# ProfessorMichaelRuzhansky

Faculty of Natural SciencesDepartment of Mathematics

Visiting Professor

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### Contact

+44 (0)20 7594 8500m.ruzhansky CV

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### Location

615Huxley BuildingSouth Kensington Campus

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## Publications

Publication Type
Year
to

192 results found

Ruzhansky M, Sugimoto M, 2019, A local-to-global boundedness argument and Fourier integral operators, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, Vol: 473, Pages: 892-904, ISSN: 0022-247X

Journal article

Daher R, Delgado J, Ruzhansky M, 2019, Titchmarsh theorems for Fourier transforms of Holder-Lipschitz functions on compact homogeneous manifolds, MONATSHEFTE FUR MATHEMATIK, Vol: 189, Pages: 23-49, ISSN: 0026-9255

Journal article

Munoz JC, Ruzhansky M, Tokmagambetov N, 2019, Wave propagation with irregular dissipation and applications to acoustic problems and shallow waters, Journal de Mathématiques Pures et Appliquées, Vol: 123, Pages: 127-147, ISSN: 0021-7824

In this paper we consider an acoustic problem of wave propagation through adiscontinuous medium. The problem is reduced to the dissipative wave equationwith distributional dissipation. We show that this problem has a so-called veryweak solution, we analyse its properties and illustrate the theoretical resultsthrough some numerical simulations by approximating the solutions to the fulldissipative model for a particular synthetic piecewise continuous medium. Inparticular, we discover numerically a very interesting phenomenon of theappearance of a new wave at the singular point. For the acoustic problem thiscan be interpreted as an echo effect at the discontinuity interface of themedium.

Journal article

Ruzhansky M, Yessirkegenov N, 2019, RELLICH INEQUALITIES FOR SUB-LAPLACIANS WITH DRIFT, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 147, Pages: 1335-1349, ISSN: 0002-9939

Journal article

Ruzhansky M, Verma D, Hardy inequalities on metric measure spaces, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, ISSN: 1364-5021

In this note we give several characterisations of weights for two-weightHardy 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'soriginal inequality. We give examples obtaining new weighted Hardy inequalities on $\mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds. We note that doubling conditions are not required for our analysis.

Journal article

Esposito M, Ruzhansky M, 2019, Pseudo-differential operators with nonlinear quantizing functions, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, ISSN: 0308-2105

In this paper we develop the calculus of pseudo-differential operatorscorresponding to the quantizations of the form $$Au(x)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{i(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)dyd\xi,$$ where $\tau:\mathbb{R}^n\to\mathbb{R}^n$ is a general function. Inparticular, for the linear choices $\tau(x)=0$, $\tau(x)=x$, and$\tau(x)=\frac{x}{2}$ this covers the well-known Kohn-Nirenberg,anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations ofsuch type appear naturally in the analysis on nilpotent Lie groups forpolynomial functions $\tau$ and here we investigate the corresponding calculusin the model case of $\mathbb{R}^n$. We also give examples of nonlinear $\tau$appearing on the polarised and non-polarised Heisenberg groups, inspired by therecent joint work with Marius Mantoiu.

Journal article

Ruzhansky M, Yessirkegenov N, 2019, New progress on weighted Trudinger–Moser and Gagliardo–Nirenberg, and critical hardy inequalities on stratified groups, Applied and Numerical Harmonic Analysis, Pages: 277-289

© Springer Nature Switzerland AG 2019. In this paper, we present a summary of our recent research on local and global weighted (singular) Trudinger–Moser inequalities with remainder terms, critical Hardy-type and weighted Gagliardo–Nirenberg inequalities on general stratified groups. These include the cases of ℝ n and Heisenberg groups. Moreover, the described critical Hardy-type inequalities give the critical case of the Hardy-type inequalities from [4].

