Commutators of fractional maximal operator on generalized Orlicz–Morrey spaces

In the present paper, we shall give necessary and sufficient conditions for the Spanne and Adams type boundedness of the commutators of fractional maximal operator on generalized Orlicz–Morrey spaces, respectively. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators.     

A characterization for Adams-type boundedness of the fractional maximal operator on generalized Orlicz–Morrey spaces

In the present paper, we shall give a characterization for weak/strong Adams-type boundedness of the fractional maximal operator on generalized Orlicz–Morrey spaces.     

Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces

A boundedness result is established for multilinear singular integral operators on the homogeneous Morrey–Herz spaces. As applications, two corollaries about interesting cases of the boundedness of the considered operators on the homogeneous Morrey–Herz spaces are obtained.     

Boundedness of fractional integral operators on α-modulation spaces

In this paper, we obtain the boundedness of the fractional integral operators, the bilinear fractional integral operators and the bilinear Hilbert transform on α-modulation spaces.     

Sharp weighted bounds for the Hardy–Littlewood maximal operators on Musielak–Orlicz spaces

Let \({\varphi}\) be a Musielak–Orlicz function satisfying that, for any \({(x,\,t)\in{\mathbb R}^n \times [0, \infty)}\), \({\varphi(\cdot,\,t)}\) belongs to the Muckenhoupt weight class \({A_\infty({\mathbb R}^n)}\) with the critical weight exponent \({q(\varphi) \in [1,\,\infty)}\) and \({\varphi(x,\,\cdot)}\) is an Orlicz function with uniformly lower type \({p^{-}_{\varphi}}\) and uniformly upper type \({p^+_\varphi}\) satisfying \({q(\varphi) < p^{-}_{\varphi}\le p^{+}_{\varphi} < \infty}\). In this paper, the author obtains a sharp weighted bound involving \({A_\infty}\) constant for the Hardy–Littlewood maximal operator on the Musielak–Orlicz space \({L^{\varphi}}\). This result recovers the known sharp weighted estimate established by Hytönen et al. in [J. Funct. Anal. 263:3883–3899, 2012].     

Compactness and essential norms of composition operators on Orlicz–Lorentz spaces

In this work, we present necessary and sufficient conditions for compactness of the composition operator on Orlicz–Lorentz spaces and determine upper and lower estimates for the essential norm of the composition operator on these spaces.     

Sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces

The aim of this paper is to establish the boundedness of certain sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. The Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.     

Characterization of image spaces of Riemann-Liouville fractional integral operators on Sobolev spaces Wm,p (Ω)

Science China Mathematics - Fractional operators are widely used in mathematical models describing abnormal and nonlocal phenomena. Although there are extensive numerical methods for solving the...     

Singular integral operators on tent spaces over spaces of homogeneous type: Fefferman–Stein box maximal functions and pointwise Carleson type estimates

In this article we generalize the singular integral operator theory on weighted tent spaces to spaces of homogeneous type. This generalization of operator theory is in the spirit of C. Fefferman and Stein since we use some auxiliary functionals on tent spaces which play roles similar to the Fefferman–Stein sharp and box maximal functions in the Lebesgue space setting. Our contribution in this operator theory is twofold: for singular integral operators (including maximal regularity operators) on tent spaces pointwise Carleson type estimates are proved and this recovers known results; on the underlying space no extra geometrical conditions are needed and this could be useful for future applications to parabolic problems in rough settings.     

Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces

In this paper, the boundedness of a large class of sublinear commutator operators T _b generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces \({M_{p,\varphi }}(w)\) with the weight function w belonging to Muckenhoupt’s class A _p is studied. When 1 < p < ∞ and b ∈ BMO, sufficient conditions on the pair (φ ₁, φ ₂) which ensure the boundedness of the operator T _b from \({M_{p,\varphi 1}}(w)\) to \({M_{p,\varphi 2}}(w)\) are found. In all cases the conditions for the boundedness of T _b are given in terms of Zygmund-type integral inequalities on (φ ₁, φ ₂), which do not require any assumption on monotonicity of φ ₁(x, r), φ ₂(x, r) in r. Then these results are applied to several particular operators such as the pseudo-differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.     

Composition operators in Orlicz–Sobolev spaces

We obtain necessary and sufficient conditions for a homeomorphism of domains in a Euclidean space to generate a bounded embedding operator of the Orlicz–Sobolev spaces defined by a special class of N-functions.     

Interpolation operators in Orlicz–Sobolev spaces

We study classical interpolation operators for finite elements, like the Scott–Zhang operator, in the context of Orlicz–Sobolev spaces. Furthermore, we show estimates for these operators with respect to quasi-norms which appear in the study of systems of p-Laplace type.     

Some Characterizations of Lipschitz Spaces via Commutators on Generalized Orlicz–Morrey Spaces

In this paper, we give some new characterizations of the Lipschitz spaces via the boundedness of commutators associated with the fractional maximal operator, Riesz potential and Calderón–Zygmund operator on generalized Orlicz–Morrey spaces.     

Boundedness of Monge–Ampère singular integral operators on Besov spaces

ABSTRACT

The purpose of this paper is to establish a theory of Besov spaces associated with sections under only the doubling condition on the measure and prove that Monge–Ampère singular integral operators are bounded on these spaces.     

Elliptic operators and maximal regularity on periodic little-H?lder spaces

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-H?lder spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions. We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of the paper hold for elliptic differential operators with operator-valued coefficients, in the setting of vector-valued functions.