Weak Lebesgue Spaces Research Articles

Weak‐type Fefferman–Stein inequality and commutators on weak Orlicz–Morrey spaces

AbstractWe consider the Fefferman–Stein inequality for weak Orlicz–Morrey spaces and the commutators and on weak Orlicz–Morrey spaces, where T is a Calderón–Zygmund operator, is a generalized fractional integral operator and b is a function in generalized Campanato spaces. We give a necessary and sufficient condition for the boundedness from of and from a weak Orlicz Morrey space to another weak Orlicz–Morrey space. We use the Fefferman–Stein inequality to prove the boundedness of the commutators. Since weak Orlicz–Morrey spaces contain the weak Lebesgue, weak Orlicz and weak Morrey spaces as special cases, our results contain the bounedness on these function spaces which are also new results.

Anisotropic elliptic system with variable exponents and degenerate coercivity with Lm data

In a bounded open domain , where , with Lipschitz boundary , we consider the Dirichlet problem for the elliptic system given by here, , , represents a vector-valued function, denotes the partial derivative of u with respect to , and the vector fields and are Carathédory functions. In this paper, we focus on nonlinear degenerate anisotropic elliptic systems with variable growth and data, where m is small. Specifically, we consider the case where the right-hand side term f belongs to with 1<m<N. To analyze this problem, we work with an appropriate functional setting that involves anisotropic Sobolev spaces with variable exponents and weak Lebesgue (Marcinkiewicz) spaces with variable exponents.

Characterizations of the BMO and Lipschitz Spaces via Commutators on Weak Lebesgue and Morrey Spaces

Characterizations of the BMO and Lipschitz Spaces via Commutators on Weak Lebesgue and Morrey Spaces

Estimation of non-uniqueness and short-time asymptotic expansions for Navier–Stokes flows

There is considerable evidence that solutions to the three-dimensional Navier–Stokes equations in the natural energy space are not unique. Assuming this is the case, it becomes important to quantify how non-uniqueness evolves. In this paper we provide an algebraic estimate for how rapidly two possibly non-unique solutions can separate over a compact spatial region in which the initial data has sub-critical regularity. Outside of this compact region, the data is only assumed to be in the scaling critical weak Lebesgue space and can be large. To establish this upper bound on the separation rate, we develop a new spatially local, short-time asymptotic expansion which is of independent interest.

Regularity criteria for weak solutions to the 3d co-rotational Beris-Edwards system via the pressure

We investigate regularity criteria for weak solutions to the Cauchy problem of the 3d co-rotational Beris-Edwards system for nematic liquid crystals, which couples the Navier–Stokes equations for the fluid velocity u with an evolution-diffusion equations for the Q-tenser. Our results yield that for any positive constant γ>0, if either the negative part of the associated pressure Π satisfiesΠ−[ln⁡(1+Π−)]1+γ∈L∞(R+;L32,∞(R3)), or the quantity 2Π+|u|2+|∇Q|2 satisfies(2Π++|u|2+|∇Q|2)[ln⁡(1+2Π++|u|2+|∇Q|2)]1+γ∈L∞(R+;L32,∞(R3)), then the weak solution (u,Q), to the 3d co-rotational Beris-Edwards system, is global-in-time smooth. Here, the subscript “−” and “+” denote the negative and the nonnegative part, respectively. L32,∞(R3) denotes the standard weak Lebesgue space. If Q≡0, then our results extend some previous known results from the theory of the 3d Navier–Stokes equations.

Anisotropic parabolic problem with variable exponent and regular data

In this paper, we study the existence of weak solutions for a class of nonlinear parabolic equations with regular data in the setting of variable exponent Sobolev spaces. We prove a "version" of a weak Lebesgue space estimate that goes back to "Lions J. L. Quelques méthodes de résolution des problèmes aux limites. Dunod, Paris (1969)" for parabolic equations with anisotropic constant exponents (pi(⋅)=pi).

