arXiv subjects

Michael Ruzhansky

Publications and source records attributed to Michael Ruzhansky.

At least 19 recordsLinked to original sources

arXiv2026

Fourier analysis on a class of Schwartz spaces and applications

The study of regularity and solvability of partial differential operators on non-compact manifolds requires the construction of adapted functional spaces. In this work we introduce $\mathcal{S}(\mathfrak{M}\times\mathbb{R}^n)$, the Schwartz space on the product of a homogeneous space $\mathfrak{M}$ and the Euclidean space $\mathbb{R}^n$. We characterize $\mathcal{S}(\mathfrak{M}\times\mathbb{R}^n)$ and its topological dual by employing discrete Fourier analysis in terms of eigenfunction expansions associated with a class of anisotropic elliptic operators on $\mathbb{R}^n$. We provide a characterization via partial Fourier coefficients and via full coefficients, that is, with respect to both components of $\mathfrak{M} \times \mathbb{R}^n$ simultaneously, which is achieved by studying the Fourier analysis on $\mathfrak{M}$. Our results provide new tools for the analysis of operators in this broader setting. As an application, we investigate the global existence and regularity of solutions for a class of partial differential equations on $\mathfrak{M}\times \mathbb{R}^n$.

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arXiv2026

Sharp lifespan estimates for heat equations with Hörmander operators

In this paper, we study the Cauchy problem $u_t-\mathcal L u=h(t)u^p$, $u(0)=\varepsilon u_0\ge0$, where $p>1$, $h\ge0$ is only locally integrable, and $\mathcal L=\sum_{j=1}^mX_j^2$ is a homogeneous Hörmander sum of squares on $\mathbb R^n$ of homogeneous dimension $q$. Diffusion spreads the solution out and weakens the nonlinearity, while the reaction $h(t)u^p$ may force blow-up. We prove that the outcome is decided by the single quantity $\mathcal{H}_{p,h}(T)=\int_0^Th(t)(1+t)^{-\frac q2(p-1)}\,d t$, in which the reaction at time $t$ is weighted by the factor by which diffusion has weakened the solution: small data are global if $\mathcal{H}_{p,h}(\infty)<\infty$, while otherwise the lifespan $T_\varepsilon$ satisfies $c\,\varepsilon^{-(p-1)}\le\mathcal{H}_{p,h}(T_\varepsilon)\le C\,\varepsilon^{-(p-1)}$ for small $\varepsilon$. Our technique of proof is based on a backward-semigroup functional combined with a Gaussian-envelope energy and an ODE cap telescoped along interlaced geometric grids. The result contains the Euclidean, Heisenberg, Carnot and Grushin cases, yields explicit lifespan laws for power, logarithmic, exponential, periodic and intermittent coefficients, and exhibits effects with no autonomous counterpart, such as a lifespan with no power law in $\varepsilon$. We also prove regularity and a priori estimates for classical solutions and discuss what remains true for a general nonlinearity $f(u)$.

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arXiv2026

Nonrelativistic limit of nonlinear Dirac ground states on noncompact quantum graphs

We study the nonrelativistic limit of nonlinear Dirac ground states on finite noncompact metric graphs with Kirchhoff-type conditions. For a nondecreasing nonlinearity of arbitrary finite superquadratic growth, a sufficiently long pendant path guarantees ground states whose nonrelativistic limits are Schrödinger ground states. Every choice of these states is relatively compact in $H^1$ and satisfies a uniform exponential decay estimate. On general graphs, we prove a compactness alternative: loss of compactness requires at least the action of a full-line ground state. Translated line profiles show that the loss constants are sharp. The proof combines an $H^1$ estimate in terms of the action with convergence of Dirac fiber maxima to Schrödinger ray maxima. For power nonlinearities, we give an integral criterion in terms of the dimensionless pendant length and obtain the same conclusions for arbitrary edge lengths on four graph topologies.

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arXiv2026

Multiplicity of bound states for a class of nonlinear Dirac equations on generalized star graphs with potentials

In this paper we investigate the existence and multiplicity of bound states for a class of nonlinear Dirac equations on a generalized star graph $\mathcal{G}$: \[ -ic\,σ_1\,\partial_xu+mc^2σ_3u+V(x)u=f(|u|)u,\qquad x\in\mathcal{G}, \] where $V:\mathcal{G}\to\mathbb{R}$ is continuous, $f\in C^1([0,\infty))$, $m>0$, $c>0$, and $σ_1,σ_3$ are the Pauli matrices. Under suitable monotonicity and growth assumptions on $f$, a uniform spectral gap and sufficiently long constant-potential intervals on $k$ terminal edges yield at least $2^k-1$ bound states, distinct up to a constant phase. The constant potentials on these intervals may differ. The construction uses half-line solutions and a uniform inverse estimate. For potentials tending to a constant at infinity, we prove compactness below the full-line ground level. In the flat-well setting, the solutions localized near individual terminal vertices lie below this level and provide a ground state through the generalized Nehari minimization. In particular, one localized solution is obtained for each minimum component containing a long terminal interval.

