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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 $p$-Laplacian operator $(-\Delta)^s_p$. We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of $\gamma,\mu,m$ and $q$. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle., Comment: 33 pages

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

In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincaré inequality, established in Ruzhansky and Suragan (J Differ Eq 262:1799–1821, 2017) for stratified groups.

In this paper we consider the space-fractional Schr\"odinger equation with a singular potential for a wide class of fractional hypoelliptic operators. Such analysis can be conveniently realised in the setting of graded Lie groups. The paper is a continuation and extension of a previous one where the classical Schr\"odinger equation on $\mathbb R^n$ with singular potentials was considered., Comment: This work is a continuation and an extension of the previous one, see arXiv:2004.10182v2, on $\mathbb R^n$. arXiv admin note: substantial text overlap with arXiv:2105.12862

In this note, we obtain a reverse version of the integral Hardy inequality on metric measure spaces. Moreover, we give necessary and sufficient conditions for the weighted reverse Hardy inequality to be true. The main tool in our proof is a continuous version of the reverse Minkowski inequality. Also, we present some consequences of the obtained reverse Hardy inequality on the homogeneous groups, hyperbolic spaces and Cartan-Hadamard manifolds., 23 pages

We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the Hörmander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.

This paper presents a novel intrinsic image transfer (IIT) algorithm for illumination manipulation, which creates a local image translation between two illumination surfaces. This model is built on an optimization-based framework consisting of three photo-realistic losses defined on the sub-layers factorized by an intrinsic image decomposition. We illustrate that all losses can be reduced without the necessity of taking an intrinsic image decomposition under the well-known spatial-varying illumination illumination-invariant reflectance prior knowledge. Moreover, with a series of relaxations, all of them can be directly defined on images, giving a closed-form solution for image illumination manipulation. This new paradigm differs from the prevailing Retinex-based algorithms, as it provides an implicit way to deal with the per-pixel image illumination. We finally demonstrate its versatility and benefits to the illumination-related tasks such as illumination compensation, image enhancement, and high dynamic range (HDR) image compression, and show the high-quality results on natural image datasets., Accepted by IEEE Transactions on Pattern Analysis and Machine Intelligence (T-PAMI)

In this paper, following the works on non-harmonic analysis of boundary value problems by Tokmagambetov, Ruzhansky and Delgado, we use Operator Ideals Theory and Gershgorin Theory to obtain explicit information concerning the spectrum of pseudo-differential operators, on a smooth manifold $\Omega$ with boundary $\partial \Omega$, in the context of the non-harmonic analysis of boundary value problems, introduced by Tokmagambetov and Ruzhansky in terms of a model operator $\mathfrak{L} $. Under certain assumptions about the eigenfunctions of the model operator, for symbols in the Hormander class $S^0_{1,0} (\overline{\Omega} \times \mathcal{I} )$, we provide a "non-harmonic version" of Gohberg's Lemma, and a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is a compact operator in $L^2(\Omega)$. Also, for pseudo-differential operators with symbols satisfying some integrability condition, one defines its associated matrix in terms of the biorthogonal system associated to $\mathfrak{L} $, and this matrix is used to give necessary and sufficient conditions for the $L^2(\Omega)$-boundedness, and to locate the spectrum of some operators. After that, we extend to the context of the non-harmonic analysis of boundary value problems the well known theorems about the exact domain of elliptic operators, and discuss some applications of the obtained results to evolution equations. Specifically we provide sufficient conditions to ensure the smoothness and stability of solutions to a generalised version of the heat equation.

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 arguments on the real line, the new multidimensional version of the Hardy inequality is given. Some consequences are also discussed., 15 pages; revised version with corrected formulations and arguments; to replace the previous version

The purpose of this paper is to establish the solvability results to direct and inverse problems for time-fractional pseudo-parabolic equations with the self-adjoint operators. We are especially interested in proving existence and uniqueness of the solutions in the abstract setting of Hilbert spaces., 18 pages

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 {L^{p}} -Sobolev and L p {L^{p}} -Caffarelli–Kohn–Nirenberg inequalities on homogeneous Lie groups.

