Search

We show the existence and uniqueness of local strong solutions of Keller-Segel system of parabolic-parabolic type for arbitrary initial data in the homogeneous Besov space which is scaling invariant. We also construct global strong solutions for small initial data, where the solutions belong to the Lorentz space in time direction. The proof is based on the maximal Lorentz regularity theorem of heat equations.

We refine some earlier results by Flores, Hernandez, Semenov, and Tradacete on compactness of the square of strictly singular endomorphisms and identifying general Banach lattices with the Kato property in the setting of rearrangement invariant spaces on [0, 1]. A Banach space X is said to have the Kato property if every strictly singular operator acting in X is compact. We show that each strictly singular operator bounded in a disjointly homogeneous rearrangement invariant space with the non-trivial Boyd indices has compact square, and that the Kato property is shared by a 2-disjointly homogeneous rearrangement invariant space X whenever $$X\supset G$$ , where G is the closure of $$L_\infty $$ in the Orlicz space, generated by the function $$e^{u^2}-1$$ . Moreover, a partial converse to the latter result is given under the assumption that $$X\subset L\log ^{1/2}L$$ . As an application we find rather sharp conditions, under which a Lorentz space $$\Lambda (2,\psi )$$ possesses the Kato property. In particular, $$\Lambda (2,\log ^{-\alpha }(e/u))$$ , with $$0

Let m, p, q ∈ (0, ∞) and let u, v, w be nonnegative weights. We characterize validity of the inequality $$\left(\int_{0}^{\infty} w(t)\left(f^{*}(t)\right)^{q} \mathrm{d} t\right)^{\frac{1}{q}} \leq C\left(\int_{0}^{\infty} v(t)\left(\int_{t}^{\infty} u(s)\left(f^{*}(s)\right)^{m} \mathrm{d} s\right)^{\frac{p}{m}} \mathrm{d} t\right)^{\frac{1}{p}}$$ for all measurable functions f defined on ℝn and provide equivalent estimates of the optimal constant C > 0 in terms of the weights and exponents. The obtained conditions characterize the embedding of the Copson—Lorentz space CLm,p(u, v), generated by the functional $$\|f\|_{C L^{m, p}(u, v)}:=\left(\int_{0}^{\infty} v(t)\left(\int_{t}^{\infty} u(s)\left(f^{*}(s)\right)^{m} \ \mathrm{d} s\right)^{\frac{p}{m}} \mathrm{d} t\right)^{\frac{1}{p}},$$ into the Lorentz space Λq(w). Moreover, the results are applied to describe the associated space of the Copson—Lorentz space CLm,p(u, v) for the full range of exponents m, p ∈ (0, ∞).

Conditions under which the system of dilations and translations of a function f in a symmetric space X is a representing system in X are found. Previously a similar result was known only for the spaces Lp, 1 ⩽ p < ∞. In particular, each function f with $$\int_0^1 {f(t)dt \ne 0} $$ in a Lorentz space Λϕ generates an absolutely representing system of dilations and translations in this space if and only if the function ϕ(t) is submultiplicative. The key role in the proof is played by the notion of the multiplier space with respect to tensor product.

Let (𝒳,d,μ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimensionω. Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spacesH*p,q(𝒳)H_*^{p,q}\left( \mathcal{X} \right)with the optimal rangep∈(ωω+η,∞)p \in \left( {{\omega \over {\omega + \eta }},\infty } \right)andq ∈(0, ∞]. When andp∈(ωω+η,1]p \in ({\omega \over {\omega + \eta }},1]q ∈(0, ∞], the authors establish its real-variable characterizations, respectively, in terms of radial maximal functions, non-tangential maximal functions, atoms, molecules, and various Littlewood–Paley functions. The authors also obtain its finite atomic characterization. As applications, the authors establish a real interpolation theorem on Hardy–Lorentz spaces, and also obtain the boundedness of Calderón–Zygmund operators on them including the critical cases. The novelty of this article lies in getting rid of the reverse doubling assumption ofμby fully using the geometrical properties of 𝒳 expressed via its dyadic reference points and dyadic cubes and, moreover, the results in the caseq ∈(0, 1) of this article are also new even when 𝒳 satisfies the reverse doubling condition.

