Searcharxiv⌕ Search

arXiv subjects

Pablo Rocha

Publications and source records attributed to Pablo Rocha.

At least 19 recordsLinked to original sources
arXiv2026

A note on variable (Hardy-)Lorentz spaces

The purpose of this note is to establish further properties of the variable Lorentz spaces $\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$ introduced by L. Ephremidze, V. Kokilashvili and S. Samko, which will allow us to apply the theory of Hardy spaces associated with ball quasi-Banach function spaces, and so define the variable Hardy-Lorentz spaces associated with $\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$. Then, the finite and infinite atomic decompositions for these spaces will be deduced immediately. We also obtain the boundedness of singular integrals and fractional type operators in variable Hardy-Lorentz spaces, and provide a Fefferman-Stein vector-valued inequality for the fractional maximal operator on the $r$-convexification of variable Lorentz spaces.

math.FA↗
arXiv2026

On potentials of distributions in Orlicz-Hardy type spaces on the Heisenberg group

In this work, we introduce Orlicz-Hardy type spaces and Orlicz-Calder\'on Hardy type spaces on the Heisenberg group $\mathbb{H}^{n}$ and study the relationship between them by means of the Heisenberg sub-Laplacian $\mathcal{L}$. More precisely, we show, under suitable assumptions, that every distribution in the Orlicz-Hardy space $H^{\Phi}(\mathbb{H}^{n})$ can be represented uniquely as the sub-Laplacian of a function in an appropriate Orlicz-Calder\'on Hardy space. In this way, for any $f \in H^{\Phi}(\mathbb{H}^{n})$, we obtain a uniqueness and solvability result for the equation $\mathcal{L}F=f$.

math.CA↗
arXiv2026

Fractional type operators on Hardy spaces associated with ball quasi-Banach function spaces

For $0 \leq \alpha < n$ and $m \in \mathbb{N} \cap \left(1 - \frac{\alpha}{n}, +\infty \right)$, we consider certain fractional type operators $T_{\alpha, m}$ generated by $m$-orthogonal matrices and prove that, for $0 < \alpha < n$, $T_{\alpha, m}$ can be extended to a bounded operator $H_X \to Y$ and, for $\alpha = 0$, $T_{0, m}$ can be extended to a bounded operator $H_X \to X$, where $X$ and $Y$ are certain ball quasi-Banach spaces related to each other and $H_X$ is the Hardy space associated with $X$. In particular, our results apply to weighted Lebesgue spaces, variable Lebesgue spaces, Lorentz spaces and Orlicz spaces, the last two are new. Our proofs rely on the ssumption that $X$ is $\mathcal{O}(n)$-invariant, the theory of weighted Hardy spaces, the Rubio de Francia iteration algorithm and the finite atomic decomposition of $H_X$.

math.FA↗
arXiv2025

Estimates for convolution operators on Hardy spaces associated with ball quasi-Banach function spaces

Let $0 \leq \alpha < n$, $N \in \mathbb{N}$, and let $X$ and $Y$ be ball quasi-Banach function spaces on $\mathbb{R}^n$. We consider operators $T_{\alpha}$ defined by convolution with kernels of type $(\alpha, N)$. Assuming that the powered Hardy-Littlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on $X$ and is bounded on the associated space, we prove that $T_0$, $\alpha = 0$, extends to a bounded operator $H_{X}(\mathbb{R}^n) \to X$ and $H_{X}(\mathbb{R}^n) \to H_{X}(\mathbb{R}^n)$; and, under certain additional assumptions on $X$ and $Y$, $T_{\alpha}$, $0 < \alpha < n$, extends to a bounded operator $H_{X}(\mathbb{R}^n) \to Y$ and $H_{X}(\mathbb{R}^n) \to H_{Y}(\mathbb{R}^n)$. In particular, from these results, it follows that singular integrals and the Riesz potential satisfy such estimates, respectively. We also provide an off-diagonal Fefferman-Stein vector-valued inequality for the fractional maximal operator on the $p$-convexification of ball quasi-Banach function spaces.