For a separable finite diffuse measure space $${\mathcal {M}}$$ M and an orthonormal basis $$\{\varphi _n\}$$ { φ n } of $$L^2({\mathcal {M}})$$ L 2 ( M ) consisting of bounded functions $$\varphi _n\in L^\infty ({\mathcal {M}})$$ φ n ∈ L ∞ ( M ) , we find a measurable subset $$E\subset {\mathcal {M}}$$ E ⊂ M of arbitrarily small complement $$|{\mathcal {M}}{\setminus } E| | M \ E | < ϵ , such that every measurable function $$f\in L^1({\mathcal {M}})$$ f ∈ L 1 ( M ) has an approximant $$g\in L^1({\mathcal {M}})$$ g ∈ L 1 ( M ) with $$g=f$$ g = f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of $${\mathcal {M}}=G/H$$ M = G / H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

In this work we extend the $L^1$-Bj\"ork-Sj\"olin theory of strongly singular convolution operators to arbitrary graded Lie groups. Our criteria are presented in terms of the oscillating H\"ormander condition due to Bj\"ork and Sj\"olin of the kernel of the operator, and the decay of its group Fourier transform is measured in terms of the infinitesimal representation of an arbitrary Rockland operator. The historical result by Bj\"ork and Sj\"olin is re-obtained in the case of the Euclidean space., Comment: 22 Pages. arXiv admin note: text overlap with arXiv:2201.12881

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 a previous work , where the classical heat equation on $\mathbb R^n$ was considered., Comment: This work is a continuation of the work arXiv:2004.11255v3. It is 21 pages

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann–Liouville derivatives is les...

In this paper the space-fractional Schrodinger equations with singular potentials are studied. Delta like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of ‘weakened’ solutions, calling them very weak solutions. The existence, uniqueness and consistency results are proved in an appropriate sense. Numerical simulations are done, and a particles accumulating effect is observed in the singular cases. From the mathematical point of view a “splitting of the strong singularity” phenomena is also observed.

In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.

In this paper, we define and study Gevrey spaces associated with a Hormander family of (globally defined) vector fields and its corresponding sub‐Laplacian. We show some natural relations between the various Gevrey spaces in this setting on general manifolds, and more particular properties on Lie groups with polynomial growth of the volume. In the case of the Heisenberg group and of SU(2), we show that all our descriptions coincide.

The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions E-alpha,E-beta(i lambda phi(x)), x is an element of R-N and E-alpha,E-beta(i(alpha)lambda phi(x)), x is an element of R-N for the various range of alpha and beta. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrodinger equations are considered.

In this paper we continue the study of non-diagonalisable hyperbolic systems with variable multiplicity started by the authors in [1] . In the case of space dependent coefficients, we prove a representation formula for solutions that allows us to derive results of regularity and propagation of singularities.

In this paper we present a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups. Moreover, using the comparison principle we obtain blow-up type results and global in $t$-boundedness of solutions of nonlinear equations for the heat $p$-sub-Laplacian on stratified Lie groups. In particular, this paper generalises and extends previous results obtained by the first author and Suragan in [RS18]., 14 pages

In [24, Theorem 2'] Charles Fefferman 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{equation} T_{\theta}(-\Delta):= (1-\Delta)^{-\frac{n\theta}{4}}e^{i (1-\Delta)^{\frac{\theta}{2}}},\,0\leq \theta, Comment: 21 pages. The proof of the main theorem has been simplified. Final version to appear in Math. Z

The $(k,a)$-generalised Fourier transform is the unitary operator defined using the $a$-deformed Dunkl harmonic oscillator.The main aim of this paper is to prove $L^p$-$L^q$ boundedness of $(k, a)$-generalised Fourier multipliers. To show the boundedness we first establish Paley inequality and Hausdorff-Young-Paley inequality for $(k, a)$-generalised Fourier transform. We also demonstrate applications of obtained results to study the well-posedness of nonlinear partial differential equations., 20 pages. arXiv admin note: text overlap with arXiv:2108.01146

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 fractional $p$-sub-Laplacian. Moreover, we deduce that the first eigenvalue is simple and isolated. Secondly, utilising established properties, we prove the existence of at least two weak solutions via the Nehari manifold technique to a class of subelliptic singular problems associated with the fractional $p$-sub-Laplacian on stratified Lie groups. We also investigate the boundedness of positive weak solutions to the considered problem via the Moser iteration technique. The results obtained here are also new even for the case $p=2$ with $\mathbb{G}$ being the Heisenberg group., Comment: 40 pages. To appear in "Mathematische Annalen"