The main purpose of this paper is to establish sharp Trudinger–Moser type inequalities on harmonic A N groups S for functions whose gradient is in the Lorentz space L ( n , q ) . Our results show that even if S is not with strictly negative sectional curvature, one can also replace the norm ∫ S | ∇ S u | n d V + ∫ S | u | n d V by ∫ S | ∇ S u | n d V in the Trudinger–Moser type inequalities.

We consider two types of maximal operators. We prove that, under some conditions, each maximal operator is bounded from the classical dyadic martingale Hardy space $H_{p}$ to the classical Lebesgue space $L_{p}$ and from the variable dyadic martingale Hardy space $H_{p(\cdot )}$ to the variable Lebesgue space $L_{p(\cdot )}$ . Using this, we can prove the boundedness of the Cesàro and Riesz maximal operator from $H_{p(\cdot )}$ to $L_{p(\cdot )}$ and from the variable Hardy–Lorentz space $H_{p(\cdot ),q}$ to the variable Lorentz space $L_{p(\cdot ),q}$ . As a consequence, we can prove theorems about almost everywhere and norm convergence.

We show existence and uniqueness theorem of local strong solutions to the Navier–Stokes equations with arbitrary initial data and external forces in the homogeneous Besov space with both negative and positive differential orders which is an invariant space under the change of scaling. If the initial data and external forces are small, then the local solutions can be extended globally in time. Our solutions also belong to the Serrin class in the usual Lebesgue space. The method is based on the maximal Lorentz regularity theorem of the Stokes equations in the homogeneous Besov spaces. As an application, we may handle such singular data as the Dirac measure and the single layer potential supported on the sphere.

In this paper, we consider the generalized Lorentz space of periodic functions of several variables and the Nikol’skii–Besov space of functions. The article establishes a sufficient condition for a function to belong from one generalized Lorentz space to another space in terms of the difference of the partial sums of the Fourier series of a given function. Exact in order estimates of the best approximation by trigonometric polynomials of functions of the Nikol’skii–Besov class are obtained.

It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality \[\mathcal{H}^{\beta}_{\infty}(\{x\in \Omega:|I_\alpha f(x)|>t\})\leq Ce^{-ct^{q'}}\] for all $\|f\|_{L^{N/\alpha,q}(\Omega)}\leq 1$ and any $\beta \in (0,N]$, where $\Omega \subset \mathbb{R}^N$, $\mathcal{H}^{\beta}_{\infty}$ is the Hausdorff content, $L^{N/\alpha,q}(\Omega)$ is a Lorentz space with $q \in (1,\infty]$, $q'=q/(q-1)$ is the H\"older conjugate to $q$, and $I_\alpha f$ denotes the Riesz potential of $f$ of order $\alpha \in (0,N)$., Comment: 25 pages

Given a bounded measurable function $\sigma$ on $\mathbb{R}^n$, we let $T_\sigma $ be the operator obtained by multiplication on the Fourier transform by $\sigma $. Let $0

A criterion for the continuity of weakly monotone functions is proposed in terms of the decreasing rearrangement of their gradient. Its use in the proof of the continuity of weakly monotone functions with gradient in rearrangement-invariant spaces is also demonstrated. In particular, weakly monotone functions whose gradient belongs to an Orlicz space or to a Lorentz space are discussed, and results available in the literature in this regard are extended and refined.

It is known (see [14] ) that, for every Lipschitz domain Ω on the plane Ω = { x + i y : y > ν ( x ) } , with ν a real valued Lipschitz function, there exists 1 ≤ p 0 2 so that the Dirichlet problem has a solution for every function f ∈ L p ( d s ) and every p ∈ ( p 0 , ∞ ) . Moreover, if p 0 > 1 , the result is false for every p ≤ p 0 . The purpose of this paper is to study in more detail what happens at the endpoint p 0 ; that is, we want to find spaces X ⊂ L p 0 so that the Dirichlet problem is solvable for every f ∈ X . These spaces X will be either the Lorentz space L p 0 , 1 ( d s ) or some type of logarithmic Orlicz space. Our results will be applied to the special case of Schwarz–Christoffel Lipschitz domains, among others, for which we explicitly compute the value of p 0 .

