Weighted L2 Spaces Research Articles

Strichartz and Smoothing Estimates for Dispersive Semi-Group e−itP(D) in Weighted L2 Spaces and Their Applications

Strichartz and Smoothing Estimates for Dispersive Semi-Group e−itP(D) in Weighted L2 Spaces and Their Applications

Limiting dynamics of stochastic complex Ginzburg–Landau lattice systems with long-range interactions in weighted space

This paper deals with the limiting dynamics of stochastic complex Ginzburg–Landau lattice systems with long-range interactions driven by nonlinear noise in a weighted space L2(Ω,lη2). We first consider the well-posedness of solutions for considered stochastic systems in the weighted space and then establish the existence and uniqueness of weak pullback mean random attractor in the weighted space.

Commutators on Fock spaces

Given a weighted ℓ2 space with weights associated with an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a calculus for the algebra of these commutators and apply it to the general case of Gelfond–Leontiev derivatives. This general class of operators includes many known examples, such as classic fractional derivatives and Dunkl operators. This allows us to establish a general framework, which goes beyond the classic Weyl–Heisenberg algebra. Concrete examples for its application are provided.

On the generalized Ornstein-Uhlenbeck operators perturbed by regular potentials and inverse square potentials in weighted L²-spaces

Generation of a quasi-contractive semigroup by generalized Ornstein-Uhlenbeck operators ℒ = −∆ + ∇Φ · ∇ − G · ∇ + V + c|x|−2 in the weighted space L2(RN, e−Φ(x)dx) is proven, where Φ ∈ C2(RN, R), G ∈ C1(RN, RN), 0 ≤ V ∈ C1(RN) and c > 0. The proofs are carried out by an application of L2-weighted Hardy inequality and bilinear form techniques.

Scattering theory in weighted L2 space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

In this paper, we consider the following inhomogeneous nonlinear Schrodinger equation (INLS) $$$i\partial_t u + \Delta u + \mu$$$ |$$$x$$$|$$$^{-b}$$$|$$$u$$$|$$$^\alpha u = 0, \quad (t,x)\in ℝ \times ℝ^d$$$ with $$$b, \alpha$$$ > 0. First, we revisit the local well-posedness in $$$H^1(ℝ^d)$$$ for (INLS) of Guzman [Nonlinear Anal. Real World Appl. 37 (2017), 249-286] and give an improvement of this result in the two and three spatial dimensional cases. Second, we study the decay of global solutions for the defocusing (INLS), i.e. $$$\mu=-1$$$ when 0 < $$$\alpha$$$ < $$$\alpha^\star$$$ where $$$\alpha^\star = \frac{4-2b}{d-2}$$$ for $$$d\geq 3$$$, and $$$\alpha^\star = \infty$$$ for $$$d=1, 2$$$ by assuming that the initial data belongs to the weighted $$$L^2$$$ space $$$\Sigma =\{u \in H^1(ℝ^d) :$$$ |$$$x$$$|$$$ u \in L^2(ℝ^d) \}$$$. Finally, we combine the local theory and the decaying property to show the scattering in $$$\Sigma$$$ for the defocusing (INLS) in the case $$$\alpha_\star$$$ < $$$\alpha$$$ < $$$\alpha^\star$$$, where $$$\alpha_\star = \frac{4-2b}{d}$$$.

Weak-strong uniqueness in weighted L2 spaces and weak suitable solutions in local Morrey spaces for the MHD equations

We consider here the magneto-hydrodynamics (MHD) equations on the whole space. For the 3D case, in the setting of the weighted L2 spaces we obtain a weak-strong uniqueness criterion provided that the velocity field and the magnetic field belong to a fairly general multipliers space. On the other hand, we study the local and global existence of weak suitable solutions for intermittent initial data, which is characterized through a local Morrey space. This large initial data space was also exhibit in a contemporary work [4] in the context of 3D Navier-Stokes equations. Finally, we make a discussion on the local and global existence problem in the 2D case.

Average case weighted [formula omitted] Markov factors with doubling weights

For weighted L2 spaces with doubling weights w on [−1,1] and norms ‖⋅‖2,w, the Markov factor on a polynomial P is defined by ‖P′‖2,w‖P‖2,w. We study this Markov factor on random polynomials with independent N(0,1) coefficients, and show that the upper bound of the average (expected) Markov factor is order degree to the 3∕2, as compared to the degree squared worst case upper bound.

