Abstract Let $$d\ge 2$$ d ? 2 . In this paper we give a simple proof of the endpoint Besov-Lorentz estimate $$\begin{aligned} \Vert I_\alpha F\Vert _{{\dot{B}}^{0,1}_{d/(d-\alpha ),1}(\mathbb {R}^d;\mathbb {R}^k)} \le C F _{L^1(\mathbb \end{aligned}$$ <...

This paper is devoted to the study of solutions with critical regularity for two-dimensional Muskat equation. We prove that Cauchy problem well-posed on endpoint Sobolev space $$L^2$$ functions three-half derivative in . result optimal respect scaling One well-known difficulty one cannot define a flow map such lifespan bounded from below subsets this space. To overcome this, we estimate norm wh...

In this paper, we establish some new endpoint regularity properties of multilinear maximal operators and their fractional variants, both in the centered uncentered cases. The main results obtain not only answer a question Liu, Wu, Xue Yabuta, but also provide Sobolev boundedness continuity for operators. More importantly, present approach to deal with above

BACKGROUND An intermediate endpoint is hypothesized to be in the middle of the causal sequence relating an independent variable to a dependent variable. The intermediate variable is also called a surrogate or mediating variable and the corresponding effect is called the mediated, surrogate endpoint, or intermediate endpoint effect. Clinical studies are often designed to change an intermediate o...