In this paper, by applying the well-known method for dealing with [Formula: see text]-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schrödinger equations, under the weak regularity assumption on the boundary of domains. As applications, the gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained.

Let $$p(\cdot ):\ {{\mathbb {R}}}^n\rightarrow (0,\infty ]$$ be a variable exponent function satisfying the globally log-Holder continuous condition, $$q\in (0,\infty ]$$ and A be a general expansive matrix on $${\mathbb {R}}^n$$ . Let $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ by means of the Littlewood–Paley g-function or the Littlewood–Paley $$g_\lambda ^*$$ -function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space $$L^{p(\cdot ),q}({\mathbb {R}}^n)$$ . Moreover, the finite atomic characterization of $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from $$H^{p(\cdot ),q}_A({\mathbb {R}}^n)$$ into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from $$H^{p(\cdot ),q}_A({\mathbb {R}}^n)$$ to $$L^{p(\cdot ),q}({\mathbb {R}}^n)$$ and, as consequences, some almost everywhere and norm convergences of these Bochner–Riesz and Weierstrass means are also obtained. These results on the Bochner–Riesz and the Weierstrass means are new even in the isotropic case.

Using the local approach to the global structure of a symmetric space $E$ we establish a relationship between strict $K$- monotonicity, lower (resp. upper) local uniform $K$-monotonicity, order continuity and the Kadec-Klee property for global convergence in measure. We also answer the question under which condition upper local uniform $K$-monotonicity concludes upper local uniform monotonicity. Finally, we present a correlation between $K$-order continuity and lower local uniform $K$-monotonicity in a symmetric space $E$ under some additional assumptions on $E$., Comment: 15 pages

We show existence theorem of global mild solutions with small initial data and external forces in Lorentz spaces with scaling invariant norms. If the initial data have more regularity in another scaling invariant class, then our mild solution is actually the strong solution. The result on local existence of solutions for large data is also discussed. Our method is based on the maximal regularity theorem on the Stokes equations in Lorentz spaces. Then we apply our theorem to prove existence of self-similar solutions provided both initial data and external forces are homogeneous functions. Since we construct the global solution by means of the implicit function theorem, as a byproduct, its stability with respect to the given data is necessarily obtained.

We study Kadec–Klee properties with respect to global (local) convergence in measure in quasi-Banach function spaces. First, we prove some general results which can be of independent interest. Next, we investigate these properties in symmetrizations $$E^{\left( *\right) }$$ . Finally, we apply general results to study these properties in Marcinkiewicz and Lorentz spaces.

In this paper, we investigate Navier-Stokes-Oseen equation describing flows of incompressible viscous fluid passing a translating and rotating obstacle. The existence, uniqueness, and polynomial stability of bounded and almost periodic weak mild solutions to Navier-Stokes-Oseen equation in the solenoidal Lorentz space \begin{document}$ L^{3}_{σ, w} $\end{document} are shown. Moreover, we also prove the unique existence of time-local mild solutions to this equation in the solenoidal Lorentz spaces \begin{document}$ L^{3,q}_{σ} $\end{document} .

Let ( Ω , F , P ) be a complete probability space. We introduce variable Lorentz space L p ( ⋅ ) , q ( Ω ) defined by rearrangement functions and its related properties. Then, we establish martingale inequalities among these martingale Hardy-Lorentz spaces H p ( ⋅ ) , q ( Ω ) by applying the interpolation theorem. Furthermore, we study the boundedness of the fractional integral operator in variable martingale Hardy spaces H p ( ⋅ ) M ( Ω ) and H p ( ⋅ ) , q M ( Ω ) .

