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Given m ∈ N ∖ { 0 } and a compact Riemannian manifold N , we construct for every map u in the critical Sobolev space W m / (m + 1) , m + 1 (S m , N) , a map U : B 1 m + 1 → N whose trace is u and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we present a refined version of the (classical) Stein inequality for the Fourier transform, elevating it to a new level of accuracy. Furthermore, we establish extended analogues of a more precise version of the Stein inequality for the Fourier transform, broadening its applicability from the range 1 < p < 2 to 2 ≤ p < ∞. [ABSTRACT FROM AUTHOR]

We consider the time periodic solutions to the Keller-Segel (KS) system coupled to Navier-Stokes equation in R N (N ≥ 4). Based on the theory of real interpolation as Meyer and Yamazaki used, we first consider the existence of time periodic solution in B C (R , L s , ∞ (R N)) when S (n) = n − 1 2 . Next for the case S (n) = 0 , we study the mild solution and point out it can become strong solution in B C (R , L s , ∞ (R N)) if the external force g satisfies a natural condition which comes from the strong solvability of the inhomogeneous Stokes equations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study an inhomogeneous Dirichlet problem of a superquadratic nonlinear elliptic system with VMO coefficients in Reifenberg flat domains. We derive pointwise potential estimates for weak solution to the Dirichlet problem. As a consequence, we derive global integrability estimates for solutions with respect to Lorentz and Morrey norms. [ABSTRACT FROM AUTHOR]

Abstract The main purpose of this paper is to establish sharp Trudinger–Moser type inequalities on hyperbolic spaces B n for functions whose hyperbolic gradient is in the Lorentz space L (n , q) , n ≥ 2. Namely, if Ω ⊂ B n is bounded and 1 < q < ∞ , we will show that there exists a constant C = C (n , q) such that for all u ∈ C 0 ∞ (Ω) with ‖ ∇ H u ‖ L n , q (Ω) ≤ 1 , we have 1 | Ω | ∫ Ω e β q | u (x) | q ′ d V ≤ C , where β q = (n ω n 1 / n) q ′ and ∇ H is the hyperbolic gradient; if Ω = B n , we use only the norm ‖ ∇ H u ‖ L n , q (B n) rather than ‖ ∇ H u ‖ L n , q (B n) + ‖ u ‖ L n , q (B n) (unlike the case of Euclidean spaces) and will show that there exists a constant C = C (n , q) such that for all u ∈ C 0 ∞ (B n) with ‖ ∇ H u ‖ L n , q (B n) ≤ 1 , we have ∫ B n Φ q (β q | u (x) | q ′ ) d V ≤ C , where Φ q (t) = e t − ∑ j = 0 j q − 2 t j j ! , j q = min ⁡ { j ∈ N : j ≥ 1 + n / q ′ }. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study semilinear elliptic systems with critical nonlinearity of the form (0.1) Δ u = Q (x , u , ∇ u) , for u : R n → R K , Q has quadratic growth in ∇ u. Our work is motivated by elliptic systems for harmonic map and biharmonic map. When n = 2 , such a system does not have smooth regularity in general for W 1 , 2 weak solutions, by a well-known example of J. Frehse. Classical results of harmonic map, proved by F. Hélein (for n = 2) and F. Béthuel (for n ≥ 3), assert that a W 1 , n weak solution of harmonic map is always smooth. We extend Béthuel's result to general system (0.1) , that a W 1 , n weak solution of the system is smooth for n ≥ 3. For a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map, we prove a similar result, that a W 2 , n / 2 weak solution of such system is always smooth, for n ≥ 5. We also construct various examples, and these examples show that our regularity results are optimal in various sense. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 . [ABSTRACT FROM AUTHOR]

We investigate K -order continuity in a symmetric space E using the fundamental function ϕ of E . We also show a connection between reflexivity and K -order continuity in symmetric spaces. Next, we present several results devoted to a characterization of uniform K -monotonicity and decreasing (increasing) uniform K -monotonicity in symmetric spaces. We also discuss a relationship between decreasing (resp. increasing) uniform monotonicity and decreasing (resp. increasing) uniform K -monotonicity. Next, we deliberate a correlation between uniform K -monotonicity and uniform rotundity in symmetric spaces. Finally, employing K -monotonicity properties and K -order continuity we provide solvability and stability of the best approximation problem in the sense of the Hardy–Littlewood–Pólya relation ≺ in symmetric spaces. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper criteria for non-squareness properties (non-squareness, local uniform non-squareness and uniform non-squareness) of Orlicz–Lorentz sequence spaces and of their n-dimensional subspaces () as well as of the subspaces of all order continuous elements in are given. Since degenerate Orlicz functions φ and degenerate weight sequences ω are also admitted, these investigations concern the most possible wide class of Orlicz–Lorentz sequence spaces. Finally, as immediate consequences, criteria for all non-squareness properties of Orlicz sequence spaces, which complete the results of Sundaresan (1966) [53], Hudzik (1985) [23], Hudzik (1985) [24], are deduced. It is worth recalling that uniform non-squareness is an important property, because it implies super-reflexivity as well as the fixed point property (see James (1964) [31], James (1972) [33] and García-Falset et al. (2006) [19]). [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence and uniqueness questions on weak solutions of the stationary Navier–Stokes equations in an exterior domain in , where the external force is given by with . First, we prove the existence and uniqueness of a weak solution for with and provided is sufficiently small. Here denotes the well-known Lorentz space. We next show that weak solutions satisfying the energy inequality are unique for under the same smallness condition on . This result provides a complete answer to the uniqueness question of weak solutions satisfying the energy inequality, the existence of which was proved by Leray in 1933. Finally, we establish the existence of weak solutions for data in a very large class, for instance, in , which generalizes Leray’s existence result. [Copyright &y& Elsevier]

Abstract: In this paper, we study the structure of closed algebraic ideals in the algebra of operators acting on a Lorentz sequence space. [Copyright &y& Elsevier]

Abstract: We study subspaces of Lorentz spaces and provide an easy-to-check characterization of strictly singular operators defined on these spaces. As an application we obtain stability under duality for the class of strictly singular operators on spaces, extending a theorem of L. Weis for operators on spaces. [Copyright &y& Elsevier]

Abstract: In this paper, the necessary and sufficient conditions are found for the boundedness of the rough B-fractional integral operators from the Lorentz spaces to , , , and from to , , . As a consequence of this, the same results are given for the fractional B-maximal operator and B-Riesz potential. [Copyright &y& Elsevier]

Abstract: We study the optimal domain for the Hardy operator considered with values in a rearrangement invariant space. In particular, this domain can be represented as the space of integrable functions with respect to a vector measure defined on a δ-ring. A precise description is given for the case of the minimal Lorentz spaces. [Copyright &y& Elsevier]

Abstract: We associate to every function space, and to every entropy function E, a scale of spaces similar to the classical Lorentz spaces . Necessary and sufficient conditions for they to be normed spaces are proved, their role in real interpolation theory is analyzed, and a number of applications to functional and interpolation properties of several variants of Lorentz spaces and entropy spaces are given. [Copyright &y& Elsevier]

Abstract: In this paper, we give a characterization of best constant approximants in Lorentz spaces , , and we establish a way to obtain the best constant approximants maximum and minimum. We also study monotony of the best constant approximation operator. [Copyright &y& Elsevier]