Book chapter

Ruzhansky M, Suragan D, 2018, A NOTE ON STABILITY OF HARDY INEQUALITIES, ANNALS OF FUNCTIONAL ANALYSIS, Vol: 9, Pages: 451-462, ISSN: 2008-8752

Journal article

Ozawa T, Ruzhansky M, Suragan D, Lp-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups, Quarterly Journal of Mathematics, ISSN: 0033-5606

We prove $L^{p}$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneousgroups, which is one of most general subclasses of nilpotent Lie groups, allwith sharp constants. We also discuss some of their consequences. Already inthe abelian case of $\mathbb{R}^{n}$ our results provide new insights in viewof the arbitrariness of the choice of the not necessarily Euclidean quasi-norm.

Journal article

Ruzhansky M, Suragan D, 2018, A comparison principle for nonlinear heat Rockland operators on graded groups, Bulletin of the London Mathematical Society, Vol: 50, Pages: 753-758, ISSN: 0024-6093

In this note we show a comparison principle for nonlinear heat Rockland operators on graded groups. We give a simple proof for it using purely algebraic relations. As an application of the established comparison principle we prove the global in t‐boundedness of solutions for a class of nonlinear equations for the heat p‐sub‐Laplacian on stratified groups.

Journal article

Ruzhansky M, Tokmagambetov N, 2018, Convolution, Fourier analysis, and distributions generated by Riesz bases, Monatshefte für Mathematik, Vol: 187, Pages: 147-170, ISSN: 0026-9255

In this note we discuss notions of convolutions generated by biorthogonal systems of elements of a Hilbert space. We develop the associated biorthogonal Fourier analysis and the theory of distributions, discuss properties of convolutions and give a number of examples.

Journal article

Akylzhanov R, Majid S, Ruzhansky M, 2018, Smooth dense subalgebras and Fourier multipliers on compact quantumgroups, Communications in Mathematical Physics, Vol: 362, Pages: 761-799, ISSN: 0010-3616

We define and study dense Frechet subalgebras of compact quantum groups realised as smooth domains associated with a Dirac type operator with compact resolvent. Further, we construct spectral triples on compact matrix quantum groups in terms of Clebsch–Gordon coefficients and the eigenvalues of the Dirac operator D. Grotendieck’s theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on them. It is also shown that regular pseudo-differential operators are closed under compositions. As a by-product, we develop elements of the distribution theory and corresponding Fourier analysis. We give applications of our construction to obtain sufficient conditions for Lp − Lq boundedness of coinvariant linear operators. We provide necessary and sufficient conditions for algebraic differential calculi on Hopf subalgebras of compact quantum groups to extend to our proposed smooth subalgebra C∞D. We check explicitly that these conditions hold true on the quantum SU2q for both its 3-dimensional and 4-dimensional calculi.

Journal article

Ruzhansky MV, Suragan D, 2018, Elements of Potential Theory on Carnot Groups, FUNCTIONAL ANALYSIS AND ITS APPLICATIONS, Vol: 52, Pages: 158-161, ISSN: 0016-2663

We propose and study elements of potential theory for the sub-Laplacian on homogeneous Carnot groups. In particular, we show the continuity of the single-layer potential and establish Plemelj-type jump relations for the double-layer potential. As a consequence, we derive a formula for the trace on smooth surfaces of the Newton potential for the sub-Laplacian. Using this, we construct a sub-Laplacian version of Kac’s boundary value problem.

Journal article

Ruzhansky M, Tokmagambetov N, 2018, Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups, Journal of Differential Equations, ISSN: 0022-0396

