Radial Weight Research Articles

On weighted generalized fractional and Hardy-type operators acting between Morrey-type spaces

Abstract We study weighted generalized Hardy and fractional operators acting from generalized Morrey spaces L p,φ ,(ℝ n ) into Orlicz-Morrey spaces L Φ,φ ,(ℝ n ). We deal with radial quasi-monotone weights and assumptions imposed on weights are given in terms of Zigmund-type integral conditions. We find conditions on φ,Φ, the weight w and the kernel of the fractional operator, which insures such a boundedness. We prove some pointwise estimates for weighted generalized fractional operators via generalized Hardy operators, which allow to obtain the weighted boundedness for fractional operators from those for Hardy operators. We provide also some easy to check numerical inequalities to verify the obtained conditions.

Integral operators with homogeneous kernels in grand Lebesgue spaces

Sufficient conditions on the kernel and the grandizer that ensure the boundedness of integral operators with homogeneous kernels in grand Lebesgue spaces on ℝ n as well as an upper bound for their norms are obtained. For some classes of grandizers, necessary conditions and lower bounds for the norm of these operators are also obtained. In the case of a radial kernel, stronger estimates are established in terms of one-dimensional grand norms of spherical means of the function. A sufficient condition for the boundedness of the operator with homogeneous kernel in classical Lebesgue spaces with arbitrary radial weight is obtained. As an application, boundedness in grand spaces of the one-dimensional operator of fractional Riemann–Liouville integration and of a multidimensional Hilbert-type operator is studied.

Pitt inequalities and restriction theorems for the Fourier transform

We prove new Pitt inequalities for the Fourier transforms with radial and non-radial weights using weighted restriction inequalities for the Fourier transform on the sphere. We also prove new Riemann–Lebesgue estimates and versions of the uncertainty principle for the Fourier transform.

On unconditional bases of reproducing kernels in Fock-type spaces

The existence of unconditional bases of reproducing kernels in the Fock-type spaces F φ with radial weights φ is studied. It is shown that there exist functions φ(r) of arbitrarily slow growth for which ln r = o(φ(r)) as r → ∞ and there are no unconditional bases of reproducing kernels in the space F φ . Thus, a criterion for the existence of unconditional bases cannot be given only in terms of the growth of the weight function.

Singular Integral Operators in Generalized Morrey Spaces on Curves in the Complex Plane

We study the boundedness of the Cauchy singular integral operators on curves in complex plane in generalized Morrey spaces. We also consider the weighted case with radial weights. We apply these results to the study of Fredholm properties of singular integral operators in weighted generalized Morrey spaces.

Adaptive backstepping radial basis function neural network controller design for a Mars lander during the powered descent phase

When a Mars lander is guided to follow a predetermined reference trajectory during the powered descent phase, large tracking errors occur due to strong perturbations caused by enormous external disturbances, such as the Martian atmosphere and wind and dust storms, as well as considerable uncertainties. The tracking performance is determined directly by the accuracy of the system model, especially with regard to nonlinear terms. In this paper, an adaptive backstepping radial basis function neural network controller is developed for a Mars lander to achieve precise tracking to a reference trajectory during the powered descent phase. The main part of the controller is designed with backstepping, and a radial basis function neural network with an online adaptive law for the weight vector is used as an auxiliary part to approximate unknown nonlinear functions, including the gravitational force, Coriolis force, centrifugal force, atmospheric drag force, atmospheric lift force, wind force, and unknown uncertainties. The proposed adaptive backstepping radial basis function neural network controller guarantees that tracking errors and radial basis sunction neural network weight estimation errors eventually converge to the uniformly ultimately bounded values according to the Lyapunov stability theory. Additionally, this study presents an online adaptive law for the weight vector. The simulation results show that the adaptive backstepping radial basis function neural network controller has an excellent tracking performance in the severe environmental conditions of Mars with strong external disturbances and large variations in uncertainties. Furthermore, this study reveals that the radial basis function neural network has an outstanding capability to approximate unknown nonlinear functions.

Expected Number of Real Roots for Random Linear Combinations of Orthogonal Polynomials Associated with Radial Weights

In this note, we obtain asymptotic expected number of real zeros for random polynomials of the form $$f_n(z)=\sum_{j=0}^na^n_jc^n_jz^j$$ where $a^n_j$ are independent and identically distributed real random variables with bounded $(2+\delta)$th absolute moment and the deterministic numbers $c^n_j$ are normalizing constants for the monomials $z^j$ within a weighted $L^2$-space induced by a radial weight function satisfying suitable smoothness and growth conditions.

