Michael Ruzhansky

Ghent University and Queen Mary University of London · Mathematics

Subject: In this article, convolutional networks of one, two, and three dimensions are compared with respect to their ability to distinguish between the drawing tests produced by Parkinson's disease patients and healthy control subjects. Motivation: The application of deep learning techniques for the analysis of drawing tests to support the diagnos...

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...

In this paper, we pursue our series of papers aiming to show the applicability of the concept of very weak solutions. We consider a wave model with irregular position dependent mass and dissipation terms, in particular, allowing for δ-like coefficients and prove that the problem has a very weak solution. Furthermore , we prove the uniqueness in an...

Let ∆ be the Laplace-Beltrami operator on a non-compact symmetric space of any rank, and denote the bottom of its L^2-spectrum as −|ρ|^2. In this paper, we provide a comprehensive characterization of both the sufficient and necessary conditions ensuring the validity of the Stein-Weiss inequality for the entire family of operators {(−∆ + b)^{− σ/2 }...

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...

We establish the $L^p$-$L^q$-boundedness of subelliptic pseudo-differential operators on a compact Lie group $G$. Effectively, we deal with the $L^p$-$L^q$-bounds for operators in the sub-Riemmanian setting because the subelliptic classes are associated to a H\"ormander sub-Laplacian. The Riemannian case associated with the Laplacian is also includ...

In this article, we explore the boundedness properties of pseudo-differential operators on radial sections of line bundles over the Poincaré upper half plane, even when dealing with symbols of limited regularity. We first prove the boundedness of these operators when the symbol is smooth. To achieve this, we establish a connection between the opera...

The main purpose of this paper is to prove Hörmander’s $L^p$ – $L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff–Young–Paley inequality for commutative hypergroups. We show the $L^p$ – $L^q$ boundedness of the spectral multipliers for the generalised r...

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...

In this paper, we consider a wave equation on a bounded domain with a Sturm–Liouville operator with a singular intermediate coefficient and a singular potential. To obtain and evaluate the solution, the method of separation of variables is used, then the expansion in the Fourier series in terms of the eigenfunctions of the Sturm–Liouville operator...

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb {Z}^n$ . We establish the well-posedness of such Cauchy problems in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in s...

The analogue of polar coordinates in the Euclidean space, a polar decomposition in a metric space, if well-defined, can be very useful in dealing with integrals with respect to a sufficiently regular measure. In this note we handle the technical details associated with such polar decompositions.

We establish Adams type Stein-Weiss inequality on global Morrey spaces on general homogeneous groups. Special properties of homogeneous norms and some boundedness results on global Morrey spaces play key roles in our proofs. As consequence, we obtain fractional Hardy, Hardy-Sobolev, Rellich and Gagliardo-Nirenberg inequalities on Morrey spaces on s...

Image structure-texture decomposition is a long-standing and fundamental problem in both image processing and computer vision fields. In this paper, we propose a generalized semi-sparse regularization framework for image structural analysis and extraction, which allows us to decouple the underlying image structures from complicated textural backgro...

In this paper we establish the $L^p$-$L^q$ estimates for global pseudo-differential operators on graded Lie groups. We provide both necessary and sufficient conditions for the $L^p$-$L^q$ boundedness of pseudo-differential operators associated with the global H\"ormander symbol classes on graded Lie groups, within the range $1<p\leq 2 \leq q<\infty...

We prove the blow-up of solutions of the semilinear damped Klein–Gordon equation in a finite time for arbitrary positive initial energy on the Heisenberg group. This work complements the paper Ruzhansky and Tokmagambetov (J Differ Equ 265(10):5212–5236, 2018), where the global in time well-posedness was proved for the small energy solutions.

In this paper we investigate Besov spaces on graded Lie groups. We prove a Nikolskii type inequality (or the Reverse Hölder inequality) on graded Lie groups and as consequence we obtain embeddings of Besov spaces. We prove a version of the Littlewood-Paley theorem on graded Lie groups. The results are applied to obtain embedding properties of Besov...