In this paper we study the Cauchy problem for the semilinear damped wave equation for the sub-Laplacian on the Heisenberg group. In the case of the positive mass, we show the global in time well-posedness for small data for power like nonlinearities. We also obtain similar well-posedness results for the wave equations for Rockland operators on general graded Lie groups. In particular, this includes higher order operators on and on the Heisenberg group, such as powers of the Laplacian or the sub-Laplacian. In addition, we establish a new family of Gagliardo-Nirenberg inequalities on graded Lie groups that play a crucial role in the proof but which are also of interest on their own: if $G$ is a graded Lie group of homogeneous dimension $Q$ and $a>0$, $1<r<\frac{Q}{a},$ and $1\leq p\leq q\leq \frac{rQ}{Q-ar},$ then we have the following Gagliardo-Nirenberg type inequality $$\|u\|_{L^{q}(G)}\lesssim \|u\|_{\dot{L}_{a}^{r}(G)}^{s} \|u\|_{L^{p}(G)}^{1-s}$$ for $s=\left(\frac1p-\frac1q\right)\left(\frac{a}Q+\frac1p-\frac1r\right)^{-1}\in [0,1]$ provided that$\frac{a}Q+\frac1p-\frac1r\not=0$, where $\dot{L}_{a}^{r}$ is the homogeneous Sobolev space of order $a$ over $L^r$. If $\frac{a}Q+\frac1p-\frac1r=0$, we have $p=q=\frac{rQ}{Q-ar}$, and then the above inequality holds for any $0\leqs\leq 1$.

Journal article

Ruzhansky M, Sabitbek B, Suragan D, Weighted $L^p$-Hardy and $L^p$-Rellich inequalities with boundary terms on stratified Lie groups, Revista Matemática Complutense, ISSN: 1139-1138

In this paper, generalised weighted $L^p$-Hardy,$L^p$-Caffarelli-Kohn-Nirenberg, and $L^p$-Rellich inequalities with boundaryterms are obtained on stratified Lie groups. As consequences, most of the Hardytype inequalities and Heisenberg- Pauli-Weyl type uncertainty principles onstratified groups are recovered. Moreover, a weighted $L^2$-Rellich typeinequality with the boundary term is obtained.

Journal article

Delgado Valencia JC, Ruzhansky M, 2018, Fourier multipliers, symbols and nuclearity on compact manifolds, Journal d'Analyse Mathematique, Vol: 135, Pages: 757-800, ISSN: 0021-7670

The notion of invariant operators, or Fourier multipliers, is discussedfor densely defined operators on Hilbert spaces, with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. As a consequence, given a compact manifoldM endowed with a positive measure, we introduce a notion ofthe operator's full symbol adapted to the Fourier analysis relative to a fixed elliptic operator E. We give a description of Fourier multipliers, or of operators invariant relative to E. We apply these concepts to study Schatten classes of operators on L2 (M) and to obtain a formula for the trace of trace class operators. We also apply it to provide conditions for operators between Lp-spaces to be r-nuclear in the sense of Grothendieck.

Journal article

Dasgupta A, Ruzhansky M, 2018, Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations, Transactions of the American Mathematical Society, Series B, Vol: 5, Pages: 81-101, ISSN: 0002-9947

In this paper we analyse the structure of the spaces of coefficients ofeigenfunction expansions of functions in Komatsu classes on compact manifolds,continuing the research in our previous paper. We prove that such spaces ofFourier coefficients are perfect sequence spaces. As a consequence we describethe tensor structure of sequential mappings on spaces of Fourier coefficientsand characterise their adjoint mappings. In particular, the considered classesinclude spaces of analytic and Gevrey functions, as well as spaces ofultradistributions, yielding tensor representations for linear mappings betweenthese spaces on compact manifolds.

Journal article

Ruzhansky MV, Tokmagambetov NE, 2018, On a Very Weak Solution of the Wave Equation for a Hamiltonian in a Singular Electromagnetic Field, MATHEMATICAL NOTES, Vol: 103, Pages: 856-858, ISSN: 0001-4346

Journal article

Ruzhansky M, Suragan D, Yessirkegenov N, Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces, Fractional Calculus and Applied Analysis, ISSN: 1311-0454

We analyse Morrey spaces, generalised Morrey spaces and Campanato spaces on homogeneous groups. The boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalised Bessel-Riesz operators and generalised fractional integral operators in generalised Morrey spaces on homogeneous groups is shown. Moreover, we prove the boundedness of the modified version of the generalised fractional integral operator and Olsen type inequalities in Campanato spaces and generalised Morrey spaces on homogeneous groups, respectively. Our results extend results known in the isotropic Euclidean settings, however, some of them are new already in the standard Euclideancases.

