arXiv subjects

Alberto Debernardi Pinos

Publications and source records attributed to Alberto Debernardi Pinos.

5 recordsLinked to original sources

arXiv2026

Fractional $\frac{1}{2}$-Laplacian of holomorphic functions via Jacobi polynomials

We give a distributional definition of the $\frac{1}{2}$-fractional Laplacian for smooth functions with power-type singularities at the origin, including negative monomials and certain holomorphic functions with isolated singularities. The construction is based on a suitable space of test functions related to the Lizorkin space of test functions, for which both the test functions and their fractional Laplacians belong to the Schwarz class and vanish to infinite order at the origin, and thus compensate for any singularity of power type at the origin. This allows the $\frac{1}{2}$-fractional Laplacian to be defined by duality without requiring the underlying function to satisfy the usual integrability assumptions. The considered distributional framework is shown to be invariant under the $\frac{1}{2}$-fractional Laplacian, yielding a natural semigroup property. By applying the construction term by term to Laurent series we obtain new series that, remarkably, involve Jacobi polynomials. With this procedure we define the fractional Laplacian for a class of holomorphic functions and derive several applications, with particular emphasis on the recently developed theory of the quaternionic fine structures of the spectral theory on the $S$-spectrum, describing the functional calculi extending the classical holomorphic functional calculus in the various classes of holomorphic-type functions arising from the Fueter-Sce-Qian extension theorem.

math.FA↗

arXiv2024

Weighted norm inequalities for integral transforms with splitting kernels

We obtain necessary and sufficient conditions on weights for a wide class of integral transforms to be bounded between weighted $L^p-L^q$ spaces, with $1\leq p\leq q\leq \infty$. The kernels $K(x,y)$ of such transforms are only assumed to satisfy upper bounds given by products of two functions, one in each variable. The obtained results are applicable to a number of transforms, some of which are included here as particular examples. Some of the new results derived here are the characterization of weights for the boundedness of the $\mathscr{H}_α$ (or Struve) transform in the case $α>\frac{1}{2}$, or the characterization of power weights for which the Laplace transform is bounded in the limiting cases $p=1$ or $q=\infty$.

math.CA↗

arXiv2024

Nuclearity and Grothendieck-Lidskii formula for quaternionic operators

We introduce an appropriate notion of trace in the setting of quaternionic linear operators, arising from the well-known companion matrices. We then use this notion to define the quaternionic Fredholm determinant of trace-class operators in Hilbert spaces, and show that an analog of the classical Grothendieck-Lidskii formula, relating the trace of an operator with its eigenvalues, holds. We then extend these results to the so-called $\frac{2}{3}$-nuclear (Fredholm) operators in the context of quaternionic locally convex spaces. While doing so, we develop some results in the theory of topological tensor products of noncommutative modules, and show that the trace defined ad hoc in terms of companion matrices, arises naturally as part of a canonical trace.

math.CA↗

arXiv2023

Fourier inequalities in Morrey and Campanato spaces

We study norm inequalities for the Fourier transform, namely, \begin{equation}\label{introduction} \|\widehat f\|_{X_{p,q}^λ} \lesssim \|f\|_{Y}, \end{equation} where $X$ is either a Morrey or Campanato space and $Y$ is an appropriate function space. In the case of the Morrey space we sharpen the estimate $ \|\widehat f\|_{M_{p,q}^λ} \lesssim \|f\|_{L_{s',q}},$ $ s\geq 2,$ $\frac{1}{s} = \frac{1}{p}-\fracλ{n}.$ We also show that \eqref{introduction} does not hold when both $X$ and $Y$ are Morrey spaces. If $X$ is a Campanato space, we prove that \eqref{introduction} holds for $Y$ being the truncated Lebesgue space.

math.CA↗

arXiv2022

Gabor orthonormal bases, tiling and periodicity

We show that if the Gabor system $\{ g(x-t) e^{2πi s x}\}$, $t \in T$, $s \in S$, is an orthonormal basis in $L^2(\mathbb{R})$ and if the window function $g$ is compactly supported, then both the time shift set $T$ and the frequency shift set $S$ must be periodic. To prove this we establish a necessary functional tiling type condition for Gabor orthonormal bases which may be of independent interest.

math.CA↗