On the generalized parabolic Hardy-Hénon equation: Existence, blow-up, self-similarity and large-time asymptotic behavior

This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in $C_{b}([0,\infty);L^{q_c,\infty}(\mathbb R^{n}))$. As a direct consequence, global existence for data in strong critical Lebesgue $L^{q_c}(\mathbb R^{n})$ follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time behavior of globally defined solutions. The zero solution $u\equiv 0$ is shown to be asymptotically stable in $L^{q_c}(\mathbb R^{n})$ -- it is the only self-similar solution which is initially small in $L^{q_c}(\mathbb R^{n})$. Moreover, blow-up results are obtained under mild assumptions on the initial data and the corresponding Fujita critical exponent is found.

Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator

Abstract In this paper, the boundedness of the Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate the weak bounds of the p-adic fractional Hausdorff operator on weighted p-adic weak Lebesgue spaces. We also obtain the sufficient condition of commutators of the p-adic fractional Hausdorff operator by taking symbol function from Lipschitz spaces. Moreover, strong type estimates for fractional Hausdorff operator and its commutator on weighted p-adic Lorentz spaces are also acquired.

On Function Spaces with Mixed Norms — A Survey

The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixed Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one is to provide a detailed proof for a useful inequality about mixed Lebesgue norms and the Hardy--Littlewood maximal operator and also to improve some known results on the maximal function characterizations of anisotropic mixed-norm Hardy spaces and the boundedness of Calderon--Zygmund operators from these anisotropic mixed-norm Hardy spaces to themselves or to mixed Lebesgue spaces. The last one is to correct some errors and seal some gaps existing in the known articles.

Weighted estimates for θ-type Calderón-Zygmund operator and its commutator on metric measure spaces

Let be a non-homogeneous metric measure space satisfying the geometrically doubling condition and the upper doubling condition. In this setting, the author proves that the θ-type Calderón-Zygmund operator is bounded on the weighted Lebesgue space and the weighted Morrey space. Furthermore, the boundedness of the commutator generated by and on weighted weak Lebesgue space and on weighted weak Morrey space is also obtained.

EXISTENCE OF FIXED POINTS AND BEST PROXIMITY POINTS OF p-CYCLIC BOYD-WONG CONTRACTIONS

We introduce a new contraction map called p-cyclic Boyd-Wongcontraction, defined on the union of p (p ≥ 2) non empty subsets of ametric space. We give sufficient conditions for the existence of a uniquefixed point, best proximity point or periodic point for the map and aniterative method is used to approximate the fixed point and the bestproximate point.variable exponents where the right-hand side f is in L q(·) , q(·) : Ω → (1, +∞). The functional setting involves anisotropic Sobolev spaces with variable exponents as well as weak Lebesgue (Marcinkiewicz) spaces with variable exponents.

EXISTENCE OF FIXED POINTS AND BEST PROXIMITY POINTS OF p-CYCLIC BOYD-WONG CONTRACTIONS

We introduce a new contraction map called p-cyclic Boyd-Wongcontraction, defined on the union of p (p ≥ 2) non empty subsets of ametric space. We give sufficient conditions for the existence of a uniquefixed point, best proximity point or periodic point for the map and aniterative method is used to approximate the fixed point and the bestproximate point.variable exponents where the right-hand side f is in L q(·) , q(·) : Ω → (1, +∞). The functional setting involves anisotropic Sobolev spaces with variable exponents as well as weak Lebesgue (Marcinkiewicz) spaces with variable exponents.

