arXiv subjects

Lucas A. Vallejos

Publications and source records attributed to Lucas A. Vallejos.

3 recordsLinked to original sources

arXiv2026

Weighted boundedness of maximal operator associated with invertible linear transformations on variable Lebesgue spaces

In this paper, we study the boundedness of $M_{A^{-1}}$, the maximal operator involves an invertible matrix, on the weighted variable spaces $L^{p(\cdot)}(w)$, where the weights are in an appropriate class. In general, we obtain the necessary conditions for the weight classes and exponents. However, for certain types of invertible matrices, we characterize the weights such that the boundedness holds.

math.CA↗

arXiv2026

Weighted $H^p-L^q$ boundedness of integral operators with rough kernels

In this paper, we study integral operators \begin{equation*} T_αf(x)=\int_{\mathbb{R}^{n}}K(x,y) f(y)dy, \end{equation*} with kernels $K(x,y)= k_1( x- A_1y)...k_m( x-A_my),$ where $k_i(x)=\frac{Ω_i(x)}{|x|^{n/q_i}}$ and $Ω_i: \mathbb{R}^n\to \mathbb{R}$ are homogeneous functions of degree zero, satisfying a size and a Dini condition, $A_{i}$ are certain invertible matrices, and $\frac n{q_1}+\dots+\frac n{q_m}=n-α,$ $0\leq α<n.$ We obtain the $H^{p}_{w^p}(\mathbb{R}^{n})-L^{q}_{w^q}(\mathbb{R}^{n})$ boundedness of these operators, for a class of Muckenhoupt weights $w$, satisfying the condition \begin{equation*} w(A_ix)\leq cw(x), \end{equation*} $a.e.x\in\mathbb{R} ^n $, $1\leq i\leq m$.

math.CA↗

arXiv2024

Existence of positive solutions for a semipositone $p(\cdot)$-Laplacian problem

In this paper we find a positive weak solution for a semipositone $p(\cdot )$- Laplacian problem. More precisely, we find a solution for the problem \[ \left\{ \begin{array}{cc} -Δ_{p(\cdot )}u=f(u)-λ& \text{in }Ω\\ u>0 & \text{in }Ω\\ u=0 & \text{on }\partial Ω\end{array}% \right. , \] where $Ω\subset \mathbb{R}^{N}$, $N\geq 2$ is a smooth bounded domain, $f$ is a contiuous function with subcritical growth, $λ>0$ and $Δ_{p(\cdot )}u=\text{div}(\left\vert \nabla u\right\vert ^{p(\cdot )-2}\nabla u)$. Also, we assume an Ambrosetti-Rabinowitz type of condition and using the Mountain Pass arguments, comparision principles and regularity principles we prove the existence of positive weak solution for $λ$ small enough.

math.AP↗