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Lenka Slavíková

Publications and source records attributed to Lenka Slavíková.

At least 19 recordsLinked to original sources
arXiv2026

A blueprint for the formalization of norm-variation of multiple ergodic averages for commuting transformations

This blueprint serves as a companion to a forthcoming, shorter traditional mathematical paper. The purpose of this blueprint is two-fold: first, it has served as the foundation for a formalization in Lean 4 of these results. This formalization has been completed largely automatically, making essential use of current frontier large language models. Second, it will serve as a resource to readers of the main paper who are interested in further technical details of the proofs. The main result concerns norm-variation estimates for multiple ergodic averages associated with $n\ge 2$ commuting measure preserving transformations, providing a quantitative strengthening of Tao's norm-convergence theorem and answering an open question of Avigad and Rute. At the core of the analysis lies an explicit real-variable estimate for twisted multilinear averages that is closely related to certain singular Brascamp--Lieb inequalities.

math.DS↗
arXiv2026

Rough averages of triangular Hilbert transforms

We study a bilinear singular integral operator obtained by taking rough averages of certain directional variants of the triangular Hilbert transform. This operator can be interpreted as the twisted paraproduct with a rough homogeneous kernel. Under a balance condition on the $L^q$ integrability of the kernel on the unit sphere, we establish a range of $L^{p_1} \times L^{p_2} \rightarrow L^p$ bounds for this operator. Our results are optimal within the known boundedness range for the twisted paraproduct with a smooth kernel.

math.CA↗
arXiv2025

Fractional Orlicz-Sobolev embeddings into Campanato type spaces

Optimal embeddings for fractional Orlicz-Sobolev spaces into (generalized) Campanato spaces on the Euclidean space are exhibited. Embeddings into vanishing Campanato spaces are also characterized. Sharp embeddings into $\operatorname{BMO}(\mathbb R^n)$ and $\operatorname{VMO}(\mathbb R^n)$ are derived as special instances. Dissimilarities to corresponding embeddings for classical fractional Sobolev spaces are pointed out.

math.FA↗
arXiv2025

Norm-variation of triple ergodic averages for commuting transformations

We prove an $r$-variation estimate, $r>4$, in the norm for ergodic averages with respect to three commuting transformations. It is not known whether such estimates hold for all $r\ge 2$ as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any $r<\infty$ for more than three commuting transformations.

math.CA↗
arXiv2025

Bilinear singular integral operators with kernels in weighted spaces

We establish the full quasi-Banach range of $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \rightarrow L^p(\mathbb R)$ bounds for one-dimensional bilinear singular integral operators with homogeneous kernels whose restriction $Ω$ to the unit sphere $\mathbb S^1$ is supported away from the degenerate line $θ_1=θ_2$, belongs to $L^q(\mathbb S^1)$ for some $q>1$ and has vanishing integral. In fact, a more general result is obtained by dropping the support condition on $Ω$ and requiring that $Ω\in L^q(\mathbb S^1,u^q)$, where $u(θ_1,θ_2)=|θ_1-θ_2|^{-1}$ for $(θ_1,θ_2)\in \mathbb S^1$. In addition, we provide counterexamples that show the failure of the $n$-dimensional version of the previous result when $n\geq 2$, as well as the failure of its $m$-linear variant in dimension one when $m\geq 3$. The relationship of these results to (un)boundedness properties of higher-dimensional multilinear Hilbert transforms is also discussed.

