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Duván Cardona

Publications and source records attributed to Duván Cardona.

At least 19 recordsLinked to original sources
arXiv2026

Kernel estimates for the Weyl-Hörmander calculus, Weak (1,1) boundedness and other regularity properties

We provide kernel estimates for pseudo-differential operators in the $S(m,g)$ Weyl-Hörmander calculus classes. These new results lead to boundedness theorems like the weak (1,1) type for these classes and the boundedness of these operators from Hardy spaces to Lebesgue spaces $L^p$. We prove that under a suitable hypoelliticity condition on the weigh $m,$ one can get the weak (1,1) inequality for the Fefferman order established previously by the second author in analogy to the classical $L^p$-boundedness theorem due to Fefferman. Our approach makes an extension to $S(m,g)$ classes of previous estimates proved by Álvarez and Hounie, Nagase, Kumano-Go, etc, for the Hörmander classes including new regularity properties in this framework from the Hardy space $H^p$ to $L^p$ and also on Lorentz spaces. We have identified that the {\it Uniform Hörmander Condition} plays a fundamental role in some of our boundedness theorems.

math.AP↗
arXiv2026

Quasilinear equations with exponential growth for Hörmander $q$-sub-Laplacians on stratified Lie groups

We prove the existence of positive weak solutions for a quasilinear equation with exponential nonlinearity on arbitrary stratified Lie groups for \emph{horizontal $q$-Laplacians,} also called $q$-sub-Laplacians on these groups. The nonlinearity combines a concave term with an exponential growth that can be subcritical, critical, or supercritical with respect to the Trudinger--Moser inequality for Lorentz spaces on stratified Lie groups. We prove a version of this inequality in this paper. Since the problem is not variational in the natural energy space, classical minimisation or critical-point techniques do not apply directly. Positive solutions to the resulting semilinear equation are then obtained via a carefully designed Galerkin method applied to this setting. Our main theorem recovers previous known results on $\mathbb{R}^n$ in the complete range $q\in (1,\infty),$ and the previous known result on the Heisenberg group $\mathbb{H}^n$ for $q=2n+2,$ and is extended here in the full range $q\in (1,\infty)$. Other particular and fundamental cases, including the Engel group, are discussed.

math.AP↗
arXiv2026

Local smoothing estimates for bilinear Fourier integral operators

We formulate a local smoothing conjecture for bilinear Fourier integral operators in every dimension $d \ge 2,$ derived from the celebrated linear case due to Sogge, which we refer to as the \emph{bilinear smoothing conjecture}. We show that the linear local smoothing conjecture implies this bilinear version. As a consequence of our approach and due to the recent progress on the subject, we establish local smoothing estimates for Fourier integral operators in dimension $d=2,$ that is, on $\mathbb{R}^2_x \times \mathbb{R}_t$. Also, a partial progress is presented for the high-dimensional case $d\geq 3.$ In particular, our method allows us to deduce that the bilinear local smoothing conjecture holds for all odd dimensions $d$.

math.AP↗
arXiv2026

The weak (1,1) boundedness of Fourier integral operators with complex phases

Let $T$ be a Fourier integral operator of order $-(n-1)/2$ associated with a canonical relation locally parametrised by a real-phase function. A fundamental result due to Seeger, Sogge, and Stein proved in the 90's, gives the boundedness of $T$ from the Hardy space $H^1$ into $L^1.$ Additionally, it was shown by T. Tao the weak (1,1) type of $T$. In this work, we establish the weak (1,1) boundedness of a Fourier integral operator $T$ of order $-(n-1)/2$ when it has associated a canonical relation parametrised by a complex phase function. This result in the complex-valued setting, cannot be derived from its counterpart in the real-valued case.

math.AP↗
arXiv2026

Boundedness of Fourier Integral Operators with complex phases on Fourier Lebesgue spaces

In this paper, we develop boundedness estimates for Fourier integral operators on Fourier Lebesgue spaces when the associated canonical relation is parametrised by a complex phase function. Our result constitutes the complex analogue of those obtained for real canonical relations by Rodino, Nicola, and Cordero. We prove that, under the spatial factorization condition of rank $\varkappa$, the corresponding Fourier integral operator is bounded on the Fourier Lebesgue space $\mathcal{F}L^p,$ provided that the order $m$ of the operator satisfies that $ m \leq -\varkappa\left|\frac{1}{p}-\frac{1}{2}\right|, 1 \leq p \leq \infty. $ This condition on the order $m$ is sharp.

