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
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189 results found

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

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

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

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

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Akylzhanov R, Majid S, Ruzhanskye M, 2018, Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups, COMMUNICATIONS IN MATHEMATICAL PHYSICS, Vol: 362, Pages: 761-799, ISSN: 0010-3616

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Esposito M, Ruzhansky M, 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.

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Munoz JC, Ruzhansky M, Tokmagambetov N, Wave propagation with irregular dissipation and applications to acoustic problems and shallow waters, Journal de Mathématiques Pures et Appliquées, 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, 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

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, 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 J, Ruzhansky M, 2018, Fourier multipliers, symbols, and nuclearity on compact manifolds, JOURNAL D ANALYSE MATHEMATIQUE, Vol: 135, Pages: 757-800, ISSN: 0021-7670

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

Karimov E, Mamchuev M, Ruzhansky M, Non-local initial problem for second order time-fractional and space-singular equation

In this work, we consider an initial problem for second order partialdifferential equations with Caputo fractional derivatives in the time-variableand Bessel operator in the space-variable. For non-local boundary conditions,we present a solution of this problem in an explicit form representing it bythe Fourier-Bessel series. The obtained solution is written in terms ofmultinomial Mittag-Leffler functions and first kind Bessel functions.

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

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, 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, 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

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

JOURNAL ARTICLE

Ruzhansky M, Tokmagambetov N, On nonlinear damped wave equations for positive operators. I. Discrete spectrum

In this paper we study a Cauchy problem for the nonlinear damped waveequations for a general positive operator with discrete spectrum. We derive theexponential 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

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, Suragan D, 2017, Geometric maximizers of Schatten norms of some convolution type integral operators, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, Vol: 456, Pages: 444-456, ISSN: 0022-247X

JOURNAL ARTICLE

Ruzhansky M, Suragan D, 2017, Anisotropic L-2-weighted Hardy and L-2-Caffarelli-Kohn-Nirenberg inequalities, COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, Vol: 19, ISSN: 0219-1997

JOURNAL ARTICLE

Ruzhansky M, Tokmagambetov N, 2017, Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed, ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, Vol: 226, Pages: 1161-1207, ISSN: 0003-9527

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

Ruzhansky M, Suragan D, Yessirkegenov N, 2017, Caffarelli-Kohn-Nirenberg and Sobolev type inequalities on stratified Lie groups, NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, Vol: 24, ISSN: 1021-9722

JOURNAL ARTICLE

Garetto C, Ruzhansky M, 2017, On C-infinity well-posedness of hyperbolic systems with multiplicities, ANNALI DI MATEMATICA PURA ED APPLICATA, Vol: 196, Pages: 1819-1834, ISSN: 0373-3114

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: 0001-8708

JOURNAL ARTICLE

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