Weighted Norm Inequalities Research Articles

An Optimal Control Problem for the Navier–Stokes Equations with Point Sources

We analyze, in two dimensions, an optimal control problem for the Navier–Stokes equations where the control variable corresponds to the amplitude of forces modeled as point sources; control constraints are also considered. This particular setting leads to solutions to the state equation exhibiting reduced regularity properties. We operate under the framework of Muckenhoupt weights, Muckenhoupt-weighted Sobolev spaces, and the corresponding weighted norm inequalities and derive the existence of optimal solutions and first- and, necessary and sufficient, second-order optimality conditions.

Weighted norm inequalities for multilinearfractional integral operators on metric measure spaces

Let $({\mathcal~X},~d,~\mu)$ be a metric measurespace satisfying the geometrically doubling condition. We establish some weak type weighted norm inequalitiesfor the multilinear fractional integral operatorand the multilinear fractional maximal function.As an application, we also establish a weak type weighted norm inequalityon Morrey spaces for the multilinear fractional integral.

On weighted norm inequalities for oscillatory integral operators

We prove weighted norm inequalities with Muckenhoupt’s A_p-weights, for a wide class of oscillatory integral operators. As a consequence, one also obtains the boundedness of commutators of the aforementioned operators with functions in BMO.

Ap Weights in Directionally (γ,m) Limited Space and Applications

Let (X,d) be a directionally (γ,m)-limited space with every γ∈(0,∞). In this setting, we aim to study an analogue of the classical theory of Ap(μ) weights. As an application, we establish some weighted estimates for the Hardy–Littlewood maximal operator. Then, we introduce the relationship between directionally (γ,m)-limited spaceand geometric doubling. Finally, we obtain the weighted norm inequalities of the Calderón–Zygmund operator and commutator in non-homogeneous space.

Weighted norm inequalities for the Opdam–Cherednik transform

In this paper, we study several weighted norm inequalities for the Opdam–Cherednik transform. We establish different versions of the Heisenberg–Pauli–Weyl inequality for this transform. In particular, we give an extension of this inequality using weights with different exponents and present a variation of the inequality that incorporates [Formula: see text]-norms for the Opdam–Cherednik transform. Also, we prove a version of the Hardy–Littlewood inequality for this transform. Finally, we give other variations of the Heisenberg–Pauli–Weyl inequality such as the Nash-type and Clarkson-type inequalities for the Opdam–Cherednik transform.

Weighted norm inequalities for the maximal functions associated to a critical radius function on variable Lebesgue spaces

In this work we obtain boundedness on weighted variable Lebesgue spaces of some maximal functions that come from the localized analysis considering a critical radius function. This analysis appears naturally in the context of the Schrödinger operator L=−Δ+V in Rd, where V a non-negative potential which satisfying a some specific reverse Hölder condition. We consider new classes of weights that locally behave as the Muckenhoupt class for Lebesgue spaces with variable exponents considered in [6] and actually including them.

Commutators in the two scalar and matrix weighted setting

In this paper, we approach the two weighted boundedness of commutators via matrix weights. This approach provides both a sufficient and a necessary condition for the two weighted boundedness of commutators with an arbitrary linear operator in terms of one matrix weighted norm inequalities for this operator. Furthermore, using this approach, we surprisingly provide conditions that almost characterize the two matrix weighted boundedness of commutators with CZOs and completely arbitrary matrix weights, which is even new in the fully scalar one weighted setting. Finally, our method allows us to extend the two weighted Holmes/Lacey/Wick results to the fully matrix setting (two matrix weights and a matrix symbol), completing a line of research initiated by the first two authors.

Two matrix weighted inequalities for commutators with fractional integral operators

In this paper we prove two matrix weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a matrix symbol. More precisely, we extend the recent results of the second author, Pott, and Treil on two matrix weighted norm inequalities for commutators of Calderon-Zygmund operators and multiplication by a matrix symbol to the fractional integral operator setting. In particular, we completely extend the fractional Bloom theory of Holmes, Rahm, and Spencer to the two matrix weighted setting with a matrix symbol and also provide new two matrix weighted norm inequalities for commutators of fractional integral operators with a matrix symbol and two arbitrary matrix weights. These results are new even in the scalar one weighted setting with a scalar symbol.

