Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

Visiting Professor



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Publication Type

351 results found

Cardona D, Ruzhansky M, 2022, Björk–Sjölin condition for strongly singular convolution operators on graded Lie groups, Mathematische Zeitschrift, Vol: 302, Pages: 1957-1981, ISSN: 0025-5874

In this work we extend the L1-Björk-Sjölin theory of strongly singular convolution operators to arbitrary graded Lie groups. Our criteria are presented in terms of the oscillating Hörmander condition due to Björk and Sjölin 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 Björk and Sjölin is re-obtained in the case of the Euclidean space.

Journal article

Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang Het al., 2022, Norms of certain functions of a distinguished Laplacian on the ax+ b groups, Mathematische Zeitschrift, Vol: 302, Pages: 2327-2352, ISSN: 0025-5874

The aim of this paper is to find new estimates for the norms of functions of a (minus) distinguished Laplace operator L on the ‘ax+ b’ groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type ψ(L)exp(itL), with ψ∈ C(R). We show that for t→ + ∞, the convolution kernel kt of this operator satisfies ‖kt‖1≍t,‖kt‖∞≍1,so that the upper estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary component, we recall the Plancherel density of L and spend certain time presenting and comparing different approaches to its calculation. Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami operator Δ ~ , closely related to L. The functions include in particular exp (- tΔ ~ γ) , t> 0 , γ> 0 , and (Δ ~ - z) s, with complex z, s.

Journal article

Ruzhansky M, Safarov AR, Khasanov GA, 2022, Uniform estimates for oscillatory integrals with homogeneous polynomial phases of degree 4, Analysis and Mathematical Physics, Vol: 12, ISSN: 1664-2368

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 the analyticity assumption.

Journal article

Cardona D, Delgado J, Ruzhansky M, 2022, Drift diffusion equations with fractional diffusion on compact Lie groups, Journal of Evolution Equations, Vol: 22, ISSN: 1424-3199

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 classes associated to a sub-Riemannian structure on the group (encoded by a Hörmander system of vector fields). Applications to diffusion equations for fractional sub-Laplacians, fractional powers of more general subelliptic operators and the corresponding quasi-geostrophic model with drift D are investigated. Examples on SU (2) for diffusion problems with fractional diffusion are analysed.

Journal article


Let G1 and G2 be compact Lie groups, X1 ∈ g1, X2 ∈ g2 and consider the operator Laq = X1 + a(x1)X2 + q(x1, x2), where a and q are ultradifferentiable functions in the sense of Komatsu, and a is real-valued. Assuming certain condition on a and q we characterize completely the global hypoellipticity and the global solvability of Laq in the sense of Komatsu. For this, we present a conjugation between Laq and a constant-coefficient operator that preserves these global properties in Komatsu classes. We also present examples of globally hypoelliptic and globally solvable operators on T1 × S3 and S3 × S3 in the sense of Komatsu. In particular, we give examples of differential operators which are not globally C∞-solvable, but are globally solvable in Gevrey spaces.

Working paper

Cardona D, Delgado J, Ruzhansky M, 2022, Determinants and Plemelj-Smithies formulas, Monatshefte fur Mathematik, Vol: 199, Pages: 459-482, ISSN: 0026-9255

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 and deduce formulas for determinants of periodic pseudo-differential operators in terms of the symbol. On the other hand, by applying a recently introduced notion of invariant operators relative to fixed decompositions of Hilbert spaces we also obtain formulae for determinants with respect to the trace class. The analysis makes use of the corresponding notion of full matrix-symbol. We also derive explicit formulas for determinants associated to elliptic operators on compact manifolds, compact Lie groups, and on homogeneous vector bundles over compact homogeneous manifolds.

Journal article

Chatzakou M, Delgado J, Ruzhansky M, 2022, On a class of anharmonic oscillators II. General case, Bulletin des Sciences Mathematiques, Vol: 180, ISSN: 0007-4497

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 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 [1], where the case of polynomial A and V has been analysed.

