Class Of Weights Research Articles

Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms

LetL=-Δ+Vbe a Schrödinger operator, whereΔis the Laplacian onRdand the nonnegative potentialVbelongs to the reverse Hölder classRHqforq≥d. The Riesz transform associated with the operatorL=-Δ+Vis denoted byR=∇(-Δ+V)-1/2and the dual Riesz transform is denoted byR⁎=(-Δ+V)-1/2∇. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder classRHqforq≥d. Then we will establish the mapping properties of the operatorRand its adjointR⁎on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators[b,R]and[b,R⁎]are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger thanAp,BMO(Rd), andLp,κ(w)corresponding to the classical Riesz transforms (V≡0).

Quantitative a priori estimates for fast diffusion equations with Caffarelli–Kohn–Nirenberg weights. Harnack inequalities and Hölder continuity

We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation ut=|x|γ∇⋅(|x|−β∇um), with 0<m<1 posed on cylinders of (0,T)×RN. The weights |x|γ and |x|−β, with γ<N and γ−2<β≤γ(N−2)/N can be both degenerate and singular and need not belong to the class A2, a typical assumption for this kind of problems. This range of parameters is optimal for the validity of a class of Caffarelli–Kohn–Nirenberg inequalities, which play the role of the standard Sobolev inequalities in this more complicated weighted setting.The weights that we consider are not translation invariant and this causes a number of extra difficulties and a variety of scenarios: for instance, the scaling properties of the equation change when considering the problem around the origin or far from it. We therefore prove quantitative – with computable constants – upper and lower estimates for local weak solutions, focussing our attention where a change of geometry appears. Such estimates fairly combine into forms of Harnack inequalities of forward, backward and elliptic type. As a consequence, we obtain Hölder continuity of the solutions, with a quantitative (even if non-optimal) exponent. Our results apply to a quite large variety of solutions and problems. The proof of the positivity estimates requires a new method and represents the main technical novelty of this paper.Our techniques are flexible and can be adapted to more general settings, for instance to a wider class of weights or to similar problems posed on Riemannian manifolds, possibly with unbounded curvature. In the linear case, m=1, we also prove quantitative estimates, recovering known results in some cases and extending such results to a wider class of weights.

Joint Reducing Subspaces of Multiplication Operators and Weight of Multi-variable Bergman Spaces

This paper mainly concerns a tuple of multiplication operators defined on the weighted and unweighted multi-variable Bergman spaces, their joint reducing subspaces and the von Neumann algebra generated by the orthogonal projections onto these subspaces. It is found that the weights play an important role in the structures of lattices of joint reducing subspaces and of associated von Neumann algebras. Also, a class of special weights is taken into account. Under a mild condition it is proved that if those multiplication operators are defined by the same symbols, then the corresponding von Neumann algebras are *-isomorphic to the one defined on the unweighted Bergman space.

Hyponormality on general Bergman spaces

A bounded operator T on a Hilbert space is hyponormal if T*T-TT* is positive. We give a necessary condition for the hyponormality of Toeplitz operators on weighted Bergman spaces, for a certain class of radial weights, when the symbol is of the form f+g?, where both functions are analytic and bounded on the unit disk. We give a sufficient condition when f is a monomial.

Properties of the weighted log-rank test in the design of confirmatory studies with delayed effects.

Proportional hazards are a common assumption when designing confirmatory clinical trials in oncology. This assumption not only affects the analysis part but also the sample size calculation. The presence of delayed effects causes a change in the hazard ratio while the trial is ongoing since at the beginning we do not observe any difference between treatment arms, and after some unknown time point, the differences between treatment arms will start to appear. Hence, the proportional hazards assumption no longer holds, and both sample size calculation and analysis methods to be used should be reconsidered. The weighted log-rank test allows a weighting for early, middle, and late differences through the Fleming and Harrington class of weights and is proven to be more efficient when the proportional hazards assumption does not hold. The Fleming and Harrington class of weights, along with the estimated delay, can be incorporated into the sample size calculation in order to maintain the desired power once the treatment arm differences start to appear. In this article, we explore the impact of delayed effects in group sequential and adaptive group sequential designs and make an empirical evaluation in terms of power and type-I error rate of the of the weighted log-rank test in a simulated scenario with fixed values of the Fleming and Harrington class of weights. We also give some practical recommendations regarding which methodology should be used in the presence of delayed effects depending on certain characteristics of the trial.

Operators on Fock-type and weighted spaces of entire functions

Let us denote by $\mathcal F_1^\phi(\mathbb C)$ the space of entire functions $f\in \mathcal H(\C)$ such that $\int_\mathbb C |f(z)|e^{-\phi(|z|)}dm(z)<\infty$, where $\phi:(0,\infty)\to \mathbb R^+$ is assumed to be continuous and non-decreasing. Also given a continuous non-increasing function $v:(0,\infty)\to \mathbb R^+$ and a complex Banach space $X$, we write $H^\infty_v(\mathbb C,X)$ for the space of $X$-valued entire functions $F$ such that $\sup_{z\in \C} v(z)\|F(z)\|<\infty$. We find a very general class of weights $\phi$ and $v$ for which the space of bounded operators $\mathcal L(\mathcal F_1^\phi(\mathbb C),X)$ can be identified with $H^\infty_v(\mathbb C, X)$.

Weighted gradient estimates for higher order elliptic systems with non-smooth coefficients

In this paper we prove the regularity estimates on the weighted Lorentz spaces for the higher-order elliptic system of equations with non-smooth coefficients on a Lipschitz domain. As a by product, we obtain the regularity estimate on the Lorentz–Morrey spaces. Our results extend previous results to the larger class of Muckenhoupt weights.

Persistent Decay of Solutions to the k-abc Equation in Weighted $$L^p$$Lp Spaces

In this paper, we study a k-abc equation with $$(k+1)$$-degree nonlinearities, which is a 4-parameter family of nonlocal evolution equations and contains some famous ones. By moderate weight functions from time-frequency analysis, we show some persistence results for solutions of the equation in weighted $$L^p_\phi := L^p({\mathbf {R}}, \phi ^p(x)dx)$$ spaces for a large class of moderate weights. To overcome the difficulty caused by the higher nonlinear term, we take full advantage of features of the admissible weight function $$\phi $$. Our results cover some works on persistence properties of the Camassa–Holm type equations.

Weighted Calderón-Zygmund Estimates for Weak Solutions of Quasi-Linear Degenerate Elliptic Equations

This paper studies regularity estimates in Sobolev spaces for weak solutions of a class of degenerate and singular quasi-linear elliptic problems of the form div[A(x,u,∇u)] = div[F] with non-homogeneous Dirichlet boundary conditions over bounded non-smooth domains. The coefficients A could be be singular, degenerate or both in x in the sense that they behave like some weight function μ, which is in the A2 class of Muckenhoupt weights. Global and interior weighted W1,p(Ω,ω)-regularity estimates are established for weak solutions of these equations with some other weight function ω. The results obtained are even new for the case μ = 1 because of the dependence on the solution u of A. In case of linear equations, our W1,p-regularity estimates can be viewed as the Sobolev’s counterpart of the Holder’s regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni.

On weighted isoperimetric inequalities with non-radial densities

ABSTRACTWe consider a class of isoperimetric problems on , where the volume and the area element carry two different weights of the type . We solve them in a special case while a more detailed study is contained in Alvino et al. (The isoperimetric problem for a class of non-radial weights and applications, in preparation). Our results imply a weighted Pólya–Szëgo principle and a priori estimates for weak solutions to a class of boundary value problems for degenerate elliptic equations.

Singular weighted Sobolev spaces and diffusion processes: an example (due to V.V. Zhikov)

ABSTRACTWe consider the Sobolev space over of square integrable functions whose gradient is also square integrable with respect to some positive weight. It is well known that smooth functions are dense in the weighted Sobolev space when the weight is uniformly bounded from below and above. This may not be the case when the weight is unbounded. In this paper, we focus on a class of two-dimensional weights where the density of smooth functions does not hold. This class was originally introduced by V.V. Zhikov; such weights have a unique singularity point of non-zero capacity. Following V.V. Zhikov, we first give a detailed analytical description of the weighted Sobolev space. Then we explain how to use Dirichlet forms theory to associate a diffusion process to such a degenerate non-regular space.

Solid hulls of weighted Banach spaces of entire functions

Given a continuous, radial, rapidly decreasing weight v on the complex plane, we study the solid hull of its associated weighted space H_v^\infty(\mathbb C) of all the entire functions f such that v|f| is bounded. The solid hull is found for a large class of weights satisfying the condition (B) of Lusky. Precise formulations are obtained for weights of the form v(r)= exp (-ar^p), a &gt; 0, p &gt; 0 . Applications to spaces of multipliers are included.

Solid hulls and cores of weighted $$H^\infty $$ H ∞ -spaces

We determine the solid hull and solid core of weighted Banach spaces $$H_v^\infty $$ of analytic functions functions f such that v|f| is bounded, both in the case of the holomorphic functions on the disc and on the whole complex plane, for a very general class of radial weights v. Precise results are presented for concrete weights on the disc that could not be treated before. It is also shown that if $$H_v^\infty $$ is solid, then the monomials are an (unconditional) basis of the closure of the polynomials in $$H_v^\infty $$ . As a consequence $$H_v^\infty $$ does not coincide with its solid hull and core in the case of the disc. An example shows that this does not hold for weighted spaces of entire functions.