We prove that the noncommutative Lorentz norm (associated to a semifinite von Neumann algebra) of a propagator of the form $\varphi(|\mathscr{L}|)$ can be estimated if the Borel function $\varphi$ is bounded by a positive monotonically decreasing vanishing at infinity continuous function $\psi$. As a consequence we obtain the $L^p-L^q$ $(1<p\leqsla...

Given a smooth manifold M (with or without boundary), in this paper we establish a global functional calculus, without the standard assumption that the operators are classical pseudo-differential operators, and the Gårding inequality for global pseudo-differential operators associated with boundary value problems. The analysis that we follow is fre...

We call an element $U$ conditionally universal for a sequential convergence space $\mathbf{\Omega}$ with respect to a minimal system $\{\varphi_n\}_{n=1}^\infty$ in a continuously and densely embedded Banach space $\mathcal{X}\hookrightarrow\mathbf{\Omega}$ if the partial sums of its phase-modified Fourier series is dense in $\mathbf{\Omega}$. We w...

In this paper we consider the semiclassical version of pseudo-differential operators on the lattice space $\hbar \mathbb{Z}^n$. The current work is an extension of a previous work and agrees with it in the limit of the parameter $\hbar \rightarrow 1$. The various representations of the operators will be studied as well as the composition, transpose...

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...

In this paper, we establish the sharp fractional subelliptic Sobolev inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The best constants are given in terms of a ground state solution of a fractional subelliptic equation involving the fractional $p$-sublaplacian ($1<p<\infty$) on stratified Lie groups. We also prove the ex...

In this paper, we first prove the weighted Levin-Cochran-Lee type inequalities on homogeneous Lie groups for arbitrary weights, quasi-norms, and-and-norms. Then, we derive a sharp weighted inequality involving specific weights given in the form of quasi-balls in homogeneous Lie groups. Finally, we also calculate the sharp constants for the aforemen...

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 H\"{o}rmander symbol c...

In this paper, we investigate direct and inverse problems for the time-fractional heat equation with a time-dependent diffusion coefficient for positive operators. First, we consider the direct problem, and the unique existence of the generalized solution is established. We also deduce some regularity results. Here, our proofs are based on the eige...

In this paper, we consider the inverse problem of determining the time-dependent source term in the general setting of Hilbert spaces and for general additional data. We prove the well-posedness of this inverse problem by reducing the problem to an operator equation for the source function.

In this article, we investigate the semiclassical version of the wave equation for the discrete Schr\"{o}dinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $\mathcal{L}_{\hbar}$ is the discrete Laplacian, and $V$ is a non-negative multiplication operator. We prove that $\mathcal{H}...

The purpose of this paper is twofold: first we study an eigenvalue problem for the fractional p-sub-Laplacian over the fractional Folland–Stein–Sobolev spaces on stratified Lie groups. We apply variational methods to investigate the eigenvalue problems. We conclude the positivity of the first eigenfunction via the strong minimum principle for the f...

We present a semi-sparsity model for 3D triangular mesh denoising, which is motivated by the success of semi-sparsity regularization in image processing applications. We demonstrate that such a regularization model can be also applied for graphic processing and gives rise to similar simultaneous-fitting results in preserving sharp features and piec...

In this paper, we investigate the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the Hörmander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We l...

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb{Z}^n$. We establish the well-posedeness of such Cauchy equations in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in s...

This note gives a wide-ranging update on the multiplier theorems by Akylzhanov and the second author [J. Funct. Anal., 278 (2020), 108324]. The proofs of the latter crucially rely on \(L^p\)-\(L^q\) norm estimates for spectral projectors of left-invariant weighted subcoercive operators on unimodular Lie groups, such as Laplacians, sub-Laplacians, a...

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods...

In this paper, we study the nonlinear Sobolev type equations on the Heisenberg group. We show that the problems do not admit nontrivial local weak solutions, i.e. "instantaneous blow up" occurs, using the nonlinear capacity method. Namely, by choosing suitable test functions, we will prove an instantaneous blow up for any initial conditions $u_0,\,...