Journal article

Ruzhansky M, Yessirkegenov N, 2022, Critical Gagliardo-Nirenberg, Trudinger, Brezis-Gallouet-Wainger inequalities on graded groups and ground states, Communications in Contemporary Mathematics, Vol: 24, ISSN: 0219-1997

In this paper, we investigate critical Gagliardo-Nirenberg, Trudinger-type and Brezis-Gallouet-Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which include the cases of n, Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo-Nirenberg inequality, the existence of least energy solutions of nonlinear Schrödinger type equations is obtained. We also express the best constant in the critical Gagliardo-Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The obtained results are already new in the setting of general stratified Lie groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland's analysis of Hölder spaces from stratified Lie groups to general homogeneous Lie groups.

Journal article

Cardona D, Ruzhansky M, 2022, Fourier multipliers for Triebel–Lizorkin spaces on compact Lie groups, Collectanea Mathematica, Vol: 73, Pages: 477-504, ISSN: 0010-0757

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.

Journal article

Kassymov A, Ruzhansky M, Suragan D, 2022, Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups, Forum Mathematicum, Vol: 34, Pages: 1147-1158, ISSN: 0933-7741

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, Lp-Sobolev and Lp-Caffarelli–Kohn–Nirenberg inequalities on homogeneous Lie groups.

Journal article

Cardona D, Ruzhansky M, 2022, Sharpness of Seeger-Sogge-Stein orders for the weak (1,1) boundedness of Fourier integral operators, Archiv der Mathematik, Vol: 119, Pages: 189-198, ISSN: 0003-889X

Let X and Y be two smooth manifolds of the same dimension. It was proved by Seeger et al. in (Ann Math 134(2): 231–251, 1991) that the Fourier integral operators with real non-degenerate phase functions in the class I1μ(X,Y;Λ),μ≤ - (n- 1) / 2 , are bounded from H1 to L1. The sharpness of the order - (n- 1) / 2 , for any elliptic operator was also proved in (Seeger et al. Ann Math 134(2): 231–251, 1991) and extended to other types of canonical relations in (Ruzhansky Hokkaido Math J 28(2): 357–362, 1992). That the operators in the class I1μ(X,Y;Λ),μ≤ - (n- 1) / 2 , satisfy the weak (1,1) inequality was proved by Tao (J Aust Math Soc 76(1):1–21, 2004). In this note, we prove that the weak (1,1) inequality for the order - (n- 1) / 2 is sharp for any elliptic Fourier integral operator, as well as its versions for canonical relations satisfying additional rank conditions.

Journal article

Chatzakou M, Ruzhansky M, Tokmagambetov N, 2022, The Heat Equation with Singular Potentials. II: Hypoelliptic Case, ACTA APPLICANDAE MATHEMATICAE, Vol: 179, ISSN: 0167-8019

Journal article

Ruzhansky M, Sabitbek B, Torebek B, 2022, Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified groups, MANUSCRIPTA MATHEMATICA, ISSN: 0025-2611

Journal article

Dasgupta A, Ruzhansky M, Tushir A, 2022, Discrete time-dependent wave equations I. Semiclassical analysis, Journal of Differential Equations, Vol: 317, Pages: 89-120, ISSN: 0022-0396

In this paper we consider a semiclassical version of the wave equations with singular Hölder time-dependent propagation speeds on the lattice ħZn. We allow the propagation speed to vanish leading to the weakly hyperbolic nature of the equations. Curiously, very much contrary to the Euclidean case considered by Colombini, de Giorgi and Spagnolo [2] and by other authors, the Cauchy problem in this case is well-posed in ℓ2(ħZn). However, we also recover the well-posedness results in the intersection of certain Gevrey and Sobolev spaces in the limit of the semiclassical parameter ħ→0.

