Operators In Lebesgue Spaces Research Articles

A generalization of Bihari’s lemma to the case of Volterra operators in Lebesgue spaces

For operators acting in the Lebesgue space Lq(Π), 1 < q < ∞, an abstract analog of Bihari’s lemma is stated and proved. We show that it can be used to derive a uniform pointwise estimate of the increment of the solution of a controlled functional-operator equation in the Lebesgue space. The procedure of reducing controlled initial boundary-value problems to this equation is illustrated by the Goursat-Darboux problem.

Poincaré and Opial inequalities for vector-valued convolution products

Various Lp form Poincaré and Opial inequalities are given for vector-valued convolution products. We apply our results to infinitesimal generators of C0-semigroups and cosine functions. Typical examples of these operators are differential operators in Lebesgue spaces.

On the numerical radius of operators in Lebesgue spaces

We show that the absolute numerical index of the space L p ( μ ) is p − 1 p q − 1 q (where 1 p + 1 q = 1 ). In other words, we prove that sup { ∫ | x | p − 1 | T x | d μ : x ∈ L p ( μ ) , ‖ x ‖ p = 1 } ⩾ p − 1 p q − 1 q ‖ T ‖ for every T ∈ L ( L p ( μ ) ) and that this inequality is the best possible when the dimension of L p ( μ ) is greater than one. We also give lower bounds for the best constant of equivalence between the numerical radius and the operator norm in L p ( μ ) for atomless μ when restricting to rank-one operators or narrow operators.

On boundedness of a certain class of Hardy--Steklov type operators in Lebesgue spaces

Lp Lq-boundedness of the map f ! w(x) R b(x) a(x) k(x,y)f(y)v(y)dy is described by dierent types of criteria expressed in terms of given parameters 0 < p,q < 1, strictly increasing boundaries a(x) and b(x), locally integrable weight functions v,w and a positive continuous kernel k(x,y) satisfying some growth conditions.

Maximal inequalities for dual Sobolev spaces $W^{-1,p}$ and applications to interpolation

We firstly describe a maximal inequality for dual spaces W^{-1,p}. This one corresponds to a Sobolev version of usual properties of the Hardy-Littlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one seems to be new and we develop arguments in the general framework of Riemannian manifold. Then we present an application to obtain interpolation results for spaces.

Alternative criteria for the boundedness of Volterra integral operators in Lebesgue spaces

Three different criteria for Lp -Lq boundedness of Volterra integral operator with locally integrable weight functions w,v and a non-negative kernel k(x,y) satisfying Oinarov’s condition are given. Relations between components of the boundedness constants are described.

Compactness of integral operators in Lebesgue spaces with mixed norm

Compactness of certain double sized multidimensional integral operators is characterized in terms of the corresponding two-dimensional operators in the framework of weighted Lebesgue spaces with mixed norm. Mathematics subject classification (2000): 46E35, 26D10.

The maximal operator in Lebesgue spaces with variable exponent near 1

AbstractThis article contains results about the boundedness of the Hardy–Littlewood maximal operator in variable exponent Lebesgue spaces. We study the situation where the exponent approaches one in some parts of the domain. We show that the boundedness depends on how fast the exponent approaches one and give nearoptimal bounds for necessary and sufficient growths. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Continuous invertibility of minimal Sturm–Liouville operators in Lebesgue spaces

Using a standard theory of differential operators in Lebesgue spaces, we re-prove and generalize some results of Chernyavskaya and Shuster, giving (mostly sufficient) conditions that minimal operators determined by expressions of the form −(ry′)′ + qy with domain and range in possibly different Lp spaces on intervals with at least one singular endpoint have bounded inverses.

On the tensor products of operators in lebesgue spaces

The following question concerning the computation of the norms of the tensor products of operators in the Lebesgue spaces is studied: Is it true that the norm of the tensor product A⊗B: Lp(μ⊗μ)→Lq(ν⊗ν) of operators A: Lp(μ)→Lq(ν) and B: Lp(μ)→Lq(ν) coincides with the product ‖A‖ ‖B‖ of their norms? An answer is positive if and only if 1≤p≤q≤+∞. Bibliography: 26 titles.

On the $\mathcal L$-Characteristic of the Superposition Operator in Lebesgue Spaces with Mixed Norm

We consider the superposition operator Fx(t,s) = f(t,s,x(t,s)) of functions of two variables in spaces with mixed norm [L_p \to L_q] . After establishing a necessary and sufficient acting condition, we get some conclusions on the \mathcal L -characteristic of F . We also prove some theorems, which imply that F is uniformly continuous on balls in the interior of its \mathcal L -characteristic.

Interpolation of linear operators in Lebesgue spaces with mixed norm

Interpolation of linear operators in Lebesgue spaces with mixed norm

Interpolation of operators in Lebesgue spaces with mixed norm and its applications to Fourier analysis

Publisher Summary This chapter discusses an interpolation method of linear operators on functions in a product measure space and applies it to some problems arising in Fourier analysis on Euclidean space. For a function f on the d-dimensional Euclidean space Rd and ɛ ≥ 0 let sɛ (f) be the Riesz-Bochner mean of order, which is defined by the Fourier transform.

Extension theorems of Hahn-Banach type for nonlinear disjointly additive functionals and operators in Lebesgue spaces

Let (Ω, τ, M) be a nonatomic separable finite measure space. Every continuous functional N on L p ( m), 1 ⩽ p < ∞, which is disjointly additive in the sense N( u + v) = N( u) + N( v) whenever uv = 0, is known to be representable by an integral with a nonlinear Caratheodory kernel. Such functionals share several regularity properties with continuous linear functionals. Here we study the question of whether every continuous, disjointly additive functional defined on a closed subspace of L p ( m) possesses an extension to L p ( m) with these same properties. This question has applications to the study of nonlinear functionals on Sobolev spaces. It is shown that for a class of subspaces, including those of finite codimension, such an extension always exists, but there are also closed subspaces not possessing this extension property. Analogous results are obtained for disjointly additive mappings from closed subspaces of L p ( m) into L 1( m) and for functionals defined on subspaces of L ∞( m). The techniques depend heavily on the utilization of Lyapunov vector measures.