In this paper, we establish Liouville type results for semilinear subelliptic systems associated with the sub-Laplacian on the Heisenberg group involving two different kinds of general nonlinearities. The main technique of the proof is the method of moving planes combined with some integral inequalities replacing the role of maximum principles. As...

We prove existence and uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a nonlocal (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\leqsla...

In this paper, we propose a semi-sparsity smoothing method based on a new sparsity-induced minimization scheme. The model is derived from the observations that semi-sparsity prior knowledge is universally applicable in situations where sparsity is not fully admitted such as in the polynomial-smoothing surfaces. We illustrate that such priors can be...

In this article, we address sparse bounds for a class of spectral multipliers that include oscillating multipliers on stratified Lie groups. Our results can be applied to obtain weighted bounds for general Riesz means and for solutions of dispersive equations.

Note: Please see pdf for full abstract with equations. In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schr¨odinger operator (I − Δ)log corresponding to the logarithmic symbol log(1 + |ξ|²), which is a singular integral operator given by (I − Δ)logu(x) = cNP.V.∫R...

In this paper, we study the heat equation with an irregular spatially dependent thermal conductivity coefficient. We prove that it has a solution in an appropriate very weak sense. Moreover, the uniqueness result and consistency with the classical solution if the latter exists are shown. Indeed, we allow the coefficient to be a distribution with a...

We consider the Schr\"odinger equation with singular position dependent effective mass and prove that it is very weakly well posed. A uniqueness result is proved in an appropriate sense, moreover, we prove the consistency with the classical theory. In particular, this allows one to consider Delta-like or more singular masses.

In this paper, in the cylindrical domain, we consider a fractional elliptic operator with Dirichlet conditions. We prove, that the first eigenvalue of the fractional elliptic operator is minimised in a circular cylinder among all cylindrical domains of the same Lebesgue measure. This inequality is called the Rayleigh–Faber–Krahn inequality. Also, w...

In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on $\mathbb{R}^n$ with an explicit expression for the constant. Namely, we show that for all $0<s<\frac{n}{2}$ and $a>0$ we have the inequality \[ \int_{\mathbb{R}^n}|f(x)|^2 \log \left( \frac{|f(x)|^2}{\|f\|^{2}_{L^2(\mathbb{R}^n)}}\right)\,dx+\frac...

We establish the Kato-type smoothing property, i.e., global-in-time smoothing estimates with homogeneous weights, for the Schr\"odinger equation on Riemannian symmetric spaces of non-compact type and general rank. These form a rich class of manifolds with nonpositive sectional curvature and exponential volume growth at infinity, e.g., hyperbolic sp...

We prove Strichartz estimates on any compact connected simple Lie group. In the diagonal case of Bourgain's exponents $p=q,$ we substantially improve the regularity orders showing the existence of indices $s<s_{0}(d)$ below the Sobolev exponent $s_{0}(d)=\frac{d}{2}-\frac{d+2}{p}.$ Motivated by the recent progress in the field, in the spirit of the...

In this work we investigate a class of degenerate Schr\"odinger equations associated to degenerate elliptic operators with irregular potentials on $\Ran$ by introducing a suitable H\"ormander metric $g$ and a $g$-weight $m$. We establish the well-posedness for the corresponding degenerate Schr\"odinger and degenerate parabolic equations. When the s...

In recent years, deep learning methods have achieved great success in various fields due to their strong performance in practical applications. In this paper, we present a light-weight neural network for Parkinson's disease diagnostics, in which a series of hand-drawn data are collected to distinguish Parkinson's disease patients from healthy contr...

The study of the Mittag-Leffler function and its various generalizations has become a very popular topic in mathematics and its applications. In the present paper we prove the following estimate for the $q$-Mittag-Leffler function: \begin{eqnarray*} \frac{1}{1+\Gamma_q\left(1-\alpha\right)z}\leq e_{\alpha,1}\left(-z;q\right)\leq\frac{1}{1+\Gamma_q\...

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...

Let $A$ and $B$ be invariant linear operators with respect to a decomposition $\{H_{j}\}_{j\in \mathbb{N}}$ of a Hilbert space $\mathcal{H}$ in subspaces of finite dimension. We give necessary and sufficient conditions for the controllability of the Cauchy problem $$ u_t=Au+Bv,\,\,u(0)=u_0,$$ in terms of the (global) matrix-valued symbols $\sigma_A...