We discuss the time periodic problem to the incompressible Navier–Stokes equations on the whole space R n , n ≥ 3 , with the external forces of non-divergence form. Firstly, we consider the existence of time periodic solutions in B C ( R ; L n , ∞ ( R n ) ) assuming the smallness of external forces in B C ( R ; L 1 ( R 3 ) ) and B C ( R ; L n 3 , ∞ ( R n ) ) in the case n ≥ 4 . Next, we show that the mild solution above becomes a strong solution in the topology of L n , ∞ ( R n ) with a natural condition of the external force, derived from the strong solvability of the inhomogeneous Stokes equations in L n , ∞ ( R n ) . For this aim, we re-construct a strong solvability of an abstract evolution equation where the associated semigroup is not strongly continuous at t = 0 .

Let p ( ⋅ ) : R n → ( 0 , ∞ ] be a variable exponent function satisfying the globally log-Holder continuous condition, q ∈ ( 0 , ∞ ] and A be a general expansive matrix on R n . In this article, the authors first introduce the anisotropic variable Hardy–Lorentz space H A p ( ⋅ ) , q ( R n ) associated with A, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain characterizations of H A p ( ⋅ ) , q ( R n ) , respectively, in terms of the atom and the Lusin area function. As an application, the authors prove that the anisotropic variable Hardy–Lorentz space H A p ( ⋅ ) , q ( R n ) serves as the intermediate space between the anisotropic variable Hardy space H A p ( ⋅ ) ( R n ) and the space L ∞ ( R n ) via the real interpolation. This, together with a special case of the real interpolation theorem of H. Kempka and J. Vybiral on the variable Lorentz space, further implies the coincidence between H A p ( ⋅ ) , q ( R n ) and the variable Lorentz space L p ( ⋅ ) , q ( R n ) when ess inf x ∈ R n p ( x ) ∈ ( 1 , ∞ ) .

We give an explicit formula to factorize functions in the Lorentz spaces L ( p , q ) . Some applications to centralizers, twisted sums and interpolation theory are included.

This paper studies the stability of a stationary solution of the Navier–Stokes system with a constant velocity at infinity in an exterior domain. More precisely, this paper considers the stability of the Navier–Stokes system governing the stationary solution which belongs to the weak L 3 -space L 3 , ∞ . Under the condition that the initial datum belongs to a solenoidal L 3 , ∞ -space, we prove that if both the L 3 , ∞ -norm of the initial datum and the L 3 , ∞ -norm of the stationary solution are sufficiently small then the system admits a unique global-in-time strong L 3 , ∞ -solution satisfying both L 3 , ∞ -asymptotic stability and L ∞ -asymptotic stability. Moreover, we investigate L 3 , r -asymptotic stability of the global-in-time solution. Using L p – L q type estimates for the Oseen semigroup and applying an equivalent norm on the Lorentz space are key ideas to establish both the existence of a unique global-in-time strong (or mild) solution of our system and the stability of our solution.

Let X be an arbitrary separable symmetric function space on [0, 1]. By using a combination of the frame approach and the notion of the multiplicator space $$\mathscr {M}(X)$$ of X with respect to the tensor product, we investigate the problem when the sequence of dyadic dilations and translations of a function $$f\in X$$ is a representing system in the space X. The main result reads that this holds whenever $$\int _0^1 f(t)\,dt\ne 0$$ and $$f\in \mathscr {M}(X).$$ Moreover, the condition $$f\in \mathscr {M}(X)$$ turns out to be sharp in a certain sense. In particular, we prove that a decreasing nonnegative function f, $$f\ne 0,$$ from a Lorentz space $$\varLambda _{\varphi }$$ generates an absolutely representing system of dyadic dilations and translations in $$\varLambda _{\varphi }$$ if and only if $$f\in \mathscr {M}(\varLambda _{\varphi }).$$