Journal article

Ruzhansky M, Yessirkegenov N, 2022, Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds with negative curvature, Journal of Mathematical Analysis and Applications, Vol: 507, ISSN: 0022-247X

In this paper we obtain Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants on Riemannian manifolds with non-positive sectional curvature and, in particular, a variety of new estimates on hyperbolic spaces. Moreover, in some cases we also show their equivalence with Trudinger-Moser inequalities. As consequences, the relations between the constants of these inequalities are investigated yielding asymptotically best constants in the obtained inequalities. We also obtain the corresponding uncertainty type principles.

Journal article

Chatzakou M, Ruzhansky M, Tokmagambetov N, 2022, Fractional SchrÖdinger Equations with Singular Potentials of Higher Order. II: Hypoelliptic Case, Reports on Mathematical Physics, Vol: 89, Pages: 59-79, ISSN: 0034-4877

In this paper we consider the space-fractional Schrödinger 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 the first part [2] where the classical Schrödinger equation on Rn with singular potentials was considered.

Journal article

Ruzhansky M, Yessirkegenov N, 2022, A comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups, Nonlinear Analysis, Theory, Methods and Applications, Vol: 215, ISSN: 0362-546X

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.

Journal article

Ruzhansky M, Yessirkegenov N, 2022, Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups, Journal of Differential Equations, Vol: 308, Pages: 455-473, ISSN: 0022-0396

In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold M: {ut−LMu=f(u),x∈M,t>0,u(0,x)=u0(x),x∈M, for u0≥0, where LM is a sub-Laplacian of M. In the case when M is a connected unimodular Lie group G, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with u0≢0, blow up in finite time if and only if 1<p≤pF:=1+2/D when f(u)≃up, where D is the global dimension of G. In the case 1<p<pF and when f:[0,∞)→[0,∞) is a locally integrable function such that f(u)≥K2up for some K2>0, we also show that the differential inequality ut−LMu≥f(u) does not admit any nontrivial distributional (a function u∈Llocp(Q) which satisfies the differential inequality in D′(Q)) solution u≥0 in Q:=(0,∞)×G. Furthermore, in the case when G has exponential volume growth and f:[0,∞)→[0,∞) is a continuous increasing function such that f(u)≤K1up for some K1>0, we prove that the Cauchy problem has a global, classical solution for 1<p<∞ and some positive u0∈Lq(G) with 1≤q<∞. Moreover, we also discuss all these results in more general settings of sub-Riemannian manifolds M.

Journal article

Karimov E, Ruzhansky M, Toshtemirov B, 2022, Solvability of the boundary-value problem for a mixed equation involving hyper-Bessel fractional differential operator and bi-ordinal Hilfer fractional derivative, Mathematical Methods in the Applied Sciences, ISSN: 0170-4214

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 found the explicit solution of initial-boundary problems for the heat equation with the regularized Caputo-like counterpart of the hyper-Bessel differential operator with the non-zero starting point.

Journal article

Kumar V, Ruzhansky M, 2022, Hardy-Littlewood Inequality and L<sup>p</sup>-L<sup>q</sup> Fourier Multipliers on Compact Hypergroups, Journal of Lie Theory, Vol: 32, Pages: 475-498, ISSN: 0949-5932

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 Hörmander’s Lp-Lq Fourier multiplier theorem on compact hypergroups for 1 < p ≤ 2 ≤ q < ∞ as an application of the Hausdorff-Young-Paley inequality. We examine our results for the hypergroups constructed from the conjugacy classes of compact Lie groups and for a class of countable compact hypergroups.

Journal article

Bin-Saad MG, Hasanov A, Ruzhansky M, 2022, Some properties relating to the Mittag–Leffler function of two variables, Integral Transforms and Special Functions, Vol: 33, Pages: 400-418, ISSN: 1065-2469

An attempt is made here to study the Mittag–Leffler function with two variables. Its various properties including integral and operational relationships with other known Mittag–Leffler functions of one variable, pure and differential recurrence relations, Euler transform, Laplace transform, Mellin transform, Whittaker transform, Mellin–Barnes integral representation, and its relationship with Wright hypergeometric function are investigated and established. Also, properties of the Mittag–Leffler function of two variables associated with fractional calculus operators are considered.