Sharp relaxation rates for plane waves of general reaction-diffusion models

It is a well-known and classical result that spectrally stable traveling waves of a general reaction-diffusion system in one spatial dimension are asymptotically stable with exponential relaxation rates. In a series of works in the 1990s, Goodman [Trans. Am. Math. Soc. 311, 683 (1989)], Kapitula [Trans. Am. Math. Soc. 349, 1901 (1997)], and Xin [Commun. Partial Differ. Equations 17, 1889 (1992)] considered plane traveling waves for such systems and they have succeeded in showing asymptotic stability for such objects. Interestingly, the (estimates for the) relaxation rates that they have exhibited are all algebraic and dimension dependent. It was heuristically argued that as the spectral gap closes in dimensions n ≥ 2, algebraic rates are the best possible. In this paper, we revisit this issue. We rigorously calculate the sharp relaxation rates in L∞ based spaces, both for the asymptotic phase and the radiation terms. They are, indeed, algebraic, but about twice better than the best ones obtained in these early works although this can be mostly attributed to the inefficiencies of using Sobolev embeddings to control L∞ norms by high order L2 based Sobolev spaces. Finally, we explicitly construct the leading order profiles, both for the phase and for the radiation terms. Our approach relies on the method of scaling variables, as introduced in the work of Gallay and Wayne [Arch. Ration. Mech. Anal. 163, 209 (2002) and Gallay and Wayne, Philos. Trans. R. Soc. A 360, 2155 (2002)] and, in fact, provides sharp relaxation rates in a class of weighted L2 spaces as well.

A note on the Schrödinger smoothing effect

AbstractThe Kato–Yajima smoothing estimate is a smoothing weighted L2 estimate with a singular power weight for the Schrödinger propagator. The weight has been generalized relatively recently to Morrey–Campanato weights. In this paper we make this generalization more sharp in terms of the so‐called Kerman–Sawyer weights. Our result is based on a more sharpened Fourier restriction estimate in a weighted L2 space. Obtained results are also extended to the fractional Schrödinger propagator.

Weighted Littlewood–Paley inequalities for heat flows in RCD spaces

We establish inequalities on vertical Littlewood–Paley square functions for heat flows in the weighted L2 space over metric measure spaces satisfying the RCD⁎(0,N) condition with N∈[1,∞) and the maximum volume growth assumption. In the noncompact setting, the later assumption can be removed by showing that the volume of the ball growths at least linearly. The estimates are sharp on the growth of the 2-heat weight and the 2-Muckenhoupt weight considered. The p-Muckenhoupt weight and the p-heat weight are also compared for all p∈(1,∞).

On the Estimates of Widths of the Classes of Functions Defined by the Generalized Moduli of Continuity and Majorants in the Weighted Space L2,x(0, 1)

The upper and lower estimates for the Kolmogorov, linear, Bernstein, Gelfand, projective, and Fourier widths are obtained in the space L2,x(0, 1) for the classes of functions $$ {W}_2^r\left({\Omega}_{m,x}^{(v)};\Psi \right), $$ where r ∈ ℤ+, m ∈ ℕ, v ≥ 0, and $$ {\Omega}_{m,x}^{(v)} $$ and Ψ are the m th order generalized modulus of continuity and the majorant, respectively. The upper and lower estimates for the suprema of Fourier–Bessel coefficients are also established for these classes. We also present the conditions for majorants under which it is possible to find the exact values of the indicated widths and the upper bounds of the Fourier–Bessel coefficients.

Closed range of $$\overline \partial $$ on unbounded domains in ℂn

In this article, we establish a general sufficient condition for closed range of the Cauchy-Riemann operator \(\overline \partial\) in appropriately weighted L2 spaces on (0, q)-forms for a fixed q on domains in ℂn. The domains we consider may be neither bounded nor pseudoconvex, and our condition is a generalization of the classical Z(q) condition that we call weak Z(q). We provide examples that explain the necessity of working in weighted spaces for closed range in L2.

Average case tractability of multivariate approximation with Gaussian kernels

We consider d-variate approximation problems in the average case setting in the weighted L2 space with the standard Gaussian weight. This space is equipped with a zero-mean Gaussian measure whose covariance kernel is a product of univariate Gaussian kernels with non-increasing positive shape parameters γj2 for j=1,2,…,d. We study the average case error of algorithms that use finitely many arbitrary continuous linear functionals. We define the information complexity n(ε,d) to be the minimal number of linear functionals which are needed to find an algorithm whose average case error is at most ε. We investigate different notions of tractability or exponential convergence-tractability (EC-tractability), and get necessary and sufficient conditions in terms of the shape parameters. In particular, for fixed s>0 and t∈(0,1) we obtain that the sufficient and necessary condition on γj2 for which limd+ε−1→∞n(ε,d)ε−s+dt=0holds is limj→∞j1−tγj2ln+γj−2=0,where ln+x=max(1,lnx).