Maximal Non-compactness of Sobolev Embeddings

It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity of spaces $L^{p,\infty}$ has been open, too. In this paper, we solve both these problems. We first show that weak Lebesgue spaces are never disjointly superadditive, so the suggested technique is ruled out. But then we show that, perhaps somewhat surprisingly, the measure of non-compactness of a sharp Sobolev embedding coincides with the embedding norm nevertheless, at least as long as $p<\infty$. Finally, we show that if the target space is $L^{\infty}$ (which formally is also a weak Lebesgue space with $p=\infty$), then the things are essentially different. To give a comprehensive answer including this case, too, we develop a new method based on a rather unexpected combinatorial argument and prove thereby a general principle, whose special case implies that the measure of non-compactness, in this case, is strictly less than its norm. We develop a technique that enables us to evaluate this measure of non-compactness exactly.

Weighted estimates for commutators of anisotropic Calderón-Zygmund operators

Let T be an anisotropic Calderón-Zygmund operator and with and being an anisotropic weighted Lipschitz space. The goal of the paper is to give five boundedness theorems of the commutator . Precisely, is bounded from to , where , , and ; is bounded from to , when or , , and ; is bounded from anisotropic weighted Hardy space to , if or , and ; is bounded from to weak Lebesgue space with , , which are extensions of isotropic settings and new even for the isotropic weighted and anisotropic unweighted settings.

On the volume of unit balls of finite-dimensional Lorentz spaces

We study the volume of unit balls Bp,qn of finite-dimensional Lorentz sequence spaces ℓp,qn. We give an iterative formula for vol(Bp,qn) for the weak Lebesgue spaces with q=∞ and explicit formulas for q=1 and q=∞. We derive asymptotic results for the nth root of vol(Bp,qn) and show that [vol(Bp,qn)]1∕n≍p,qn−1∕p for all 0<p<∞ and 0<q≤∞. We study further the ratio between the volume of unit balls of weak Lebesgue spaces and the volume of unit balls of classical Lebesgue spaces. We conclude with an application of the volume estimates and characterize the decay of the entropy numbers of the embedding of the weak Lebesgue space ℓp,∞n into ℓpn.

On Generalized Holder's Inequality in Weak Morrey Spaces

In this paper we reprove generalized H\"{o}lder's inequality in weak Morrey spaces.In particular, we get sharper bounds than those in \cite{gunawan2}. Thebounds are obtained through the relation of weak Morrey spaces and weakLebesgue spaces.

Pointwise Multipliers on Weak Morrey Spaces

Abstract We consider generalized weak Morrey spaces with variable growth condition on spaces of homogeneous type and characterize the pointwise multipliers from a generalized weak Morrey space to another one. The set of all pointwise multipliers from a weak Lebesgue space to another one is also a weak Lebesgue space. However, we point out that the weak Morrey spaces do not always have this property just as the Morrey spaces not always.

Anisotropic nonlinear elliptic systems with variable exponents and degenerate coercivity

In this paper, we prove existence results for distributional solutions of anisotropic nonlinear elliptic systems with variable growth and degenerate coercivity. The functional setting involves anisotropic Sobolev spaces with variable exponents as well as weak Lebesgue (Marcinkiewicz) spaces with variable exponents.

On regularity and singularity for $$L^{\infty }(0,T;L^{3,w}(\mathbb {R}^3))$$ solutions to the Navier–Stokes equations

We study local regularity properties of a weak solution u to the Cauchy problem of the incompressible Navier–Stokes equations. We present a new regularity criterion for the weak solution u satisfying the condition $$L^{\infty }(0,T;L^{3,w} (\mathbb {R}^3))$$ without any smallness assumption on that scale, where $$L^{3,w}(\mathbb {R}^3)$$ denotes the standard weak Lebesgue space. As an application, we conclude that there are at most a finite number of blowup points at any singular time t.

Weak Triebel–Lizorkin Spaces with Variable Integrability, Summability and Smoothness

Weak Triebel–Lizorkin spaces with variable integrability, summability and smoothness are first introduced. Then we establish a vector estimate for weak Lebesgue spaces with variable exponent. As an application we give equivalent quasi-norms in these new spaces by means of Peetre's maximal functions. Finally, we obtain the boundedness of the \varphi transform on these new spaces and their atomic and molecular decompositions.