arXiv subjects
Publications and source records attributed to Michael Ruzhansky.
We introduce abstract net spaces on directed sets and prove their embedding and interpolation properties. Typical examples of interest are lattices of irreducible unitary representations of compact Lie groups and of class I representations with respect to a subgroup. As an application, we prove Hardy-Littlewood type inequalities and their converses on compact Lie groups and on compact homogeneous manifolds.
In this paper we study the well-posedness of weakly hyperbolic systems with time dependent coefficients. We assume that the eigenvalues are low regular, in the sense that they are Hölder with respect to $t$. In the past these kind of systems have been investigated by Yuzawa \cite{Yu:05} and Kajitani \cite{KY:06} by employing semigroup techniques (Tanabe-Sobolevski method). Here, under a certain uniform property of the eigenvalues, we improve the Gevrey well-posedness result of \cite{Yu:05} and we obtain well-posedness in spaces of ultradistributions as well. Our main idea is a reduction of the system to block Sylvester form and then the formulation of suitable energy estimates inspired by the treatment of scalar equations in \cite{GR:11}
In this paper we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. We give applications of the developed analysis to obtain a-priori estimates for solutions of operators that are elliptic within the constructed calculus.
This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper, where dispersive equations were treated. For operators $a(D_x)$ of order $m$ satisfying the dispersiveness condition $\nabla a(ξ)\neq0$ for $ξ\not=0$, the global smoothing estimate $$ \|\langle x\rangle^{-s}|D_x|^{(m-1)/2}e^{ita(D_x)} φ(x)\|_{L^2(\mathbb R_t\times\mathbb R^n_x)} \leq C\|φ\|_{L^2(\mathbb R^n_x)} \quad {\rm(}s>1/2{\rm)} $$ is well-known, while it is also known to fail for non-dispersive operators. For the case when the dispersiveness breaks, we suggest the estimate in the form $$ \|{\langle{x}\rangle^{-s}|\nabla a(D_x)|^{1/2} e^{it a(D_x)}φ(x)}\|_{L^2({\mathbb R_t\times\mathbb R^n_x})} \leq C\|φ\|_{L^2({\mathbb R^n_x})}\quad{\rm(}s>1/2{\rm)} $$ which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator $a(D_x)$. We show that this estimate and its variants do continue to hold for a variety of non-dispersive operators $a(D_x)$, where $\nabla a(ξ)$ may become zero on some set. Moreover, other types of such estimates, and the case of time-dependent equations are also discussed.
Let $G$ be a unimodular type I second countable locally compact group and $\hat G$ its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on $G\times\hat G$, and its relations to suitably defined Wigner transforms and Weyl systems. We also unveil its connections with crossed products $C^*$-algebras associated to certain $C^*$-dynamical systems, and apply it to the spectral analysis of covariant families of operators. Applications are given to nilpotent Lie groups, in which case we relate quantizations with operator-valued and scalar-valued symbols.
In this note we construct an integral boundary condition for the Kohn Laplacian in a given domain on the Heisenberg group extending to the setting of the Heisenberg group M. Kac's "principle of not feeling the boundary". This also amounts to finding the trace on smooth surfaces of the Newton potential associated to the Kohn Laplacian. We also obtain similar results for higher powers of the Kohn Laplacian.
In this paper we prove Lp multiplier theorems for invariant and non-invariant operators on compact Lie groups in the spirit of the well-known Hormander-Mikhlin theorem on Rn and its variants on tori Tn. We also give applications to a-priori estimates for non-hypoelliptic operators. Already in the case of tori we get an interesting refinement of the classical multiplier theorem.
Hadamard's global inverse theorem provides conditions for a function to be globally invertible on Rn. In this note we show that the conditions are robust enough for the conclusion to hold even if we relax the conditions by removing the assumption at a finite number of points. As a consequence, we get a global inverse function theorem for homogeneous functions.
This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior domains with compact boundary. Also, the known results on large data problems will be reviewed together with open problems.
In this note we present criteria on both symbols and integral kernels ensuring that the corresponding operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given. A notion of an invariant operator and its full symbol associated to an elliptic operator are introduced. Some applications to the study of $r$-nuclearity on $L^{p}$ spaces are also obtained.
In this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given.
In this paper we prove Nikolskii's inequality on general compact Lie groups and on compact homogeneous spaces with the constant interpreted in terms of the eigenvalue counting function of the Laplacian on the space, giving the best constant for certain indices, attained on the Dirichlet kernel. Consequently, we establish embedding theorems between Besov spaces on compact homogeneous spaces, as well as embeddings between Besov spaces and Wiener and Beurling spaces. We also analyse Triebel-Lizorkin spaces and $β$-versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces.
In this paper we investigate the well-posedness of the Cauchy problem for the wave equation for sums of squares of vector fields on compact Lie groups. We obtain the loss of regularity for solutions to the Cauchy problem in local Sobolev spaces depending on the order to which the Hörmander condition is satisfied, but no loss in globally defined spaces. We also establish Gevrey well-posedness for equations with irregular coefficients and/or multiple characteristics. As in the Sobolev spaces, if formulated in local coordinates, we observe well-posedness with the loss of local Gevrey order depending on the order to which the Hörmander condition is satisfied.
In this note we present a symbolic pseudo-differential calculus on the Heisenberg group. We particularise to this group our general construction [4,3,2] of pseudo-differential calculi on graded groups. The relation between the Weyl quantization and the representations of the Heisenberg group enables us to consider here scalar-valued symbols. We find that the conditions defining the symbol classes are similar but different to the ones in [1]. Applications are given to Schwartz hypoellipticity and to subelliptic estimates on the Heisenberg group.
In this paper we develop the functional calculus for elliptic operators on compact Lie groups without the assumption that the operator is a classical pseudo-differential operator. Consequently, we provide a symbolic descriptions of complex powers of such operators. As an application, we give a constructive symbolic proof of the Gårding inequality for operators in $(ρ,δ)$-classes in the setting of compact Lie groups.
This paper is a survey article of results and arguments from several of authors' papers, and it describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on ideas of comparison principle and canonical transforms. For operators $a(D_x)$ of order $m$ satisfying the dispersiveness condition $\nabla a(ξ)\neq0$, a range of smoothing estimates is established. Especially, time-global smoothing estimates for the operator $a(D_x)$ with lower order terms are the benefit of our new method. These estimates are known to fail for general non-dispersive operators. For the case when the dispersiveness breaks, we suggest a modification of the smoothing estimate. It is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator $a(D_x)$. Moreover, it does continue to hold for a variety of non-dispersive operators $a(D_x)$, where $\nabla a(ξ)$ may become zero on some set. It is interesting that this method allows us to carry out a global microlocal reduction of equations to the translation invariance property of the Lebesgue measure.
In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means to assume that the coefficients are less regular than Hölder. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a `very weak solution' adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifiers of coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or ultradistributional solutions under conditions when such solutions also exist. In concrete applications, the dependence on the regularising parameter can be traced explicitly.
We study the Lp-properties of positive Rockland operators and define Sobolev spaces on general graded groups. This generalises the case of sub-Laplacians on stratified groups studied by G. Folland in [3]. We show that the defined Sobolev spaces are actually independent of the choice of a positive Rockland operator. Furthermore, we show that they are interpolation spaces and establish duality and Sobolev embedding theorems in this context.