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 Rockland operators. By relating spectral projectors to heat kernels, explicit estimates of the $L^p$-$L^q$ norms can be immediately exploited for a much wider range of (connected unimodular) Lie groups and operators than previously known. The comparison with previously established bounds by authors show that the heat kernel estimates are sharp. As an application, it is shown that several consequences of the multiplier theorems, such as time asymptotics for the $L^p$-$L^q$ norms of the heat kernels and Sobolev-type embeddings, are then automatic for the considered operators., Comment: 10 pages

In this work we study a class of anharmonic oscillators on $\mathbb{R}^n$ corresponding to Hamiltonians of the form $A(D)+V(x)$, where $A(\xi)$ and $V(x)$ are $C^{\infty}$ functions enjoying some regularity conditions. Our class includes fractional relativistic Schr\"odinger operators and anharmonic oscillators with fractional potentials. By associating a H\"ormander metric we obtain spectral properties in terms of Schatten-von Neumann classes for their negative powers and derive from them estimates on the rate of growth for the eigenvalues of the operators $A(D)+V(x)$. This extends the analysis in the first part of our work, where the case of polynomial $A$ and $V$ has been analysed., Comment: 19 pages

This paper deals with the inequalities comparing the norm of a function on a compact hypergroup and the norm of its Fourier coefficients. We prove the classical Paley inequality in the setting of compact hypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley inequalities in the noncommutative context. We establish Hormander's L-p- L-q Fourier multiplier theorem on compact hypergroups for 1 < p

In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solution are also proved, under suitable considerations. This extends and improves the results obtained in the first part [Altybay et al. Fractional Klein-Gordon equation with singular mass. Chaos Solitons Fractals. 2021;143:Article ID 110579] which was devoted to the classical Euclidean Klein-Gordon equation.

In this paper, we propose a hybrid parallel programming approach for a numerical solution of a two-dimensional acoustic wave equation using an implicit difference scheme for a single computer. The calculations are carried out in an implicit finite difference scheme. First, we transform the differential equation into an implicit finite-difference equation and then using the alternating direction implicit (ADI) method, we split the equation into two sub-equations. Using the cyclic reduction algorithm, we calculate an approximate solution. Finally, we change this algorithm to parallelize on graphics processing unit (GPU), GPU + Open Multi-Processing (OpenMP), and Hybrid (GPU + OpenMP + message passing interface (MPI)) computing platforms. The special focus is on improving the performance of the parallel algorithms to calculate the acceleration based on the execution time. We show that the code that runs on the hybrid approach gives the expected results by comparing our results to those obtained by running the same simulation on a classical processor core, Compute Unified Device Architecture (CUDA), and CUDA + OpenMP implementations.

When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. In this study, we construct self-similar solutions of some model degenerate partial differential equations of the second, third, and fourth order. These self-similar solutions are expressed in terms of hypergeometric functions.

In this paper we present $L^2$ and $L^p$ versions of the geometric Hardy inequalities in half-spaces and convex domains on stratified (Lie) groups. As a consequence, we obtain the geometric uncertainty principles. We give examples of the obtained results for the Heisenberg and the Engel groups., 18 pages

This paper deals with the multi-term generalisation of the time-fractional diffusion-wave equation for general operators with discrete spectrum, as well as for positive hypoelliptic operators, with homogeneous multi-point time-nonlocal conditions. Several examples of the settings where our nonlocal problems are applicable are given. The results for the discrete spectrum are also applied to treat the case of general homogeneous hypoelliptic left-invariant differential operators on general graded Lie groups, by using the representation theory of the group. For all these problems, we show the existence, uniqueness, and the explicit representation formulae for the solutions., 32 pages

Background: In this study, we used scanning electron microscopy (SEM) and atomic force microscopy (AFM) to examine the changes in morphology of red blood cells (RBCs) and to investigate the nanoparticles (NPs) found on their surface in cervical cancer (CC) patients undergoing radiation therapy (RT). Methods and Results: We obtained smears of venous blood from 12 CC patients at the start, midway and at the end of external beam RT and then midway and at the end of brachytherapy. It was found that in CC patients, the number of RBCs with abnormal morphology increased and NPs appeared on their surface. During RT, the total number of abnormally shaped RBCs and the number and size of NPs increased. The RBC diameter was 8.38 +/- 0.36 mu m in the control group and 9.41 +/- 0.47 mu m in CC patients. The average diameter of NPs on the RBC surface was 69.91 +/- 12.15 nm and their average height 23.75 +/- 3.70 nm. After RT, the morphology of RBCs was restored, and the formation of NPs decreased. Conclusion: The changes observed could serve as the basis for developing efficacy indicators of cancer radiation therapy.