Journal article

Ruzhansky M, Karimov E, Mamchuev M, Non-local initial problem for second order time-fractional and space-singular equation, Hokkaido Mathematical Journal, ISSN: 0018-3482

In this work, we consider an initial problem for second order partialdifferential equations with Caputo fractional derivatives in the time-variable andBessel operator in the space-variable. For non-local boundary conditions, we presenta solution of this problem in an explicit form representing it by the Fourier-Besselseries. The obtained solution is written in terms of multinomial Mittag-Lefflerfunctions and first kind Bessel functions.

Journal article

Delgado J, Ruzhansky M, 2018, The bounded approximation property of variable lebesgue spaces and nuclearity, Mathematica Scandinavica, Vol: 122, Pages: 299-319, ISSN: 0025-5521

In this paper we prove the bounded approximation property for vari-able exponent Lebesgue spaces, study the concept of nuclearity on such spaces andapply it to trace formulae such as the Grothendieck-Lidskii formula. We apply theobtained results to derive criteria for nuclearity and trace formulae for periodicoperators onRnin terms of global symbols.

Journal article

Garetto C, Jäh C, Ruzhansky M, 2018, Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I. Well-posedness, Mathematische Annalen, ISSN: 0025-5831

In this paper we analyse the well-posedness of the Cauchy problem for arather general class of hyperbolic systems with space-time dependentcoefficients and with multiple characteristics of variable multiplicity. First,we establish a well-posedness result in anisotropic Sobolev spaces for systemswith upper triangular principal part under interesting natural conditions onthe orders of lower order terms below the diagonal. Namely, the terms below thediagonal at a distance $k$ to it must be of order $-k$. This setting alsoallows for the Jordan block structure in the system. Second, we give conditionsfor the Schur type triangularisation of general systems with variablecoefficients for reducing them to the form with an upper triangular principalpart for which the first result can be applied. We give explicit details forthe appearing conditions and constructions for $2\times 2$ and $3\times 3$systems, complemented by several examples.

Journal article

Ruzhansky M, Suragan D, Yessirkegenov N, 2018, Sobolev type inequalities, Euler-Hilbert-Sobolev and Sobolev-Lorentz-Zygmund spaces on homogeneous groups, Integral Equations and Operator Theory, Vol: 90, ISSN: 0378-620X

We define Euler–Hilbert–Sobolev spaces and obtain embed-ding results on homogeneous groups using Euler operators, which arehomogeneous differential operators of order zero. Sharp remainder termsofLpand weighted Sobolev type and Sobolev–Rellich inequalities onhomogeneous groups are given. Most inequalities are obtained withbest constants. As consequences, we obtain analogues of the generalisedclassical Sobolev type and Sobolev–Rellich inequalities. We also dis-cuss applications of logarithmic Hardy inequalities to Sobolev–Lorentz–Zygmund spaces. The obtained results are new already in the anisotropicRnas well as in the isotropicRndue to the freedom in the choice of anyhomogeneous quasi-norm.

