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Michael Ruzhansky

Publications and source records attributed to Michael Ruzhansky.

At least 235 records · Page 13Linked to original sources
arXiv2018

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III. Hilbert spaces and Universality

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.

math.FA↗
arXiv2018

Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups

In this paper, we present the geometric Hardy inequality for the sub-Laplacian in the half-spaces on the stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space on the Heisenberg group with a sharp constant \begin{equation*} \int_{\mathbb{H}^+} |\nabla_{H}u|^p dξ\geq \left(\frac{p-1}{p}\right)^p \int_{\mathbb{H}^+} \frac{\mathcal{W}(ξ)^p}{dist(ξ,\partial \mathbb{H}^+)^p} |u|^p dξ, \,\, p>1, \end{equation*} which solves the conjecture in the paper \cite{Larson}. Also, we obtain a version of the Hardy-Sobolev inequality in a half-space on the Heisenberg group \begin{equation*} \left(\int_{\mathbb{H}^+} |\nabla_{H} u|^p dξ- \left(\frac{p-1}{p}\right)^p \int_{\mathbb{H}^+} \frac{\mathcal{W}(ξ)^p}{dist(ξ,\partial \mathbb{H}^+)^p} |u|^p dξ\right)^{\frac{1}{p}} \geq C \left(\int_{\mathbb{H}^+} |u|^{p^*} dξ\right)^{\frac{1}{p^*}}, \end{equation*} where $dist(ξ,\partial \mathbb{H}^+)$ is the Euclidean distance to the boundary, $p^* := Qp/(Q-p)$, $2\leq p<Q$, and $$\mathcal{W}(ξ)=\left(\sum_{i=1}^{n}\langle X_i(ξ), ν\rangle^2+\langle Y_i(ξ), ν\rangle^2\right)^{\frac{1}{2}},$$ is the angle function. For $p=2$, this gives the Hardy-Sobolev-Maz'ya inequality on the Heisenberg group.

math.AP↗
arXiv2018

Very weak solutions to hypoelliptic wave equations

In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, Hölder, and distributional. For Hölder coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of "very weak solutions" to the Cauchy problem, that was already successfully used in similar contexts in [GR15] and [RT17b]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$-evolution equations for higher order operators on $\mathbb{R}^{n}$ or on groups, the results already being new in all these cases.

math.AP↗
arXiv2018

Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups

In this note we prove the Stein-Weiss inequality on general homogeneous Lie groups. The obtained results extend previously known inequalities. Special properties of homogeneous norms play a key role in our proofs. Also, we give a simple proof of the Hardy-Littlewood-Sobolev inequality on general homogeneous Lie groups.

math.AP↗
arXiv2018

Approximations in $L^1$ with convergent Fourier series

For a separable finite diffuse measure space $\mathcal{M}$ and an orthonormal basis $\{φ_n\}$ of $L^2(\mathcal{M})$ consisting of bounded functions $φ_n\in L^\infty(\mathcal{M})$, we find a measurable subset $E\subset\mathcal{M}$ of arbitrarily small complement $|\mathcal{M}\setminus E|<ε$, such that every measurable function $f\in L^1(\mathcal{M})$ has an approximant $g\in L^1(\mathcal{M})$ 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 $\mathcal{M}=G/H$ 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.

math.FA↗
arXiv2018

Hörmander condition for pseudo-multipliers associated to the harmonic oscillator

In this paper we prove Hörmander-Mihlin multiplier theorems for pseudo-multipliers associated to the harmonic oscillator (also called the Hermite operator). Our approach can be extended to also obtain the $L^p$-boundedness results for multilinear pseudo-multipliers. By using the Littlewood-Paley theorem associated to the harmonic oscillator we also give $L^p$-boundedness and $L^p$-compactness properties for multipliers. $(L^p,L^q)$-estimates for spectral pseudo-multipliers also are investigated.

