arXiv subjects
Publications and source records attributed to Guillermo García-Sáez.
In this short communication we answer negatively to a question posed by the authors in their work \cite[Section 1.4]{Brue2022} about the complex interpolation nature of the spaces $BV^{s}(\R^n)$, introduced by G.E. Comi and G. Stefani in \cite{ComiStefani2019}.
Bessel potential spaces, introduced in the 1960s, are derived through complex interpolation between Lebesgue and Sobolev spaces, making them intermediate spaces of fractional differentiability order. Bessel potential spaces have recently gained attention due to their identification with the Riesz fractional gradient. This paper explores Bessel potential spaces as complex interpolation spaces, providing original proofs of fundamental properties based on abstract interpolation theory. Main results include a direct proof of norm equivalence, continuous embeddings, and the relationship with Gagliardo spaces.
We study the nonlocal $ρ$-Laplacian, defined as the composition of the nonlocal divergence and gradient operators associated with a general radial kernel $ρ$: $Δ_ρu=\mbox{div}_ρ\left(D_ρu\right)$. Our first main contribution is to establish a precise connection between this operator and the class of integro-differential elliptic operators studied by Fernández-Real and Ros-Oton (\cite{FernandezRos}), identifying explicit conditions on the kernel $ρ$ that guarantee membership in this class. Our second main contribution concerns maximum and comparison principles for the $ρ$-Laplacian. We establish both a strong and a weak maximum principle under conditions on $ρ$ that are strictly weaker than those required for membership in the integro-differential class, thereby covering a genuinely broader family of operators. The results require only minimal assumptions on the kernel, and in particular do not rely on any fractional-type comparability condition.
In this work we establish that the recently introduced fractional Sobolev spaces based on the Riesz fractional gradient of Musielak-Orlicz functions by one of the authors, coincide with the space of Bessel potentials of functions on such generalized Orlicz setting. Moreover, we identify them as complex interpolation spaces, and exploiting the well known properties for interpolation of operators we obtain several structural properties for those spaces.
In this work, we introduce a new family of functions spaces, the weighted fractional Sobolev spaces $X^{s,p}_{0,w}(Ω)$, where $w$ is a weight in the Muckenhoupt class $A_p$. This space is a natural extension of the fractional Sobolev spaces $H^{s,p}_0$, obtained by means of the Riesz fractional gradient $D^s$, to the setting of the weighted Lebesgue spaces $L^p_w$. As it happened in the unweighted space, the spaces $X^{s,p}_{0,w}(Ω)$ coincide with the weighted version of the Bessel potential space. We obtaien several structural properties for these spaces, as well as continuous and compact embeddings. We conclude with the study of a family of degenerate fractional elliptic partial differential equations.
From the recent developing of nonlocal gradients with finite horizon $δ>0$ based on general kernels, we introduce a new nonlocal $p$-Laplacian and study the eigenvalue problem associated with it. Furthermore, by virtue of $Γ$-convergence arguments, we establish stability results of the solutions for varying horizon in the extreme cases $δ\to 0^+$ and $δ\to\infty$, recovering the solutions for the local eigenvalue problem associated with the $p$-Laplacian, and the ones associated with the $H^{s,p}$-Laplacian, respectively.
Bessel potential spaces have gained renewed interest due to their robust structural properties and applications in fractional partial differential equations (PDEs). These spaces, derived through complex interpolation between Lebesgue and Sobolev spaces, are closely related to the Riesz fractional gradient. Recent studies have demonstrated continuous and compact embeddings of Bessel potential spaces into Lebesgue spaces. This paper extends these findings by addressing the compactness of continuous embeddings from the perspective of abstract interpolation theory. We present three distinct proofs, leveraging compactness results, translation estimates, and the relationship between Gagliardo and Bessel spaces. Our results provide a deeper understanding of the functional analytic properties of Bessel potential spaces and their applications in fractional PDEs.