math.AP↗
arXiv2026

Kernel estimates and weak (1,1)-boundedness of pseudo-differential operators on compact Lie groups

Given a compact Lie group $G$ and its unitary dual $\widehat{G}$, we establish the weak (1,1) continuity for pseudo-differential operators in the global Hörmander classes of order $-n(1-ρ)/2$ on $G\times \widehat{G}$. Our approach consists of proving suitable estimates for the kernel of such operators. Furthermore, we use these kernel estimates to give an alternative proof for the $H^1(G)$-$L^1(G)$-continuity of these classes now allowing the full range $0\leqδ\leqρ\leq1, \;ρ\neq0,\;δ\neq1$. The conditions for the operators are formulated using the Hörmander classes $S^m_{ρ,δ}(G):=S^m_{ρ,δ}(G\times \widehat{G})$ of symbols in the non-commutative phase space $G\times \widehat{G}$, which are extensions of the well-known $(ρ,δ)$-classes in the Euclidean space. Our results are formulated in the complete range $0\leq δ\leq ρ\leq 1,$ $ρ\neq0,\;$$δ\neq 1$. As an application of this boundedness result we provide end-point a-priori $L^1$-estimates for the sub-Laplacian $\mathcal{L}_{sub}=X^2+Y^2,$ and for the heat type operator $T=Z-X^2-Y^2$ on $SU(2)\cong \mathbb{S}^3$ that cannot be obtained by application of the standard pseudo-differential calculus due to Hörmander. More precisely, we prove that if one considers the subelliptic problem, \begin{equation}\label{IVP:abstract} \begin{cases}Tu=f ,& \text{ } \\u,f\in \mathscr{D}'(SU(2)):=(C^\infty(SU(2)))', & \text{ } \end{cases} \end{equation} then, for $f\in W^{1,-\frac{1}{4}}(SU(2)),$ one has that $u\in L^{1,\infty}(SU(2)).$

math.AP↗
arXiv2026

Hyperbolic partial differential equations with complex characteristics on Fourier Lebesgue spaces

The aim of this paper is to establish well-posedness properties for hyperbolic PDEs on Fourier Lebesgue spaces. We consider hyperbolic operators with complex characteristics. Since our approach comes from harmonic analysis, we establish boundedness properties of Fourier integral operators with complex-valued phase functions on Fourier Lebesgue spaces, Besov spaces and Triebel-Lizorkin spaces. Indeed, these classes of operators serve as propagators of the considered PDE problems. In terms of the boundedness properties, we prove new results in the case where the canonical relation of the operator is assumed to satisfy the {\it spatial smooth factorization condition}

math.AP↗
arXiv2025

Estimates for pseudo-differential operators on the torus revisited. III

This paper finishes the goal of the authors started in two previous manuscripts dedicated to revisiting the continuity properties of toroidal pseudo-differential operators with symbols in the Hörmander classes. Here we prove pointwise estimates in terms of the Fefferman-Stein sharp maximal function and of the Hardy-Littlewood maximal function. Combining these estimates with the properties of Muckenhoupt's weight class $A_p$ we obtain boundedness theorems for pseudo-differential operators between weighted Lebesgue spaces on the torus $L^p(w)$. These results are given in the context of the global symbolic analysis defined on $\mathbb{T}^n\times \mathbb{Z}^n$ as developed by Ruzhansky and Turunen by using discrete Fourier analysis, and extend those of Park and Tomita available in the Euclidean case. Moreover, we include continuity results on Sobolev spaces $W^s_p$ and on Besov spaces $B^s_{p,q}$ on the torus. Our techniques are taken from Park and Tomita \cite{park-tomita} and we consider its toroidal extension here for the completeness of the boundedness of toroidal pseudo-differential operators with respect to the current literature.