Let $1<p\leq \infty$ and let $n\geq 2.$ It was proved independently by C. Calder\'on, R. Coifman and G. Weiss that the dyadic maximal function \begin{equation*} \mathcal{M}^{d\sigma}_Df(x)=\sup_{j\in\mathbb{Z}}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-2^jy)d\sigma(y)\right| \end{equation*} is a bounded operator on $L^p(\mathbb{R}^n)$ where $d\si...

In this paper we consider the problem of estimation of oscillatory integrals with Mittag-Leffler functions in two variables. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory type integrals.

In this paper, we obtain a reverse version of the integral Hardy inequality on metric measure space with two negative exponents. For applications we show the reverse Hardy–Littlewood–Sobolev and the Stein–Weiss inequalities with two negative exponents on homogeneous Lie groups and with arbitrary quasi-norm, the result of which appears to be new in...

Let $G$ be a compact Lie group of dimension $n.$ In this work we characterise the membership of classical pseudo-differential operators on $G$ in the trace class ideal $S_{1}(L^2(G)),$ as well as in the setting of the Schatten ideals $S_{r}(L^2(G)),$ for all $r>0.$ In particular, we deduce Schatten characterisations of elliptic pseudo-differential...

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<= 2<= q < +∞....

Fefferman (Acta Math 24:9–36, 1970, Theorem 2\('\)) has proved the weak (1,1) boundedness for a class of oscillating singular integrals that includes the oscillating spectral multipliers of the Euclidean Laplacian \(\Delta ,\) namely, operators of the form $$\begin{aligned} T_{\theta }(-\Delta ):= (1-\Delta )^{-\frac{n\theta }{4}}e^{i (1-\Delta )^{...

In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case $p=1$ and $1 \leq q <\infty.$ This result complements the Hardy inequalities obtained in \cite{RV} in the case $1< p\le q<\infty.$ The case $p=1$ requires a different argument and does not follow as the limit of known...

In this paper we explore the weak solutions of the Cauchy problem and an inverse source problem for the heat equation in the quantum calculus, formulated in abstract Hilbert spaces. For this we use the Fourier series expansions.

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein-Weiss inequality, also known as a weighted Hardy-Littlewood-Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is ach...

In the present paper, we study the Cauchy-Dirichlet problem to a nonlocal nonlinear diffusion equation with polynomial nonlinearities D0|tαu+(-Δ)psu=γ|u|m-1u+μ|u|q-2u,γ,μ∈R,m>0,q>1, involving time-fractional Caputo derivative D0|tα and space-fractional p-Laplacian operator (-Δ)ps. We give a simple proof of the comparison principle for the considere...

In this paper, we have introduced the Prabhakar fractional q-integral and q-differential operators. We first study the semi-group property of the Prab-hakar fractional q-integral operator, which allowed us to introduce the corresponding q-differential operator. Formulas for compositions of q-integral and q-differential operators are also presented....

In this paper, we obtain a reverse version of the integral Hardy inequality on metric measure space with two negative exponents. Also, as for applications we show the reverse Hardy-Littlewood-Sobolev and the Stein-Weiss inequalities with two negative exponents on homogeneous Lie groups and with arbitrary quasi-norm, the result which appears to be n...

This paper presents a novel intrinsic image transfer (IIT) algorithm for image illumination manipulation, which creates a local image translation between two illumination surfaces. This model is built on an optimization-based framework composed of illumination, reflectance and content photo-realistic losses, respectively. Each loss is firstly defin...

In this paper, we investigate the $H^{p}(G) \rightarrow L^{p}(G)$ , $0< p \leq 1$ , boundedness of multiplier operators defined via group Fourier transform on a graded Lie group $G$ , where $H^{p}(G)$ is the Hardy space on $G$ . Our main result extends those obtained in [Colloq. Math. 165 (2021), 1–30], where the $L^{1}(G)\rightarrow L^{1,\infty }(...

