In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new “almost” Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson’s equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.

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This paper establishes the hyper-contractivity in L∞ (ℝd) (it's known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the dif;fusion exponent m > 1 -2/d. The results show that for the supercritical and critical case 1 - 2/d < m ≤ 2 - 2/d, if || U0||d(2-m)/2) < Cd, m where Cd, m is a universal constant, then for any t > 0, || u (⋅, t)|| ... is bounded and decays as t goes to infinity, For the subcritical case m > 2 - 2/d, the solution u (⋅, t) ∈ L∞ (ℝd) with any initial data U0 ∈ L1+ (ℝd) for any positive time.

We present a unique global solvability and flocking estimate of an entropic weak solution to the one-dimensional pressureless Euler system with a flocking dissipation in all-to-all coupling setting. This model appears naturally as a quasi-equilibrium model for hydrodynamic description of CuckerSmale flocking. For the unique global solvability, we adopt the variation approach from Ding and Huang's work [19] on the inhomogeneous pressureless gas dynamic model. When initial mass and velocity are locally integrable and bounded measurable functions, respectively, we give explicit representations for the global entropic weak solutions. Our results do not require any smallness of initial data except that initial mass density is almost everywhere positive.

In this paper, we generalize the$A$_{∞}extrapolation theorem ($Cruz-Uribe–Martell–Pérez$, Extrapolation from$A$_{∞}weights and applications,$J. Funct. Anal.$$213$(2004), 412–439) and the$A$_{$p$}extrapolation theorem of Rubio de Francia to Schrödinger settings. In addition, we also establish weighted vector-valued inequalities for Schrödinger-type maximal operators by using weights belonging to $A_{p}^{\rho,\infty }$ which includes$A$_{$p$}. As applications, we establish weighted vector-valued inequalities for some Schrödinger-type operators.