math.CA↗
arXiv2024

Dimension-free estimates for low degree functions on the Hamming cube

The main result of this paper are dimension-free $L^p$ inequalities, $1 2,$ $\varepsilon>0,$ and $θ=θ(\varepsilon,p)\in (0,1)$ satisfying \[ \frac{1}{p}=\fracθ{p+\varepsilon}+\frac{1-θ}{2} \] we obtain, for any function $f:\{-1,1\}^n\to \mathbb{C}$ whose spectrum is bounded from above by $d,$ the Bernstein-Markov type inequalities \[\|Δ^k f\|_{p} \le C(p,\varepsilon)^k \,d^k\, \|f\|_{2}^{1-θ}\|f\|_{p+\varepsilon}^θ,\qquad k\in \mathbb{N}.\] Analogous inequalities are also proved for $p\in (1,2)$ with $p-\varepsilon$ replacing $p+\varepsilon.$ As a corollary, if $f$ is Boolean-valued or $f\colon \{-1,1\}^n\to \{-1,0,1\},$ we obtain the bounds \[\|Δ^k f\|_{p} \le C(p)^k \,d^k\, \|f\|_p,\qquad k\in \mathbb{N}.\] At the endpoint $p=\infty$ we provide counterexamples for which a linear growth in $d$ does not suffice when $k=1$. We also obtain a counterpart of this result on tail spaces. Namely, for $p>2$ we prove that any function $f:\{-1,1\}^n\to \mathbb{C}$ whose spectrum is bounded from below by $d$ satisfies the upper bound on the decay of the heat semigroup $$ \|e^{-tΔ}f\|_{p} \le \exp(-c(p,\varepsilon) td) \|f\|_{2}^{1-θ}\|f\|_{p+\varepsilon}^θ,\qquad t>0,$$ and an analogous estimate for $p\in (1,2).$ The constants $c(p,\varepsilon)$ and $C(p,\varepsilon)$ depend only on $p$ and $\varepsilon$; crucially, they are independent of the dimension $n$.

math.FA↗
arXiv2024

Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space

Existence and global regularity results for boundary-value problems of Robin type for harmonic and polyharmonic functions in $n$-dimensional half-spaces are offered. The Robin condition on the normal derivative on the boundary of the half-space is prescribed by a nonlinear function $\mathcal N$ of the relevant harmonic or polyharmonic functions. General Orlicz type growths for the function $\mathcal N$ are considered. For instance, functions $\mathcal N$ of classical power type, their perturbations by logarithmic factors, and exponential functions are allowed. New sharp boundedness properties in Orlicz spaces of some classical operators from harmonic analysis, of independent interest, are critical for our approach.

math.AP↗
arXiv2024

Higher-order Sobolev embeddings into spaces of Campanato and Morrey type

Necessary and sufficient conditions are offered for Sobolev type spaces built on rearrangement-invariant spaces to be continuously embedded into (generalized) Campanato and Morrey spaces on open subsets of the $n$-dimensional Euclidean space. As a consequence, the optimal target and domain spaces in the relevant embeddings are identified. Our general criteria are implemented to derive sharp embeddings in the class of Orlicz-Sobolev spaces.

math.FA↗
arXiv2024

On the modulus of continuity of fractional Orlicz-Sobolev functions

Necessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on $\rn$ to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these conditions are fulfilled. These results pertain to the supercritical Sobolev regime and complement earlier sharp embeddings into rearrangement-invariant spaces concerning the subcritical setting. Classical embeddings for fractional Sobolev spaces into Hölder spaces are recovered as special instances. Proofs require novel strategies, since customary methods fail to produce optimal conclusions.

math.FA↗
arXiv2023

Multilinear singular integrals with homogeneous kernels near $L^1$

We obtain the optimal open range of $L^{p_1}(\mathbb R^n)\times\cdots\times L^{p_m}(\mathbb R^n)\to L^p(\mathbb R^n)$ bounds for multilinear singular integral operators with homogeneous kernels of the form $Ω(\frac{y}{|y|})|y|^{-mn}$, where $Ω$ is a function in $L^{q}(\mathbb S^{mn-1})$ with vanishing integral and $q>1$.

math.CA↗
arXiv2022

Local bounds for singular Brascamp-Lieb forms with cubical structure

We prove a range of $L^p$ bounds for singular Brascamp-Lieb forms with cubical structure. We pass through sparse and local bounds, the latter proved by an iteration of Fourier expansion, telescoping, and the Cauchy-Schwarz inequality. We allow $2^{m-1}<p\le \infty$ with $m$ the dimension of the cube, extending an earlier result that required $p=2^m$. The threshold $2^{m-1}$ is sharp in our theorems.

math.CA↗
arXiv2022

Boundedness of functions in fractional Orlicz-Sobolev spaces

A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^\infty(\mathbb R^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to be continuous. Improvements of this result are also offered. They provide the optimal Orlicz target space, and the optimal rearrangement-invariant target space in the embedding in question. These results complement those already available in the subcritical case, where the embedding into $L^\infty(\mathbb R^n)$ fails. They also augment a classical embedding theorem for standard fractional Sobolev spaces.

