Sharp weighted inequalities for multilinear fractional maximal operators and fractional integrals

In this paper, we study weighted inequalities for multilinear fractional maximal operators and fractional integrals. We prove sharp weighted Lebesgue space estimates for both operators when the vector of weights belongs to . In addition we prove sharp two weight mixed estimates for multilinear operators in the spirit of the linear estimates given in 3 .     

Weighted estimates for integral operators on local type spaces

We prove the weighted boundedness for a family of integral operators on Lebesgue spaces and local type spaces. To this end we show that can be controlled by the Calderón operator and a local maximal operator. This approach allows us to characterize the power weighted boundedness on Lebesgue spaces.     

Note on Nikodym maximal in the variable Lebesgue spaces

In this paper we consider the k‐plane Nikodym maximal estimates in the variable Lebesgue spaces . We first formulate the problem about the boundedness of the k‐plane Nikodym maximal and show that the maximal estimate in is equivalent to that in for . So, the optimal Nikodym maximal estimate in follows from Cordoba's estimate.     

Weighted L<Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-inequalities for multi-parameter Riesz type potentials and strong fractional maximal operators

We prove weighted L ^p -inequalities for multi-parameter Riesz type potentials, strong fractional maximal operators and their dyadic counterparts. Our proofs avoid the Good-λ inequalities used earlier in the R ^m -case and are based on our integrated multi-parameter summation by parts lemma, that might be of independent interest.     

Weighted norm inequalities for the commutators of multilinear singular integral operators

In this paper, the authors consider the weighted estimates for the commutators of multilinear Calderón-Zygmund operators.By introducing an operator which shifts the commutation, and establishing the weighted estimates for this new operator, the authors prove that, if p₁ ∈ （1,∞）, p₂,…,p_m ∈(1,∞], p ∈ （0,∞） with 1/p =Σ1≤k≤ m 1/pk, then for any weight w, the commutators of m-linear Galderón-Zygmund operator are bounded from L^P1（Rⁿ,M_l（logL）^σw）× p2（Rn,M^w）×...×Lpm（Rn,Mw） to Lp（Rn,w）with σ to be a constant depending only on p₁ and the order of commutator     

Some maximal inequalities on Triebel–Lizorkin spaces for

In this work we give some maximal inequalities in Triebel–Lizorkin spaces, which are “‐variants” of Fefferman–Stein vector‐valued maximal inequality and Peetre's maximal inequality. We will give some applications of the new maximal inequalities and discuss sharpness of some results.     

Some characterizations of upper doubling conditions on metric measure spaces

We provide several equivalent characterizations for the upper doubling condition introduced in the framework of T. Hytönen for non‐homogeneous metric measure spaces. We also introduce the “smooth strong upper doubling” condition and provide equivalent characterizations, which is related to the development of Littlewood–Paley theory on this non‐homogeneous setting.     

Weighted boundedness of the 2‐fold product of Hardy–Littlewood maximal operators

We study new weighted estimates for the 2‐fold product of Hardy–Littlewood maximal operators defined by . This operator appears very naturally in the theory of bilinear operators such as the bilinear Calderón–Zygmund operators, the bilinear Hardy–Littlewood maximal operator introduced by Calderón or in the study of pseudodifferential operators. To this end, we need to study Hölder's inequality for Lorentz spaces with change of measures Unfortunately, we shall prove that this inequality does not hold, in general, and we shall have to consider a weaker version of it.     

Hardy spaces for Bessel–Schrödinger operators

Consider the Bessel operator with a potential on , namely We assume that and is a nonnegative function. By definition, a function belongs to the Hardy space if Under certain assumptions on V we characterize the space in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to for with no additional assumptions on the potential V.     

Generalized weighted Morrey spaces and classical operators

We introduce the notion of generalized weighted Morrey spaces and investigate the boundedness of some operators in these spaces, such as the Hardy–Littlewood maximal operator, generalized fractional maximal operators, generalized fractional integral operators, and singular integral operators. We also study their boundedness in the vector‐valued setting.     

Littlewood–Paley characterizations of higher‐order Sobolev spaces via averages on balls

In this article, the authors characterize higher‐order Sobolev spaces , with , and , or with , and , via the Lusin area function and the Littlewood–Paley ‐function in terms of ball averages, where denotes the maximal integer not greater than . Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order .     

A compactness criterion and application to the commutators associated with Schrödinger operators

Let T be an integral operator. In this paper, we introduce a ‐compactness criterion of , where . As an application, we apply this criterion to deal with ‐compactness of commutators associated to Schrödinger operators with potentials in the reverse Hölder's class.     

