Analog of the Riesz theorem and the basis property in L p of a system of root functions of a differential operator: I

An ordinary differential operator of arbitrary order is considered. We find necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of a system of root functions of this operator in L _p .     

A note on Faber operator

For a rectifable Jordan curve Γ with complementary domainsD and D,Anderson conjectured that the Faber operator is a bounded isomorphism between the Besov spaces Bp(1 p ∞) of analytic functions in the unit disk and in the inner domain D,respectively.We point out that the conjecture is not true in the special case p=2,and show that in this case the Faber operator is a bounded isomorphism if and only if Γ is a quasi-circle.     

On the error of multidimensional quadrature formulas for certain classes of functions

We obtain regular (with respect to the power scale) estimates of the errors of multidimensional optimal quadrature formulas in spaces of periodic functions with constraints on Fourier coefficients in the ? _p -norm for 1 < p < 2.     

On an analog of the Riesz theorem and the basis property of the system of root functions of a differential operator in L p : II

We consider an ordinary differential operator of arbitrary order, obtain necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of the system of root functions of the given operator in L _p .     

Spectral properties of the complex airy operator on the half-line

We prove a theorem on the completeness of the system of root functions of the Schrödinger operator L = ?d ²/dx ² + p(x) on the half-line R₊ with a potential p for which L appears to be maximal sectorial. An application of this theorem to the complex Airy operator L _c = ?d ²/dx ² + cx, c = const, implies the completeness of the system of eigenfunctions of L _c for the case in which |arg c| < 2π/3.We use subtler methods to prove a theorem stating that the system of eigenfunctions of this special operator remains complete under the condition that |arg c| < 5π/6.     

On the asymmetry of norms of convolution operators. I

Let G be a compact group and π be a monomial representation of G which is irreducible. For a certain class of π-representative functions we obtain the exact bound of the function as a left-convolution operator on L_p(G) for 1 ? p ? 2 and good estimates when p > 2. This information is sufficient to conclude that for every noncommutative finite group, the L_p and L_p′-convolution norms are not the same when $\text{1 < p < 2,}\text{1}\text{p}\text{+}\text{1}\text{p′}\text{= 1}$.     

On local summability of Riesz potentials in the case Reα>0

The rangeI ^α (L _p ) of the Riesz potential operator, defined in the sense of distributions in the casep≥n/α, is shown to consist of regular distributions. Moreover, it is shown thatI ^α (L _p ) ?L _p ^loc (R ⁿ ) for all 1≤p<∞ and 0<α<∞. The distribution space used is that of Lizorkin, which is invariant with respect to the Riesz operator.     

Littlewood-Paley and Lusin functions on nilpotent Lie groups

In this paper, we define the Littlewood-Paley and Lusin functions associated to the sub-Laplacian operator on nilpotent Lie groups. Then we prove the Lp (1<p<∞) boundedness of Littlewood-Paley and Lusin functions.     

Projected gradient iteration for nonlinear operator equation

In this paper, we use a projected gradient algorithm to solve a nonlinear operator equation with ?_p-norm (1<p≤2) constraint. Gradient iterations with ?_p-norm constraints have been studied recently both in the context of inverse problem and of compressed sensing. In this paper, the constrained gradient iteration is implemented via a projected operator. We establish the ?₂-norm convergence of sequence constructed by the constrained gradient iteration when p∈(1,2]. The performance of the method is testified by a numerical example.     

Bounds for Maximal Functions Associated with Rotational Invariant Measures in High Dimensions

In recent articles (A. Criado in Proc. R. Soc. Edinb. Sect. A 140(3):541–552, 2010; Aldaz and Pérez Lázaro in Positivity 15:199–213, 2011) it was proved that when μ is a finite, radial measure in ? ⁿ with a bounded, radially decreasing density, the L ^p (μ) norm of the associated maximal operator M _μ grows to infinity with the dimension for a small range of values of p near 1. We prove that when μ is Lebesgue measure restricted to the unit ball and p<2, the L ^p operator norms of the maximal operator are unbounded in dimension, even when the action is restricted to radially decreasing functions. In spite of this, this maximal operator admits dimension-free L ^p bounds for every p>2, when restricted to radially decreasing functions. On the other hand, when μ is the Gaussian measure, the L ^p operator norms of the maximal operator grow to infinity with the dimension for any finite p>1, even in the subspace of radially decreasing functions.     

Basis properties in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> of systems of root functions of a spectral problem with spectral parameter in a boundary condition

We consider a spectral problem for a fourth-order ordinary differential equation with spectral parameter in a boundary condition. We study the structure of root spaces and analyze the basis properties in the space L _p (0, l), 1 < p < ∞, of systems of root functions of that problem.     

Spectral properties of some regular boundary value problems for fourth order differential operators

In this paper we consider the problem $\begin{gathered} y^{iv} + p_2 (x)y'' + p_1 (x)y' + p_0 (x)y = \lambda y,0 < x < 1, \hfill \\ y^{(s)} (1) - ( - 1)^\sigma y^{(s)} (0) + \sum\limits_{l = 0}^{s - 1} {\alpha _{s,l} y^{(l)} (0) = 0,} s = 1,2,3, \hfill \\ y(1) - ( - 1)^\sigma y(0) = 0, \hfill \\ \end{gathered} $ where λ is a spectral parameter; p _j (x) ∈ L ₁(0, 1), j = 0, 1, 2, are complex-valued functions; α _s;l , s = 1, 2, 3, $l = \overline {0,s - 1} $ , are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α _3,2 + α _1,0 ≠ α _2,1. It is proved that the system of root functions of this spectral problem forms a basis in the space L _p (0, 1), 1 < p < ∞, when α _3,2+α _1,0 ≠ α _2,1, p _j (x) ∈ W ₁ ^j (0, 1), j = 1, 2, and p ₀(x) ∈ L ₁(0, 1); moreover, this basis is unconditional for p = 2.     

Fonctions lipschitziennes et espaces de Sobolev fractionnaires

We consider a semigroup of Markovian and symmetric operators to which we associate fractional Sobolev spaces D^α_p (0 < α < 1 and 1 < p < ∞) defined as domains of fractional powers (−A_p)^α/2, where A_p is the generator of the semigroup in L^p. We show under rather general assumptions that Lipschitz continuous functions operate by composition on D^α_p if p ≥ 2. This holds in particular in the case of the Ornstein-Uhlenbeck semigroup on an abstract Wiener space.     

Generalized Hilbert operator on the Dirichlet-type space

The boundedness of the generalized Hilbert operator on the Dirichlet-type space S_p when 0<p<1 is investigated.     

Une méthode de point fixe pour le p-laplacien

Due to the compactness of the operator (-Δ_p)¹ N_ƒ, 1 < p < ∞, where Δ_p is the p-Laplacian and N_ƒ is the Nemytskii operator corresponding to a Caratheodory function ƒ: Ω × R → R, which satisfies a particular growth condition, the homotopy invariance of Leray-Schauder degree can be used in order to prove the existence of a W₀^1,p (Ω)-solution for the equation -Δ_p u = N_ƒ u.     

Some properties of root functions of ordinary second-order differential operators in the absence of the carleman condition

On a finite interval G of the real line, we consider the root functions of an ordinary second-order differential operator without any boundary conditions for the case in which the imaginary part of the spectral parameter is unbounded.We refine the estimates for the C-and L _p -norms of a root function and its first derivative on a compact set contained in the interior of G for the case in which the Carleman condition fails.A sufficient condition is obtained for the root functions of an ordinary second-order differential operator to have the Bessel property, assuming that the Carleman condition fails. We show that, under certain conditions, this problem can be reduced to analyzing the Bessel property of systems of exponentials.     

Parabolic Schrödinger operators

We characterize the domain of the parabolic Schrödinger operator t_∂−Δ+V in Lp(Rn+1), 1<p<∞, where the potential V is nonnegative and belongs to the Parabolic Reverse Hölder class p(PB).     

On the divergence of expansions in systems of root functions of the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville operator with an arbitrary complex-valued potential q(x) ?? L ₁(0, ??) and with degenerate boundary conditions. We show that, under some additional condition, the system of root functions of that operator is not a basis in the space L ₂(0, ??).     

Sharp general local estimates for dyadic-like maximal operators and related Bellman functions

For each 1?q<p we precisely evaluate the main Bellman functions associated with the local Lp→Lq estimates of the dyadic maximal operator on Rn. Actually we do that in the more general setting of tree-like maximal operators and with respect to general convex and increasing growth functions. We prove that these Bellman functions equal to analogous extremal problems for the Hardy operator which can be viewed as a symmetrization principle for such operators. Under certain mild conditions on the growth functions we show that for the latter extremals exist (although for the original Bellman functions do not) and analyzing them we give a determination of the corresponding Bellman function.     

Conjugate function theory in weak1 Dirichlet algebras

The fundamental theorems on conjugate functions are shown to be valid for weak¹ Dirichlet algebras. In particular the conjugation operator is shown to be a continuous map of L^p to L^p for 1 < p < ∞, to be a continuous map of L¹ to L^p, 0 < p < 1, and to map functions in L^∞ to exponentially integrable functions. These results allow a number of results for Dirichlet algebras to be extended to weak¹ Dirichlet algebras.