Functional Analysis and Its Applications最新文献

We suggest a general construction of continuous Banach bundles of holomorphic function algebras on subvarieties of the closed noncommutative ball. These algebras are of the form (mathcal{A}_d/overline{I_x}), where (mathcal{A}_d) is the noncommutative disc algebra defined by G. Popescu, and (overline{I_x}) is the closure in (mathcal{A}_d) of a graded ideal (I_x) in the algebra of noncommutative polynomials, depending continuously on a point (x) of a topological space (X). Moreover, we construct bundles of Fréchet algebras (mathcal{F}_d/overline{I_x}) of holomorphic functions on subvarieties of the open noncommutative ball. The algebra (mathcal{F}_d) of free holomorphic functions on the unit ball was also introduced by G. Popescu, and (overline{I_x}) stands for the closure in (mathcal{F}_d) of a graded ideal (I_x) in the algebra of noncommutative polynomials, depending continuously on a point (xin X).

Some identities that involve the elliptic version of the Cauchy matrices are presented and proved. They include the determinant formula, the formula for the inverse matrix, the matrix product identity and the factorization formula.

In this paper we study the everywhere Hölder continuity of the minima of the following class of vectorial integral funcionals:

with some general conditions on the density (G).

We make the following assumptions about the function (G). Let (Omega) be a bounded open subset of (mathbb{R}^{n}), with (ngeq 2), and let (G colon Omega timesmathbb{R}^{m}timesmathbb{R}_{0,+}^{m}to mathbb{R}) be a Carathéodory function, where (mathbb{R}_{0,+}=[0,+infty)) and (mathbb{R} _{0,+}^{m}=mathbb{R}_{0,+}times dots timesmathbb{R}_{0,+}) with (mgeq 1). We make the following growth conditions on (G): there exists a constant (L>1) such that

for (mathcal{L}^{n}) a.e. (xin Omega ), for every (s^{alpha}in mathbb{R}) and every (xi^{alpha}inmathbb{R}) with (alpha=1,dots,m), (mgeq 1) and with (a(x) in L^{sigma}(Omega)), (a(x)geq 0) for (mathcal{L}^{n}) a.e. (xin Omega), (sigma >{n}/{p}), (1leq q<{p^{2}}/{n}) and (1<p<n).

Assuming that the previous growth hypothesis holds, we prove the following regularity result. If (u,{in}, W^{1,p}(Omega,mathbb{R}^{m})) is a local minimizer of the previous functional, then (u^{alpha}in C_{mathrm{loc}}^{o,beta_{0}}(Omega) ) for every (alpha=1,dots,m), with (beta_{0}in (0,1) ). The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude Hölder continuity.

Quasiderivations of the universal enveloping algebra (Umathfrak{gl}_n) were first introduced by D. Gurevich, P. Pyatov, and P. Saponov in their study of reflection equation algebras; they are linear operators on (Umathfrak{gl}_n) that satisfy certain algebraic relations, which generalise the usual Leibniz rule. In this note, we show that the iterated action of the operator equal to a linear combination of the quasiderivations on a certain set of generators of the center of (Umathfrak{gl}_n) (namely on the symmetrised coefficients of the characteristic polynomial) produces commuting elements. The resulting algebra coincides with the quantum Mischenko–Fomenko algebra in (Umathfrak{gl}_n), introduced earlier by Tarasov, Rybnikov, Molev, and others.

We consider a third-order non-self-adjoint operator which is an (L)-operator in the Lax pair for the Boussinesq equation on the circle. We construct a mapping from the set of operator coefficients to the set of spectral data, similar to the corresponding mapping for the Hill operator constructed by E. Korotyaev. We prove that, in a neighborhood of zero, our mapping is analytic and one-to-one.

It is shown how certain recent results in the theory of determinants and traces can be applied to obtain new theorems on the distribution of eigenvalues of nuclear operators on Banach spaces and to prove the equality of the spectral and nuclear traces of such operators. As an example, we consider a new class of operators: the class of generalized Lapresté nuclear operators.

We consider the problem of the motion of a dynamically symmetric heavy rigid body about a fixed point and give a detailed proof that the problem has no additional real-analytic first integral in all but the well-known classical cases.

Let (mathbb H) be a quaternion algebra generated by (I,J) and (K). We say that a hypercomplex nilpotent Lie algebra (mathfrak g) is (mathbb H)-solvable if there exists a sequence of (mathbb H)-invariant subalgebras containing (mathfrak g_{i+1}=[mathfrak g_i,mathfrak g_i]),

such that ([mathfrak g_i^{mathbb H},mathfrak g_i^{mathbb H}]subsetmathfrak g^{mathbb H}_{i+1}) and (mathfrak g_{i+1}^{mathbb H}=mathbb H[mathfrak g_i^{mathbb H},mathfrak g_i^{mathbb H}] ). Let (N=Gammasetminus G) be a hypercomplex nilmanifold with the flat Obata connection and (mathfrak g=operatorname{Lie}(G)). We prove that the Lie algebra (mathfrak g) is (mathbb H)-solvable.

Wang and Ma introduced the notion of asymmetric (L_p)-difference bodies. They further gave the extrema of volumes for the asymmetric (L_p)-difference body and its polar. Thereafter, Shi and Wang obtained their versions of quermassintegrals and dual quermassintegrals. In this paper, we determine the extrema of the (q)-quermassintegrals and dual (q)-quermassintegrals for the asymmetric (L_p)-difference bodies.