Journal article

Ruzhansky M, Suragan D, Yessirkegenov N, 2018, Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for Lp-weighted Hardy inequalities, Transactions of the American Mathematical Society, Vol: 5, Pages: 32-62, ISSN: 0002-9947

In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for 1<p,q<∞,0<r<∞with p+q≥r, δ∈[0,1]∩[r−qr, pr] with δrp+(1−δ) rq=1and a, b, c∈R with c=δ(a−1) +b(1−δ), and for all functions f∈C∞0(Rn\{0})we have‖|x|cf‖Lr(Rn)≤∣∣∣∣pn−p(1−a)∣∣∣∣δ‖|x|a∇f‖δLp(Rn)∥∥∥|x|bf∥∥∥1−δLq(Rn) forn=p(1−a), where the constant ∣∣∣pn−p(1−a I∣∣δ is sharp for p=q with a−b=1or p=q with p(1−a)+bq= 0. In the critical case n=p(1−a) we have‖|x|f‖Lr(Rn)≤pδ‖|x| a log |x|∇f‖δLp(Rn)∥∥∥|x|bf∥∥∥1−δLq(Rn). Moreover, we also obtain anisotropic versions of these inequalities which canbe conveniently formulated in the language of Folland and Stein’s homogeneous groups. Consequently, we obtain remainder estimates forLp-weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of Rn. The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version ofLp-weighted Hardy inequalities involving a distance and stability estimate. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is also investigated. We also establish sharpHardy type inequalities in Lp,1<p<∞, with superweights, i.e., with the weights of the form (a+b)|x|α)βp|x|m allowing for different choices ofα and β. There are two reasons why we call the appearingweights the superweights: the arbitrariness of the choice of any homogeneous quasi-norm and a wide range of parameters.

Journal article

Ruzhansky M, 2018, OPTIMIZATION OF CONSTANTS FOR SUBELLIPTIC EMBEDDINGS, 6th International Conference on Control and Optimization with Industrial Applications (COIA), Publisher: BAKU STATE UNIV, INST APPLIED MATHEMATICS, Pages: 25-27

Conference paper

Mantoiu M, Ruzhansky M, 2017, Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups, Documenta Mathematica, Vol: 22, Pages: 1539-1592, ISSN: 1431-0635

Let G be a unimodular type I second countable locally compactgroup and let Gb be its unitary dual. We introduce and study a globalpseudo-differential calculus for operator-valued symbols defined on G × Gb ,and its relations to suitably defined Wigner transforms and Weyl systems.We also unveil its connections with crossed products C∗-algebras associatedto certain C∗-dynamical systems, and apply it to the spectral analysisof covariant families of operators. Applications are given to nilpotentLie groups, in which case we relate quantizations with operator-valued andscalar-valued symbols.

Journal article

Ruzhansky M, Tokmagambetov N, On nonlinear damped wave equations for positive operators. I. Discrete spectrum, Differential and Integral Equations, ISSN: 0893-4983

In this paper we study a Cauchy problem for the nonlinear dampedwave equations for a general positive operator with discrete spectrum. We derivethe exponential in time decay of solutions to the linear problem with decay ratedepending on the interplay between the bottom of the operator’s spectrum andthe mass term. Consequently, we prove global in time well-posedness results forsemilinear and for more general nonlinear equations with small data. Examplesare given for nonlinear damped wave equations for the harmonic oscillator, forthe twisted Laplacian (Landau Hamiltonian), and for the Laplacians on compactmanifolds.

Journal article

Ruzhansky M, Suragan D, A note on stability of Hardy inequalities, Annals of Functional Analysis, ISSN: 2008-8752

In this note we formulate recent stability results for Hardy inequalitiesin the language of Folland and Stein’s homogeneous groups. Consequently, weobtain remainder estimates for Rellich type inequalities on homogeneous groups.Main differences from the Eucledian results are that the obtained stability estimateshold for any homogeneous quasi-norm.

Journal article

Kanguzhin B, Ruzhansky M, Tokmagambetov N, 2017, On convolutions in Hilbert spaces, Functional Analysis and Its Applications, Vol: 51, Pages: 221-224, ISSN: 0016-2663

A convolution in a Hilbert space is defined, and its basic properties are studied.

Journal article

Ruzhansky M, Suragan D, 2017, Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups, Advances in Mathematics, Vol: 317, Pages: 799-822, ISSN: 1090-2082

We give sharp remainder terms of Lp and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy–Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.

Journal article

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