Morrey-Campanato Spaces Research Articles

Integral operators on B σ -Morrey-Campanato spaces

We show the boundedness of the Hardy-Littlewood maximal operator, singular and fractional integral operators, and more general sublinear operators on Bσ-Morrey-Campanato spaces. These function spaces have been introduced recently to unify central Morrey spaces, λ-central mean oscillation spaces and usual Morrey-Campanato spaces. Using the Bσ-Morrey-Campanato spaces, we can study both local and global regularities of functions simultaneously, and unify a series of results on the boundedness of operators on several classical function spaces.

Some characterizations for weighted Morrey‐Campanato spaces

AbstractIn this paper, we establish some new characterizations for weighted Morrey‐Campanato spaces by using the convolution \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\varphi _{t_B}*f$\end{document} to replace the mean value fB of a function f in the weighted Morrey‐Campanato norm, where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\varphi \in {\cal S}({{\mathbb R}^n})$\end{document} is an appropriate Schwartz function. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Elementary characterizations of generalized weighted Morrey-Campanato spaces

Let α ∈ (0,∞), p, q ∈ [1,∞), s be a nonnegative integer, and ω ∈ A1(Rn) (the class of Muckenhoupt’s weights). In this paper, we introduce the generalized weighted Morrey-Campanato space L(α, p, q, s, ω; Rn) and obtain its equivalence on different p ∈ [1, β) and integers s ≥ └nα┘ (the integer part of nα), where β = (\( \frac{1} {q} \) − α)−1 when α < \( \frac{1} {q} \) or β = ∞ when α ≥ \( \frac{1} {q} \). We then introduce the generalized weighted Lipschitz space Λ(α, q, ω; Rn) and prove that L(α, p, q, s, ω; Rn) ⊂ Λ(α, q, ω; Rn) when α ∈ (0,∞), s ≥ └nα┘, and p ∈ [1, β).

Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators

AbstractLet be a space of homogeneous type in the sense of Coifman and Weiss, and let be a collection of balls in . The authors introduce the localized atomic Hardy space the localized Morrey-Campanato space and the localized Morrey-Campanato-BLO (bounded lower oscillation) space with α ∊ ℝ and p ∊ (0, ∞) , and they establish their basic properties, including and several equivalent characterizations for In particular, the authors prove that when α &gt; 0 and p ∊ [1, ∞), then and when p ∈(0,1], then the dual space of is Let ρ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces and , respectively, by and when is determined by ρ. The authors then obtain the boundedness from of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley g-function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators

AbstractLetbe a space of homogeneous type in the sense of Coifman and Weiss, and letbe a collection of balls in. The authors introduce the localized atomic Hardy spacethe localized Morrey-Campanato spaceand the localized Morrey-Campanato-BLO (bounded lower oscillation) spacewithα∊ ℝ andp∊ (0, ∞) , and they establish their basic properties, includingand several equivalent characterizations forIn particular, the authors prove that when α &amp;gt; 0 andp∊ [1, ∞), thenand whenp∈(0,1], then the dual space ofisLetρbe an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spacesand, respectively, byandwhenis determined byρ. The authors then obtain the boundedness fromof the radial and the Poisson semigroup maximal functions and the Littlewood-Paleyg-function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field

In this paper, we consider regularity criterion for solutions to the 3D viscous incompressible MHD equations in Morrey–Campanato spaces. It is proved that if the vorticity field ω = ∇ × u belongs to M ̇ 2 , 3 r for 0 < r ≤ 1 on [ 0 , T ] , then the solution remains smooth on [ 0 , T ] .

A Note on Weighted Estimates for the Schrödinger Operator

We study two problems closely related to each other. The first one is concerned with some smoothing weighted estimates with weights in a certain Morrey-Campanato spaces, for the solution of the free Schrodinger equation. The second one is a weighed trace inequality.