Journal article

Karimov E, Ruzhansky M, Tokmagambetov N, 2022, Cauchy type problems for fractional differential equations, Integral Transforms and Special Functions, Vol: 33, Pages: 47-64, ISSN: 1065-2469

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 less understood. In this paper, we propose new type initial, inner, and inner-boundary value problems for fractional differential equations with the Riemann–Liouville derivatives. The results on the existence and uniqueness are proved, and conditions on the solvability are found. The well-posedness of the new type of initial, inner, and inner-boundary conditions is also discussed. Moreover, we give explicit formulas for the solutions. As an application fractional partial differential equations for general positive operators are studied.

Journal article

Chatzakou M, Ruzhansky M, Tokmagambetov N, 2022, Fractional Klein-Gordon equation with singular mass. II: hypoelliptic case, Complex Variables and Elliptic Equations, Vol: 67, Pages: 615-632, ISSN: 1747-6933

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.

Journal article

Ruzhansky M, Serikbaev D, Torebek BT, Tokmagambetov Net al., 2022, Direct and inverse problems for time-fractional pseudo-parabolic equations, Quaestiones Mathematicae, Vol: 45, Pages: 1071-1089, ISSN: 1607-3606

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.

Journal article

Avetisyan Z, Grigoryan M, Ruzhansky M, 2021, Approximations in L<sup>1</sup> with convergent Fourier series, Mathematische Zeitschrift, Vol: 299, Pages: 1907-1927, ISSN: 0025-5874

For a separable finite diffuse measure space M and an orthonormal basis { φn} of L2(M) consisting of bounded functions φn∈ L∞(M) , we find a measurable subset E⊂ M of arbitrarily small complement | M\ E| < ϵ, such that every measurable function f∈ L1(M) has an approximant g∈ L1(M) with 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 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.

Journal article

Cardona D, Delgado J, Ruzhansky M, 2021, L<sup>p</sup> -Bounds for Pseudo-differential Operators on Graded Lie Groups, Journal of Geometric Analysis, Vol: 31, Pages: 11603-11647, ISSN: 1050-6926

In this work we obtain sharp Lp-estimates for pseudo-differential operators on arbitrary graded Lie groups. The results are presented within the setting of the global symbolic calculus on graded Lie groups by using the Fourier analysis associated to every graded Lie group which extends the usual one due to Hörmander on Rn. The main result extends the classical Fefferman’s sharp theorem on the Lp-boundedness of pseudo-differential operators for Hörmander classes on Rn to general graded Lie groups, also adding the borderline ρ= δ case.

Journal article

Kumar V, Ruzhansky M, 2021, <i>L</i> <i>p</i>-<i>L</i> <i>q</i> Boundedness of <i>(k, a)</i>-Fourier Multipliers with Applications to Nonlinear Equations, International Mathematics Research Notices, ISSN: 1073-7928

<jats:title>Abstract</jats:title> <jats:p>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.</jats:p>

Journal article

Ruzhansky M, Torebek BT, 2021, Van der Corput lemmas for Mittag-Leffler functions. II. α–directions, Bulletin des Sciences Mathematiques, Vol: 171, ISSN: 0007-4497

The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory integrals appearing in the analysis of time-fractional partial differential equations. More specifically, we study integral of the form Iα,β(λ)=∫REα,β(iαλϕ(x))ψ(x)dx, for the range 0<α≤2,β>0. This extends the variety of estimates obtained in the first part, where integrals with functions Eα,β(iλϕ(x)) have been studied. Several generalisations of the van der Corput lemmas are proved. As an application of the above results, the generalised Riemann-Lebesgue lemma, the Cauchy problem for the time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

Journal article

Delgado J, Ruzhansky M, 2021, Schatten-von Neumann classes of integral operators, Journal des Mathematiques Pures et Appliquees, Vol: 154, Pages: 1-29, ISSN: 0021-7824

In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in Rn of finite measure, and conditions for operators on Rn given in terms of anharmonic oscillators. We also give examples in the settings of compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups.

Journal article

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