arXiv subjects

Pedro Fernández-Martínez

Publications and source records attributed to Pedro Fernández-Martínez.

6 recordsLinked to original sources

arXiv2022

Some examples of equivalent rearrangement-invariant quasi-norms defined via f* or f**

We consider Lorentz-Karamata spaces, small and grand Lorentz-Karamata spaces, and the so-called $\mathcal{L}$, $\mathcal{R}$, $\mathcal{LL}$, $\mathcal{RL}$, $\mathcal{RL}$, and $\mathcal{RR}$ spaces. The quasi-norms for a function $f$ in each of these spaces can be defined via the non-increasing rearrangement $f^*$ or via the maximal function $f^{**}$. We investigate when these quasi-norms are equivalent. Most of the proofs are based on Hardy-type inequalities. As application we demonstrate how our general results can be used to establish interpolation formulae for the grand and small Lorentz-Karamata spaces.

math.FA↗

arXiv2022

Reiteration Formulae for the Real Interpolation Method Including limiting ${\mathcal L}$ or ${\mathcal R}$ Spaces

We consider K-interpolation methods involving slowly varying functions. Let $\overline{A}_{θ,*}^{\mathcal{L}}$ and $\overline{A}_{θ,*}^{\mathcal{R}}$ $(0\leqθ\leq1)$ be the so called ${\mathcal{L}}$ or ${\mathcal{R}}$ limiting interpolation spaces which arise naturally in reiteration formulae for the limiting cases. We characterize the interpolation spaces $\Big(\overline{A}_{θ_0,*}^{\mathcal{L}}, *\Big)_{η,r,a}$, $\Big(\overline{A}_{θ_0,*}^{\mathcal{R}}, *\Big)_{η,r,a}$, $\Big(*, \overline{A}_{θ_1,*}^{\mathcal{L}}\Big)_{η,r,a}$, and $\Big(*, \overline{A}_{θ_1,*}^{\mathcal{R}}\Big)_{η,r,a}$ $(0\leqη\leq1)$ for the limiting cases $θ_0=0$ and $θ_1=1$. This supplements the earlier papers of the authors, which only considered the case $0<θ_0<θ_1<1$. The proofs of most reiteration formulae are based on Holmstedt-type formulae. Applications to grand and small Lorentz spaces as well as to Lorentz-Karamata spaces are given.

math.FA↗

arXiv2021

Reiteration Theorem for ${\mathcal R}$ and ${\mathcal L}$-spaces with the same parameter

Let $E, F, E_0, E_1$ be rearrangement invariant spaces; let $a, \mathrm{b}, \mathrm{b}_0, \mathrm{b}_1$ be slowly varying functions and $0< θ_0,θ_1<1$. We characterize the interpolation spaces $$\Big(\overline{X}^{\mathcal R}_{θ_0,\mathrm{b}_0,E_0,\mathrm{a},F}, \overline{X}^{\mathcal L}_{θ_1,\mathrm{b}_1,E_1,\mathrm{a},F}\Big)_{η,\mathrm{b},E}\:, \quad 0\leqη\leq1,$$ when the parameters $θ_0$ and $θ_1$ are equal (under appropriate conditions on $\mathrm{b}_i(t)$, $i=0,1$). This completes the study started in \cite{Do2020,FMS-RL3}, which only considered the case $θ_0<θ_1$. As an application we recover and generalize interpolation identities for grand and small Lebesgue spaces.

math.FA↗

arXiv2021

General Reiteration Theorems for $\mathcal{R}$ and $\mathcal{L}$ Clases: Mixed Interpolation of $\mathcal{R}$ and $\mathcal{L}$-spaces

Given $E_0, E_1, F_0, F_1, E$ rearrangement invariant function spaces, $a_0$, $a_1$, $b_0$, $b_1$, $b$ slowly varying functions and $0< θ_0<θ_1<1$, we characterize the interpolation spaces $$(\overline{X}^{\mathcal R}_{θ_0,b_0,E_0,a_0,F_0}, \overline{X}^{\mathcal R}_{θ_1, b_1,E_1,a_1,F_1})_{θ,b,E},\quad (\overline{X}^{\mathcal L}_{θ_0, b_0,E_0,a_0,F_0}, \overline{X}^{\mathcal L}_{θ_1,b_1,E_1,a_1,F_1})_{θ,b,E}$$ and $$(\overline{X}^{\mathcal R}_{θ_0,b_0,E_0,a_0,F_0}, \overline{X}^{\mathcal L}_{θ_1, b_1,E_1,a_1,F_1})_{θ,b,E},\quad (\overline{X}^{\mathcal L}_{θ_0, b_0,E_0,a_0,F_0}, \overline{X}^{\mathcal R}_{θ_1,b_1,E_1,a_1,F_1})_{θ,b,E},$$ for all possible values of $θ\in[0,1]$. Applications to interpolation identities for grand and small Lebesgue spaces, Gamma spaces and $A$ and $B$-type spaces are given.

math.FA↗

arXiv2020

General Reiteration Theorems for R and L Classes: Case of Right R-Spaces and Left L-Spaces

Given $E_0, E_1, E, F$ rearrangement invariant spaces, $a, b, b_0, b_1$ slowly varying functions and $0\leq θ_0<θ_1\leq 1$, we characterize the interpolation space $$(\overline{X}_{θ_0,b_0,E_0}, \overline{X}^{\mathcal R}_{θ_1, b_1,E_1,a,F})_{θ,b,E} \quad \text{and} \quad (\overline{X}^{\mathcal L}_{θ_0, b_0,E_0,a,F}, \overline{X}_{θ_1,b_1,E_1})_{θ,b,E},$$ for all possible values of $θ\in[0,1]$. Applications to interpolation identities for grand and small Lebesgue spaces, Gamma spaces and $A$ and $B$-type spaces are given.

math.FA↗

arXiv2017

New Young inequalities and applications

We establish upper bounds for the convolution operator acting between interpolation spaces. This will provide several examples of Young Inequalities in different families of function spaces. We use this result to prove a bilinear interpolation theorem and we show applications to the study of bilinear multipliers.

math.FA↗