A type theorem on the weighted Herz‐type Hardy spaces and applications

This paper gives a type theorem, which is a boundedness criterion for singular integral operators from the weighted Herz‐type Hardy spaces into the weighted local Herz‐type Hardy spaces. As applications, the corresponding mapping properties for the Cauchy integral and Calderón's commutators are obtained. In addition, a counter example is shown that neither is a Calderón–Zygmund singular integral operator bounded on the homogeneous local Herz‐type Hardy space, nor bounded on the classical local Hardy space.     

Spectral radius properties for layer potentials associated with the elastostatics and hydrostatics equations in nonsmooth domains

By producing a L² convergent Neumann series, we prove the invertibility of the elastostatics and hydrostatics boundary layer potentials on arbitrary Lipschitz domains with small Lipschitz character and 3D polyhedra with large dihedral angles.     

Weak Musielak–Orlicz Hardy spaces and applications

Let satisfy that , for any given , is an Orlicz function and is a Muckenhoupt weight uniformly in . In this article, the authors introduce the weak Musielak–Orlicz Hardy space via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of , respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or ‐function. All these characterizations for weighted weak Hardy spaces (namely, and with and ) are new and part of these characterizations even for weak Hardy spaces (namely, and with ) are also new. As an application, the boundedness of Calderón–Zygmund operators from to in the critical case is presented.     

Lindelöf theorems for monotone Sobolev functions in Orlicz spaces on uniform domains

In this paper, we are concerned with Lindelöf type theorems for monotone (in the sense of Lebesgue) Sobolev functions u on a uniform domain satisfying where ? denotes the gradient, denotes the distance from z to the boundary , φ is of log‐type and ω is a weight function satisfying the doubling condition.     

Sobolev‐type inequalities for potentials in grand variable exponent Lebesgue spaces

We introduce a new scale of grand variable exponent Lebesgue spaces denoted by . These spaces unify two non‐standard classes of function spaces, namely, grand Lebesgue and variable exponent Lebesgue spaces. The boundedness of integral operators of Harmonic Analysis such as maximal, potential, Calderón–Zygmund operators and their commutators are established in these spaces. Among others, we prove Sobolev‐type theorems for fractional integrals in . The spaces and operators are defined, generally speaking, on quasi‐metric measure spaces with doubling measure. The results are new even for Euclidean spaces.     

Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half‐space with a cavity C. Zero normal derivative is assumed at the boundary of the half‐space; differently, at ?C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ?C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half‐space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g, which describes a constant pressure at ?C, we recover a simplified representation based on a polarization tensor. Copyright © 2016 John Wiley & Sons, Ltd.     

Real‐variable characterizations of Musielak–Orlicz–Hardy spaces associated with Schrödinger operators on domains

Let n≥3, Ω be a strongly Lipschitz domain of and L_Ω:=?Δ+V a Schrödinger operator on L²(Ω) with the Dirichlet boundary condition, where Δ is the Laplace operator and the nonnegative potential V belongs to the reverse Hölder class for some q₀>n/2. Assume that the growth function satisfies that ?(x,·) is an Orlicz function, (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index , where and μ₀∈(0,1] denotes the critical regularity index of the heat kernels of the Laplace operator Δ on Ω. In this article, the authors first show that the heat kernels of L_Ω satisfy the Gaussian upper bound estimates and the Hölder continuity. The authors then introduce the ‘geometrical’ Musielak–Orlicz–Hardy space via , the Hardy space associated with on , and establish its several equivalent characterizations, respectively, in terms of the non‐tangential or the vertical maximal functions or the Lusin area functions associated with L_Ω. All the results essentially improve the known results even on Hardy spaces with p∈(n/(n + δ),1] (in this case, ?(x,t):=t^p for all x∈Ω and t∈[0,∞)). Copyright © 2016 John Wiley & Sons, Ltd.     

Characterizing the Phragmén‐Lindelöf condition for evolution on algebraic curves

For an algebraic curve V in ${\mathbb C}^k \times {\mathbb C}^n$ it is investigated when it satisfies the Phragmén‐Lindelöf condition PL(ω) of evolution in certain classes of ultradifferentiable functions. Necessary as well as sufficient conditions are obtained which lead to a complete characterization for curves in ${\mathbb C}\times {\mathbb C}^n$. Some examples illustrate how these results can be applied. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim