Hölder estimates for magnetic Schrödinger semigroups in $${\mathbb {R}}^{d}$$ from mirror coupling

We use the mirror coupling of Brownian motion to show that under a $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) -dependent Kato-type assumption on the possibly nonsmooth electromagnetic potential, the corresponding magnetic Schrödinger semigroup in $${\mathbb {R}}^d$$ R d has a global $$L^{p}$$ L p -to-$$C^{0,\beta }$$ C 0 , β Hölder smoothing property for all $$p\in [1,\infty ]$$ p ∈ [ 1 , ∞ ] ; in particular, his all eigenfunctions are uniformly $$\beta $$ β -Hölder continuous. This result shows that the eigenfunctions of the Hamilton operator of a molecule in a magnetic field are uniformly $$\beta $$ β -Hölder continuous under weak $$L^q$$ L q -assumptions on the magnetic potential.

Kato smoothing properties of a class of nonlinear dispersive wave equations on a periodic domain

The solutions of the Cauchy problem of the KdV equation on a periodic domain $\T$, \[ u_t +uu_x +u_{xxx} =0, \quad u(x,0)= \phi (x), \quad x\in \T, \ t\in \R,\] possess neither the sharp Kato smoothing property, \[ \phi \in H^s (\T) \implies \partial ^{s+1}_xu \in L^{\infty}_x (\T, L^2 (0,T)),\] nor the Kato smoothing property, \[ \phi \in H^s (\T) \implies u\in L^2 (0,T; H^{s+1} (\T)).\] Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain $\T$, \[ u_t +uu_x +u_{xxx} - g(x) (g(x) u)_{xx} =0, \qquad u(x,0)= \phi (x), \quad x\in \T, \ t>0 \, ,\ \qquad (1) \] where $g\in C^{\infty} (\T)$ is a real value function with the support \[ \mbox{$\omega = \{ x\in \T, \ g(x) \ne 0\}$.}\] It is shown that \begin{itemize} \item[(1)] if $\omega\ne \emptyset$, then the solutions of the Cauchy problem (1) possess the Kato smoothing property; \item[(2)] if $g$ is a nonzero constant function, then the solutions of the Cauchy problem (1) possess the sharp Kato smoothing property. \end{itemize}

Well-posedness of the Muskat problem in subcritical L p -Sobolev spaces

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an L p -setting with p ∈ (1, ∞). The Sobolev space $W_p^s(\mathbb R)$ with s = 1+1/p is a critical space for this problem. We prove, for each s ∈ (1+1/p, 2) that the Rayleigh–Taylor condition identifies an open subset of $W_p^s(\mathbb R)$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

From curves to currents

Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.

A note on smoothing properties of the Bergman projection

Recently Herbig, McNeal, and Straube have showed that the Bergman projection of conjugate holomorphic functions is smooth up to the boundary on smoothly bounded domains that satisfy condition R. We show that a further smoothing property holds on a family of Reinhardt domains; namely, the Bergman projection of conjugate holomorphic functions is holomorphic past the boundary.