math.AP↗
arXiv2025

Riesz-means bounds for functional-difference operators for mirror curves

Let \( P \) and \( Q \) be the quantum-mechanical momentum and position operators on \( L^2(\R) \). Let $ζ>0.$ We provide estimates for the {\it Riesz means} $\varkappa(λ)$ associated with the system of eigenvalues of the operator \begin{align} H(ζ) = \e^{-bP} + \e^{bP} + \e^{2πb Q} + ζ\e^{-2πb Q} = U + U^{-1} + V + ζV^{-1}, \end{align} when $λ\rightarrow\infty.$ This operator arises in the quantisation of the local {\it del Pezzo Calabi-Yau threefold}, defined as the total space of the anti-canonical bundle over the {\it Hirzebruch surface} \( S = \mathbb{P}^{1} \times \mathbb{P}^{1} \). Our approach is motivated by the spectral analysis of $\varkappa(λ)$ in the framework developed by Laptev, Schimmer and Takhtajan in [13].

math-ph↗
arXiv2025

On the Sobolev boundedness of vector fields on compact Riemannian manifolds

We analyze the sharpness of the Sobolev order for left-invariant vector fields on compact Riemannian manifolds. Utilizing techniques from pseudo-differential operator theory and microlocal analysis, we investigate the asymptotic behavior of eigenvalues associated with these vector fields. As an application, we demonstrate the ill-posedness of a class of Cauchy problems involving left-invariant vector fields on compact Lie groups.

math.AP↗
arXiv2025

Oscillating singular integral operators on graded Lie groups revisited

In this work, we extend the Euclidean theory of oscillating singular integrals due to Fefferman and Stein in \cite{Fefferman1970,FeffermanStein1972} to arbitrary graded Lie groups. Our approach reveals the strong compatibility between the geometric measure theory of a graded Lie group and the Fourier analysis associated with Rockland operators. Our criteria are presented in terms of the oscillating Fefferman condition of the kernel of the operator and its group Fourier transform. One of the novelties of this work is that we use the infinitesimal representation of a Rockland operator to measure the decay of the Fourier transform of the kernel.

math.FA↗
arXiv2025

Boundedness of pseudo-differential operators on the torus revisited. II

In this paper we continue our program of revisiting the new aspects about the boundedness properties of pseudo-differential operators on the torus. Here we prove $H^p$-$L^p$ and $H^p$-estimates for Hörmander classes of pseudo-differential operators on the torus $\mathbb{T}^n$ for $p\leq 1$. The results are presented in the context of the global symbolic analysis developed by Ruzhansky and Turunen on $\mathbb{T}^n \times \mathbb{Z}^n$ by using the discrete Fourier analysis, which extends the $(ρ, δ)$-Hörmander classes on $\mathbb{T}^n$ defined by local coordinate systems. These results extend those proved by Álvarez and Hounie for the Euclidean case, considering even the case $ρ\leqδ$.

math.AP↗
arXiv2025

Estimates for pseudo-differential operators on the torus revisited. I

In this paper we prove $L^p$-estimates for Hörmander classes of pseudo-differential operators on the torus $\mathbb{T}^n$. The results are presented in the context of the global symbolic calculus of Ruzhansky and Turunen on $\mathbb{T}^n\times\mathbb{Z}^n$ by using the discrete Fourier analysis on the torus which extends the usual ($ρ,δ$)-Hörmander classes on $\mathbb{T}^n$. The main results extend Álvarez and Hounie's method for $\mathbb{R}^n$ to the torus, and Fefferman's $L^p$-boundedness theorem in the toroidal setting allowing the condition $δ\geqρ$. When $δ\leq ρ$, our results recover the available estimates in the literature.

math.AP↗
arXiv2024

Nuclearity, Schatten-von Neumann classes, distribution of eigenvalues and $L^p$-$L^q$-boundedness of Fourier integral operators on compact manifolds