math.FA↗
arXiv2025

Estimates for Riesz potential on weighted variable Hardy spaces revisited

In [Math. Ineq. \& appl., Vol 26 (2) (2023), 511-530] and [Period. Math. Hung., 89 (1) (2024), 116-128], the present author proved that the Riesz potential $I_{\alpha}$ extends to a bounded operator $H^{p(\cdot)}_{\omega}(\mathbb{R}^n) \to L^{q(\cdot)}_{\omega}(\mathbb{R}^n)$ and $H^{p(\cdot)}_{\omega}(\mathbb{R}^n) \to H^{q(\cdot)}_{\omega}(\mathbb{R}^n)$ respectively, under the following two assumptions: $A1)$ $\omega \in \mathcal{W}_{q(\cdot)}$ with $q(\cdot) \in \mathcal{P}^{\log}(\mathbb{R}^{n})$ and $\frac{1}{p(\cdot)} := \frac{1}{q(\cdot)} + \frac{\alpha}{n}$; $A2)$ for every cube $Q \subset \mathbb{R}^{n}$, $\| \chi_Q \|_{L^{q(\cdot)}_{\omega}} \approx |Q|^{-\alpha/n} \| \chi_Q \|_{L^{p(\cdot)}_{\omega}}$. In this note, we re-establish such estimates for $I_{\alpha}$ without assuming the hypothesis $A2)$. These proofs are simpler than the previous ones.

math.CA↗
arXiv2025

The $H^p(\mathbb{Z}^n)-H^q(\mathbb{Z}^n)$ boundedness of the discrete Riesz potential

In [J. Class. Anal., vol. 26 (1) (2025), 63-76], we proved that the discrete Riesz potential $I_{\alpha}$ is a bounded operator $H^p(\mathbb{Z}^n) \to H^q(\mathbb{Z}^n)$ for $\frac{n-1}{n} < p \leq 1$, $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$ and $0 < \alpha < n$. In this note, we extend such boundedness on the full range $0 < p \leq 1$.

math.CA↗
arXiv2025

Fractional series operators on $\mathbb{Z}^n$

For $0 \leq \alpha < n$ and $m \in \mathbb{N} \cap (1 - \frac{\alpha}{n}, \, \infty)$, we introduce a class of fractional series operators $T_{\alpha, m}$ defined on $\mathbb{Z}^n$ which are generated by certain $m$-invertible matrices with integer coefficients. In this note, we prove that $T_{\alpha, m}$ is a bounded operator $H^p(\mathbb{Z}^n) \to \ell^q(\mathbb{Z}^n)$ for $0 < p < \frac{n}{\alpha}$ and $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$. This generalizes the results obtained by the author in [Acta Math. Hungar., 168 (1) (2022), 202-216].

math.CA↗
arXiv2025

Variable Calder\'on-Hardy spaces on the Heisenberg group

Let $\mathbb{H}^{n}$ be the Heisenberg group and $Q = 2n+2$. For $1 < q < \infty$, $\gamma > 0$ and an exponent function $p(\cdot)$ on $\mathbb{H}^n$, which satisfy log-H\"older conditions, with $0 < p_{-} \leq p_{+} < \infty$, we introduce the variable Calder\'on-Hardy spaces $\mathcal{H}^{p(\cdot)}_{q, \gamma}(\mathbb{H}^{n})$, and show for every $f \in H^{p(\cdot)}(\mathbb{H}^{n})$ that the equation \[ \mathcal{L} F = f \] has a unique solution $F$ in $\mathcal{H}^{p(\cdot)}_{q, 2}(\mathbb{H}^{n})$, where $\mathcal{L}$ is the sublaplacian on $\mathbb{H}^{n}$, $1 < q < \frac{n+1}{n}$ and $Q (2 + \frac{Q}{q})^{-1} < \underline{p}$.

