Homogeneous operators and homogeneous integral operators

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure - measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples - classes of homogeneous integral operators on various metric spaces, and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article - various specific cases - illustrate general statements and results given in the paper and at the same time are of interest in their own way.

Characterization of Generalized Young Measures Generated by $${\mathcal {A}}$$-free Measures

We give two characterizations, one for the class of generalized Young measures generated by $${{\,\mathrm{{\mathcal {A}}}\,}}$$ A -free measures and one for the class generated by $${\mathcal {B}}$$ B -gradient measures $${\mathcal {B}}u$$ B u . Here, $${{\,\mathrm{{\mathcal {A}}}\,}}$$ A and $${\mathcal {B}}$$ B are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The first characterization places the class of generalized $${\mathcal {A}}$$ A -free Young measures in duality with the class of $${{\,\mathrm{{\mathcal {A}}}\,}}$$ A -quasiconvex integrands by means of a well-known Hahn–Banach separation property. The second characterization establishes a similar statement for generalized $${\mathcal {B}}$$ B -gradient Young measures. Concerning applications, we discuss several examples that showcase the failure of $$\mathrm {L}^1$$ L 1 -compensated compactness when concentration of mass is allowed. These include the failure of $$\mathrm {L}^1$$ L 1 -estimates for elliptic systems and the lack of rigidity for a version of the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set $$\Omega $$ Ω , the inclusions $$\begin{aligned} \mathrm {L}^1(\Omega ) \cap \ker {\mathcal {A}}&\hookrightarrow {\mathcal {M}}(\Omega ) \cap \ker {{\,\mathrm{{\mathcal {A}}}\,}}\,,\\ \{{\mathcal {B}}u\in \mathrm {C}^\infty (\Omega )\}&\hookrightarrow \{{\mathcal {B}}u\in {\mathcal {M}}(\Omega )\} \end{aligned}$$ L 1 ( Ω ) ∩ ker A ↪ M ( Ω ) ∩ ker A , { B u ∈ C ∞ ( Ω ) } ↪ { B u ∈ M ( Ω ) } are dense with respect to the area-functional convergence of measures.

Functional calculus on non-homogeneous operators on nilpotent groups

We study the functional calculus associated with a hypoelliptic left-invariant differential operator $$\mathcal {L}$$ L on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator $$\mathcal {L}_0$$ L 0 on the ‘local’ contraction $$G_0$$ G 0 of G, as well as of the corresponding Rockland operator $$\mathcal {L}_\infty $$ L ∞ on the ‘global’ contraction $$G_\infty $$ G ∞ of G. We provide asymptotic estimates of the Riesz potentials associated with $$\mathcal {L}$$ L at 0 and at $$\infty $$ ∞ , as well as of the kernels associated with functions of $$\mathcal {L}$$ L satisfying Mihlin conditions of every order. We also prove some Mihlin–Hörmander multiplier theorems for $$\mathcal {L}$$ L which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the ‘Plancherel measure’ associated with $$\mathcal {L}$$ L from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.

Correlators in the Gaussian and chiral supereigenvalue models in the Neveu-Schwarz sector

We analyze the Gaussian and chiral supereigenvalue models in the Neveu-Schwarz sector. We show that their partition functions can be expressed as the infinite sums of the homogeneous operators acting on the elementary functions. In spite of the fact that the usual W-representations of these matrix models can not be provided here, we can still derive the compact expressions of the correlators in these two supereigenvalue models. Furthermore, the non-Gaussian (chiral) cases are also discussed.

Spaces generated by the cone of sublinear operators

This paper deals with a study on classes of non linear operators. Let $SL(X,Y)$ be the set of all sublinear operators between two Riesz spaces $X$ and $Y$. It is a convex cone of the space $H(X,Y)$ of all positively homogeneous operators. In this paper we study some spaces generated by this cone, therefore we study several properties, which are well known in the theory of Riesz spaces, like order continuity, order boundedness etc. Finally, we try to generalise the concept of adjoint operator. First, by using the analytic form of Hahn-Banach theorem, we adapt the notion of adjoint operator to the category of positively homogeneous operators. Then we apply it to the class of operators generated by the sublinear operators.