We link Sogge's type $L^p$-estimates for eigenfunctions of the Laplacian on compact manifolds with the problem of providing criteria for the $r$-nuclearity of Fourier integral operators. The classes of Fourier integral operators $I^μ_{ρ,1-ρ}(X,Y;C)$ considered here are associated with complex canonical relations $C$, i.e. they are parametrised by a complex-valued phase function. Our analysis also includes the case of real canonical relations, namely, the class of Fourier integral operators with real-valued phase functions. The nuclear trace in the sense of Grothendieck is investigated for these operators as well as the validity of the Grothendieck-Lidskii formula on Lebesgue spaces. Criteria are presented in terms of the factorisation condition for the complex canonical relation. Necessary and sufficient conditions for the membership of Fourier integral operators in Schatten-von Neumann classes are presented in the case where the Schatten index $r>0$ belongs to the set $\mathbb{N} $ and sharp sufficient conditions are presented in the general case $r>0$. In particular, we establish necessary and sufficient conditions for the membership of Fourier integral operators to the ideal of trace class operators and to the ideal of Hilbert-Schmidt operators on $L^2(X)$. The rate of decay of eigenvalues and the trace of Fourier integral operators is also investigated in both settings, in the Hilbert space case of $L^2(X)$ using Schatten-von Neumann properties and in the context of the Banach spaces $L^p(X),$ $1<p<\infty,$ utilising the notion of $r$-nuclearity.

math.AP↗
arXiv2024

Pseudo-differential operators on Homogeneous vector bundles over compact homogeneous manifolds

In this work, we introduce a global theory of subelliptic pseudo-differential operators on arbitrary homogeneous vector bundles over orientable compact homogeneous manifolds. We will show that a global pseudo-differential calculus can be associated to the operators acting on any pair of homogeneous vector-bundles with base space $M,$ if the compact Lie group $G$ that acts on $M=G/K$ is endowed with a (Riemannian or) sub-Riemannian structure. This is always possible if we choose on $G$ a sub-Laplacian associated to a Hörmander system of vector-fields or we fix the Laplace-Beltrami operator on $G$. We begin with developing a global subelliptic symbolic calculus for vector-valued pseudo-differential operators on $G$ and then, we show that this vector-valued calculus induces a pseudo-differential calculus on homogeneous vector bundles, which, among other things, is stable under the action of the complex functional calculus. We prove global versions of the Calderón-Vaillancourt theorem, Fefferman theorem and also, of the Gårding inequality. We present applications of the obtained Gårding inequality to the wellposedness of evolution problems. We characterise the classes of pseudo-differential operators on homogeneous vector bundles in the sense of Hörmander (which are defined by using local coordinate systems) in terms of their global symbols. Finally, using this formalism, we compute the global symbol of the exterior derivative, its adjoint, and the symbol of the Dirac operator on the vector bundle of differential forms. We hope that this work will provide a solid foundation for further research using the global quantisation of operators on (vector-bundles over) compact homogeneous manifolds.

math.AP↗
arXiv2024

Anharmonic semigroups and applications to global well-posedness of nonlinear heat equations

In this work we consider the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^γ}$ for $γ>0$ associated to an anharmonic oscillator of the form $ \mathcal{A}_{k,\,\ell}=(-Δ)^{\ell}+|x|^{2k}$ where $k,\ell$ are integers $\geq 1$. By introducing a suitable Hörmander metric on the phase-space we analyse the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^γ}$ within the framework of Hörmander $S(M,g)$ classes and obtain mapping properties in the scale of modulation spaces $M^{p,q},\, 0<p,q\leq \infty,$ with respect to an anharmonic modulation weight. As an application, we apply the obtained bounds to establish the well-posedness for the nonlinear heat equation associated with $\mathcal{A}_{k,\,\ell}^γ$. It is worth noting that the results presented in this paper are novel, even in the case where $γ=1.$

math.AP↗
arXiv2024

$L^2$-Maximal functions on graded Lie groups

Bourgain in his seminal paper [2] about the analysis of maximal functions associated to convex bodies, has estimated in a sharp way the $L^2$-operator norm of the maximal function associated to a kernel $K\in L^1,$ with differentiable Fourier transform $\widehat{K}.$ We formulate the extension to Bourgain's $L^2$-estimate in the setting of maximal functions on graded Lie groups. Our criterion is formulated in terms of the group Fourier transform of the kernel. We discuss the application of our main result to the $L^p$-boundedness of maximal functions on graded Lie groups.

math.FA↗