Let $X(\mathbb{R})$ be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$. The algebra $C_X(\dot{\mathbb{R}})$ of continuous Fourier multipliers on $X(\mathbb{R})$ is defined as the closure of the set of continuous functions of bounded variation on $\dot{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$ with respect to the multiplier norm. It was proved by C. Fernandes, Yu. Karlovich and the first author \cite{FKK19} that if the space $X(\mathbb{R})$ is reflexive, then the ideal of compact operators is contained in the Banach algebra $\mathcal{A}_{X(\mathbb{R})}$ generated by all multiplication operators $aI$ by continuous functions $a\in C(\dot{\mathbb{R}})$ and by all Fourier convolution operators $W^0(b)$ with symbols $b\in C_X(\dot{\mathbb{R}})$. We show that there are separable and non-reflexive Banach function spaces $X(\mathbb{R})$ such that the algebra $\mathcal{A}_{X(\mathbb{R})}$ does not contain all rank one operators. In particular, this happens in the case of the Lorentz spaces $L^{p,1}(\mathbb{R})$ with $1, Comment: To appear in Integral Equations and Operator Theory

We study the second spatial derivatives of suitable weak solutions to the incompressible Navier-Stokes equations in dimension three. We show that it is locally $L ^{\frac43, q}$ for any $q > \frac43$, which improves from the current result $L ^{\frac43, \infty}$. Similar improvements in Lorentz space are also obtained for higher derivatives of the vorticity for smooth solutions. We use a blow-up technique to obtain nonlinear bounds compatible with the scaling. The local study works on the vorticity equation and uses De Giorgi iteration. In this local study, we can obtain any regularity of the vorticity without any a priori knowledge of the pressure. The local-to-global step uses a recently constructed maximal function for transport equations., 39 pages

In this paper we study local regularity properties of weak solutions to a class of nonlinear noncoercive elliptic Dirichlet problems with $$L^1$$ datum. The model example is $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p(w)+b(x)|Dw|^{p-1}=f(x)&{}\text {in }\Omega ,\\ w=0&{}\text {on }\partial \Omega . \end{array}\right. } \end{aligned}$$Here $$\Omega \subset {\mathbb {R}}^N$$ is a bounded open subset, $$N>1$$, $$-\Delta _p$$ is the well known p-Laplace operator, $$1

In this paper, we investigate the Cauchy problem associated to a system of PDEs of Oldroyd type. The considered model describes the evolution of certain viscoelastic fluids within a corotational framework. The non-corotational setting is also addressed in dimension two. We show that some widespread results concerning the incompressible Navier-Stokes equations can be extended to the considered systems. In particular we show the existence and uniqueness of global-in-time classical solutions for large data in dimension two. This result is supported by suitable condition on the initial data to provide a global-in-time Lipschitz regularity for the flow, which allows to overcome specific challenging due to the lack of a decay in time of the main forcing terms. Secondly, we address the global-in-time well-posedness in dimension d ≥ 3 . We prove the propagation of Lipschitz regularities for the flow. For this result, we just assume the initial data to be sufficiently small in a critical Lorentz space.

We prove the continuity of the gradient of weak solutions to nonhomogeneous p ( ⋅ ) -Laplace systems with coefficients. We assume that the nonhomogeneous term belongs to the Lorentz space L n , 1 , the coefficient is Dini continuous and the variable exponent p ( ⋅ ) is logarithmic Dini type continuous.

We study the local structure of the separated point x in the space E ( ⁎ ) with respect to properties of its nonincreasing rearrangement x ⁎ in the space E . In particular, we get the characterizations of local structure in the Lorentz and Marcinkiewicz spaces. It is worth to mention that the local structure of a separated point can be applicable to the local best dominated approximation problems in Banach lattices.

Let M 1 3 ( c ) be an 3-dimensional Lorentz space form and C ( M 1 3 ( c ) ) denote the conformal transformation group of M 1 3 ( c ) . A spacelike surface x : M 2 → M 1 3 ( c ) is called a conformal homogeneous spacelike surface. If there exists a subgroup G ⊂ C ( M 1 3 ( c ) ) such that the orbit G ( p ) = x ( M 2 ) , p ∈ x ( M 2 ) . In this paper, we classify completely conformal homogeneous spacelike surfaces up to a conformal transformation of M 1 3 ( c ) .

