Dyadic John–Nirenberg space

We discuss the dyadic John–Nirenberg space that is a generalization of functions of bounded mean oscillation. A John–Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John–Nirenberg space and provide a method to construct nontrivial functions in the dyadic John–Nirenberg space. Moreover, we prove that the John–Nirenberg space is complete. Several open problems are also discussed.

Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups

Let X0,X1,…,Xq(q<N) be real vector fields, which are left invariant on homogeneous group G, provided that X0 is homogeneous of degree two and X1,…,Xq are homogeneous of degree one. We consider the following nondivergence degenerate operator with drift L=∑i,j=1qaij(x)XiXj+a0(x)X0, where the coefficients aij(x), a0(x) belonging to vanishing mean oscillation space are bounded measurable functions. Furthermore, aij(x) satisfies the uniform ellipticity condition on Rq and a0(x)≠0. We obtain the local weighted Sobolev–Morrey estimates by applying the boundedness of commutators and interpolation inequalities on weighted Morrey spaces.

Caratheodory theorem about prime ends on Riemann surfaces

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio--Ryazanov--Srebro--Yakubov at 2004 and then included in the known monograph of these authors in the modern mapping theory at 2009. As it was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasi--isometries. We obtain here a series of criteria in terms of dilatations for the homeomorphic extension of the mappings with finite length distortion between domains on Riemann surfaces to the completions of the domains by prime ends of Caratheodory. Here we start from the general criterion in Lemma 1 in terms of singular functional parameters and then derive on this basis many other criteria. In particular, Lemma 1 implies Theorem 1 with a criterion of the Lehto type and Corollary 1 shows that the conclusion holds, if the dilatation grows not quickly than logarithm of the hyperbolic distance at every boundary point. The next consequence in Theorem 2 gives an integral criterion of the Orlicz type and Corollary 2 says on simple integral conditions of the exponential type. Further, Theorem 3 and Remark 2 contain criteria in terms of singular integrals of the Calderon--Zygmund type. The other criterion in Theorem 4 is the existence of a dominant for the dilatation in the class FMO (functions with finite mean oscillation), i.e., having a finite mean deviation from its mean value over infinitesimal discs centered at boundary points. In other words, the latter means that such a dominant has a finite dispersion over the given infinitesimal discs. In particular, the latter leads to Corollary 3 on a dominant in the well--known class BMO (bounded mean oscillation) by John--Nirenberg and to a simple criterion in Corollary 4 on finiteness of the average of the dilatation over infinitesimal disks centered at boundary points.

Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

The Uniformly Parabolic Equations of Higher Order with Discontinuous Data in Generalized Morrey Spaces and Elliptic Equations in Unbounded Domains

We study the regularity of the solutions of the Cauchy-Dirichlet problem for linear uniformly parabolic equations of higher order with vanishing mean oscillation (VMO) coefficients. We prove continuity in generalized parabolic Morrey spaces Mp,φ of sublinear operators generated by the parabolic Calderon-Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. We obtain strong solution belongs to the generalized Sobolev-Morrey space Wp,φm,1∘Q. Also we consider elliptic equation in unbounded domains.

Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System

The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on R2. The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) for compressible Navier–Stokes and compressible Navier–Stokes–Korteweg systems. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space–time L2 of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by L2 on space, decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz’s energy estimate, and the Fefferman–Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.