arXiv subjects

Luboš Pick

Publications and source records attributed to Luboš Pick.

At least 19 recordsLinked to original sources

arXiv2026

Operators on Orlicz sequence spaces and $Δ_2$-fundamentality

A classical result states that the Hardy--Littlewood maximal operator is bounded on an Orlicz space $L^A(\mathbb{R}^n)$ if and only if its conjugate Young function $\tilde{A}$ satisfies the $Δ_2$-condition. The same condition also characterizes the boundedness on $L^A(0,\infty)$ of the Hardy averaging operator. We consider a discrete analogue of the problem, extended to a general interpolation framework. We offer several characterizing conditions for the boundedness of discrete maximal and average operators on Orlicz spaces. Although the principal result is as expected, for its proof some new techniques have to be developed. To this end, we introduce a new notion of the so-called $Δ_2$-fundamental sequence, and give its interesting characterization by a simple condition involving only a limes superior of the ratio of two subsequent terms. We also prove a dual statement concerning operators of Copson type.

math.FA↗

arXiv2026

Non-improvability of sharp endpoint estimates

For an integer $n$ and the parameter $γ\in(0,n)$, the Riesz potential $I_γ$ is known to take boundedly $L^1(\mathbb{R}^n)$ into $L^{\frac{n}{n-γ},\infty}(\mathbb{R}^n)$, and also that the target space is the smallest possible among all rearrangement-invariant Banach function spaces. We study the natural question whether the target space can be improved when the domain space is replaced with a (smaller) Lorentz space $L^{1,q}(\mathbb{R}^n)$ with $q\in(0,1)$. The classical methods cannot be used because the spaces $L^{1,q}(\mathbb{R}^n)$ are not equivalently normable. We develop two new abstract methods, establishing rather general results, a particular consequence of each (albeit achieved through completely different means) being the negative answer to this question. The methods are based on special functional properties of endpoint spaces. The results can be applied to a wide field of operators satisfying certain minimal requirements.

math.FA↗

arXiv2026

Potential trace inequalities via a Calderón-type theorem

In this paper we develop a general theoretical tool for the establishment of the boundedness of notoriously difficult operators (such as potentials) on certain specific types of rearrangement-invariant function spaces from analogous properties of operators that are easier to handle (such as fractional maximal operators). A principal example of the new results one obtains by our analysis is the following inequality, which generalizes a result of Korobkov and Kristensen (who had treated the case $μ=\mathcal{L}^n$, the Lebesgue measure on $\mathbb{R}^n$): There exists a constant $C>0$ such that \[\int_{\mathbb{R}^n} |I_α^μf|^p dν\leq C \|f\|_{L^{p,1}(\mathbb{R}^n,μ)}^p\] for all $f$ in the Lorentz space $L^{p,1}(\mathbb{R}^n,μ)$, where $μ, ν$ are Radon measures such that \[\sup_{Q} \frac{μ(Q)}{l(Q)^{d}} < \infty \quad \text{and} \quad \sup_{μ(Q)>0} \frac{ν(Q)}{\quadμ(Q)^{1-\frac{αp}{d}}} < \infty,\] and $I_α^μ$ is the Riesz potential defined with respect to $μ$ of order $α\in (0,d)$. More broadly, we obtain inequalities in this spirit in the context of rearrangement-invariant spaces through a result of independent interest, an extension of an interpolation theorem of Calderón where the target space in one endpoint is a space of bounded functions.

math.FA↗

arXiv2025

Interpolation of classical Lorentz spaces measuring oscillation

We obtain an explicit characterization of the $K$-functional of a pair of weighted classical Lorentz spaces of type $S$. We develop a method for obtaining such characterization based on a relation between the desired quantity and the $K$-functional of a specific couple of spaces of type $Λ$, which are substantially more manageable than their companions of type $S$. The core of our techniques is a subtle manipulation with respective fundamental functions. We present several applications, in particular we nail down a formula for the $K$-functional of a Lebesgue space and a classical Lorentz space of type $S$ with a power weight, and using this formula we establish an inequality of a reverse Marchaud type.

math.FA↗

arXiv2025

The Persson--Stepanov theorem revisited

We develop a new proof of the result of L.-E.~Persson and V.D.~Stepanov \cite[Theorems 1 and 3]{Per:02}, which provides a characterization of a Hardy integral inequality involving two weights, and which can be applied to an effective treatment of the geometric mean operator. Our approach enables us to extend their result to the full range of parameters, in particular involving the critical case $p=1$, which was excluded in the original work. Our proof avoids all duality steps and discretization techniques and uses solely elementary means.

