Operators In Lp Research Articles

L2 to Lp bounds for spectral projectors on the Euclidean two-dimensional torus

AbstractWe consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include $\ell^2$ decoupling, small cap decoupling and estimates of exponential sums.

Correction to: On the Takai duality for Lp operator crossed products

Correction to: On the Takai duality for Lp operator crossed products

On the boundedness and norm of certain generalized Hilbert operators in ℓp$\ell ^p$

AbstractThe Hilbert matrix has been extensively studied in the previous literature. In this paper, we look at generalized Hilbert operators arising from measures on the interval [0,1], such that the Hilbert matrix is obtained by the Lebesgue measure. We provide a necessary and sufficient condition for these operators to be bounded in and calculate their norm. Interestingly, we obtain complete results in this direction for both .

Sharp [formula omitted]-error estimates for sampling operators

We study approximation properties of linear sampling operators in the spaces Lp for 1≤p<∞. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of a function in Lp and discrete information on the behaviour of a function at sampling points. The new measure of smoothness enables us to improve and extend several classical results of approximation theory to the case of linear sampling operators. In particular, we obtain matching direct and inverse approximation inequalities for sampling operators in Lp, find the exact order of decay of the corresponding Lp-errors for particular classes of functions, and introduce a special K-functional and its realization suitable for studying smoothness properties of sampling operators.

APPROXIMATION PROPERTIES OF MODIFIED GAUSS-WEIERSTRASS INTEGRAL OPERATORS IN EXPONENTIALLY WEIGHTED Lp SPACES

In here, we use modi ed Gauss-Weierstrass operators and givesome approximation results in the exponential weighted Lp spaces. Theseoperators are reproduce not only 1 but also a certain exponential functions.Forthis purpose, rstly we consider modi ed Gauss-Weierstrass integral operatorsfrom exponentially weighted Lp;a (R) into Lp;2a (R) spaces. Then, we give rate of convergence of the operators in Lp;2a (R) : Also, we prove the convergence of operators in the exponential weighted Lp;2a (R) spaces using the Korovkin type theorem. Finally, we give pointwise convergence of the operators at a generalized Lebesgue point.

On stochastic Itô processes with drift in [formula omitted

For Itô stochastic processes in Rd with drift in Ld Aleksandrov’s type estimates are established in the elliptic and parabolic settings. They are applied to estimating the resolvent operators of the corresponding elliptic and parabolic operators in Lp and Lp+1, respectively, where p≥d.

Power-regular Bishop operators and spectral decompositions

It is proved that a wide class of Bishop-type operators Tϕ,τ are power-regular operators in Lp(Ω,μ), 1⩽p&lt;∞, computing the exact value of the local spectral radius at any function u∈Lp(Ω,μ). Moreover, it is shown that the local spectral radius at any u coincides with the spectral radius of Tϕ,τ as far as u is non-zero. As a consequence, it is proved that non-invertible Bishop-type operators are non-decomposable whenever log|ϕ|∈L1(Ω,μ) (in particular, not quasinilpotent); not enjoying even the weaker spectral decompositions \textit{Bishop property} (β) and \textit{property} (δ).

Dynamical complexity and K-theory of Lp operator crossed products

We apply quantitative (or controlled) [Formula: see text]-theory to prove that a certain [Formula: see text] assembly map is an isomorphism for [Formula: see text] when an action of a countable discrete group [Formula: see text] on a compact Hausdorff space [Formula: see text] has finite dynamical complexity. When [Formula: see text], this is a model for the Baum–Connes assembly map for [Formula: see text] with coefficients in [Formula: see text], and was shown to be an isomorphism by Guentner et al.

Separation Problem for Bi-Harmonic Differential Operators in Lp\u2212 spaces on Manifolds

Consider the bi-harmonic differential expression of the form A=triangle _{M}^{2}+q on a manifold of bounded geometry (M,g) with metric g, where △M is the scalar Laplacian on M and q≥0 is a locally integrable function on M.In the terminology of Everitt and Giertz, the differential expression A is said to be separated in Lp(M), if for all u∈Lp(M) such that Au∈Lp(M), we have qu∈Lp(M). In this paper, we give sufficient conditions for A to be separated in Lp(M),where 1<p<∞.

Lp-estimates for parabolic systems with unbounded coefficients coupled at zero and first order

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space Lp(Rd;Rm)(d,m≥1) with p∈[1,+∞). Sufficient conditions for the associated evolution operator G(t,s) in Cb(Rd;Rm) to extend to a strongly continuous operator in Lp(Rd;Rm) are given. Some Lp–Lq estimates are also established together with Lp gradient estimates.

