Local Morrey Spaces Research Articles

Doob's inequality, Burkholder–Gundy inequality and martingale transforms on martingale local Morrey spaces

Doob's inequality, Burkholder–Gundy inequality and martingale transforms on martingale local Morrey spaces

Stein–Weiss inequalities on local Morrey spaces with variable exponents

Stein–Weiss inequalities on local Morrey spaces with variable exponents

Weak type estimates of genuine Calderón–Zygmund operators on the local Morrey spaces associated with ball quasi-Banach function spaces

Abstract Weak type estimates for genuine Calderón–Zygmund operators are established on the local Morrey spaces associated with ball quasi-Banach function spaces by two different methods. Above all, we obtain weak type estimates for the operator on the local weak Morrey spaces with variable exponents.

BOUNDEDNESS OF PSEUDO-DIFFERENTIAL OPERATORS ON WEIGHTED HARDY SPACES AND VARIABLE EXPONENTS HARDY LOCAL MORREY SPACES

In this paper, we use the atomic decomposition to establish the boundedness of pseudo-differential operators belonging to Hörmander class on weighted Hardy spaces $H^p(\omega)$ and on variable exponents Hardy local Morrey spaces $HLM_{p(\cdot)}^{u(\cdot)}$.

Liouville-Type Theorems for 3D Stationary Tropical Climate Model in Mixed Local Morrey Spaces

Liouville-Type Theorems for 3D Stationary Tropical Climate Model in Mixed Local Morrey Spaces

Generalized fractional integrals in the vanishing generalized weighted local and global Morrey spaces

In this paper, we prove the boundedness of generalized fractional integral operators I? in the vanishing generalized weighted Morrey-type spaces, such as vanishing generalized weighted local Morrey spaces and vanishing generalized weighted global Morrey spaces by using weighted Lp estimates over balls. In more detail, we obtain the Spanne-type boundedness of the generalized fractional integral operators I? in the vanishing generalized weighted local Morrey spaces with wq ? A1+ q/p' for 1 < p < q < ?, and from the vanishing generalized weighted local Morrey spaces to the vanishing generalized weighted weak local Morrey spaces with w A1,q for p = 1, 1 < q < ?. We also prove the Adams-type boundedness of the generalized fractional integral operators I? in the vanishing generalized weighted global Morrey spaces with w Ap,q for 1 < p < q < ? and from the vanishing generalized weighted global Morrey spaces to the vanishing generalized weighted weak global Morrey spaces with w A1,q for p = 1, 1 < q < ?. The our all weight functions belong to Muckenhoupt-Weeden classes Ap,q.

The Liouville theorems for 3D stationary tropical climate model in local Morrey spaces

This paper studies the Liouville theorems for the 3D stationary tropical climate model in local Morrey spaces. If (u,v,θ) belongs to some local Morrey spaces, there is only trivial solution to the stationary tropical climate model. The obtained results extend and generalize the classical results for the stationary tropical climate model in Lebesgue spaces, which was established by Ding and Wu (2021).

Stein-Weiss inequality for local mixed radial-angular Morrey spaces

Abstract In this article, a generalization of the well-known Stein-Weiss inequality for the fractional integral operator on functions with different integrability properties in the radial and the angular direction in local Morrey spaces is established. We find that some conditions can be relaxed for the Stein-Weiss inequality for local mixed radial-angular Morrey spaces.

On Liouville type theorems for three-dimensional stationary MHD and Hall-MHD equations

In this paper, we study the Liouville type theorems of the MHD (magnetohydrodynamics) equation and Hall-MHD equation in three-dimensional steady states. In the framework of local Morrey space, we overcome the estimations of the pressure term with an infinite energy solution and obtain some sufficient conditions to guarantee the existence of a unique trivial solution for these two kinds of systems. Then, by using the embedding relations between the Lorentz spaces and local Morrey spaces, we give the Liouville type theorems for the two kinds of systems in the Lorentz spaces. The obtained results do not need the condition of finite Dirichlet integral, and hence generalize and improve some existing results about the two kinds of systems in the framework of Lebesgue space.

Spherical Maximal Function on Local Morrey Spaces with Variable Exponents

Spherical Maximal Function on Local Morrey Spaces with Variable Exponents

Liouville type theorems for the stationary Hall‐magnetohydrodynamic equations in local Morrey spaces

This paper is concerned with the Liouville type theorems for the 3D stationary incompressible Hall‐magnetohydrodynamic (Hall‐MHD) equations. We establish that under some sufficient conditions in local Morrey spaces, solutions of the stationary Hall‐MHD equations are identically zero. In particular, we also prove Liouville type results for the stationary incompressible MHD equations on . Our theorems extend and generalize the classical results for the stationary incompressible Navier–Stokes equations.

On interpolation and 𝐾-monotonicity for discrete local Morrey spaces

A formula is given that makes it possible to reduce the calculation of an interpolation functor on a pair of local Morrey spaces to the calculation of this functor on pairs of vector function spaces constructed from the ideal spaces involved in the definition of the Morrey spaces in question. It is shown that a pair of local Morrey spaces is K K -monotone if and only if the pair of vector function spaces mentioned above is K K -monotone. This reduction makes it possible to obtain new interpolation theorems even for classical local spaces.