math.FA↗

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

Maximal noncompactness of embeddings into Marcinkiewicz spaces

We develop a new functional-analytic technique for investigating the degree of noncompactness of an operator defined on a quasinormed space and taking values in a Marcinkiewicz space. The main result is a general principle from which it can be derived that such operators are almost always maximally noncompact in the sense that their ball measure of noncompactness coincides with their operator norm. We point out specifications of the universal principle to the case of the identity operator.

math.FA↗

arXiv2025

Embeddings between generalized weighted Lorentz spaces

We give a new characterization of a continuous embedding between two function spaces of type $GΓ$. Such spaces are governed by functionals of type \begin{equation*} \|f\|_{GΓ(r,q;w,δ)} := \left(\int_{0}^{L} \left( \frac1{Δ(t)} \int_0^t f^*(s)^r δ(s) ds \right)^{\frac{q}{r}} w(t) dt \right)^\frac1{q}, \end{equation*} in which $f^*$ is the nonincreasing rearrangement of $f$, $L\in(0,\infty]$, $r,q \in (0, \infty)$, $w, δ$ are weights on $(0,L)$ and $Δ(t)=\int_{0}^{t}δ(s)\,ds$ for $t\in(0,L)$. To characterize the embedding of such a space, say $GΓ(r_1,q_1;w_1,δ_1)$, into another, $GΓ(r_2,q_2;w_2,δ_2)$, means to find a balance condition on the four positive real parameters and the four weights in order that an appropriate inequality holds for every admissible function. We develop a new discretization technique which will enable us to get rid of restrictions on parameters imposed in earlier work such as the non-degeneracy conditions or certain relations between the $r$'s and $q$'s. Such restrictions were caused mainly by the use of duality techniques, which we avoid in this paper. On the other hand we consider here only the case when $q_1 \le q_2$, leaving the reverse case to future work.

math.FA↗

arXiv2025

Weighted inequalities for sub-monotone functionals

We establish a set of relations between several quite diverse types of weighted inequalities involving various integral operators and fairly general quasinorm-like functionals which we call sub-monotone. The main result enables one to solve a specific problem by transferring it to another one for which a solution is known. The main result is formulated in a rather surprising generality, involving previously unknown cases, and it works even for some nonlinear operators such as the geometric or harmonic mean operators. Proofs use only elementary means.

math.CA↗

arXiv2024

Almost compact embeddings between Orlicz and Lorentz spaces

We characterize when an Orlicz space $L^A$ is almost compactly (uniformly absolutely continuously) embedded into a Lorentz space $L^{p,q}$ in terms of a balance condition involving parameters $p,q\in[1,\infty]$, and a Young function $A$. In the course of the proof, we develop a new method based on an inequality of Young type involving the measure of level sets of a given function.

math.FA↗

arXiv2024

Maximal noncompactness of limiting Sobolev embeddings

We develop a new method suitable for establishing lower bounds on the ball measure of noncompactness of operators acting between considerably general quasinormed function spaces. This new method removes some of the restrictions oft-presented in the previous work. Most notably, the target function space need not be disjointly superadditive nor equipped with a norm. Instead, a property that is far more often at our disposal is exploited, namely the absolute continuity of the target quasinorm. We use this new method to prove that limiting Sobolev embeddings into spaces of Brezis--Wainger type are so-called maximally noncompact, i.e., their ball measure of noncompactness is the worst possible.

math.FA↗

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

Optimal Sobolev embeddings for the Ornstein-Uhlenbeck operator

A comprehensive analysis of Sobolev-type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space is offered. A unified approach is proposed, providing one with criteria for their validity in the class of rearrangement-invariant function norms. Optimal target and domain norms in the relevant inequalities are characterized via a reduction principle to one-dimensional inequalities for a Calderón type integral operator patterned on the Gaussian isoperimetric function. Consequently, the best possible norms in a variety of specific families of spaces, including Lebesgue, Lorentz, Lorentz-Zygmund, Orlicz and Marcinkiewicz spaces, are detected. The reduction principle hinges on a preliminary discussion of the existence and uniqueness of generalized solutions to equations, in the Gauss space, for the Ornstein-Uhlenbeck operator, with a just integrable right-hand side. A decisive role is also played by a pointwise estimate, in rearrangement form, for these solutions.