Classes of Hardy spaces associated with operators, duality theorem and applications

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2({\mathbb R}^n)$ with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space $H^1_L({\mathbb R}^n)$ and a $\textrm {BMO}_L({\mathbb R}^n)$ space associated with the operator $L$ were introduced and studied. In this paper we define a class of $H^p_L({\mathbb R}^n)$ spaces associated with the operator $L$ for a range of $p<1$ acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical $H^p({\mathbb R}^n)$ spaces. We then establish a duality theorem between the $H^p_L({\mathbb R}^n)$ spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on $H^p_L({\mathbb R}^n)$ and give the inclusion between the classical $H^p({\mathbb R}^n)$ spaces and the $H^p_L({\mathbb R}^n)$ spaces associated with operators.

The Navier–Stokes equations in the critical Morrey–Campanato space

We shall discuss various points on solutions of the 3D Navier-Stokes equations from the point of view of Morrey-Campanato spaces (global solutions, strong-weak uniqueness, the role of real interpolation, regularity).

Duals of Hardy spaces on homogeneous groups

AbstractHardy spaces on homogeneous groups were introduced and studied by Folland and Stein [3]. The purpose of this note is to show that duals of Hardy spaces Hp , 0 &lt; p ≤ 1, on homogeneous groups can be identified with Morrey–Campanato spaces. This closes a gap in the original proof of this fact in [3]. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

New function spaces of Morrey-Campanato type on spaces of homogeneous type

In the context of spaces of homogeneous type, we introduce and develop some new function spaces of Morrey-Campanato type. The new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity, and they generalize the classical Morrey-Campanato spaces. We show that the John-Nirenberg inequality holds on these spaces. We also establish the endpoint boundedness of fractional integrals.

A characterization of the Morrey-Campanato spaces

In this paper, we give a new characterization of the Morrey–Campanato spaces by using the convolution φ t B *f(x) to replace the minimizing polynomial P B f of a function f in the Morrey-Campanato norm, where is an appropriate Schwartz function.

Some Function Spaces Relative to Morrey-Campanato Spaces on Metric Spaces

In this paper, the author introduces the Morrey-Campanato spaces Lsp(X) and the spaces Cps(X) on spaces of homogeneous type including metric spaces and some fractals, and establishes some embedding theorems between these spaces under some restrictions and the Besov spaces and the Triebel-Lizorkin spaces. In particular, the author proves that Lsp(X) = Bs∞,∞(X) if 0 &lt; s &lt; ∞ and µ(X) &lt; ∞. The author also introduces some new function spaces Asp(X) and Bsp(X) and proves that these new spaces when 0 &lt; s &lt; 1 and 1 &lt; p &lt; ∞ are just the Triebel-Lizorkin space Fsp,∞(X) if X is a metric space, and the spaces A1p(X) and B1p(X) when 1 &lt; p &lt; ∞ are just the Hajłasz-Sobolev spaces W1p(X). Finally, as an application, the author gives a new characterization of the Hajłasz-Sobolev spaces by making use of the sharp maximal function.

On Coercive Estimates for Solutions of Linear Systems of Hydrodynamic Type

Local and nonlocal coercive estimates in Lebesgue, Holder, and Morrey–Campanato spaces are proved for solutions of linear stationary systems of hydrodynamic type. For this purpose, the technique proposed by Italian mathematicians for the study of solutions to equations and systems of elliptic and parabolic types is developed. Bibliography: 14 titles.

Unicité dans $L^3 (\mathbb R^3)$ et d'autres espaces fonctionnels limites pour Navier-Stokes

The main result of this paper is the proof of uniqueness for mild solutions of the Navier-Stokes equations in L^3 (\mathbb R^3) . This result is extended as well to some Morrey-Campanato spaces.

Global existence and local continuity of solutions for semilinear parabolic equations

We study the semilinear parabolic equations . The potentialis allowed to be in a new and general functional class characterized by one integrability condition, which controls both the global growth and local singularity of V while allowing global eistence and local continuity for solutions of the equation. Locally, the class we are proposing contains certain parabolic Morrey—Campanato spaces. Globally, the class is also general enoug to allow V under some grwoth conditions proposed by several other authors. Examples of V includeand all functions in which includes, for example where ε>0 and α>1. In fact the codition we are proposing is essentially optimal under certain conditions.

Local Approximation Spaces

The paper deals with characterizations of spaces F^s_{pq} and B^s_{pq} on \mathbb R^n, \mathbb R^n_+ and bounded domains via local oscillations and differences of functions. Furthermore Morrey-Campanato spaces are treated.