Necessary and Sufficient Conditions for the Solvability of the Neumann and the Regularity Problems in L p of Some Schrödinger Equations on Lipschitz Domains

Let n ≥ 3and Ω be a bounded Lipschitz domain in $\mathbb {R}^{n}$ . Assume that the non-negative potential V belongs to the reverse Holder class $RH_{n}(\mathbb {R}^{n})$ and p ∈ (2, ∞). In this article, two necessary and sufficient conditions for the unique solvability of the Neumann and the Regularity problems of the Schrodinger equation − Δu + V u = 0 in Ω with boundary data in L p , in terms of a weak reverse Holder inequality with exponent p and the unique solvability of the Neumann and the Regularity problems with boundary data in some weighted L 2 space, are established. As applications, for any p ∈ (1, ∞), the unique solvability of the Regularity problem for the Schrodinger equation − Δu + V u = 0 in the bounded (semi-)convex domain Ω with boundary data in L p is obtained.

On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation

ABSTRACTWe consider time global behavior of solutions to the focusing mass-subcritical NLS equation in a weighted L2 space. We prove that there exists a threshold solution such that (i) it does not scatter; (ii) with respect to a certain scale-invariant quantity, this solution attains minimum value in all nonscattering solutions. In the mass-critical case, it is known that ground states are this kind of threshold solution. However, in our case, it turns out that the above threshold solution is not a standing wave solution.

Quantitative weighted estimates for rough homogeneous singular integrals

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L 2(w), we obtain a bound that is quadratic in A 2 constant $${\left[ w \right]_{{A_2}}}$$ . We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.

Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces

In this paper we study the hydrodynamic (small mass approximation) limit of a Fokker-Planck equation. This equation arises in the kinetic description of the evolution of a particle system immersed in a viscous Stokes flow. We discuss two different methods of hydrodynamic convergence. The first method works with initial data in a weighted L2 space and uses weak convergence and the extraction of convergent subsequences. The second uses entropic initial data and gives an L1 convergence to the solution of the limit problem via the study of the relative entropy.

Representation of singular integrals by dyadic operators, and the [formula omitted] theorem

This exposition presents a self-contained proof of the A2 theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space L2(w). The strategy of the proof is a streamlined version of the author’s original one, based on a probabilistic Dyadic Representation Theorem for singular integral operators. While more recent non-probabilistic approaches are also available now, the probabilistic method provides additional structural information, which has independent interest and other applications. The presentation emphasizes connections to the David–Journé T(1) theorem, whose proof is obtained as a byproduct. Only very basic Probability is used; in particular, the conditional probabilities of the original proof are completely avoided.

The fractional Laplacian in power-weighted Lp spaces: Integration-by-parts formulas and self-adjointness

We consider the fractional Laplacian operator (−Δ)s (let s∈(0,1)) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the L2(Rd) scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space H˙s(Rd). More precisely, we focus on functions belonging to some weighted L2 space whose fractional Laplacian belongs to another weighted L2 space: we prove and disprove the validity of the integration-by-parts formula depending on the behaviour of the weight ρ(x) at infinity. The latter is assumed to be like a power both near the origin and at infinity (the two powers being possibly different). Our results have direct consequences for the self-adjointness of the linear operator formally given by ρ−1(−Δ)s. The generality of the techniques developed allows us to deal with weighted Lp spaces as well.

On second order hyperbolic equations with coefficients degenerating at infinity and the loss of derivatives and decays

In this paper, we study well-posedness issues in the weighted L2 space for the Cauchy problem on [0,T]×Rx for wave equations of the form ∂t2u−a(t,x)∂x2u=0. We shall give the condition a(t,x)>0 for all (t,x)∈[0,T]×Rx which is between the strictly hyperbolic condition and weakly hyperbolic one, and allows the decaying coefficient a(t,x) such that lim|x|→∞⁡a(t,x)=0 for all t∈[0,T]. Our concerns are the loss of derivatives and decays of the solutions.