math.FA↗

arXiv2022

Optimality problems in Orlicz spaces

In mathematical modelling, the data and solutions are represented as measurable functions and their quality is oftentimes captured by the membership to a certain function space. One of the core questions for an analysis of a model is the mutual relationship between the data and solution quality. The optimality of the obtained results deserves a special focus. It requires a careful choice of families of function spaces balancing between their expressivity, i.e. the ability to capture fine properties of the model, and their accessibility, i.e. its technical difficulty for practical use. This paper presents a unified and general approach to optimality problems in Orlicz spaces. Orlicz spaces are parametrized by a single convex function and neatly balance the expressivity and accessibility. We prove a general principle that yields an easily verifiable necessary and sufficient condition for the existence or the non-existence of an optimal Orlicz space in various tasks. We demonstrate its use in specific problems, including the continuity of Sobolev embeddings and boundedness of integral operators such as the Hardy--Littlewood maximal operator and the Laplace transform.

math.FA↗

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↗

arXiv2022

Weighted inequalities for a superposition of the Copson operator and the Hardy operator

We study a three-weight inequality for the superposition of the Hardy operator and the Copson operator, namely \begin{equation*} \bigg(\int_a^b \bigg(\int_t^b \bigg(\int_a^s f(τ)^p v(τ) \,dτ\bigg)^\frac{q}{p} u(s) \,ds \bigg)^{\frac{r}{q}} w(t) \,dt \bigg)^{\frac{1}{r}} \leq C \int_a^b f(t)\,dt, \end{equation*} in which $(a,b)$ is any nontrivial interval, $q,r$ are positive real parameters and $p\in(0,1]$. A simple change of variables can be used to obtain any weighted $L^p$-norm with $p\ge1$ on the right-hand side. Another simple change of variables can be used to equivalently turn this inequality into the one in which the Hardy and Copson operators swap their positions. We focus on characterizing those triples of weight functions $(u,v,w)$ for which this inequality holds for all nonnegative measurable functions $f$ with a constant independent of $f$. We use a new type of approach based on an innovative method of discretization which enables us to avoid duality techniques and therefore to remove various restrictions that appear in earlier work. This paper is dedicated to Professor Stefan Samko on the occasion of his 80th birthday.

math.FA↗

arXiv2022

Weighted inequalities involving Hardy and Copson operators

We characterize a four-weight inequality involving the Hardy operator and the Copson operator. More precisely, given $p_1, p_2, q_1, q_2 \in (0, \infty)$, we find necessary and sufficient conditions on nonnegative measurable functions $u_1, u_2, v_1, v_2$ on $(0,\infty)$ for which there exists a positive constant $c$ such that the inequality \begin{align*} &\bigg(\int_0^{\infty} \bigg(\int_0^t f(s)^{p_2} v_2(s)^{p_2} ds \bigg)^{\frac{q_2}{p_2}} u_2(t)^{q_2} dt \bigg)^{\frac{1}{q_2}} \notag \\ & \hspace{3cm} \leq c \bigg(\int_0^{\infty} \bigg(\int_t^{\infty} f(s)^{p_1} v_1(s)^{p_1} ds \bigg)^{\frac{q_1}{p_1}} u_1(t)^{q_1} dt \bigg)^{\frac{1}{q_1}} \end{align*} holds for every non-negative measurable function $f$ on $(0, \infty)$. The proof is based on discretizing and antidiscretizing techniques. The principal innovation consists in development of a new method which carefully avoids duality techniques and therefore enables us to obtain the characterization in previously unavailable situations, solving thereby a long-standing open problem. We then apply the characterization of the inequality to the establishing of criteria for embeddings between weighted Copson spaces $\operatorname{Cop}_{p_1,q_1} (u_1, v_1)$ and weighted Cesàro spaces $\operatorname{Ces}_{p_2, q_2} (u_2, v_2)$, and also between spaces $S^q(w)$ equipped with the norm $\|f\|_{S^q(w)}= \bigg(\int_0^\infty [f^{**}(t)-f^*(t)]^q w(t)\,dt\bigg)^{{1}/{q}}$ and classical Lorentz spaces of type $Λ$.

math.FA↗