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Joel E. Restrepo

Publications and source records attributed to Joel E. Restrepo.

16 recordsLinked to original sources
arXiv2026

Direct and inverse abstract Cauchy problems with fractional powers of almost sectorial operators

We derive the explicit solution operator of an abstract Cauchy problem involving a time-variable coefficient and a fractional power of an almost sectorial operator. The time-variable coefficient is recovered by solving the inverse abstract Cauchy problem using the solution operator representation. As a complement, we also study similar problems by considering almost sectorial operators that depend on a time-variable.

math.AP↗
arXiv2026

Boundedness of spectral multipliers on locally compact groups and applications

We prove that the noncommutative Lorentz norm (associated to a semifinite von Neumann algebra) of a propagator of the form $φ(|\mathscr{L}|)$ can be estimated if the modulus of the Borel function $φ$ is bounded by a continuous positive monotonically decreasing function that vanishes at infinity $ψ$. As a consequence, we obtain the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates for the solutions of heat, wave, and Schrödinger type equations (new in this setting) on a locally compact separable unimodular group $G$ by using a non-local integro-differential operator in time and any positive left invariant operator (maybe unbounded and with discrete or continuous spectrum) on $G$. We also provide asymptotic estimates (large-time behavior) for the solutions, which in some cases can be claimed to be sharp. Illustrative examples are given for several groups.

math.AP↗
arXiv2025

Abstract Volterra integral equations of wave type with almost sectorial operators

We study classical solutions (existence, uniqueness, and explicit solution operator) for homogeneous, linear, and semilinear abstract Volterra integral equations of wave type with almost sectorial operators. We use a functional calculus for the latter type of operators to construct a general class of bounded linear operators that in particular contains the propagators (solution operators) of the considered equations. Some properties of this family of operators are also given.

math.AP↗
arXiv2025

On projection from $A^p_ω$ to the Hardy spaces $H^p$

This paper describes the known results on the projection from the most general holomorphic spaces $A^p_ω$, which depend on a functional parameter $ω$ and are over the unit disc, upper half-plane and the finite complex plane, to the classical Hardy spaces $H^p.$ The paper can be considered as a survey in the mentioned topic. A new result on the projection of the half-plane $A^p_ω$ to the half-plane Hardy space $H^p$ is obtained.

math.CV↗
arXiv2025

Anomaly of the fractional heat propagator in abstract settings

We study the following time-fractional heat equation: \begin{equation*} ^{C}\partial_{t}^αu(t)+\mathscr{L}u(t)=0,\quad u(0)=u_0\in X, \quad t\in[0,T],\quad T>0,\quad 0<α<1, \end{equation*} where $^{C}\partial_{t}^α$ is the Djrbashian-Caputo fractional derivative, $X$ is a complex Banach space and $\mathscr{L}:\mathcal{D}(\mathscr{L})\subset X\to X$ is a closed linear operator. The solution operator of the equation above is given by the strongly continuous operator $E_α(-t^α\mathscr{L})$ for any $t\geqslant0$, closely related with the Mittag-Leffler function $E_α(-x)$ for $x\geqslant0.$ There are different ways to present explicitly this operator and one of the most popular is given in terms of the $C_0$-semigroup generated by $-\mathscr{L}$ $\big(\{e^{-t\mathscr{L}}\}_{t\geqslant0}\big)$ as follows: \[ E_α(-t^α\mathscr{L})=\int_0^{+\infty}M_α(s)e^{-st^α\mathscr{L}}{\rm d}s,\quad t\geqslant0, \] where $M_α(s)$ is a Wright-type function. We will see that the latter expression is not always optimal (regarding restrictions: endpoint lost) to estimate different norms. An additional restriction appears while bounding the above integral, which can be avoided by using directly the function itself and its well-known uniform bound $|E_α(-x)|\leqslant \frac{C}{1+x},$ $x\geqslant0.$

math.AP↗
arXiv2024

Integro-differential diffusion equations on graded Lie groups

We first study the existence, uniqueness and well-posedness of a general class of integro-differential diffusion equation on $L^p(\mathbb{G})$ $(1<p<+\infty$, $\mathbb{G}$ is a graded Lie group). We show the explicit solution of the considered equation by using the Fourier analysis of the group. The equation involves a nonlocal in time operator (with a general kernel) and a positive Rockland operator acting on $\mathbb{G}.$ Also, we provide $L^p(\mathbb{G})-L^q(\mathbb{G})$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates and time decay rate for the solutions. In fact, by using some contemporary results, one can translate the latter regularity problem to the study of boundedness of its propagator which strongly depends on the traces of the spectral projections of the Rockland operator. Moreover, in many cases, we can obtain time asymptotic decay for the solutions which depends intrinsically on the considered kernel. As a complement, we give some norm estimates for the solutions in terms of a homogeneous Sobolev space in $L^{2}(\mathbb{G})$ that involves the Rockland operator. We also give a counterpart of our results in the setting of compact Lie groups. Illustrative examples are also given.

math.AP↗
arXiv2024

Evolutionary integro-differential equations of scalar type on locally compact groups