In this work we study a class of anharmonic oscillators on Rn corresponding to Hamiltonians of the form A(D)+V(x), where A(ξ) and V(x) are C∞ functions enjoying some regularity conditions. Our class includes fractional relativistic Schrödinger operators and anharmonic oscillators with fractional potentials. By associating a Hörmander metric we obta...

In this article, we present a notion of the harmonic oscillator on the Heisenberg group H n , which, under a few reasonable assumptions, forms the natural analog of a harmonic oscillator on [Formula: see text]: a negative sum of squares of operators on H n , which is essentially self-adjoint on L ² (H n ) with purely discrete spectrum and whose eig...

In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schrödinger operator. The proof hinges on a direct method of moving planes for the logarithmic Schrödinger operator.

We study the Lane-Emden system involving the logarithmic Laplacian. By using a direct method of moving planes for the logarithmic Laplacian, we obtain the symmetry and monotonicity of the positive solutions to the Lane-Emden system. We also establish some key ingredients needed in order to apply the method of moving planes such as the maximum princ...

The aim of this paper is the investigation of the existence and uniqueness of solutions to Cauchy-type problems for fractional q-difference equations with the bi-ordinal Hilfer fractional q-derivative which is an extension of the Hilfer fractional q-derivative. An approach is based on the equivalence of the nonlinear Cauchy-type problem with a nonl...

In this work we investigate the well-posed for diffusion equations associated to subelliptic pseudo-differential operators on compact Lie groups. The diffusion by strongly elliptic operators is considered as a special case and in particular the fractional diffusion with respect to the Laplacian. The general case is studied within the Hörmander clas...

In this paper we consider the uniform estimates for oscillatory integrals with homogeneous polynomial phases of degree 4 in two variables. The obtained estimate is sharp and the result is an analogue of the more general theorem of Karpushkin (Proc I.G.Petrovsky Seminar 9:3–39, 1983) for sufficiently smooth functions, thus, in particular, removing t...

In this paper, we consider a wave equation on a bounded domain with a Sturm-Liouville operator with a singular intermediate coefficient and a singular potential. To obtain and evaluate the solution, the method of separation of variables is used, then the expansion in the Fourier series in terms of the eigenfunctions of the Sturm-Liouville operator...

In this note we investigate a local Weyl type formula for an arbitrary pseudo-differential operator on a compact Lie group in terms of its matrix-valued symbol. We link our formula to the principal symbol of the operator, to the representation theory of the group, and to the problem of the quantum unique ergodicity.

The aim of this paper is to find new estimates for the norms of functions of a (minus) distinguished Laplace operator \({\mathcal {L}}\) on the ‘\(ax+b\)’ groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type \(\psi (\sqrt{{\mathcal {L}}})\exp (it \sqrt{{\mathcal {L}}})\), with \(\psi \in C_0({...

This note gives a wide-ranging update on the multiplier theorems by Akylzhanov and the second author [J. Funct. Anal., 278 (2020), 108324]. The proofs of the latter crucially rely on $L^p$-$L^q$ norm estimates for spectral projectors of left-invariant weighted subcoercive operators on unimodular Lie groups, such as Laplacians, sub-Laplacians and Ro...

We extend the estimates proved by Donnelly and Fefferman and by Lebeau and Robbiano for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the...

In this work we characterise the H ̈ormander classes S^m_{ρ,δ} (G, H ̈or) on the open manifold G = (−1, 1)n. We show that by endowing the open manifold G = (−1, 1)n with a group structure, the corresponding global Fourier analysis on the group allows one to define a global notion of symbol on the phase space G × Rn. Then, the class of pseudo-differ...

We establish a new improvement of the classical $L^p$-Hardy inequality on the multidimensional Euclidean space in the supercritical case. Recently, in [14], there has been a new kind of development of the one dimensional Hardy inequality. Using some radialisation techniques of functions and then exploiting symmetric decreasing rearrangement argumen...

The paper is denoted to the initial-boundary value problem for the wave equation with the Sturm-Liouville operator with irregular (distributive) potentials. To obtain a solution to the equation, the separation method and asymptotics of the eigenvalues and eigenfunctions of the Sturm-Liouville operator are used. Homogeneous and inhomogeneous cases o...