Vanishing Bergman Kernels on the Disk

We discuss topics related to zeroes of the Bergman kernels, and present a method for generating Bergman kernels with arbitrarily, but finitely, many zeroes. It is also shown that a Bergman kernel induced by a radial weight on the unit disk cannot have infinitely many zeroes. Similar questions for the Segal–Bargmann spaces of the complex plane are briefly discussed.

A Note on Completeness of Weighted Normed Spaces of Analytic Functions

Given a non-negative weight $v$, not necessarily bounded or strictly positive, defined on a domain $G$ in the complex plane, we consider the weighted space $H_v^\infty(G)$ of all holomorphic functions on $G$ such that the product $v|f|$ is bounded in $G$ and study the question of when is such a space complete under the canonical sup-seminorm. We obtain both some necessary and some sufficient conditions in terms of the weight $v$, exhibit several relevant examples, and characterize completeness in the case of spaces with radial weights on balanced domains.

Approximation by Proper Holomorphic Maps and Tropical Power Series

Let $$w$$ be an unbounded radial weight on the complex plane. We study the following approximation problem: find a proper holomorphic map $$f: \mathbb {C}\rightarrow \mathbb {C}^n$$ such that |f| is equivalent to $$w$$ . We give several characterizations of those $$w$$ for which the problem is solvable. In particular, a constructive characterization is given in terms of tropical power series. Moreover, the following natural objects and properties are involved: essential weights on the complex plane, approximation by power series with positive coefficients, and approximation by the maximum of a holomorphic function modulus. Extensions to several complex variables and approximation by harmonic maps are also considered.

Intrinsic alignments of galaxies in the Illustris simulation

We study intrinsic alignments (IA) of galaxy image shapes within the Illustris cosmic structure formation simulations. We investigate how IA correlations depend on observable galaxy properties such as stellar mass, apparent magnitude, redshift and photometric type, and on the employed shape measurement method. The correlations considered include the matter density–intrinsic ellipticity (mI), galaxy density–intrinsic ellipticity (dI), gravitational shear–intrinsic ellipticity (GI) and intrinsic ellipticity–intrinsic ellipticity (II) correlations. We find stronger correlations for more massive and more luminous galaxies, as well as for earlier photometric types, in agreement with observations. Moreover, the correlations significantly depend on the choice of shape estimator, even if calibrated to serve as unbiased shear estimators. In particular, shape estimators that down-weight the outer parts of galaxy images produce much weaker IA signals on intermediate and large scales than methods employing flat radial weights. The expected contribution of IA to the observed ellipticity correlation in tomographic cosmic shear surveys may be below one percent or several percent of the full signal depending on the details of the shape measurement method. A comparison of our results to a tidal alignment model indicates that such a model is able to reproduce the IA correlations well on intermediate and large scales, provided the effect of varying galaxy density is correctly taken into account. We also find that the GI contributions to the observed ellipticity correlations could be inferred directly from measurements of galaxy density–intrinsic ellipticity correlations, except on small scales, where systematic differences between mI and dI correlations are large.

Wilt incidence and cultural variability of Fusarium oxysporum f.sp. udum collected from different districts of Uttar Pradesh

Pigeonpea (Cajanus cajan) L. Millsp. is an important legume crop widely used as food grain as it is rich source of protein, carbohydrate, essential amino acids, minerals and fibres. India is reknowned as a major pigeonpea producer country all over the world. Pigeonpea is susceptible to a number of pathogens, among which Fusarium oxysporum f.sp. udum is considered as the most important fungal pathogen causing considerable economic loss in India and all over the world. Among different states in India, Uttar Pradesh is the major pigeonpea growing state having most of the wilt susceptible pigeonpea growing areas causing considerable yield losses. In the present study, collection of diseased samples from wilt affected areas of different districts, isolation of test fungi, test of pathogenicity in pots under wirenet house condition was conducted. Further test of wilt incidence of the selected strains of Fusarum oxysporum f.sp. udum through root dip method and soil inoculation method was undertaken in earthen pots under wirenet house to make a precise comparison between the two methods. Cultural variabilities like radial growth, growth rate and mycelia dry weight among the selected isolates were studied under laboratory conditions.

