arXiv subjects

Michael Ruzhansky

Publications and source records attributed to Michael Ruzhansky.

At least 289 records · Page 16Linked to original sources

arXiv2016

Nonharmonic analysis of boundary value problems without WZ condition

In this work we continue our research on nonharmonic analysis of boundary value problems as initiated in our recent paper (IMRN 2016). There, we assumed that the eigenfunctions of the model operator on which the construction is based do not have zeros. In this paper we have weakened this condition extending the applicability of the developed pseudo-differential analysis. Also, we do not assume that the underlying set is bounded.

math.FA↗

arXiv2016

Fourier multipliers and group von Neumann algebras

In this paper we establish the $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices $1<p\leq 2 \leq q<\infty$. Our approach is based on the operator algebras techniques. The result depends on a version of the Hausdorff-Young-Paley inequality that we establish on general locally compact separable unimodular groups. In particular, the obtained result implies the corresponding Hörmander's Fourier multiplier theorem on $\mathbb{R}^{n}$ and the corresponding known results for Fourier multipliers on compact Lie groups.

math.OA↗

arXiv2016

On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and $p$-sub-Laplacian inequalities on stratified groups

In this paper, we present a version of horizontal weighted Hardy-Rellich type and Caffarelli-Kohn-Nirenberg type inequalities on stratified groups and study some of their consequences. Our results reflect on many results previously known in special cases. Moreover, a new simple proof of the Badiale-Tarantello conjecture [2] on the best constant of a Hardy type inequality is provided. We also show a family of Poincaré inequalities as well as inequalities involving the weighted and unweighted $p$-sub-Laplacians.

math.FA↗

arXiv2016

Hardy-Littlewood inequalities and Fourier multipliers on SU(2)

In this paper we prove a noncommutative version of Hardy-Littlewood inequalities relating a function and its Fourier coefficients on the group $SU(2)$. As a consequence, we use it to obtain lower bounds for the $L^p-L^q$ norms of Fourier multipliers on the group $SU(2)$, for $1 < p \leq 2 \leq q < 1$. In addition, we give upper bounds of a similar form, analogous to the known results on the torus, but now in the noncommutative setting of $SU(2)$.

math.FA↗

arXiv2016

Grothendieck-Lidskii trace formula for mixed-norm and variable Lebesgue spaces

In this note we present the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ and variable exponent Lebesgue type spaces. As a consequence, this also implies the same property for modulation and Wiener-Amalgam spaces. We then characterise nuclear operators on such spaces and state the corresponding Grothendieck-Lidskii trace formulae. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb R^n$ and functions of the harmonic oscillator in terms of the global symbol.

math.FA↗

arXiv2016

Local Hardy and Rellich inequalities for sums of squares of vector fields

We prove local refined versions of Hardy's and Rellich's inequalities as well as of uncertainty principles for sums of squares of vector fields on bounded sets of smooth manifolds under certain assumptions on the vector fields. We also give some explicit examples, in particular, for sums of squares of vector fields on Euclidean spaces and for sub-Laplacians on stratified Lie groups.

math.AP↗

arXiv2016

On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

In this note we prove an analogue of the Rayleigh-Faber-Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere $\mathbb{S}^{n}$ and on the real hyperbolic space $\mathbb{H}^{n}$. It completes the study of such question for complete, connected, simply connected Riemannian manifolds of constant sectional curvature. We also discuss an extremum problem for the second eigenvalue on $\mathbb{H}^{n}$ and prove the Hong-Krahn-Szegö type inequality. The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space.

math.SP↗

arXiv2016

Approximation property and nuclearity on mixed-norm $L^p$, modulation and Wiener amalgam spaces

In this paper we first prove the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ spaces. Using Gabor frame representation this implies that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck's theory becomes applicable, and we give criteria for nuclearity and $r$-nuclearity for operators acting on these space as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces.

math.FA↗

arXiv2016

Schatten classes and traces on compact groups

In this paper we present symbolic criteria for invariant operators on compact topological groups $G$ characterising the Schatten-von Neumann classes $S_{r}(L^{2}(G))$ for all $0<r\leq\infty$. Since it is known that for pseudo-differential operators criteria in terms of kernels may be less effective (Carleman's example), our criteria are given in terms of the operators' symbols defined on the noncommutative analogue of the phase space $G\times\hat{G}$, where $G$ is a compact topological (or Lie) group and $\hat{G}$ is its unitary dual. We also show results concerning general non-invariant operators as well as Schatten properties on Sobolev spaces. A trace formula is derived for operators in the Schatten class $S_{1}(L^{2}(G))$. Examples are given for Bessel potentials associated to sub-Laplacians (sums of squares) on compact Lie groups, as well as for powers of the sub-Laplacian and for other non-elliptic operators on SU(2)$\simeq\mathbb S^3$ and on SO(3).