This paper is primarily devoted to a class of interpolation inequalities of Hardy and Rellich types on the Heisenberg group $\mathbb{H}^n$. Consequently, several weighted Hardy type, Heisenberg-Pauli-Weyl uncertainty principle and Hardy-Rellich type inqualities are established on $\mathbb{H}^n$. Moreover, new weighted Sobolev type embeddings are de...

In this work we extend the L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-Björk-Sjölin theory of strongly singular convolution operators to arbit...

We establish Plemelj-Smithies formulas for determinants in different algebras of operators. In particular we define a Poincaré type determinant for operators on the torus Tn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...

In this paper we prove the fractional Gagliardo-Nirenberg inequality on homogeneous Lie groups. Also, we establish weighted fractional Caffarelli-Kohn-Nirenberg inequality and Lyapunov-type inequality for the Riesz potential on homogeneous Lie groups. The obtained Lyapunov inequality for the Riesz potential is new already in the classical setting o...

In this paper, we study a semilinear heat equation with forcing term depending on space. The local existence, the blow up and global in time solutions are obtained. Our technique of proof is based on methods of test functions and the comparison principle. In addition, we give an upper bound estimate on the lifespan of blow up solutions.

In this note, we prove the reverse Stein–Weiss inequality on general homogeneous Lie groups. The results obtained extend previously known inequalities. Special properties of homogeneous norms and the reverse integral Hardy inequality play key roles in our proofs. Also, we prove reverse Hardy, Hardy–Littlewood–Sobolev, L p -Sobolev and L p -Caffarel...

In a rectangular domain, a boundary‐value problem is considered for a mixed equation with a regularized Caputo‐like counterpart of hyper‐Bessel differential operator and the bi‐ordinal Hilfer's fractional derivative. By using the method of separation of variables a unique solvability of the considered problem has been established. Moreover, we have...

In this paper we consider the heat equation with a strongly singular potential and show that it has a very weak solution. Our analysis is devoted to general hypoelliptic operators and is developed in the setting of graded Lie groups. The current work continues and extends the work (Altybay et al. in Appl. Math. Comput. 399:126006, 2021), where the...

In the present paper, we study the Cauchy-Dirichlet problem to the nonlocal nonlinear diffusion equation with polynomial nonlinearities $$\mathcal{D}_{0|t}^{\alpha }u+(-\Delta)^s_pu=\gamma|u|^{m-1}u+\mu|u|^{q-2}u,\,\gamma,\mu\in\mathbb{R},\,m>0,q>1,$$ involving time-fractional Caputo derivative $\mathcal{D}_{0|t}^{\alpha}$ and space-fractional $p$-...

In this paper we consider the uniform estimates for oscillatory integrals with a two-order homogeneous polynomial phase. The estimate is sharp and the result is an analogue of the more general theorem of V. N. Karpushkin \cite{Karpushkin1983} for sufficiently smooth functions.

The purpose of this paper is threefold: first we prove Sobolev-Rellich-Kondrachov type embeddings for the fractional Folland-Stein-Sobolev spaces on stratified Lie groups. Secondly, we study an eigenvalue problem for the fractional $p$-sub-Laplacian over the corresponding fractional Folland-Stein-Sobolev spaces on stratified Lie groups. We apply va...

A classical theorem of Titchmarsh relates the $L^2$-Lipschitz functions and decay of the Fourier transform of the functions. In this note, we prove the Titchmarsh theorem for Damek-Ricci space (also known as harmonic $NA$ groups) via moduli of continuity of higher orders. We also prove an analogue of another Titchmarsh theorem which provides integr...

In this paper, we consider a semiclassical version of the fractional Klein-Gordon equation on the lattice, $h{\mathbb{Z}}^n.$ Contrary to the Euclidean case that was considered in [2], the discrete fractional Klein-Gordon equation is well-posed in $\ell^2(h{\mathbb{Z}}^n).$ However, we also recover the well-posedness results in the certain Sobolev...