math.CA↗
arXiv2025

Calder\'on-Hardy type spaces and the Heisenberg sub-Laplacian

For $0 < p \leq 1 < q < \infty$ and $\gamma > 0$, we introduce the Calder\'on-Hardy spaces $\mathcal{H}^{p}_{q, \gamma}(\mathbb{H}^{n})$ on the Heisenberg group $\mathbb{H}^{n}$, and show for every $f \in H^{p}(\mathbb{H}^{n})$ that the equation \[ \mathcal{L} F = f \] has a unique solution $F$ in $\mathcal{H}^{p}_{q, 2}(\mathbb{H}^{n})$, where $\mathcal{L}$ is the sublaplacian on $\mathbb{H}^{n}$, $1 < q < \frac{n+1}{n}$ and $(2n+2) \, (2 + \frac{2n+2}{q})^{-1} < p \leq 1$.

math.CA↗
arXiv2024

A molecular decomposition for $H^p(\mathbb{Z}^n)$

In this work, for the range $\frac{n-1}{n} < p \leq 1$, we give a molecular reconstruction theorem for $H^p(\mathbb{Z}^n)$. As an application of this result and the atomic decomposition developed by S. Boza and M. Carro in [Proc. R. Soc. Edinb., 132 A (1) (2002), 25-43], we prove that the discrete Riesz potential $I_{\alpha}$ defined on $\mathbb{Z}^n$ is a bounded operator $H^p(\mathbb{Z}^n) \to H^q(\mathbb{Z}^n)$ for $\frac{n-1}{n} < p < \frac{n}{\alpha}$ and $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$, where $0 < \alpha < n$.

math.CA↗
arXiv2024

Classical discrete operators on variable $\ell^{p(\cdot)}(\mathbb{Z})$ spaces

We show, by applying discrete weighted norm inequalities and the Rubio de Francia algorithm, that the discrete Hilbert transform and discrete Riesz potential are bounded on variable $\ell^{p(\cdot)}(\mathbb{Z})$ spaces whenever the discrete Hardy-Littlewood maximal is bounded on $\ell^{p'(\cdot)}(\mathbb{Z})$. We also obtain vector-valued inequalities for the discrete fractional maximal operator.

math.CA↗
arXiv2024

A note about the discrete Riesz potential on $\mathbb{Z}^n$

In this note we prove that the discrete Riesz potential $I_{\alpha}$ defined on $\mathbb{Z}^n$ is a bounded operator $H^p (\mathbb{Z}^n) \to \ell^q (\mathbb{Z}^n)$ for $0 < p \leq 1$ and $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$, where $0 < \alpha < n$.

math.CA↗
arXiv2024

Fractional type operators on the Heisenberg group

Let $\rho(\cdot)$ be the Koranyi norm on the Heisenberg group $\mathbb{H}^{n} \equiv (\mathbb{R}^{2n} \times \mathbb{R}, \, \cdot \, )$ defined by \[ \rho(x,t) = \left( |x|^{4} + 16 t^{2} \right)^{1/4}, \,\,\,\, (x,t) \in \mathbb{H}^{n}. \] For $0 \leq \alpha < Q:=2n+2$, $m \in \mathbb{N} \cap \left(1 - \frac{\alpha}{Q}, \infty \right)$, and $m$ positive constants $\alpha_1, ..., \alpha_m$ such that $\alpha_1 + \cdot \cdot \cdot + \alpha_m = Q - \alpha$, we consider the following generalization of the Riesz potential on $\mathbb{H}^{n}$ \[ T_{\alpha, \, m}f(x,t) = \int_{\mathbb{H}^{n}} f(y,s) \prod_{j=1}^{m} \rho\left((A_j y, r_j^{-2} s)^{-1} \cdot ( x, t)\right)^{-\alpha_j} \, dy \, ds, \] where, in the case $0 < \alpha < Q$, the $A_j$'s are matrices belonging to $Sp (2n, \mathbb{R}) \cap SO(2n)$ and $r_j = 1$ for every $j=1, ..., m$; for $\alpha = 0$, we consider $A_j = r_j^{-1} \, I_{2n \times 2n}$ for every $j=1, ..., m$, where the $r_j$'s are positive constants such that $r_{i}^{2} - r_{j}^{2} \neq 0$ if $i \neq j$. In this note we study the behavior of these operators on variable Hardy spaces in $\mathbb{H}^{n}$.