In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant $C=C(\alpha,d)>0$ such that \[ \|I_\alpha F \|_{L^{d/(d-\alpha),1}(\mathbb{R}^d;\mathbb{R}^d)} \leq C \|F\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)} \] for all fields $F \in L^1(\mathbb{R}^d;\mathbb{R}^d)$ such that $\operatorname*{curl} F=0$ in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime $p=1$ of the well-established results for $p>1$., Comment: 19 pages

The main aim of this paper is to give two infinite series of examples of Lorentz space forms that can be obtained from Lorentz polyhedra by identification of faces. These Lorentz space forms are bi-quotients of the form $\Gamma_1\backslash G/\Gamma_2$, where $G=\widetilde{\operatorname{SU}(1,1)}\cong\widetilde{\operatorname{SL}(2,{\mathbb R})}$ is a simply connected Lie group with the Lorentz metric given by the Killing form, $\Gamma_1$ and $\Gamma_2$ are discrete subgroups of $G$ and $\Gamma_2$ is cyclic. A construction of polyhedral fundamental domains for the action of $\Gamma_1\times\Gamma_2$ on $G$ via $(g,h)\cdot x=gxh^{-1}$ was given in the earlier work of the second author. In this paper we give an explicit description of the fundamental domains obtained by this construction for two infinite series of groups. These results are connected to singularity theory as the bi-quotients $\Gamma_1\backslash G/\Gamma_2$ appear as links of certain quasi-homogeneous $\mathbb Q$-Gorenstein surface singularities, i.e.\ the intersections of the singular variety with sufficiently small spheres around the isolated singular point., Comment: 16 pages, 6 figures, 2 tables of figures

We say that a mapping [math] satisfies the [math] - [math] -property if [math] whenever [math] , where [math] means the Hausdorff measure. We prove that every mapping [math] of Sobolev class [math] with [math] satisfies the [math] - [math] -property for every [math] with ¶ [math] ¶ We prove also that for [math] and for the critical value [math] the corresponding [math] - [math] -property fails in general. Nevertheless, this [math] - [math] -property holds for [math] if we assume in addition that the highest derivatives [math] belong to the Lorentz space [math] instead of [math] . ¶ We extend these results to the case of fractional Sobolev spaces as well. Also, we establish some Fubini-type theorems for [math] -Nproperties and discuss their applications to the Morse–Sard theorem and its recent extensions.

[EN] We show a factorization through Lorentz spaces for Banach-space-valued operators defined in Banach function spaces. Although our results are inspired in the classical factorization theorem for operators from Ls-spaces through Lorentz spaces Lq,1 due to Pisier, our arguments are different and essentially connected with Maurey's theorem for operators that factor through Lp-spaces. As a consequence, we obtain a new characterization of Lorentz Lq,1-spaces in terms of lattice geometric properties, in the line of the (isomorphic) description of Lp-spaces as the unique ones that are p-convex and p-concave., Funding was provided by Secretaria de Estado de Investigacion, Desarrollo e Innovacion and FEDER (Grant No. MTM2016-77054-c2-1-P).

Consider the Cauchy problem of the Navier-Stokes equations in R n with initial data a in the homogeneous Besov space B ˙ p , q − 1 + n p ( R n ) for n p ∞ and 1 ≦ q ≦ ∞ . We show that the Stokes flow e t Δ a can be controlled in L α , q ( 0 , ∞ ; B ˙ r , 1 0 ( R n ) ) for 2 α + n r = 1 with p ≦ r ∞ , where L α , q denotes the Lorentz space. As an application, the global existence theorem of mild solutions for the small initial data is established in the above class which is slightly stronger than Serrin's. Conversely, if the global solution belongs to the usual Serrin class L α , q ( 0 , ∞ ; L r ( R n ) ) for r and α as above with 1 q ≦ ∞ , then the initial data a necessarily belongs to B ˙ r , q − 1 + n r ( R n ) . Moreover, we prove that such solutions are analytic in the space variables. Our method for the proof of analyticity is based on a priori estimates of higher derivatives of solutions in L p ( R n ) with Holder continuity in time.

