arXiv subjects

Sorina Barza

Publications and source records attributed to Sorina Barza.

9 recordsLinked to original sources

arXiv2026

A variation on the Pólya-Segő principle in one dimension

We establish a one-dimensional Pólya-Szegő principle for the Riesz $(p,α)$-variation $\mathcal{V}_p^α$, a family of functionals that interpolates between Wiener $p$-variation and the Sobolev seminorm $\|f'\|_{L^p}$. More precisely, for every measurable function $f$, we prove that its non-increasing rearrangement $f^*$ satisfies $$ \mathcal{V}_p^α(f^*)\le\mathcal{V}_p^α(f) $$ for all $1\le p<\infty$ and $0\leα\le 1-1/p$. This result contains and extends the classical variation-diminishing property of rearrangements from Sobolev spaces to a scale of spaces admitting fractional smoothness and, in particular, applies to functions that are nowhere differentiable. Our methods also yield a sharp rearrangement inequality for a modulus of continuity defined via $p$-variation, providing an analogue of a problem posed by Ul'yanov. Finally, we sketch how our inequality can be useful in the study of nonlinear Fredholm integral equations.

math.CA↗

arXiv2026

Discrete reversed Hardy inequalities and Hölder estimates in Lorentz sequence spaces

We prove sharp reversed Hardy-type inequalities on the cone of decreasing sequences in weighted $\ell^p$-spaces with power weights, applying them to estimate standard norms in Lorentz sequence spaces. We prove that the discrete structure of these spaces yields optimal constants that are different from those in the continuous setting.

math.FA↗

arXiv2026

Eliminating positive-measure level sets by small Lipschitz perturbations

We establish a new regularity phenomenon of continuous functions. Specifically, given any continuous function $f$ and arbitrary $ε>0$, we construct a Lipschitz perturbation $g_ε$ whose Lipschitz seminorm is less than $ε$ such that every level set of $f+g_ε$ has Lebesgue measure zero.

math.CA↗

arXiv2023

End-point Norm Estimates for Cesàro and Copson Operators

For a large class of operators acting between weighted $\ell^\infty$ spaces, exact formulas are given for their norms and the norms of their restrictions to the cones of nonnegative sequences; nonnegative, nonincreasing sequences; and nonnegative, nondecreasing sequences. The weights involved are arbitrary nonnegative sequences and may differ in the domain and codomain spaces. The results are applied to the Cesàro and Copson operators, giving their norms and their distances to the identity operator on the whole space and on the cones. Simplifications of these formulas are derived in the case of these operators acting on power-weighted $\ell^\infty$. As an application, best constants are given for inequalities relating the weighted $\ell^\infty$ norms of the Cesàro and Copson operators both for general weights and for power weights.

math.FA↗

arXiv2017

Factorizations of weighted Hardy inequalities

We present factorizations of weighted Lebesgue, Ce\-s\` aro and Copson spaces, for weights satisfying the conditions which assure the boundedness of the Hardy's integral operator between weighted Lebesgue spaces. Our results enhance, among other, the best known forms of weighted Hardy inequalities.

math.CA↗

arXiv2007

Sharp constants related to the triangle inequality in Lorentz spaces

We study the Lorentz spaces $L^{p,s}(R,μ)$ in the range $1 1$.

math.FA↗

arXiv2005

Hardy's inequalities for monotone functions on partially ordered measure spaces

We characterize the weighted Hardy's inequalities for monotone functions in ${\mathbb R^n_+}.$ In dimension $n=1$, this recovers the classical theory of $B_p$ weights. For $n>1$, the result was only known for the case $p=1$. In fact, our main theorem is proved in the more general setting of partially ordered measure spaces.

math.CA↗

arXiv2004

Mixed norm and multidimensional Lorentz spaces

In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved (\cite{Sa}, \cite{CaSoA}). However, the question for multidimensional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the two-dimensional decreasing rearrangement, to be normable. To this end, it is also useful to study the mixed norm Lorentz spaces. Finally, we prove embeddings between all the classical, multidimensional, and mixed norm Lorentz spaces.

math.CA↗

arXiv2001

Multidimensional rearrangement and Lorentz spaces

We define a multidimensional rearrangement, which is related to classical inequalities for functions that are monotone in each variable. We prove the main measure theoretical results of the new theory and characterize the functional properties of the associated weighted Lorentz spaces.

math.CA↗