Rolewicz-type chaotic operators

In this article we introduce a new class of Rolewicz-type operators in ℓp, 1≤p<∞. We exhibit a collection F having the cardinality of the continuum, consisting of operators of this type which are chaotic and remain so under almost all finite linear combinations, provided that the linear combination has sufficiently large norm. As a corollary to our main result we also obtain that there exists a countable collection of such operators whose all finite linear combinations are chaotic provided that they have sufficiently large norm.

Pointwise ergodic theorems for bounded Lamperti representations of amenable groups

Dominated and Pointwise Ergodic Theorems for bounded representations of second countable locally compact amenable groups by Lamperti operators in L p ( Ω , F , m ) , L^p(\Omega ,\mathcal F, m), p &gt; 1 p&gt;1 fixed, are proved; we restrict ourselves to Cesàro averages in this paper. These theorems generalize or are closely related to well-known theorems for powers of power bounded Lamperti operators.

Modified gamma operators in L p spaces

In this paper, we consider certain positive linear operators connected with the classical gamma operators in Lp (1 ≤ p ≤ ∞) spaces. We study approximation properties of these operators, including theorems on the degree of approximation. The results are given in terms of some Ditzian–Totik modulus of smoothness.

On bounded invertibility of a class of degenerate elliptic differential operators in Lp

A result on the existence of a bounded degenerated elliptic differential operator is proved and an a priori estimate in the space () is obtained.

On the similarity of linear operators in L p to integral operators of the first and second kind

We construct an example of a compact operator of the third kind in Lp (p ≠ 2) not similar to any integral operator of the first or second kind. This example shows that not every linear equation of the third kind in Lp (p ≠ 2) can be reduced by an invertible continuous linear change to an equivalent integral equation of the first or second kind. The example also proves the impossibility of a characterization of integral and Carleman integral operators in Lp (p ≠ 2) in terms of the spectrum and its components.

Tikhonov regularization in Lp applied to inverse medium scattering

This paper presents Tikhonov- and iterated soft-shrinkage regularization methods for nonlinear inverse medium scattering problems. Motivated by recent sparsity-promoting reconstruction schemes for inverse problems, we assume that the contrast of the medium is supported within a small subdomain of a known search domain and minimize Tikhonov functionals with sparsity-promoting penalty terms based on Lp-norms. Analytically, this is based on scattering theory for the Helmholtz equation with the refractive index in Lp, 1 < p < ∞, and on crucial continuity and compactness properties of the contrast-to-measurement operator. Algorithmically, we use an iterated soft-shrinkage scheme combined with the differentiability of the forward operator in Lp to approximate the minimizer of the Tikhonov functional. The feasibility of this approach together with the quality of the obtained reconstructions is demonstrated via numerical examples.

Variation for singular integrals on Lipschitz graphs: 𝐿^{𝑝} and endpoint estimates

Let 1 ≤ n &gt; d 1\leq n&gt;d be integers and let μ \mu denote the n n -dimensional Hausdorff measure restricted to an n n -dimensional Lipschitz graph in R d \mathbb {R}^d with slope strictly less than 1 1 . For ρ &gt; 2 \rho &gt;2 , we prove that the ρ \rho -variation and oscillation for Calderón-Zygmund singular integrals with odd kernel are bounded operators in L p ( μ ) L^{p}(\mu ) for 1 &gt; p &gt; ∞ 1&gt;p&gt;\infty , from L 1 ( μ ) L^1(\mu ) to L 1 , ∞ ( μ ) L^{1,\infty }(\mu ) , and from L ∞ ( μ ) L^\infty (\mu ) to B M O ( μ ) BMO(\mu ) . Concerning the first endpoint estimate, we actually show that such operators are bounded from the space of finite complex Radon measures in R d \mathbb {R}^d to L 1 , ∞ ( μ ) L^{1,\infty }(\mu ) .

Measure-compact operators, almost compact operators, and linear functional equations in L p

Under study are the measure-compact operators and almost compact operators in Lp. We construct an example of a measure-compact operator that is not almost compact. Introducing two classes of closed linear operators in Lp, we prove that the resolvents of these operators are almost compact or measure-compact. We present methods for the reduction of linear functional equations of the second kind in Lp with almost compact or measure-compact operators to equivalent linear integral equations in Lp with quasidegenerate Carleman kernels.

Differentiation of integrals in higher dimensions

We prove a localization principle for directional maximal operators in L(p)(R(n)), with p > 1. The resulting bounds, which we conjecture hold for the largest possible class of directions, imply Lebesgue-type differentiation of integrals over tubes that point in the given directions.

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 Lp.