math.FA↗
arXiv2018

Spectral identities and smoothing estimates for evolution operators

Smoothing (and decay) spacetime estimates are discussed for evolution groups of self-adjoint operators in an abstract setting. The basic assumption is the existence (and weak continuity) of the spectral density in a functional setting. Spectral identities for the time evolution of such operators are derived, enabling results concerning "best constants" for smoothing estimates. When combined with suitable "comparison principles" (analogous to those established in our previous work), they yield smoothing estimates for classes of functions of the operators . A important particular case is the derivation of global spacetime estimates for a perturbed operator $H+V$ on the basis of its comparison with the unperturbed operator $H.$ A number of applications are given, including smoothing estimates for fractional Laplacians, Stark Hamiltonians and Schrödinger operators with potentials.

math.SP↗
arXiv2018

Anisotropic fractional Gagliardo-Nirenberg, weighted Caffarelli-Kohn-Nirenberg and Lyapunov-type inequalities, and applications to Riesz potentials and $p$-sub-Laplacian systems

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 of $\mathbb{R}^{N}$. As an application, we give two-sided estimate for the first eigenvalue of the Riesz potential. Also, we obtain Lyapunov inequality for the system of the fractional $p$-sub-Laplacian equations and give an application to estimate its eigenvalues

math.AP↗
arXiv2018

A comparison principle for nonlinear heat Rockland operators on graded groups

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.

math.AP↗
arXiv2018

Euler semigroup, Hardy-Sobolev and Gagliardo-Nirenberg type inequalities on homogeneous groups

In this paper we describe the Euler semigroup $\{e^{-t\mathbb{E}^{*}\mathbb{E}}\}_{t>0}$ on homogeneous Lie groups, which allows us to obtain various types of the Hardy-Sobolev and Gagliardo-Nirenberg type inequalities for the Euler operator $\mathbb{E}$. Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and $|\cdot|$-radial weighted Hardy-Sobolev type inequality are established.

math.FA↗
arXiv2018

Hypoelliptic functional inequalities

In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Rellich, Hardy-Littllewood-Sobolev, Galiardo-Nirenberg, Caffarelli-Kohn-Nirenberg and Trudinger-Moser inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. Moreover, we obtain several versions of local and global weighted Trudinger-Moser inequalities with remainder terms, critical Hardy and weighted Gagliardo-Nirenberg inequalities, which appear to be new also in the case of the sub-Laplacian. Curiously, we also show the equivalence of many of these critical inequalities as well as asymptotic relations between their best constants. The approach developed in this paper relies on establishing integral versions of Hardy inequalities on homogeneous groups, for which we also find necessary and sufficient conditions for the weights for such inequalities to be true. Consequently, we link such integral Hardy inequalities to different hypoelliptic inequalities by using the Riesz and Bessel kernels associated to the described hypoelliptic operators.

math.FA↗
arXiv2018

Pseudo-differential operators with nonlinear quantizing functions

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

math.FA↗
arXiv2018

Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds with negative curvature

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.

math.FA↗
arXiv2018

The bounded approximation property of variable Lebesgue spaces and nuclearity

In this paper we prove the bounded approximation property for variable exponent Lebesgue spaces, study the concept of nuclearity on such spaces and apply it to trace formulae such as the Grothendieck-Lidskii formula. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb R^n$ in terms of global symbols.

math.FA↗
arXiv2018

Sobolev type inequalities, Euler-Hilbert-Sobolev and Sobolev-Lorentz-Zygmund spaces on homogeneous groups

We define Euler-Hilbert-Sobolev spaces and obtain embedding results on homogeneous groups using Euler operators, which are homogeneous differential operators of order zero. Sharp remainder terms of $L^{p}$ and weighted Sobolev type and Sobolev-Rellich inequalities on homogeneous groups are given. Most inequalities are obtained with best constants. As consequences, we obtain analogues of the generalised classical Sobolev type and Sobolev-Rellich inequalities. We also discuss applications of logarithmic Hardy inequalities to Sobolev-Lorentz-Zygmund spaces. The obtained results are new already in the anisotropic $\mathbb R^{n}$ as well as in the isotropic $\mathbb R^{n}$ due to the freedom in the choice of any homogeneous quasi-norm.

math.FA↗
arXiv2018

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I. Well-posedness

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

math.AP↗