Integral inequalities in amalgam spaces

Weighted norm inequalities for the Volterra integral operator with the kernel are characterized in the amalgam space for certain ranges of indices.

Weighted Norm Inequalities for Commutators of the Kato Square Root of Second Order Elliptic Operators on ℝn

Weighted Norm Inequalities for Commutators of the Kato Square Root of Second Order Elliptic Operators on ℝn

Weighted norm inequalities for the maximal operator on L^p(·) over spaces of homogeneous type

Given a space of homogeneous type \((X,d,\mu)\), we prove strong-type weighted norm inequalities for the Hardy-Littlewood maximal operator over the variable exponent Lebesgue spaces \(L^{p(\cdot)}\). We prove that the variable Muckenhoupt condition \(A_{p(\cdot)}\) is necessary and sufficient for the strong type inequality if \(p(\cdot)\) satisfies log-Hölder continuity conditions and \(1 < p_- \leq p_+ < \infty\). Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved by Cruz-Uribe, Fiorenza and Neugebauer (2012).

On weighted Calderon-Zygmund singular integrals and applications

This paper studies some weighted norm inequalities related to some Calderon-Zygmund singular integrals. Applications to the Sobolev-Gagliardo-Nirenberg inequality, differential forms, and the potential equation du= f are given.

Weighted Norm Inequalities for Marcinkiewicz Integrals with Non-Smooth Kernels on Spaces of Homogeneous Type

Weighted Norm Inequalities for Marcinkiewicz Integrals with Non-Smooth Kernels on Spaces of Homogeneous Type

Weighted norm inequalities for maximal operator of Fourier series

Weighted norm inequalities for maximal operator of Fourier series

Characterizations of predual spaces to a class of Sobolev multiplier type spaces

We characterize preduals and Köthe duals to a class of Sobolev multiplier type spaces. Our results fit in well with the modern theory of function spaces of harmonic analysis and are also applicable to nonlinear partial differential equations. We make use of several tools from nonlinear potential theory, weighted norm inequalities, and the theory of Banach function spaces to obtain our results.

Weighted Norm Inequalities for Multilinear Fourier Multipliers with Mixed Norm

In this paper, weighted norm inequalities for multilinear Fourier multipliers satisfying Sobolev regularity with mixed norm are discussed. Our result can be understood as a generalization of the result by Fujita and Tomita by using theLr-based Sobolev space,1<r≤2with mixed norm.

Weak and Strong Type Estimates for the Multilinear Littlewood–Paley Operators

Let \(S_{\alpha }\) be the multilinear square function defined on the cone with aperture \(\alpha \ge 1\). In this paper, we investigate several kinds of weighted norm inequalities for \(S_{\alpha }\). We first obtain a sharp weighted estimate in terms of aperture \(\alpha \) and \(\vec {w} \in A_{\vec {p}}\). By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman–Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyer’s conjecture, for which a Coifman–Fefferman inequality with the precise \(A_{\infty }\) norm is proved. Finally, we present the local decay estimates using the extrapolation techniques and dyadic analysis respectively. All the conclusions aforementioned hold for the Littlewood–Paley \(g^*_{\lambda }\) function. Some results are new even in the linear case.

Weighted norm inequalities for positive operators restricted on the cone of λ-quasiconcave functions — Corrigendum

Weighted norm inequalities for positive operators restricted on the cone of <i>λ</i>-quasiconcave functions — Corrigendum

New Weighted Norm Inequalities for Certain Classes of Multilinear Operators on Morrey-type Spaces

New Weighted Norm Inequalities for Certain Classes of Multilinear Operators on Morrey-type Spaces

Multilinear multipliers and singular integrals with smooth kernels on Hardy spaces

We consider weighted norm inequalities for multilinear multipliers whose symbols satisfy a product-type Hörmander condition. Our approach is to consider a more general family of multilinear singular integral operators associated to a family of smooth kernels that satisfy an L r L^r -Schwartz regularity condition. We give conditions for these operators to satisfy weighted Hardy space estimates and derive our results for multipliers as a special case. As an additional application, we use Rubio de Francia extrapolation to prove multilinear estimates on the variable exponent Hardy spaces.