Weighted Lp Spaces Research Articles

Some weighted estimates for Littlewood–Paley functions and radial multipliers

We prove some weighted estimates for certain Littlewood–Paley operators on the weighted Hardy spaces Hwp (0<p⩽1) and on the weighted Lp spaces. We also prove some weighted estimates for the Bochner–Riesz operators and the spherical means.

Estimates of Integral Kernels for Semigroups Associated with Second-Order Elliptic Operators with Singular Coefficients

In this paper we obtain pointwise two-sided estimates for the integral kernel of the semigroup associated with second-order elliptic differential operators −∇⋅(a∇)+b 1⋅∇+∇⋅b 2+V with real measurable (singular) coefficients, on an open set Ω⊂R N . The assumptions we impose on the lower-order terms allow for the case when the semigroup exists on L p (Ω) for p only from an interval in [1,∞), neither enjoys a standard Gaussian estimate nor is ultracontractive in the scale L p (Ω). We show however that the semigroup is ultracontractive in the scale of weighted spaces L p (Ω,ϕ2 dx) with a suitable weight ϕ and derive an upper and lower bound on its integral kernel.

On the Representation of Band Limited Functions Using Finitely Many Bits

In this paper, we consider the question of representing an entire function of finite order and type in terms of finitely many bits, and reconstructing the function from these. Instead of making any further assumptions about the function, we measure the error in reconstruction in a suitably weighted Lp norm. The optimal number of bits in order to obtain a given accuracy is given by the Kolmogorov entropy. We determine this entropy in the case of certain compact subsets of these weighted Lp spaces and obtain constructive algorithms to determine the asymptotically optimal bit representation from finitely many samples of the function. Our theory includes both equidistant and non-uniform sampling. The reconstructions are polynomials, having several other optimality properties.

Shift Invariant Subspaces with Arbitrary Indices in ℓp Spaces

We construct right shift invariant subspaces of index n, 1⩽n⩽∞, in ℓp spaces, 2<p<∞, and in weighted ℓp spaces.

REAL FUNCTIONS IN WEIGHTED HARDY SPACES

We study the following problem: to describe weights w on the unit circle such that the analytic and antianalytic subspaces of the corresponding weighted space Lp(w) have nonzero intersection. In the special case p=2, the problem is equivalent to the well-known problem on exposed points in H 1. We show that the property in question is local, i.e., it depends on local behavior of the weight w at each point of the unit circle. Some necessary and sufficient conditions in terms of the Herglotz integrals are obtained. Bibliography: 6 titles.

New mapping properties for the resolvent of theLaplacian and recovery of singularities of a multi-dimensionalscattering potential

We prove that in dimension three and higher potential scatteringthe leading order singularities (and in some specialcases - all singularities) of unknown potential are obtained exactly bythe Born approximation. The proof is based on new estimates forthe continuous spectrum of the Laplacian in weightedL p-spaces. These estimates allow us to consider the potentials withstronger singularities than in previous publications.

Weighted Lp Inequalities for a Class of Integral Operators Including the Classical Index Transforms

In this paper we study the behaviour of certain integral operators acting on weighted Lp spaces. Particular cases include the classical integral transforms of Kontorovich and Lebedev and Mehler and Fock and the 2F1-index transform considered by González, Hayek, and Negrin.

ON THE NORM OF THE FOURIER-JACOBI PROJECTION

We extend the results of Pollard [7] and give asymptotic estimates for the norm of the Fourier-Jacobi projection operator in the appropriate weighted Lp space.

The Existence of Translation Invariant Subspaces of Symmetric Self-Adjoint Sequence Spaces on [formula omitted

We prove that if X is a reflexive translation invariant Banach space of complex sequences on Z that contains all finitely supported sequences, in which the coordinate functionals are continuous, and for every sequence {c(n)} in the space the sequences {c(n)} and {c(−n)} are also in the space, then X has a nontrivial translation invariant subspace. This provides in particular a positive solution to the translation invariant subspace problem for weighted ℓp spaces on Z with even weights, for 1<p<∞. The proof is based on an intermediate result that asserts that if A is an operator on a reflexive real Banach space of dimension greater than one, and there exist non-zero vectors, u in the space and v in the dual space, such that {〈Anu, v〉}∞n=0 is a moment sequence of a finite positive Borel measure on a bounded interval on the real line, then A has a nontrivial invariant subspace.

Lp Uniqueness of Non-symmetric Diffusion Operators with Singular Drift Coefficients: I. The Finite-Dimensional Case

Two uniqueness results for C0 semigroups on weighted Lp spaces over Rn generated by operators of type Δ+β·∇ with singular drift β are proven. A key ingredient in the proofs is the verification of some kind of “weak Kato inequality” which seems to break down exactly for those drift singularities where Lp uniqueness breaks down as well.

A Remark on Estimates for Uniformly Elliptic Operators on WeightedLp Spaces and Morrey Spaces

In this paper we consider uniformly elliptic operators with non-negative potentials V on ℝn (n ≥ 3) which belong to certain reverse Holder class. We show several estimates for VL–1, V1/2∇L–1 and ∇2L–1 on weighted Lp spaces and Morrey spaces under certain assumptions on aij(x), V and p. Our results generalize some results of J. Zhong and Z. Shen in the case of L = –Δ + V(x) and extend to uniformly elliptic operators.