Volterra type operators on growth Fock spaces

Let \(\omega \) be an unbounded radial weight on \(\mathbb {C}^d\), \(d\ge 1\). Using results related to approximation of \(\omega \) by entire maps, we investigate Volterra type and weighted composition operators defined on the growth space \(\mathcal {A}^\omega (\mathbb {C}^d)\). Special attention is given to the operators defined on the growth Fock spaces.

A note on the Kakeya maximal operator and radial weights on the plane

We obtain an estimate of the operator norm of the weighted Kakeya (Nikodým) maximal operator without dilation on $L^2(w)$. Here we assume that a radial weight $w$ satisfies the doubling and supremum condition. Recall that, in the definition of the Kakeya maximal operator, the rectangle in the supremum ranges over all rectangles in the plane pointed in all possible directions and having side lengths $a$ and $aN$ with $N$ fixed. We are interested in its eccentricity $N$ with $a$ fixed. We give an example of a non-constant weight showing that $\sqrt{\log N}$ cannot be removed.

Differentiation and integration operators on weighted Banach spaces of holomorphic functions

We obtain a new natural description of the class of radial weights for which some previous results of A. Harutyunyan and W. Lusky concerning the boundedness of differentiation and integration operators on corresponding spaces are valid. To do this, we develop an elementary approach which is essentially different from the previous one and allows us to establish several new results and new characterizations of some popular classes of radial weights.

Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy - entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN). We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy - entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincar{e} inequality which is interpreted as a linearized entropy - entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods. Consequences for the (WFD) flow will be studied in Part II of this work.

Growth of proper holomorphic maps and tropical power series

Motivated by a question of J. Globevnik, we show that a proper holomorphic immersion of the unit disk D into C2 or a proper holomorphic embedding f:D→C3 may have arbitrary growth. Also, using tropical power series, we characterize those radial weights w on the complex plane for which there exist n∈N and a proper holomorphic map f:C→Cn such that |f(z)| is equivalent to w(z).

Trace class criteria for Toeplitz and composition operators on small Bergman spaces

We characterize the Schatten class Toeplitz operators induced by a positive Borel measure on the unit disc and the reproducing kernel of the Bergman space Aω2, where ω is a radial weight satisfying the doubling property ∫r1ω(s)ds≤C∫1+r21ω(s)ds. By using this, we describe the Schatten class composition operators. We also discuss basic properties of composition operators acting from Aωp to Avq.

On Caffarelli-Kohn-Nirenberg Inequalities for Block-Radial Functions

The paper provides weighted Sobolev inequalities of the Caffarelli-Kohn-Nirenberg type for functions with multi-radial symmetry. An elementary example of such inequality is the following inequality of Hardy type for functions u = u(r1(x), r2(x)), where \(r_{1}(x) = \sqrt {{{x}_{1}^{2}} + {{x}_{2}^{2}}}\) and \(r_{2}(x) = \sqrt {{{x}_{3}^{2}} + {{x}_{4}^{2}}}\) from the subspace \({\dot {H}}_{(2,2)}^{1,3} (\mathbb {R}^{4})\) of the Sobolev space \(\dot {H}^{1,3}(\mathbb {R}^{4})\), radially symmetric in variables (x1, x2) and in variables (x3, x4): $$ {\int }_{\mathbb {R}^{4}} \frac {u^{2}}{r_{1}(x)r_{2}(x)} dx \le C {\int }_{\mathbb {R}^{4}} | \nabla u|^{2} dx, $$ Similarly to the previously studied radial case, the range of parameters in CKN inequalities can be extended, sometimes to infinity, providing a pointwise estimate similar to the radial estimate in Strauss (Comm. Math. Phys. 55, 149–162 1977). Furthermore, the “multi-radial” weights are a stronger singularity than radial weights of the same homogeneity, e.g. \(\frac {1}{r_{1}(x)r_{2}(x)} \ge \frac {1}{2|x|^{2}}\).

On a theorem of Cartwright in higher dimensions

We consider harmonic functions in the unit ball of $\mathbb{R}^{n+1}$ that are unbounded near the boundary but can be estimated from above by some (rapidly increasing) radial weight $w$. Our main result gives some conditions on $w$ that guarantee the estimate from below on the harmonic function by a multiple of this weight. In dimension two this reverse estimate was first obtained by M. Cartwright for the case of the power weights, $w_p(z)=(1-|z|)^{-p}$ for $p>1$, and then generalized to a wide class of regular weights by a number of authors.