Singular and Fractional Integral Operators on Weighted Local Morrey Spaces

We obtain a characterization of the weighted inequalities for the Riesz transforms on weighted local Morrey spaces. The condition is sufficient for the boundedness on the same spaces of all Calderón–Zygmund operators suitably defined on the functions of the space. In the case of the fractional maximal operator and the fractional integral we obtain a characterization valid for exponents satisfying the Sobolev relation. For power weights we get sharp results for these operators in the usual versions of weighted Morrey spaces, neither restricted to the Sobolev relation of the exponents nor to the one-weighted setting.

Liouville Theorems for a Stationary and Non-stationary Coupled System of Liquid Crystal Flows in Local Morrey Spaces

We consider here the simplified Ericksen–Leslie system on the whole space $$\mathbb {R}^{3}$$ . This system deals with the incompressible Navier–Stokes equations strongly coupled with a harmonic map flow which models the dynamical behavior for nematic liquid crystals. For both, the stationary (time independent) case and the non-stationary (time dependent) case, using the fairly general framework of a kind of local Morrey spaces, we obtain some a priori conditions on the unknowns of this coupled system to prove that they vanish identically. This results are known as Liouville-type theorems. As a bi-product, our theorems also improve some well-known results on Liouville-type theorems for the particular case of classical Navier–Stokes equations.

Two-type estimates for the boundedness of generalized Riesz potential operator in the generalized weighted local Morrey spaces

In this paper, we prove the Spanne-type boundedness of the generalized Riesz potential operator $I_{\rho}$ on the generalized weighted local Morrey spaces with $w^{q} \in A_{1+\frac{q}{p'}}$ for $1 \le p \le q \le \infty $ and including its the weak estimates with $w \in A_{1,q}$ for $1\le q\le\infty$. We also prove the Adams-type boundedness of the operator $I_{\rho}$ on the generalized weighted Morrey spaces with $ w \in A_{p,q}$ for $1\le p\le q \le \infty$ and also including its the weak estimates with $w \in A_{1,q}$ for $1\le q\le \infty$.

Calderón Operator on Local Morrey Spaces with Variable Exponents

In this paper, we establish the boundedness of the Calderón operator on local Morrey spaces with variable exponents. We obtain our result by extending the extrapolation theory of Rubio de Francia to the local Morrey spaces with variable exponents. The exponent functions of the local Morrey spaces with the exponent functions are only required to satisfy the log-Hölder continuity assumption at the origin and infinity only. As special cases of the main result, we have Hardy’s inequalities, the Hilbert inequalities and the boundedness of the Riemann–Liouville and Weyl averaging operators on local Morrey spaces with variable exponents.

On the Local Pressure Expansion for the Navier–Stokes Equations

We show that the pressure associated with a distributional solution of the Navier–Stokes equations on the whole space satisfies a local expansion defined as a distribution if and only if the solution is mild. This gives a new perspective on Lemarié-Rieusset’s “equivalence theorem.” Here, the Leray projection operator composed with a gradient is defined without using the Littlewood-Paley decomposition. Prior sufficient conditions for the local expansion assumed spatial decay or estimates on the gradient and imply the considered solution is mild. An important tool is an explicit description of the bounded mean oscillation solution to a Poisson equation, which we examine in detail. As applications we include an improvement of a uniqueness criteria by the authors in Morrey spaces and revisit a proof of a regularity criteria in dynamically restricted local Morrey spaces due to Grujić and Xu.

Singular integral operators and sublinear operators on Hardy local Morrey spaces with variable exponents

This paper establishes the mapping properties of the singular integral operators, the Littlewood-Paley functions and the maximal Bochner-Riesz means on the Hardy local Morrey spaces with variable exponents by using extrapolation theory.

Muckenhoupt-Type Conditions on Weighted Morrey Spaces

We define a Muckenhoup-type condition on weighted Morrey spaces using the Köthe dual of the space. We show that the condition is necessary and sufficient for the boundedness of the maximal operator defined with balls centered at the origin on weighted Morrey spaces. A modified condition characterizes the weighted inequalities for the Calderón operator. We also show that the Muckenhoup-type condition is necessary and sufficient for the boundedness on weighted local Morrey spaces of the usual Hardy–Littlewood maximal operator, simplifying the previous characterization of Nakamura–Sawano–Tanaka. For the same operator, in the case of global Morrey spaces the condition is necessary and for the sufficiency we add a local \(A_p\) condition. We can extrapolate from Lebesgue \(A_p\)-weighted inequalities to weighted global and local Morrey spaces in a very general setting, with applications to many operators.

A regularity criterion for 3D NSE in ‘dynamically restricted’ local Morrey spaces

ABSTRACT It is shown that a local-in-time strong solution u to the 3D Navier–Stokes equations remains regular on an interval provided a smallness -condition on u in a dynamically restricted local Morrey space is stipulated; more precisely, where η is a dynamic dissipation scale consistent with the turbulence phenomenology and α and p are suitable parameters. Such regularity criterion guarantees the volumetric sparseness of local spatial structure of intense vorticity components, preventing the formation of the finite-time blow up at T under the framework of -sparseness classes introduced in Bradshaw et al. (An algebraic reduction of the ‘Scaling Gap’ in the Navier–Stokes regularity problem. Arch Ration Mech Anal. 2019;231(3):1983–2005).