Weighted Lorentz Spaces Research Articles

A two-weight boundedness criterion and its applications

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, [Formula: see text], on [Formula: see text] in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley [Formula: see text]-functions, Lusin area functions, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley [Formula: see text]-functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on [Formula: see text] in the aforementioned function spaces.

Rearrangement-invariant hulls of weighted Lebesgue spaces

We characterize the rearrangement-invariant hull, with respect to a given measure μ, of weighted Lebesgue spaces. The solution leads us to first consider when this space is contained in the sum of (L1+L∞)(R,μ) and the final condition is given in terms of embeddings for weighted Lorentz spaces.

Unconditional uniqueness and non-uniqueness for Hardy–Hénon parabolic equations

We study the problems of uniqueness for Hardy–Hénon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (Hénon type) in the nonlinear term. To deal with the Hardy–Hénon type nonlinearities, we employ weighted Lorentz spaces as solution spaces. We prove unconditional uniqueness and non-uniqueness, and we establish uniqueness criterion for Hardy–Hénon parabolic equations in the weighted Lorentz spaces. The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces.

Gradient estimates of general nonlinear singular elliptic equations with measure data

We develop a global Calderón-Zygmund estimate for the gradients of renormalized solutions to the general nonlinear singular elliptic equations −divA(x,u,Du)=μ on a Reifenberg flat domain with the homogeneous Dirichlet boundary condition, while μ is a finite signed Radon measure. The associated nonlinearity behaves as the elliptic p-Laplacian with respect to Du for the singular case p∈(1,2−1/n], whose discontinuity in the x-variable is measured in terms of small BMO, and the Lipschitz continuity is required with respect to the u-variable. We prove it in two folds: the perturbation technique and the weighted Vitali type covering are first employed to establish the weighted good-λ type inequality, then such inequality is used to prove the desired global gradient estimates in weighted Lorentz spaces and Lorentz-Morrey spaces. As a direct consequence, finally we obtain a global gradient regularity in weighted Orlicz spaces.

On Discrete Weighted Lorentz Spaces and Equivalent Relations between Discrete ℓp-Classes

In this paper, we study some relations between different weights in the classes Bp,Bp*,Mp and Mp* that characterize the boundedness of the Hardy operator and the adjoint Hardy operator. We also prove that these classes generate the same weighted Lorentz space Λp. These results will be proven by using the properties of classes Bp,Bp*,Mp and Mp*, including the self-improving properties and also the properties of the generalized Hardy operator Hp, the adjoint operator Sq and some fundamental relations between them connecting their composition to their sum.

Bilinear Fourier Multipliers of Bounded Variation

Abstract In this paper, we obtain weighted estimates for bilinear Fourier multipliers of bounded variation that provide new restricted weak-type bounds. We also study their boundedness on the setting of the weighted Lorentz spaces. The results are obtained using Rubio de Francia extrapolation as the main tool.

Weighted Lorentz spaces: Sharp mixed Ap − A∞ estimate for maximal functions

We prove the sharp mixed Ap−A∞ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely‖M‖Lp,q(w)≲p,q,n[w]Ap1p[σ]A∞1min⁡(p,q), where σ=w11−p. Our method is rearrangement free and can also be used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for M in a dual setting.

Weighted estimates for Bochner–Riesz operators on Lorentz spaces

We present new estimates in the setting of weighted Lorentz spaces of operators satisfying a limited Rubio de Francia condition; namely $T$ is bounded on $L^{p}(v)$ for every $v$ in an strictly smaller class of weights than the Muckenhoupt class $A_p$. Important examples will be the Bochner–Riesz operators $BR_\lambda$ with $0<\lambda <{(n-1)}/2$, sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with $\Omega \in L^{r}(\Sigma )$, $1 < r < \infty$.

Weak martingale Hardy-type spaces associated with quasi-Banach function lattice

Abstract In this paper, the authors introduce weak martingale Hardy-type spaces associated with a quasi-Banach function lattice. The authors then establish the atomic characterizations of these weak martingale Hardy-type spaces. As applications, the authors give the sufficient conditions for the boundedness of σ-sublinear operators from weak martingale Hardy-type spaces to a quasi-Banach function lattice. Furthermore, the authors clarify the relation among different weak martingale Hardy-type spaces in the framework of a rearrangement-invariant quasi-Banach function space. Finally, the authors apply these results to the weighted Lorentz space and the generalized grand Lebesgue space.

A study on convolutions in weighted lorentz spaces using steklov means

In this study, some estimations of convolution type operators defined with the help of Steklov operator in weighted Lorentz space L_ω^(p,q) (T) are obtained. Also, some basic properties of convolution type operators in these spaces are investigated.

Calculations of Norms for Monotone Operators on Cones of Functions with Monotonicity Properties

The paper is devoted to the problem of exact calculation of the norms in ideal spaces for monotone operators on the cones of functions with monotonicity properties. We implement a general approach to this problem that covers many concrete variants of monotone operators in ideal spaces and different monotonicity conditions for functions. As applications, we calculate the norms of some integral operators on the cones, associate norms over some cones in Lebesgue spaces, the norms of the dilation operator and embedding operators on weighted Lorentz spaces with general weights. Under some more general conditions, we present order sharp estimates for Hardy-type operators on the cones.

Boundedness of Sparse and Rough Operators on Weighted Lorentz Spaces

We present new estimates in the setting of weighted Lorentz spaces for important operators in Harmonic Analysis such as sparse operators, Bochner–Riesz at the critical index, Hormander multipliers and rough singular integrals among others.

