Weighted Lebesgue Spaces Research Articles

New multilinear Littlewood–Paley gλ∗ function and commutator on weighted Lebesgue spaces

Via the new weight function , the authors introduce a new class of multilinear Littlewood–Paley functions and establish the boundedness on weighted Lebesgue spaces. In addition, the authors obtain the boundedness of the multilinear commutator and multilinear iterated commutator generated by the multilinear Littlewood–Paley function and the new function on weighted Lebesgue spaces. The results in this article include the known results in previous studies. When , that is, in the case of one linear, our conclusions are also new, further extending the results in previous work.

Exchange Formulae for the Stieltjes–Poisson Transform over Weighted Lebesgue Spaces

This paper aims to develop exchange formulae for the Stieltjes–Poisson transform by using Mellin-type convolutions in the context of weighted Lebesgue spaces. A key result is the introduction of bilinear and continuous Mellin-type convolutions, expanding the scope of the analysis to include the space of weighted L1 functions and the space of continuous functions vanishing at infinity.

Necessary and Sufficient Conditions for the Boundedness of Multiple Integral Operators with Super-Homogeneous Kernels in Weighted Lebesgue Space

Super-homogeneous functions including homogeneous functions, quasi-homogeneous functions, and several non-homogeneous functions are considered. Using the weight function method, the construction conditions of Hilbert-type multiple integral inequalities with super-homogeneous kernels are first discussed. Then, using the obtained results, the construction problem of bounded multiple integral operators with super-homogeneous kernels in weighted Lebesgue space is discussed, and the necessary and sufficient conditions for operator boundedness and the operator norm formula are obtained.

The Smirnov Property for Weighted Lebesgue Spaces

We establish lower norm bounds for multivariate functions within weighted Lebesgue spaces, characterised by a summation of functions whose components solve a system of nonlinear integral equations. This problem originates in portfolio selection theory, where these equations allow one to identify mean-variance optimal portfolios, composed of standard European options on several underlying assets. We elaborate on the Smirnov property—an integrability condition for the weights that guarantees the uniqueness of solutions to the system. Sufficient conditions on weights to satisfy this property are provided, and counterexamples are constructed, where either the Smirnov property does not hold or the uniqueness of solutions fails.

Parseval–Goldstein-Type Theorems for Lebedev–Skalskaya Transforms

This paper investigates Parseval–Goldstein-type relationships in the framework of Lebedev–Skalskaya transforms. The research also examines the continuity properties of these transforms, along with their adjoint counterparts over weighted Lebesgue spaces. Furthermore, the behavior of Lebedev–Skalskaya transforms and their adjoint transforms in the context of weighted Lebesgue spaces is analyzed. This study aims to provide deeper insights into the functional properties and applications of these transforms in mathematical analysis.

Multilinear square operators meet new weight functions

Via the new weight Ap→θ(φ), the authors introduce a new class of multilinear square operators. The boundedness on the weighted Lebesgue space and the weighted Morrey space is obtained, respectively. Our results include the known results of the standard multilinear square operator and the weight Ap→. Moreover, the results in this article seem to be new even for one-linear case.

Localized operators on weighted Herz spaces

AbstractWe introduce the notion of localized operators. We extend the boundedness of the localized operators from the weighted Lebesgue spaces to the weighted Herz spaces. The localized operators include the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy‐type operators, the geometric mean operator, and the one‐sided maximal function. Therefore, this paper extends the mapping properties of the the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy‐type operators, the geometric mean operator, and the one‐sided maximal function to the weighted Herz spaces.

Generalized Carleson embeddings of Müntz spaces

This paper establishes Carleson embeddings of Müntz spaces MΛq into weighted Lebesgue spaces Lp(dμ), where μ is a Borel regular measure on [0,1] satisfying μ([1−ε])≲εβ. In the case β⩾1 we show that such measures are exactly the ones for which Carleson embeddings Lpβ↪Lp(dμ) hold. The case β∈(0,1) is more intricate, but we characterize such measures μ in terms of a summability condition on their moments. Our proof relies on a generalization of Lp estimates à la Gurariy-Macaev in the weighted Lp spaces setting, which we think can be of interest in other contexts.

Essential norm and Schatten class of Hankel operators on doubling Fock spaces

In this paper, we obtain the essential norm of Hankel operators from a doubling Fock space Fφp to a weighted Lebesgue space Lφq for all possible 1≤p,q<∞. We also characterize the Schatten-h class of Hankel operators from Fφ2 to Lφ2.

Weighted estimates for the Bergman projection on planar domains

We investigate weighted Lebesgue space estimates for the Bergman projection on a simply connected planar domain via the domain’s Riemann map. We extend the bounds which follow from a standard change-of-variable argument in two ways. First, we provide a regularity condition on the Riemann map, which turns out to be necessary in the case of uniform domains, in order to obtain the full range of weighted estimates for the Bergman projection for weights in a Békollè-Bonami-type class. Second, by slightly strengthening our condition on the Riemann map, we obtain the weighted weak-type (1,1) estimate as well. Our proofs draw on techniques from both conformal mapping and dyadic harmonic analysis.