math.CA↗
arXiv2024

Convolution operators and variable Hardy spaces on the Heisenberg group

Let $\mathbb{H}^{n}$ be the Heisenberg group. For $0 \leq \alpha < Q=2n+2$ and $N \in \mathbb{N}$ we consider exponent functions $p(\cdot) : \mathbb{H}^{n} \to (0, +\infty)$, which satisfies H\"older conditions, such that $\frac{Q}{Q+N} < p_{-} \leq p(\cdot) \leq p_{+} < \frac{Q}{\alpha}$. In this article we prove the $H^{p(\cdot)}(\mathbb{H}^{n}) \to L^{q(\cdot)}(\mathbb{H}^{n})$ and $H^{p(\cdot)}(\mathbb{H}^{n}) \to H^{q(\cdot)}(\mathbb{H}^{n})$ boundedness of convolution operators with kernels of type $(\alpha, N)$ on $\mathbb{H}^{n}$, where $\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)} - \frac{\alpha}{Q}$. In particular, the Riesz potential on $\mathbb{H}^{n}$ satisfies such estimates.

math.CA↗
arXiv2022

A molecular reconstruction theorem for $H^{p(\cdot)}_{\omega}(\mathbb{R}^{n})$

In this article we give a molecular reconstruction theorem for $H_{\omega}^{p(\cdot)}(\mathbb{R}^{n})$. As an application of this result and the atomic decomposition developed in [5] we show that classical singular integrals can be extended to bounded operators on $H_{\omega}^{p(\cdot)}(\mathbb{R}^{n})$. We also prove, for certain exponents $q(\cdot)$ and certain weights $\omega$, that Riesz potential $I_{\alpha}$, with $0 < \alpha < n$, can be extended to a bounded operator from $H^{p(\cdot)}_{\omega}(\mathbb{R}^{n})$ into $H^{q(\cdot)}_{\omega}(\mathbb{R}^{n})$, for $\frac{1}{p(\cdot)} := \frac{1}{q(\cdot)} + \frac{\alpha}{n}$.

math.CA↗
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 $\omega$, that the Riesz potential $I_{\alpha}$, with $0 < \alpha < n$, can be extended to a bounded operator from $H^{p(\cdot)}_{\omega}(\mathbb{R}^{n})$ into $L^{q(\cdot)}_{\omega}(\mathbb{R}^{n})$, for $\frac{1}{p(\cdot)} := \frac{1}{q(\cdot)} + \frac{\alpha}{n}$.

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 $\varphi : \mathbb{R}^{2} \setminus \{ {\bf 0} \} \to \mathbb{R}^{2}$ satisfying a nonisotropic homogeneity condition, the Fourier transform $\hat{\mu}$ of the Borel measure on $\mathbb{R}^{4}$ defined by \[ \mu(E) = \int_{U} \chi_{E}(x, \varphi(x)) \, dx \] where $E$ is a Borel set of $\mathbb{R}^{4}$ and $U = \{ (t^{\alpha_1}, t^{\alpha_2}s) : c < s < d, \, 0 < t < 1 \}$. The aim of this article is to give a decay estimate for $\hat{\mu}$, for the case where the set of nonelliptic points of $\varphi$ 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 $\varphi_{U} : U \to \mathbb{R}^{2}$. We also give $L^{p}$-improving properties for the convolution operator $T_{\mu} f = \mu \ast f$.

math.CA↗