A Hardy–Littlewood maximal operator adapted to the harmonic oscillator

This paper constructs a Hardy–Littlewood type maximal operator adapted to the Schr¨odinger operator L := −∆ + |x| 2 acting on L2 (Rd). It achieves this through the use of the Gaussian grid ∆γ0, constructed by Maas, van Neerven, and Portal [Ark. Mat. 50 (2012), no. 2, 379–395] with the Ornstein-Uhlenbeck operator in mind. At the scale of this grid, this maximal operator will resemble the classical Hardy–Littlewood operator. At a larger scale, the cubes of the maximal function are decomposed into cubes from ∆γ0 nd weighted appropriately. Through this maximal function, a new class of weights is defined, A+p , with the property that for any w ∈ A+p the heat maximal operator associated with L is bounded from Lp(w) to itself. This class contains any other known class that possesses this property. In particular, it is strictly larger than Ap.

Poincaré inequalities for Sobolev spaces with matrix-valued weights and applications to degenerate partial differential equations

For bounded domains Ω, we prove that the Lp-norm of a regular function with compact support is controlled by weighted Lp-norms of its gradient, where the weight belongs to a class of symmetric non-negative definite matrix-valued functions. The class of weights is defined by regularity assumptions and structural conditions on the degeneracy set, where the determinant vanishes. In particular, the weight A is assumed to have rank at least 1 when restricted to the normal bundle of the degeneracy set S. This generalization of the classical Poincaré inequality is then applied to develop a robust theory of first-order Lp-based Sobolev spaces with matrix-valued weight A. The Poincaré inequality and these Sobolev spaces are then applied to produce various results on existence, uniqueness and qualitative properties of weak solutions to boundary-value problems for degenerate elliptic, degenerate parabolic and degenerate hyperbolic partial differential equations (PDEs) of second order written in divergence form, where A is calibrated to the matrix of coefficients of the second-order spatial derivatives. The notion of weak solution is variational: the spatial states belong to the matrix-weighted Sobolev spaces with p = 2. For the degenerate elliptic PDEs, the Dirichlet problem is treated by the use of the Poincaré inequality and Lax–Milgram theorem, while the treatment of the Cauchy–Dirichlet problem for the degenerate evolution equations relies only on the Poincaré inequality and the parabolic and hyperbolic counterparts of the Lax–Milgram theorem.

Garling sequence spaces

The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each $1\leqslant p<\infty$ and each nonincreasing weight $\textbf{w}\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous, and uniformly subprojective Banach space $g(\textbf{w},p)$. We also show that $g(\textbf{w},p)$ admits a unique subsymmetric basis despite the fact that for a wide class of weights it does not admit a symmetric basis. This provides the first known examples of Banach spaces where those two properties coexist.

Persistence properties for the two-component Novikov equation in weighted Lp spaces

ABSTRACTThis paper deals with the Cauchy problem of two-component Novikov equation, and the system was proposed by Popowicz. We mainly investigate the persistence properties of strong solutions in weighted spaces for a large class of moderate weights. Our results extends the work of Brandolese on persistence properties of the Camassa-Holm equation to more general system with cubic nonlinearity and interaction between the two components.

Blowup criterion and persistent decay for a modified Camassa-Holm system

In this paper, we study a modified two-component Camassa-Holm system modeling shallow water waves moving over a linear shear flow. We demonstrate a condition on the initial data that lead to finite time blow-up of solutions. By moderate weight functions from time-frequency analysis, we provide persistence results for solutions of the system in weighted Lp-spaces for a large class of moderate weights.

Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities

Let X be a metric space equipped with a doubling measure. We consider weights w(x) = dist(x,E)−α, where E is a closed set in X and $\alpha \in \mathbb {R}$ . We establish sharp conditions, based on the Assouad (co)dimension of E, for the inclusion of w in Muckenhoupt’s Ap classes of weights, 1 ≤ p < ∞. With the help of general Ap-weighted embedding results, we then prove (global) Hardy–Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.

Ap(ϕ) Weights, BMO(ϕ), and Calderón-Zygmund Operators of ϕ-Type

We introduce new classes of weights and BMO functions associated with a nondecreasing function ϕ of upper type β with β&gt;0 and obtain the weighted norm inequalities for Calderón-Zygmund operators of ϕ-type and their commutators.