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arXiv2026

Heat Equation driven by mixed local-nonlocal operators with non-regular space-dependent coefficients

In this paper, we study the Cauchy problem for a heat equation governed by a mixed local--nonlocal diffusion operator with spatially dependent coefficients. For bounded measurable coefficients satisfying uniform positivity conditions and initial data in $H^1(\mathbb R^d)$, we first establish existence and uniqueness of weak solutions in the natural energy space and derive an a priori energy estimate. We then extend the analysis to strictly positive distributional coefficients and compactly supported distributional initial data by means of a Friedrichs-type regularisation procedure. Under suitable moderateness assumptions on the regularising nets, we establish the existence of very weak solutions and prove uniqueness modulo negligible perturbations of admissible regularisations. Finally, under appropriate convergence assumptions on the regularised coefficients and initial data, together with preservation of uniform positivity, we prove consistency of the very weak solution with the corresponding weak solution.

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arXiv2026

Topological Effects on Bubbling in the Critical Dirichlet Problem on Hyperbolic Domains: $3\leq N \leq5$

Let $Ω\Subset\mathbb H^N$, $N\in\{3,4,5\}$, be a bounded connected $C^2$ domain. We prove that the pure critical Dirichlet problem \[ -Δ_{\mathbb H}u=u^{\frac{N+2}{N-2}} \quad\text{in }Ω, \qquad u=0 \quad\text{on }\partialΩ \] admits a positive solution whenever $H_d(Ω;\mathbb F_2)\neq0$ for some $1\le d\le N-1$. This gives a hyperbolic Bahri-Coron theorem for $3\le N\le5$ under $C^2$ boundary regularity. Under conformal reduction, the hyperbolic geometry produces a positive potential and leads to dimension-dependent bubbling mechanisms. In dimension three, the required energy drop follows from the balance between diagonal corrections and pair interactions at fixed large multiplicity. In dimensions four and five, it is obtained at the matched scale through a normalized defect estimate and an all-pairs source-transfer bound. A unified Thom-barycenter construction converts these analytic estimates into the topological contradiction. Thus nontrivial domain topology forces existence despite critical loss of compactness and the additional geometric potential.

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arXiv2026

Normalized Solutions and Semiclassical Concentration for Upper-Critical Fractional Choquard Equations

We study a fractional Choquard equation with an upper-critical Hartree term, an $L^2$-supercritical Hartree perturbation, and a semiclassical potential under a prescribed $L^2$-mass constraint. The potential is bounded and nonnegative, has a nonempty zero set, and has a positive lower limit at infinity. For every prescribed mass and all sufficiently small semiclassical parameters, we prove the existence of a pair of normalized solutions $\pm u_\varepsilon$ with a negative Lagrange multiplier. The proof combines a strict energy bound below the critical one-bubble level, compactness modulo translations for the autonomous ground-state set, a simultaneous cutoff of both Hartree terms, and a localized constrained mountain-pass argument. Moreover, suitable translates of $u_\varepsilon$ converge strongly in $H^s(\mathbb{R}^N)$ to a positive autonomous ground state, and the corresponding concentration points approach the zero set of the potential as $\varepsilon\to0$.

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arXiv2026

Mixed local-nonlocal quasilinear problems with mixed interpolated Hardy potential

This paper addresses the existence of nontrivial solutions to a class of mixed local-nonlocal problems involving a mixed interpolated Hardy potential. We first establish a concentration-compactness principle for mixed local and nonlocal operators. This result is combined with Ricceri's variational principle to obtain an existence result for quasilinear elliptic problems under different growth assumptions on the nonlinearity. Furthermore, we apply the classical mountain pass theorem to obtain a second existence result in the superlinear case.

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arXiv2026

Fujita exponents on quantum Euclidean spaces

We study the well-posedness of a non-linear heat equation with power nonlinearity with positive initial data on quantum Euclidean spaces. We prove a noncommutative analogue of the classical Fujita theorem by identifying the critical exponent separating finite-time blow-up from global existence for small initial data. Moreover, we establish a fundamental inequality in general semifinite von Neumann algebras that is of independent interest and plays a crucial role in the study of global existence and local well-posedness of solutions of nonlinear equations in noncommutative setting.

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arXiv2026

Spectral analysis, maximum principles and shape optimization for nonlinear superposition operators of mixed fractional order

The main objective of this paper is to investigate the spectral properties, maximum principles, and shape optimization problems for a broad class of nonlinear ``superposition operators" defined as continuous superpositions of operators of mixed fractional order, modulated by a signed finite Borel measure on the unit interval. This framework encompasses, as particular cases, mixed local and nonlocal operators such as $-Δ_p+(-Δ_p)^s$, finite (possibly infinite) sums of fractional $p$-Laplacians with different orders, as well as operators involving fractional Laplacians with ``wrong" signs. The main findings, obtained through variational techniques, concern the spectral analysis of the Dirichlet eigenvalue problem associated with general superposition operators with special emphasis on various properties of the first eigenvalue and its corresponding eigenfunction. We establish weak and strong maximum principles for positive superposition operators by introducing an appropriate notion of the {\it nonlocal tail} for this class of superposition operators and deriving a logarithmic estimate, both of which are of independent interest. Utilizing these newly developed tools, we further investigate the spectral properties of such superposition operators and prove that the first eigenvalue is isolated and simple. Moreover, we show that the eigenfunctions corresponding to positive eigenvalues are globally bounded and that they change sign when associated with higher eigenvalues. In addition, we demonstrate that the second eigenvalue is well-defined and provide the mountain pass characterization. Finally, we address shape optimization problems, in particular, the Faber--Krahn inequality associated with the principal frequency associated with the superposition operators.

