequality which has been used earlier in an elliptic context by T. Rivière and the last named author.

math.AP↗
arXiv2015

Anisotropic Ornstein non inequalities

We investigate existence of a priori estimates for differential operators in $L^1$ norm: for anisotropic homogeneous differential operators $T_1, \ldots , T_{\ell}$, we study the conditions under which the inequality $$ \|T_1 f\|_{L_1(\mathbb{R}^d)} \lesssim \sum\limits_{j = 2}^{\ell}\|T_j f\|_{L_1(\mathbb{R}^d)} $$ holds true. We also discuss a similar problem for martingale transforms.

math.CA↗
arXiv2014

On the equivalence between the sets of the trigonometric polynomials

In this paper we construct an injection from the linear space of trigonometric polynomials defined on $\mathbb{T}^d$ with bounded degrees with respect to each variable to a suitable linear subspace $L^1_E\subset L^1(\mathbb{T})$. We give such a quantitative condition on $L^1_E$ that this injection is a isomorphism of a Banach spaces equipped with $L^1$ norm and the norm of the isomorphism is independent on the dimension $d$.

math.CA↗