Hardy Type Inequalities in Classical and Grand Lebesgue Spaces $L_{p)}$, $0&lt;p\le 1$, for Quasi-Monotone Functions

In 2020 Rovshan A. Bandaliev et al. proved the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces $L_{p)} (0,1)$, $0&lt;p\leq1$. In particular, they established similar results for the Hardy operator in classical weighted Lebesgue spaces. Moreover, it is proved that the grand Lebesgue space $L_{p) } (0,1)$ is a quasi-Banach function space. In this work, we are interested in Hardy inequalities applied to quasi-monotonic functions in classical Lebesgue spaces and grand Lebesgue spaces. We establish the boundedness of Hardy operator for quasi-monotone functions in grand Lebesgue spaces $L_{p)}$, $w(0,1),$ $0&lt;p\leq1$. In addition some integral inequalities for the Hardy operator are proved in classical weighted Lebesgue spaces $L_{p,w}(0,1)$, $0&lt;p&lt;1,$ for quasi-monotone functions. All inequalities are proved with sharp constants. Some results of Rovshan A. Bandaliev et al. are deduced as particular cases. Also other estimates are obtained in classical Lebesgue spaces for Hardy's operator and its dual.

On a class of Hausdorff type operators on interval

The paper studies integral operators on the interval of real line which naturally arise in some problems in the theory of integral equations and mathematical physics. We study boundedness in weighted Lebesgue spaces: Sufficient and necessary conditions for boundedness are given. Also, special important particular cases of operators and spaces are considered as examples. We construct identity approximation within the framework of the class of operators under study, and two different approaches are given.

Completeness of the systems of Bessel functions of index $-5/2$

Let $L^2((0;1);x^4 dx)$ be the weighted Lebesgue space of all measurable functions $f:(0;1)\rightarrow\mathbb C$, satisfying $\int_{0}^1 t^4 |f(t)|^2\, dt&lt;+\infty$. Let $J_{-5/2}$ be the Bessel function of the first kind of index $-5/2$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system $\big\{\rho_k^2\sqrt{x\rho_k}J_{-5/2}(x\rho_k):k\in\mathbb N\big\}$ in the space $L^2((0;1);x^4 dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(\rho_k)_{k\in\mathbb N}$. In this case, we study an integral representation of some class $E_{4,+}$ of even entire functions of exponential type $\sigma\le 1$. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than $-1$, due to B. Vynnyts'kyi, V. Dilnyi, O. Shavala and the author.

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.

Orbits of the backward shifts with limit points

We show that the bilateral backward shift on ℓp(Z,ω) that has a projective orbit with a non-zero limit point is supercyclic. This phenomenon holds also for Γ-supercyclicity, which extends a result obtained for the first time by Chan and Seceleanu. Moreover, we show that if K is a compact subset of ℓp(N,ω) such that its orbit under the unilateral backward shift B on ℓp(N,ω) has a non-zero weak limit point, then B is hypercyclic. Similar results for translation semigroups on weighted Lebesgue spaces are obtained.

Construction 条件 for bounded integral operator with super-homogeneous kernel in weighted Lebesgue space and formula of the operator norm

Construction 条件 for bounded integral operator with super-homogeneous kernel in weighted Lebesgue space and formula of the operator norm

Parameter Conditions for Boundedness of Two Integral Operators in Weighted Lebsgue Space and Calculation of Operator Norms

Firstly, Hilbert-type integral inequalities with best constant factors are established for two non-homogeneous kernels. Then, by utilizing the relationship between the Hilbert-type inequality and the integral operator of same kernel, the parameter conditions for two integral operators with non-homogeneous kernels in weighted Lebesgue space to be bounded and the formula for calculating the operator norm are obtained.

Heat kernel bounds and Ricci curvature for Lipschitz manifolds

Given any d-dimensional Lipschitz Riemannian manifold (M,g) with heat kernel p, we establish uniform upper bounds on p which can always be decoupled in space and time. More precisely, we prove the existence of a constant C>0 and a bounded Lipschitz function R:M→(0,∞) such that for every x∈M and every t>0, supy∈Mp(t,x,y)≤Cmin{t,R2(x)}−d/2.This allows us to identify suitable weighted Lebesgue spaces w.r.t. the given volume measure as subsets of the Kato class induced by (M,g). In the case ∂M≠0̸, we also provide an analogous inclusion for Lebesgue spaces w.r.t. the surface measure on ∂M.We use these insights to give sufficient conditions for a possibly noncomplete Lipschitz Riemannian manifold to be tamed, i.e. to admit a measure-valued lower bound on the Ricci curvature, formulated in a synthetic sense.

Compactness of commutators of fractional integral operators on ball Banach function spaces

&lt;abstract&gt;&lt;p&gt;Let $ 0 &amp;lt; \alpha &amp;lt; n $ and $ b $ be a locally integrable function. In this paper, we obtain the characterization of compactness of the iterated commutator $ (T_{\Omega, \alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator with the homogeneous kernel $ T_{\Omega, \alpha} $ on ball Banach function spaces. As applications, we derive the characterization of compactness via the commutator $ (T_{\Omega, \alpha})_b^m $ on weighted Lebesgue spaces, and further obtain a necessary and sufficient condition for the compactness of the iterated commutator $ (T_{\alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on Morrey spaces. Moreover, we also show the necessary and sufficient condition for the compactness of the commutator $ [b, T_{\alpha}] $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on variable Lebesgue spaces and mixed Morrey spaces.&lt;/p&gt;&lt;/abstract&gt;