math.FA↗

arXiv2016

On $C^\infty$ well-posedness of hyperbolic systems with multiplicities

In this paper we study first order hyperbolic systems with multiple characteristics (weakly hyperbolic) and time-dependent analytic coefficients. The main question is when the Cauchy problem for such systems is well-posed in $C^{\infty}$ and in $\mathcal{D}'$. We prove that the analyticity of the coefficients combined with suitable hypotheses on the eigenvalues guarantee the $C^\infty$ well-posedness of the corresponding Cauchy problem. This result is an extension to systems of the analogous results for scalar equations recently obtained by Jannelli and Taglialatela in \cite{JT} and by the authors in \cite{GR:13}.

math.AP↗

arXiv2016

Eigenfunction expansions of ultradifferentiable functions and ultradistributions

In this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold $X$. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on $X$. This extends the result for analytic functions on compact manifold by Seeley, and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces by the authors.

math.FA↗

arXiv2015

Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary

Given a compact manifold $M$ with boundary $\partial M$, in this paper we introduce a global symbolic calculus of pseudo-differential operators associated to $(M,\partial M)$. The symbols of operators with boundary conditions on $\partial M$ are defined in terms of the biorthogonal expansions in eigenfunctions of a fixed operator $L$ with the same boundary conditions on $\partial M$. The boundary $\partial M$ is allowed to have (arbitrary) singularities. As an application, several criteria for the membership in Schatten classes on $L^2(M)$ and $r$-nuclearity on $L^p(M)$ are obtained. We also describe a new addition to the Grothendieck-Lidskii formula in this setting. Examples and applications are given to operators on $M=[0,1]^n$ with non-periodic boundary conditions, and of operators with non-local boundary conditions.

math.AP↗

arXiv2015

Fourier multipliers, symbols and nuclearity on compact manifolds

The notion of invariant operators, or Fourier multipliers, is discussed for 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 manifold endowed with a positive measure, we introduce a notion of the operator's full symbol adapted to the Fourier analysis relative to a fixed elliptic operator. We give a description of Fourier multipliers, or of operators invariant relative to the elliptic operator. We apply these concepts to study Schatten classes of operators 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.

math.FA↗

arXiv2015

Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and Lp-Lq Fourier multipliers on compact homogeneous manifolds

In this paper we prove new inequalities describing the relationship between the "size" of a function on a compact homogeneous manifold and the "size" of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in $L^p$ on SU(2) in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on general compact homogeneous manifolds. The latter is applied to obtain conditions for the $L^p$-$L^q$ boundedness of Fourier multipliers for $1<p\leq 2\leq q<\infty$ on compact homogeneous manifolds as well as the $L^p$-$L^q$ boundedness of general (non-invariant) operators on compact Lie groups. We also record an abstract version of the Marcinkiewicz interpolation theorem on totally ordered discrete sets, to be used in the proofs with different Plancherel measures on the unitary duals.

math.FA↗

arXiv2015

Isoperimetric inequalities for Schatten norms of Riesz potentials

In this note we prove that the ball is a maximiser of some Schatten $p$-norms of the Riesz potential operators among all domains of a given measure in $\mathbb R^{d}$. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain and isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szegö inequalities.

math.SP↗

arXiv2015

On the Gevrey well-posedness of the Kirchhoff equation

This paper is devoted to proving the almost global solvability of the Cauchy problem for the Kirchhoff equation in the Gevrey space $γ^s_{η,L^2}$. Furthermore, similar results are obtained for the initial-boundary value problems in bounded domains and in exterior domains with compact boundary.

math.AP↗

arXiv2015

Global regularity properties for a class of Fourier integral operators

While the local $L^p$-boundedness of nondegeneral Fourier integral operators is known from the work of Seeger, Sogge and Stein, not so many results are available for the global boundedness on $L^p(\mathbb R^n)$. In this paper we give a sufficient condition for the global $L^p$-boundedness for a class of Fourier integral operators which includes many natural examples. We also describe a construction that can be used to deduce global results from the local ones. An application is given to obtain global $L^p$-estimates for solutions to Cauchy problems for hyperbolic partial differential equations.

math.AP↗

arXiv2015

Isoperimetric inequalities for the logarithmic potential operator

In this paper we prove that the disc is a maximiser of the Schatten $p$-norm of the logarithmic potential operator among all domains of a given measure in $\mathbb R^{2}$, for all even integers $2\leq p<\infty$. We also show that the equilateral triangle has the largest Schatten $p$-norm among all triangles of a given area. For the logarithmic potential operator on bounded open or triangular domains, we also obtain analogies of the Rayleigh-Faber-Krahn or P{ó}lya inequalities, respectively. The logarithmic potential operator can be related to a nonlocal boundary value problem for the Laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well.

math.FA↗