Searcharxiv⌕ Search

arXiv subjects

Pablo Rocha

Publications and source records attributed to Pablo Rocha.

33 records · Page 2Linked to original sources
arXiv2022

Inequalities for weighted spaces with variable exponents

In this article we obtain an "off-diagonal" version of the Fefferman-Stein vector-valued maximal inequality on weighted Lebesgue spaces with variable exponents. As an application of this result and the atomic decomposition developed in [12] we prove, for certain exponents $q(\cdot)$ in $\mathcal{P}^{\log}(\mathbb{R}^{n})$ and certain weights $ω$, that the Riesz potential $I_α$, with $0 < α< n$, can be extended to a bounded operator from $H^{p(\cdot)}_ω(\mathbb{R}^{n})$ into $L^{q(\cdot)}_ω(\mathbb{R}^{n})$, for $\frac{1}{p(\cdot)} := \frac{1}{q(\cdot)} + \fracα{n}$.

math.CA↗
arXiv2022

Fractional series operators on discrete Hardy spaces

We estudy the $H^{p}(\mathbb{Z})$ - $\ell^{q}(\mathbb{Z})$ boundedness of the fractional series operator $T_γ$ given by \[ (T_γb)(j) = \sum_{i \neq \pm j} \frac{b(i)}{|i-j|^α|i+j|^β}, \] where $0 \leq γ< 1$, $α, β> 0$ and $α+ β= 1 -γ$. By means of a counter-example, we also show that the operator $T_γ$ is not bounded from $H^{p}(\mathbb{Z})$ into $H^{q}(\mathbb{Z})$.

math.CA↗
arXiv2022

A decay estimate for the Fourier transform of certain singular measures in $\mathbb{R}^{4}$ and applications

We consider, for a class of functions $φ: \mathbb{R}^{2} \setminus \{ {\bf 0} \} \to \mathbb{R}^{2}$ satisfying a nonisotropic homogeneity condition, the Fourier transform $\hatμ$ of the Borel measure on $\mathbb{R}^{4}$ defined by \[ μ(E) = \int_{U} χ_{E}(x, φ(x)) \, dx \] where $E$ is a Borel set of $\mathbb{R}^{4}$ and $U = \{ (t^{α_1}, t^{α_2}s) : c < s < d, \, 0 < t < 1 \}$. The aim of this article is to give a decay estimate for $\hatμ$, for the case where the set of nonelliptic points of $φ$ is a curve in $\bar{U} \setminus \{ {\bf 0} \}$. From this estimate we obtain a restriction theorem for the usual Fourier transform to the graph of $φ_{U} : U \to \mathbb{R}^{2}$. We also give $L^{p}$-improving properties for the convolution operator $T_μ f = μ\ast f$.

math.CA↗
arXiv2021

A multiplicative measure on the positive real axis

In this note we construct a measure $μ$ on a $σ$-algebra $\mathcal{M}$ of subsets of the positive real axis, $\mathbb{R}_{>0}$, with the following multiplicative property: \[ μ\left( \bigcup_j E_j \right) = \prod_j μ(E_j) \] for every countable collection $\{ E_j \}$ of pairwise disjoint sets of $\mathcal{M}$. For them, we apply the Carathéodory's procedure to the triplet $\left( \mathbb{R}_{>0}, \cdot \, , τ\right)$, where $\cdot$ is the product of R and $τ$ is the usual topology on $\mathbb{R}_{>0}$. We conclude this note describing the connection between this multiplicative measure $μ$ and the Lebesgue measure.

math.CA↗
arXiv2015

A Note on Hardy Spaces and Bounded Operators

In this note we show that if f belongs to Hp(Rn)\capLs(Rn), where 0 < p <= 1 < s < 1, then there exists a (p;infinite)-atomic decomposition which converges to f in Ls(Rn). From this fact, we prove that a bounded operator T on Ls(Rn) can be extended to a bounded operator from Hp(Rn) into Lp(Rn) if and only if T is bounded uniformly in Lp norm on all (p;infinite)-atoms. A similar result is also obtained from Hp(Rn) into Hp(Rn).

math.CA↗