Degenerate differential operators in weighted Hölder spaces

A differential operator ?, arising from the differential expression $$lv(t) \equiv ( - 1)^r v^{[n]} (t) + \sum\nolimits_{k = 0}^{n - 1} {p_k } (t)v^{[k]} (t) + Av(t),0 \leqslant t \leqslant 1,$$ , and system of boundary value conditions $$P_v [v] = \sum\nolimits_{k = 0}^{n_v } {\alpha _{vk} } r^{[k]} (1) = 0.v - 1, \ldots ,\mu ,0 \leqslant \mu< n$$ is considered in a Banach space E. Herev ^[k](t)=(a(t) d/dt) ^k v(t)a(t) being continuous fort?0, α(t) >0 for t > 0 and \(\int_0^1 {\frac{{dz}}{{a(z)}} = + \infty ;}\) the operator A is strongly positive in E. The estimates , are obtained for ?: n even, λ varying over a half plane.     

Hölder type inequalities in Lorentz spaces

We study optimal Hölder type inequalities for the Lorentz spaces L ^p,s (R, μ), in the range 1 < p < ∞, 1 ≤ s ≤ ∞, for both the maximal and the dual norms. These estimates also give sharp results for the corresponding associate norms.     

Pointwise inequalities of logarithmic type in Hardy-Hölder spaces

We prove some optimal logarithmic estimates in the Hardy space H ^∞(G) with Hölder regularity, where G is the open unit disk or an annular domain of ?. These estimates extend the results established by S.Chaabane and I.Feki in the Hardy-Sobolev space H ^k,∞ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem for the Laplace operator and the Robin inverse problem.     

Čebyšev's type inequalities for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given.     

Some trace inequalities of Čebyšev type for functions of operators in hilbert spaces

Some trace operator inequalities for synchronous functions that are related to the ?eby?ev inequality for sequences of real numbers are given.     

Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces

We show that the generator of any uniformly bounded set-valued Nemytskij composition operator acting between generalized Hölder function metric spaces, with nonempty, bounded, closed, and convex values, is an affine function.     

On Hardy-Littlewood-Pólya and Taikov type inequalities for multiple operators in Hilbert spaces

We present a unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Pólya and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemannian manifolds and derive the well-known Taikov and Hardy-Littlewood-Pólya inequalities for functions defined on the d-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of one class by elements of another class. In addition, we establish sharp Solyar type inequalities for unbounded closed operators with closed range.

     

Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces

Some Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications in relation with the celebrated Hölder–McCarthy’s inequality for positive operators and Ky Fan’s inequality for real numbers are given as well.     

Operator inequalities of Hölder type for unitarily invariant norms

The aim of this work is to present some Hölder-type inequalities for sums and products of operators related to unitarily invariant norms. These results generalize some known Hölder inequalities for operators.

     

Hölder metric subregularity for multifunctions in type Banach spaces

Using the Borwein–Preiss variational principle and in terms of the proximal coderivative, we provide a new type of sufficient conditions for the Hölder metric subregularity and Hölder error bounds in a class of smooth Banach spaces. As an application, new characterizations for the tilt stability of Hölder minimizers are established.     

Exceptional case of the linear conjugation problem in weighted Hölder classes

We study the linear conjugation problem for the case in which the coefficient of the problem may have finitely many zeros and/or pole singularities on the contour. All studies are carried out in weighted H¨older classes with complex weight. We obtain a closed-form expression for the solution and the solvability conditions.     

Hausdorff operators on p-adic linear spaces and their properties in Hardy, BMO, and Hölder spaces

Sufficient conditions for the boundedness of p-adic matrix operators in Hardy, Hölder and BMO spaces are obtained. These conditions are expressed in terms of the determinant of the matrix and its norm in a p-adic linear space.     

Hölder regularity for equations of prescribed anisotropic mean curvature type

In this note, we prove Hölder regularity for equations of prescribed anisotropic mean curvature type. As an application, we obtain the regularity of weak surfaces with prescribed anisotropic mean curvature.     

Regularity of solutions of boundary value problems for a second-order parabolic equation in weighted Hölder spaces

We consider the first boundary value problem and the oblique derivative problem for a linear second-order parabolic equation in noncylindrical not necessarily bounded domains with nonsmooth (with respect to t) and noncompact lateral boundary under the assumption that the right-hand side and the lower-order coefficients of the equation may have certain growth when approaching the parabolic boundary of the domain and all coefficients are locally Hölder with given characteristics of the Hölder property. We construct a smoothness scale of solutions of these boundary value problems in Hölder spaces of functions that admit growth of higher derivatives near the parabolic boundary of the domain.     

Some Poincaré type inequalities for quadratic matrix fields

We present some Poincaré type inequalities for quadratic matrix fields with applications e.g. in gradient plasticity or fluid dynamics. In particular, an application to the pseudostress-velocity formulation of the stationary Stokes problem is discussed. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equiconvergence of expansions in eigenfunctions of Sturm-Liouville operators with distributional potentials in Hölder spaces

We establish the equiconvergence of expansions of an arbitrary function in the class L ₂(0, π) in the Fourier series in sines and in the Fourier series in the eigenfunctions of the first boundary value problem for the one-dimensional Schrödinger operator with a nonclassical potential. The equiconvergence is studied in the norm of the Hölder space. The potential is the derivative of a function that belongs to a fractional-order Sobolev space.     

Criteria for the compactness and the Fredholm property of pseudodifferential operators in Hölder-Zygmund weighted spaces

We obtain necessary and sufficient conditions for the complete continuity (the Fredholm property) in Hölder-Zygmund spaces on ? ⁿ whose weight has a power-law behavior at infinity for pseudodifferential operators with symbols in the Hörmander class S _1,δ ^m , 0 ≤ δ < 1 (slowly varying symbols in the class S _1,0 ^m ). We show that such operators are compact operators or Fredholm operators in weighted Hölder-Zygmund spaces if and only if they are compact operators or Fredholm operators, respectively, in Sobolev spaces.