Publication’s review for February 2019

1. Akylzhanov R., Liflyand E., Ruzhansky M., Re-expansions on compact Lie groups. arxiv

Abstract. In this paper, the re-expansion problems are considered on compact Lie groups. First,  weighted versions of classical re-expansion results are establised in the setting of multi-dimensional tori. A natural extension of the classical re-expansion problem to general compact Lie groups can be formulated as follows: given a function on the maximal torus of a compact Lie group, what conditions on its (toroidal) Fourier coefficients are sufficient in order to have that the group Fourier coefficients of its central extension are summable. We derive the necessary and sufficient conditions for the above property to hold in terms of the root system of the group. Consequently, we show how this problem leads to the re-expansions of even/odd functions on compact Lie groups, giving a necessary and sufficient condition in terms of the discrete Hilbert transform and the root system. In the model case of the group SU(2) a simple sufficient condition is given.

2. Ruzhansky M., Velasquez-Rodriguez J. P., Non-harmonic Gohberg’s lemma, Gershgorin theory and heat equation on manifolds with boundary. arxiv

Abstract. In this paper, following the works on non-harmonic analysis of boundary value problems by Tokmagambetov, Ruzhansky and Delgado, we use Operator Ideals Theory and Gershgorin Theory to obtain explicit information concerning the spectrum of pseudo-differential operators, on a smooth manifold Ω with boundary Ω, in the context of the non-harmonic analysis of boundary value problems, introduced by Tokmagambetov and Ruzhansky in terms of a model operator 𝔏. Under certain assumptions about the eigenfunctions of the model operator, for symbols in the Hörmander class S01,0(Ω⎯⎯⎯⎯⎯×), we provide a “non-harmonic version” of Gohberg’s Lemma, and a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is a compact operator in L2(Ω). Also, for pseudo-differential operators with symbols satisfying some integrability condition, one defines its associated matrix in terms of the biorthogonal system associated to 𝔏, and this matrix is used to give necessary and sufficient conditions for the L2(Ω)-boundedness, and to locate the spectrum of some operators. After that, we extend to the context of the non-harmonic analysis of boundary value problems the well known theorems about the exact domain of elliptic operators, and discuss some applications of the obtained results to evolution equations. Specifically we provide sufficient conditions to ensure the smoothness and stability of solutions to a generalised version of the heat equation.

3. Ruzhansky M., Sabitbek B., Suragan D., Geometric Hardy inequalities on starshaped sets. arxiv

Abstract. This paper presents  the geometric Hardy inequalities on the starshaped sets in the Carnot groups. Also, the geometric Hardy inequalities on half-spaces are obtained for general vector fields.

4. Cardona D., Ruzhansky M., Boundedness of pseudo-differential operators in subelliptic Sobolev and Besov spaces on compact Lie groups. arxiv

Abstract. This paper investigates the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the Hörmander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We link the description of subelliptic Sobolev spaces with the matrix-valued quantisation procedure of pseudo-differential operators in order to provide subelliptic Sobolev and Besov estimates for operators in the Hörmander classes. Interpolation properties between Besov spaces and Triebel-Lizorkin spaces are also investigated.

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