In this paper we establish the subelliptic Picone type identities. As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub-Laplacians which are operators of the form $$\begin{aligned} {\mathcal {L}}_{p}f:= \sum _{i=1}^{N} X_i\left( |X_i f|^{p_i-2} X_i f \right) ,\quad 1

In this note we continue giving the characterisation of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. This is a continuation of our paper [M. Ruzhansky and D. Verma. Hardy inequalities on metric measure spaces, Proc. R. Soc. A., 475(2223):20180310, 2018] where we treated the case $p\leq q$. Here the remaining range $p>q$ is considered, namely, $0, Comment: 18 pages; this is the second part to the paper arXiv:1806.03728. Final version, to appear in Proc. Royal Soc. A

This paper deals with the fractional generalization of the integro-differential diffusion-wave equation for the Heisenberg sub-Laplacian, with homogeneous Bitsadze-Samarskii type time-nonlocal conditions. For the considered problem, we show the existence, uniqueness and the explicit representation formulae for the solution.

We propose a simple feature extraction algorithm for license plate location, which can reduce the occurrence of pseudo-licenses significantly. Our scheme arises from a novelL0-norm image smoothing, in which the multiple local textures in the complex backgrounds can be suppressed remarkably without changing the structures and edges of the license objects. Due to this “edgeaware” property, we then combine a feature filtering with an efficient binarized image, a simple multi-scale image analysis algorithm, to remove the potential false license plates. Finally, we extract license plates with a projection method. Experimental results show the proposed method provides a flexible and powerful way to the license plate location in complex backgrounds.

We give a criterion for the global boundedness of integral operators which are known to be locally bounded. As an application, we discuss the global $L^p$-boundedness for a class of Fourier integral operators. While the local $L^p$-boundedness of Fourier integral operators is known from the work of Seeger, Sogge and Stein, not so many results are available for the global boundedness on $L^p({\mathbb R}^n)$. We give several natural sufficient conditions for them., 13 pages. This paper is a substantially reworked version of our paper arXiv:1510.03807, where we now deduce the result of that paper on the global boundedness for Fourier integral operators from a much more general principle for integral operators

We propose analogues of Green’s and Picone’s identities for the p-sub-Laplacian on stratified Lie groups. In particular, these imply a generalised Diaz-Saa inequality. Using these we derive a comparison principle and uniqueness of positive solutions to nonlinear hypoelliptic equations on general stratified Lie groups extending to this setting previously known results on Euclidean and Heisenberg groups.

Questions related to “very weak” solutions of physical models of acoustic and shallow water wave propagation with singular dissipation are studied. The existence of a new type of solutions is proved. An existence theorem for a very weak solution of the problem is obtained. Finally it is shown that very weak solutions are consistent with classical ones in a certain sense, provided that the latter exist.

A class of inverse problems for restoring the right-hand side of the pseudo-parabolic equation for 1D Laplace operator is considered. The inverse problem is to be well-posed in the sense of Hadamard whenever an overdetermination condition of the final temperature is given. Mathematical statements involve inverse problems for the pseudo-parabolic equation in which, solving the equation, we have to find the unknown right-hand side depending only on the space variable. We prove the existence and uniqueness of the classical solutions. The proof of the existence and uniqueness results of the solutions is carried out by using L-Fourier analysis. The mentioned results are presented as well as for the fractional time pseudo–parabolic equation. Inverse problems of identifying the coefficients of right hand side of the pseudo-parabolic equation from the local overdetermination condition have important applications in various areas of applied science and engineering, also such problems can be modeled using common homogeneous left-invariant hypoelliptic operators on common graded Lie groups.