We construct time-periodic solutions to the Boussinesq equations in a three-dimensional exterior domain. To this end, we estimate the L 1 norm of buoyancy, and give some estimates of the heat semigroup in the Lorentz space.

For p ∈ ( 1 , ∞ ) , we consider the following weighted Neumann eigenvalue problem on B 1 c , the exterior of the closed unit ball in R N : (0.1) − Δ p ϕ = λ g | ϕ | p − 2 ϕ in B 1 c , ∂ ϕ ∂ ν = 0 on ∂ B 1 , where Δ p is the p -Laplace operator and g ∈ L l o c 1 ( B 1 c ) is an indefinite weight function. Depending on the values of p and the dimension N , we take g in certain Lorentz spaces or weighted Lebesgue spaces and show that (0.1) admits an unbounded sequence of positive eigenvalues that includes a unique principal eigenvalue. For this purpose, we establish the compact embeddings of W 1 , p ( B 1 c ) into L p ( B 1 c , | g | ) for g in certain weighted Lebesgue spaces. For N > p , we also provide an alternate proof for the embedding of W 1 , p ( B 1 c ) into the Lorentz space L p ∗ , p ( B 1 c ) . Further, we show that the set of all eigenvalues of (0.1) is closed.

The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LlogL and more general Lorentz-Zygmund spaces., Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s00041-015-9455-5

Source at http://dx.doi.org/10.7153/jmi-2018-12-47 Let Λq( ω ), q > 0, denote the Lorentz space equipped with the (quasi) norm [MATHEMATICAL FORMULA] for a function f on [0,1] and with ω positive and equipped with some additional growth properties. A generalization of Boas theorem in the form of a two-sided inequality is obtained in the case of both general regular system [MATHEMATICAL FORMULA] and generalized Lorentz Λq( ω ) spaces

A Banach lattice E is called p-disjointly homogeneous, 1 ≤ p ≤ ∞ , when every sequence of pairwise disjoint normalized elements in E has a subsequence equivalent to the unit vector basis of l p . Employing methods from interpolation theory, we clarify which r.i. spaces on [ 0 , 1 ] are p-disjointly homogeneous. In particular, for every 1 p ∞ and any increasing concave function φ on [ 0 , 1 ] , which is not equivalent to neither 1 nor t, there exists a p-disjointly homogeneous r.i. space with the fundamental function φ. Moreover, it is shown that given 1 p ∞ and an increasing concave function φ with non-trivial dilation indices, there is a unique p-disjointly homogeneous space among all interpolation spaces between the Lorentz and Marcinkiewicz spaces associated with φ.

We construct Green's function for second order elliptic operators of the form $Lu=-\nabla \cdot (\mathbf{A} \nabla u + \boldsymbol{b} u)+ \boldsymbol c \cdot \nabla u+ du$ in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients $\mathbf A$ is uniformly elliptic and bounded and the lower order coefficients $\boldsymbol{b}$, $\boldsymbol{c}$, and $d$ belong to certain Lebesgue classes and satisfy the condition $d - \nabla \cdot \boldsymbol{b} \ge 0$. In particular, we allow the lower order coefficients to be singular. We also obtain the global pointwise bounds for the gradient of Green's function in the case when the mean oscillations of the coefficients $\mathbf A$ and $\boldsymbol{b}$ satisfy the Dini conditions and the domain is $C^{1, \rm{Dini}}$., 33 pages, added changes suggested by the referee. To appear in Communications in Partial Differential Equations

We analyze the relationship of the Arino–Muckenhoupt weights with the \(K\)-spaces obtained when the real interpolation method defined by a parameter function is applied to the pairs \((L^1,L^{\infty })\) and \((L^{1,\infty },L^{\infty })\) of function spaces associated to the semivariation of a vector measure.

We establish some interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin’s criterion and Escauriza-Seregin-Sverak’s criterion. We also show that if a weak solution u satisfies $$\left\| {u( \cdot ,t)} \right\|_{L^p } \leqslant C( - t)^{(3 - p)/2p} $$ for some 3 < p < ∞, then the number of singular points is finite.