A regularity result via fractional maximal operators for p-Laplace equations in weighted Lorentz spaces

ABSTRACT We study the global regularity in weighted Lorentz spaces for the gradient of weak solutions to p-Laplace equations in an open-bounded non-smooth domain which satisfies a p-capacity condition. Moreover, the gradient estimate is presented in terms of fractional maximal operators which are related to the Riesz potential and the fractional derivative. Our main concern is to extend the results of Tran and Nguyen [New gradient estimates for solutions to quasi-linear divergence form elliptic equations with general Dirichlet boundary data. J Differ Equ. 2020;268(4):1427–1462] but in the weighted Lorentz spaces associated to the Muckenhoupt weights. The key technique that we use in this paper is the good-λ inequality and the interesting properties of the cut-off fractional maximal operators.

Modified modulus of smoothness and approximation in weighted Lorentz spaces by Borel and Euler means

Modified modulus of smoothness and approximation in weighted Lorentz spaces by Borel and Euler means

Calderón–Zygmund Operators and Commutators on Weighted Lorentz Spaces

We find necessary conditions (which are also sufficient, for some particular cases) for a pair of weights u and w such that a Calder_on-Zygmund operator T, or its commutator [b; T], with b 2 BMO, is bounded on the weighted Lorentz spaces _p u(w), for 1 < p < 1. This result completes the study already known for the Hardy-Littlewood maximal operator and the Hilbert transform, and hence unifies the weighted theories for the Ap and Bp classes.

Vector Valued Maximal Carleson Type Operators on the Weighted Lorentz Spaces

In this paper, by using the idea of linearizing maximal operators originated by Charles Fefferman and the TT* method of Stein–Wainger, we establish a weighted inequality for vector valued maximal Carleson type operators with singular kernels proposed by Andersen and John on the weighted Lorentz spaces with vector-valued functions.

The supremum-involving Hardy-type operators on Lorentz-type spaces

Given measurable functions u, \sigma on an interval (0,b) and a kernel function k(x,y) on (0,b)^2 satisfying Oinarov condition, the supremum-involving Hardy-type operators Rf(x)=\sup_{x\leq\tau &lt; b}u(\tau)\int_0^\tau k(\tau,y)\sigma (y)f(y)dy, x &gt; 0 in Orlicz-Lorentz spaces are investigated. We obtain sufficient conditions of boundedness of R: \Lambda_{u_0}^{G_0}(w_0)\\ \rightarrow \Lambda_{u_1}^{G_1}(w_1) and R: \Lambda_{u_0}^{G_0}(w_0)\rightarrow \Lambda_{u_1}^{G_1,\infty}(w_1) . Furthermore, in the case of weighted Lorentz spaces, two characterizations of the boundedness of the operator R:\Lambda_{u_0}^{p_0}(w_0)\rightarrow\Lambda_{u_1}^{p_1,q_1}(w_1) are achieved as well as the compactness of the operator R is characterized. It is notable that in the present paper the spaces are only required to be quasi-Banach spaces other than Banach spaces.

Hardy-Type Operators in Lorentz-Type Spaces Defined on Measure Spaces

Weight criteria for the boundedness and compactness of generalized Hardy-type operators (0.1) $$Tf(x) = {u_1}(x)\int_{\left\{ {\phi (y) \le \psi (x)} \right\}} {f(y){u_2}(y){v_0}(y)d\mu (y),\;\;\;\;\;x \in X},$$ in Orlicz-Lorentz spaces defined on measure spaces is investigated where the functions ϕ, ψ, u1, u2, v0 are positive measurable functions. Some sufficient conditions of boundedness of $$T:\;\Lambda _{{v_0}}^{{G_0}}({w_0}) \to \Lambda _{{v_1}}^{{G_1}}({w_1})$$ and $$T:\;\Lambda _{{v_0}}^{{G_0}}({w_0}) \to \Lambda _{{v_1}}^{{G_1},\infty }({w_1})$$ are obtained on Orlicz-Lorentz spaces. Furthermore, we achieve sufficient and necessary conditions for T to be bounded and compact from a weighted Lorentz space $$\Lambda _{{v_0}}^{{p_0}}({w_0})$$ to another $$\Lambda _{{v_1}}^{{p_1},{q_1}}({w_1})$$ . It is notable that the function spaces concerned here are quasi-Banach spaces instead of Banach spaces.

Weighted gradient estimates for general nonlinear elliptic equations involving measure data

We prove a global Calderón-Zygmund type estimate in the weighted Lorentz spaces for the gradients of solution to general nonlinear elliptic equations involving a signed Radon measure. The associated nonlinearity is assumed to depend on the solution itself and satisfy the (δ,R0)-BMO condition with respect to x-variable, while the boundary of underlying domain is assumed to be Reifenberg flat. Here, we mainly employ the perturbation technique, the modified Vitali type covering and an extrapolation argument. As its direct consequence, we also prove global gradient estimates in the Orlicz spaces and variable Lebesgue spaces, respectively.

Interpolation between weighted Lorentz spaces with respect to a vector measure

In this paper, we consider weighted Lorentz spaces with respect to a vector measure and derive some of their properties. We describe the interpolation with a parameter function of these spaces. As an application, we get a type of the generalization of Steffensen's inequality for $L^p(\|m\|)$ and interpolation spaces for couples of Lorentz-Zygmund spaces.