On the Norm of the Fourier—Gegenbauer Projection in Weighted L p Spaces

We extend the results of Pollard [4] and give asymptotic estimates for the norm of the Fourier—Gegenbauer projection operator in the appropriate weighted Lp space. In particular, we settle the question of whether the projection is bounded for p=(2λ+1)/λ and p=(2λ+1)/(λ+1) , where λ is the index for the family of Gegenbauer polynomials under consideration.

Heat Kernel Estimates and L P Spectral Independence of Elliptic Operators

Let Ω be an open subset of Rd, and let Tp for p∈[1, ∞) be consistent C0-semigroups given by kernels that satisfy an upper heat kernel estimate. Denoting their generators by Ap, we show that the spectrum σ(Ap) is independent of p∈[1, ∞). We also treat the case of weighted Lp-spaces for weights that satisfy a subexponential growth condition. An example shows that independence of the spectrum may fail for an exponential weight. 1991 Mathematics Subject Classification 47D06, 47A10, 35P05.

Cauchy Problem with Subcritical Nonlinearity

The semilinear parabolic system onRnwith subcritical nonlinearity is studied as an abstract evolutionary equation in a Banach spaceX≔Lp(Rn). As in the case of bounded domains the existence of a strongly continuous semigroup of globalXα-solutions and its dissipativeness is shown to follow from a single estimate of solutions. Despite the lack of compactness of the Sobolev embeddings inRnthe compactness of trajectories of the semigroup is proved using the auxiliary estimate of solutions in a weighted space Lp(Rn,(1+|x|2)ν), and the existence of a global attractor is also shown. Thus this paper generalizes earlier considerations and provides a basis for further applications.

On the Mehler‐Fock Transform in Lp‐Space

AbstractThe present paper is devoted to study the transform by the index of the Legendre function which is known as the Mehler‐Fock transform. Mapping properties of the Mehler‐Fodr transform in the weighted space Lp(ω(t); IR+) are given as the inversion formula. The image space is also characterized.

Some Inequalities for Derivatives of Trigonometric and Algebraic Polynomials

We investigate inequalities for derivatives of trigonometric and algebraic polynomials in weighted LP spaces with weights satisfying the Muckenhoupt Ap condition. The proofs are based on an identity of Balazs and Kilgore [1] for derivatives of trigonometric polynomials. Also an inequality of Brudnyi in terms of rth order moduli of continuity ωr will be given. We are able to give values to the constants in the inequalities.

Spline Wavelet Bases of Weighted L p Spaces, 1 ≤p &lt; ∞

We study necessary conditions on the weight w for the spline wavelet systems to be bases in the weighted space LP (w) . In this article we study some questions arising in the investigation of the prob- lem of describing those classes of weight functions w for which the wavelet systems are bases or unconditional bases in the weighted spaces LP (w) = LP(R, w). A priori these conditions depend on the concrete system. But there are some features which are common to all wavelet systems. Here we examine only the case of spline wavelet systems defined on the real line R. For a given non-negative integer m, the spline wavelets are defined in the following way: Let V0 = {f E L2(R) n Cm-l(R) such that the restriction of f to each interval )n, n + 1 ( is a polynomial of degree < m}, where by C (R) we denote the class of functions on R whose derivatives of order r are continuous and by C-' (R) we denote the class of piecewise continuous functions on R. Defining Vj+j = {f(2x) : f(x) E Vj} , we get a multiscale analysis of L2(R) in the sense of Mallat and Meyer (see (M), (D)).

Lipschitz conditions satisfied by Hankel transforms

It is shown that Hankel transforms of functions on certain weighted LP spaces satisfy Lipschitz and integral Lipschitz conditions. In particular, Fourier-cosine and Fourier-sine transforms satisfy such Lipschitz conditions on such spaces.

Positive multilinear operators acting on weighted Lp spaces

For j = 1, 2,…, n + 1, let ( X j , ∑ j , μ j ) be σ-finite measure spaces and let P ( X j ) denote the class of nonnegative measurable functions on X j . Given a positive multilinear operator M: P (X 1) × P (X 2) × … × P (X n) → P (X n+1), and fixed indices p 1, p 2,…, p n , q ϵ [1, ∞], we consider the problem of determining those nonnegative (weight) functions w 1, w 2,…, w n and v on X 1, X 2,…, X n and X n + 1 , respectively, for which ∫ X n+1 [M(f 1, f 2 …, f n) v] q dμ n+1 1 q ⩽C ∏ j=1 n ∫ X j (f jw j) p dμ j 1 p j , with C > 0 independent of f jϵ P (X j), j=1,2, …, n.

Weighted $L^p$-boundedness of Fourier series with respect to generalized Jacobi weights

Let w be a generalized Jacobi weight on the interval [-1,1] and, for each function f, let Snf denote the n-th partial sum of the Fourier series of f in the orthogonal polynomials associated to w. We prove a result about uniform boundedness of the operators Sn in some weighted Lp spaces. The study of the norms of the kernels Kn related to the operators Sn allows us to obtain a relation between the Fourier series with respect to different generalized Jacobi weights.