We study existence, uniqueness, norm estimates and asymptotic time behaviour (in some cases can be claimed to be sharp) for the solution of a general evolutionary integral (differential) equation of scalar type on a locally compact separable unimodular group $G$ governed by any positive left invariant operator (unbounded and either with discrete or continuous spectrum) on $G$. We complement our studies by proving some time-space (Strichartz type) estimates for the classical heat equation, and its time-fractional counterpart, as well as for the time-fractional wave equation. The latter estimates allow us to give some results about the well-posedness of nonlinear partial integro-differential equations. We provide many examples of the results by considering particular equations, operators and groups through the whole paper.

math.AP↗
arXiv2024

$L^p-L^q$ estimates for non-local heat and wave type equations on locally compact groups

We prove the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates for the solutions of heat and wave type equations on a locally compact separable unimodular group $G$ by using an integro-differential operator in time and any positive left invariant operator (maybe unbounded) on $G$. We complement our studies by giving asymptotic time estimates for the solutions, which in some cases are sharp.

math.AP↗
arXiv2023

Generalized fractional Dirac type operators

We introduce a class of fractional Dirac type operators with time variable coefficients by means of a Witt basis, the Djrbashian-Caputo fractional derivative and the fractional Laplacian, both operators defined with respect to some given functions. Direct and inverse fractional Cauchy type problems are studied for the introduced operators. We give explicit solutions of the considered fractional Cauchy type problems. We also use a recent method to recover a variable coefficient solution of some inverse fractional wave and heat type equations. Illustrative examples are provided.

math.CA↗
arXiv2023

Asymptotic estimates for the growth of deformed Hankel transform by modulus of continuity

We derive asymptotic estimates for the growth of the norm of the deformed Hankel transform on the deformed Hankel--Lipschitz space defined via a generalised modulus of continuity. The established results are similar in nature to the well-known Titchmarsh theorem, which provide a characterization of the square integrable functions satisfying certain Cauchy--Lipschitz condition in terms of an asymptotic estimate for the growth of the norm of their Fourier transform. We also give some necessary conditions in terms of the generalised modulus of continuity for the boundedness of the Dunkl transform of functions in Dunkl-Lipschitz spaces, improving the Hausdorff-Young inequality for the Dunkl transform in this special scenario.

math.FA↗
arXiv2023

Heat and wave type equations with non-local operators, II. Hilbert spaces and graded Lie groups

We study heat and wave type equations on a separable Hilbert space $\mathcal{H}$ by considering non-local operators in time with any positive densely defined linear operator with discrete spectrum. We show the explicit representation of the solution and analyse the time-decay rate in a scale of suitable Sobolev space. We perform similar analysis on multi-term heat and multi-wave type equations. The main tool here is the Fourier analysis which can be developed in a separable Hilbert space based on the linear operator involved. As an application, the same Cauchy problems are considered and analysed in the setting of a graded Lie group. In this case our analysis relies on the group Fourier analysis. An extra ingredient in this framework allows, in the case of heat type equations, to establish $L^p$-$L^q$ estimates for $1\leqslant p\leqslant 2\leqslant q<+\infty$ for the solutions on graded Lie group groups. Examples and applications of the developed theory are given, either in terms of self-adjoint operators on compact or non-compact manifolds, or in the case of particular settings of graded Lie groups. The results of this paper significantly extend in different directions the results of Part I, where operators on compact Lie groups were considered. We note that the results obtained in this paper are also new already in the Euclidean setting of $\mathbb{R}^n$.

math.AP↗
arXiv2023

Heat and wave type equations with non-local operators, I. Compact Lie groups

We prove existence, uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a non-local (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1<p\leqslant 2\leqslant q<+\infty$. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat type equation as well.

math.AP↗
arXiv2022

Prabhakar-type linear differential equations with variable coefficients

Linear differential equations with variable coefficients and Prabhakar-type operators featuring Mittag-Leffler kernels are solved. In each case, the unique solution is constructed explicitly as a convergent infinite series involving compositions of Prabhakar fractional integrals. We also extend these results to Prabhakar operators with respect to functions. As an important illustrative example, we consider the case of constant coefficients, and give the solutions in a more closed form by using multivariate Mittag-Leffler functions.

math.CA↗
arXiv2021

A new representation for the solutions of fractional differential equations with variable coefficients

A recent development in the theory of fractional differential equations with variable coefficients has been a method for obtaining an exact solution in the form of an infinite series involving nested fractional integral operators. This solution representation is constructive but difficult to calculate in practice. Here we show a new representation of the solution function, as a convergent series of single fractional integrals, which will be easier to use for computational work and applications. In the particular case of constant coefficients, the solution is given in terms of the Mittag-Leffler function. We also show some applications in Cauchy problems for partial differential equations involving both time-fractional and space-fractional operators and with time-dependent coefficients.

math.CA↗
arXiv2021

Explicit solutions for linear variable-coefficient fractional differential equations with respect to functions

Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite series of fractional integro-differential operators, which can be widely and efficiently used for analytic and computational purposes. In the case of constant coefficients, the solution can be expressed in terms of the multivariate Mittag-Leffler functions. In particular, the obtained result extends the Luchko-Gorenflo representation formula to a general class of linear fractional differential equations with variable coefficients, to complex fractional derivatives, and to fractional derivatives with respect to a given function.

math.CA↗