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arXiv2026

Very weak solutions of the heat equation with anisotropically singular time-dependent diffusivity

We investigate the heat equation with a time-dependent, anisotropic, and potentially singular diffusivity tensor. Since weak (in the Sobolev sense) or distributional solutions may not exist in this setting, we employ the framework of very weak solutions to establish the existence and uniqueness of solutions to the heat equation with singular, anisotropic, time-dependent diffusivity.

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arXiv2026

Weyl Calculus on Graded Groups

The aim of this paper is to establish a pseudo-differential Weyl calculus on graded nilpotent Lie groups $G$ which extends the celebrated Weyl calculus on $\mathbb{R}^n$. To reach this goal, we develop a symbolic calculus for a very general class of quantization schemes, following [Doc. Math., 22, 1539--1592, 2017], using the Hörmander symbol classes $S^m_{ρ, δ}(G)$ introduced in [Progress in Mathematics, 314. Birkhäuser/Springer, 2016]. We particularly focus on the so-called symmetric calculi, for which quantizing and taking the adjoint commute, among them the Euclidean Weyl calculus, but we also recover the (non-symmetric) Kohn-Nirenberg calculus, on $\mathbb{R}^n$ and on general graded groups [Progress in Mathematics, 314. Birkhäuser/Springer, 2016]. Several interesting applications follow directly from our calculus: expected mapping properties on Sobolev spaces, the existence of one-sided parametrices and the Gårding inequality for elliptic operators, and a generalization of the Poisson bracket for symmetric quantizations on stratified groups. In the particular case of the Heisenberg group $\mathbb{H}_n$, we are able to answer the fundamental questions of this paper: which, among all the admissible quantizations, is the natural Weyl quantization on $\mathbb{H}_n$? And which are the criteria that determine it uniquely? The surprisingly simple but compelling answers raise the question whether what is true for $\mathbb{R}^n$ and $\mathbb{H}_n$ also extends to general graded groups, which we answer in the affirmative in this paper. Among other things, we discuss and investigate an analogue of the symplectic invariance property of the Weyl quantization in the setting of graded groups, as well as the notion of the Poisson bracket for symbols in the setting of stratified groups, linking it to the symbolic properties of the commutators.

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arXiv2026

On a fractional nonlinear Schrödinger equation with irregular coefficients. case: d<2s

In the case when $d<2s$, where $d$ is the space dimension and $s$ is the fractional power of the Laplacian, we study the well-posedness for a cubic nonlinear Schrödinger equation (CNLSE) generated by the fractional Laplacian and involving distributional, or less regular, coefficients. We formulate our problem in the setting of the concept of so-called very weak solutions and prove that it has a very weak solution. Moreover, we prove the uniqueness in some adequate sense as well as the compatibility of the very weak solution with the classical one when the latter exists. Our results cover the classical case when: $d=1, s=1$. A second task in this paper is to conduct some numerical experiments where interesting behaviours of the very weak solution are observed. The obtained result is the first example of the very weak well-posedness in the setting of nonlinear partial differential equations.

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arXiv2026

Hormander-Mikhlin type theorem on non-commutative spaces

In this paper, we introduce a Fourier-type formalism on non-commutative spaces. As a result, we obtain two versions of Hormander-Mikhlin Lp-multiplier theorem: on locally compact Kac groups and on semi-finite von Neumann algebras, respectively. In the simplest case our result coincides with a sharp version of the classical Hormander Lp-multiplier theorem, which was obtained by Grafakos and Slavikova in [11]. Finally, we present some applications to the evolution equation in non-commutative setting.

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arXiv2026

Compactness and Spectral Properties of Multiplier Operators in the Walsh System

We investigate compactness and spectral properties of multiplier operators associated with the Walsh system in the spaces $L^p[0,1]$, $1<p<\infty$. Building upon previously established criteria for boundedness of Walsh multipliers, we prove an exact compactness criterion in the $L^p\to L^p$ regime for all $1<p<\infty$(assuming boundedness of the multiplier), and also in the $L^p\to L^2$ regime for $2<p<\infty$. The key result states that compactness is equivalent to the condition $a_n\to 0$ for the multiplier symbol. We also examine in detail the point spectrum and derive strict spectral inclusions; in the Hilbert space case $p=2$ we obtain a complete description of the spectrum. For $p\neq 2$, we emphasize the limitations of transferring "diagonal" arguments and formulate results in a form that does not admit incorrect generalizations.

math.FA↗

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↗

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

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 $u\in L^{1,\infty}(SU(2)).$

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