math.FA↗
arXiv2020

Bilinear Fourier multipliers and the rate of decay of their derivatives

We investigate two types of boundedness criteria for bilinear Fourier multiplier operators with symbols with bounded partial derivatives of all (or sufficiently many) orders. Theorems of the first type explicitly prescribe only a certain rate of decay of the symbol itself while theorems of the second type require, in addition, the same rate of decay of all derivatives of the symbol. We show that even though these two types of bilinear multiplier theorems are closely related, there are some fundamental differences between them which arise in limiting cases. Also, since theorems of the latter type have so far been studied mainly in connection with the more general class of bilinear pseudodifferential operators, we revisit them in the special case of bilinear Fourier multipliers, providing also some improvements of the existing results in this setting.

math.CA↗
arXiv2020

A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type

Given a bounded measurable function $σ$ on $\mathbb{R}^n$, we let $T_σ$ be the operator obtained by multiplication on the Fourier transform by $σ$. Let $0<s_1\le s_2\le \cdots \le s_n<1$ and $ψ$ be a Schwartz function on the real line whose Fourier transform $\widehatψ$ is supported in $[-2,-1/2]\cup[1/2,2]$ and which satisfies $\sum_{j \in \mathbb{Z}} \widehatψ\left(2^{-j} ξ\right)=1$ for all $ξ\neq 0$. In this work we sharpen the known forms of the Marcinkiewicz multiplier theorem by finding an almost optimal function space with the property that, if the function \begin{equation*} (ξ_1,\dots, ξ_n)\mapsto \prod_{i=1}^n (I-\partial_i^2)^{\frac {s_i}2} \Big[ \prod_{i=1}^n \widehatψ(ξ_i) σ(2^{j_1}ξ_1,\dots , 2^{j_n}ξ_n)\Big] \end{equation*} belongs to it uniformly in $j_1,\dots , j_n \in \mathbb Z$, then $T_σ$ is bounded on $ {L}^p(\mathbb R^n)$ when $ |\frac{1}{p}-\frac{1}{2} | < s_1$ and $1<p<\infty$. In the case where $s_i\neq s_{i+1}$ for all $i$, it was proved in [Grafakos, Israel J. Math., to appear] that the Lorentz space $L ^{\frac{1}{s_1},1} (\mathbb{R}^n) $ is the function space sought. In this work we address the significantly more difficult general case when for certain indices $i$ we might have $s_i=s_{i+1}$. We obtain a version of the Marcinkiewicz multiplier theorem in which the space $L ^{\frac{1}{s_1},1}$ is replaced by an appropriate Lorentz space associated with a certain concave function related to the number of terms among $s_2,\dots , s_n$ that equal $s_1$. Our result is optimal up to an arbitrarily small power of the logarithm in the defining concave function of the Lorentz space.

math.CA↗
arXiv2020

On the limit as $s\to 0^+$ of fractional Orlicz-Sobolev spaces

An extended version of the Maz'ya-Shaposhnikova theorem on the limit as $s\to 0^+$ of the Gagliardo-Slobodeckij fractional seminorm is established in the Orlicz space setting. Our result holds in fractional Orlicz-Sobolev spaces associated with Young functions satisfying the $Δ_2$-condition, and, as shown by counterexamples, it may fail if this condition is dropped.

math.FA↗
arXiv2020

Fractional Orlicz-Sobolev embeddings

The optimal Orlicz target space is exhibited for embeddings of fractional-order Orlicz-Sobolev spaces in $\mathbb R^n$. An improved embedding with an Orlicz-Lorentz target space, which is optimal in the broader class of all rearrangement-invariant spaces, is also established. Both spaces of order $s\in (0,1)$, and higher-order spaces are considered. Related Hardy type inequalities are proposed as well. An extension theorem is proved, that enables us to derive embeddings for spaces defined in Lipschitz domains. Necessary and sufficient conditions for the compactness of fractional Orlicz-Sobolev embeddings are provided.

math.FA↗
arXiv2019

Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

Sobolev embeddings, of arbitrary order, are considered into function spaces on domains of $\mathbb R^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius. Norms in arbitrary rearrangement-invariant spaces are contemplated. A comprehensive approach is proposed based on the reduction of the relevant $n$-dimensional embeddings to one-dimensional Hardy-type inequalities. Interestingly, the latter inequalities depend on the involved measure only through the power $d$. Our results allow for the detection of the optimal target space in Sobolev embeddings, for broad families of norms, in situations where customary techniques do not apply. In particular, new embeddings, with augmented target spaces, are deduced even for standard Sobolev spaces.

math.FA↗