arXiv subjects

Carlos Garzón

Publications and source records attributed to Carlos Garzón.

3 recordsLinked to original sources

arXiv2026

On the frequency function of Hardy-Littlewood maximal functions

We study the frequency function (introduced by Temur) in both the discrete and continuous settings. More precisely, we extend the definition of the frequency function to the higher-dimensional continuous setting and to the uncentered Hardy-Littlewood maximal function. We analyze the asymptotic behavior of the frequency function and the density of its small values for functions in $\ell^1(\mathbb{Z)}$ and $L^1(\mathbb{R}^d)$ answering some questions posed by Temur. Finally, we study the size of the frequency function for functions in $\ell^p(\mathbb{Z})$ with $p>1$, showing that this case differs significantly from the case $p=1$.

math.CA↗

arXiv2026

On Kato's smoothing effects for KdV and Benjamin type equations

We analyze how the interaction between local and nonlocal dispersions, combined with different types of nonlinearities, influences the smoothing effects of solutions. To achieve this goal, we consider a model that generalizes the KdV and Benjamin equations and demonstrate that its solutions exhibit Kato's smoothing effect and satisfy the propagation of regularity principle. As a result, we confirm that the higher-order dispersive term determines the local gain of fractional regularity of solutions. Our results are general; they not only recover known results for the KdV and Benjamin equations, but also provide new insights for a broader family of models of physical and mathematical interest with polynomial dispersions of arbitrary order.

math.AP↗

arXiv2026

The smoothest average and some extremal problems for polynomials

We study the problem of finding the "smoothest'' local average of a function $f \in \ell^2(\mathbb{Z})$ when we consider its convolution with suitable kernels $u$. The measurement of smoothness is as follows: Given a positive integer $k$, we aim to minimize the constant \begin{equation*} \sup_{0 \neq f \in \ell^2(\mathbb{Z})} \frac{\|\nabla^{k}(u\ast f)\|_{\ell^2(\mathbb{Z})}}{\|f\|_{\ell^2(\mathbb{Z})}} \end{equation*} among all symmetric kernels $u : \{-n,\dots,n\} \to \mathbb{R}$ with normalization $\sum_{j=-n}^{n}u(j) = 1$. We are also interested in finding the kernel for which the least constant is attained. For $k=1$ and $k=2$, the sharp constants and optimal kernels were obtained by Kravitz-Steinerberger, and Richardson. In this paper, we provide alternative proofs for $k\in \{1,2\}$ by using complex analysis tools. Moreover, we establish the case $k=3$, and also the cases $k\in \{4,6\}$ when the kernels are restricted to have non-negative Fourier transform. These are the first results in the literature for $k>2$. Finally, we deduce a general relation between the sharp constants and optimal kernels corresponding to $\nabla^k$ and $\nabla^{2k}$.

math.CA↗