Morrey-type Spaces Research Articles

Some Remarks on Spaces of Morrey Type

We deepen the study of some Morrey type spaces, denoted by Mp,λ(Ω), defined on an unbounded open subset Ω of ℝn. In particular, we construct decompositions for functions belonging to two different subspaces of Mp,λ(Ω), which allow us to prove a compactness result for an operator in Sobolev spaces. We also introduce a weighted Morrey type space, settled between the above‐mentioned subspaces.

Hölder continuity for a class of X-elliptic equations with singular lower order term

The regularity for a class of X-elliptic equations with lower order term $$ Lu + vu = - \sum\limits_{i,j = 1}^m {X_j *} (a_{ij} (x)X_i u) + vu = \mu $$ is studied, where X = {X1, ..., Xm} is a family of locally Lipschitz continuous vector fields, v is in certain Morrey type space and µ a nonnegative Radon measure. The Holder continuity of the solution is proved when µ satisfies suitable growth condition, and a converse result on the estimate of µ is obtained when u is in certain Holder class.

Necessary and Sufficient Conditions for the Boundedness of the Riesz Potential in Local Morrey-type Spaces

The problem of the boundedness of the Riesz potential Iα, 0 < α < n, in local Morrey-type spaces is reduced to the boundedness of the Hardy operator in weighted Lp-spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for the boundedness in local Morrey-type spaces for all admissible values of the parameters. Moreover, for a certain range of the parameters, these sufficient conditions coincide with the necessary ones.

Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces

This paper considers the boundedness of Calderon– Zygmund singular integral operators in local and global Morrey-type spaces. We formulate sufficient conditions for the boundedness of Calderon-Zygmund singular integral operators for all admissible parameter values. In the case of local Morrey-type spaces for a certain range of parameters, these sufficient conditions are also necessary for genuine Calderon–Zygmund singular integral operators.

L ∞ estimates for variational solutions of boundary value problems in unbounded domains

We state L ∞ estimates for weak solutions of boundary value problems when the coefficients of the differential operators are discontinuous and belong to Morrey type spaces. We deduce also some a priori bounds for strong solutions.

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp,λ*(ℝn) on the basis of Lorentz space Lp,∞ = Lp*(ℝn) (in particular, Mp,0*(ℝn) = Lp,∞, if p > 1), and study some fundamental properties of them; Second, we prove that the heat operator U(t) = etΔ. and Calderon-Zygmund singular integral operators are bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato’s method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp,λ*(ℝn) (1 < p ⩽ n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp,n−p*(ℝn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato’s results.

Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces

Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces

Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces

The problem of the boundedness of the fractional maximal operator M α , 0 < α < n , in local and global Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones.

Necessary and sufficient conditions for the boundedness of the fractional maximal operator in local Morrey-type spaces

Necessary and sufficient conditions for the boundedness of the fractional maximal operator in local Morrey-type spaces

Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces

It is proved that the boundedness of the maximal operator M from a Lebesgue space L-p1 (R-n) to a general local Morrey-type space LMp2 theta,w(R-n) is equivalent to the boundedness of the embedding operator from L-p1(R-n) to LMp2 theta,(w)(R-n) and in its turn to the boundedness of the Hardy operator from L-p1/p2 (0,infinity) to the weighted Lebesgue space L-theta/p2,L-v (0,infinity) for a certain weight function v determined by the functional parameter w. This allows obtaining necessary and sufficient conditions on the function w ensuring the boundedness of M from L-p1 (R-n) to LMp2 theta,w(R-n) for any 0 1. These conditions with p(1) = p(2) = 1 are necessary and sufficient for the boundedness of M from L-1(R-n) to the weak local Morreytype space WLM1 theta,w(R-n).

Homogeneity Criterion for the Navier-Stokes Equations in the Whole Spaces

This paper is concerned with the Navier-Stokes flows in the homogeneous spaces of degree -1, the critical homogeneous spaces in the study of the existence of regular solutions for the Navier-Stokes equations by means of linearization. In order to narrow the gap for the existence of small regular solutions in \( \dot B^{-1}_{\infty,\infty}(R^n)^n \), the biggest critical homogeneous space among those embedded in the space of tempered distributions, we study small solutions in the homogeneous Besov space \( \dot B^{-1+n/p}_{p,\infty}(R^n)^n \) and a homogeneous space defined by \( \hat M_n(R^n)^n \), which contains the Morrey-type space of measures \( \tilde M_n(R^n)^n \) appeared in Giga and Miyakawa [20]. The earlier investigations on the existence of small regular solutions in homogeneous Morrey spaces, Morrey-type spaces of finite measures, and homogeneous Besov spaces are strengthened. These results also imply the existence of small forward self-similar solutions to the Navier-Stokes equations. Finally, we show alternatively the uniqueness of solutions to the Navier-Stokes equations in the critical homogeneous space \( C([0,\infty);L_n(R^n)^n) \) by applying Giga-Sohr's \( L_p(L_q) \) estimates on the Stokes problem.