Lp + Lq and Lp ∩ Lq are not isomorphic for all 1 ≤ p,q ≤ ∞, p ≠ q

We prove that if $1\le p,q\le \infty$, then the spaces $L_p+L_q$ and $L_p\cap L_q$ are isomorphic if and only if $p=q$. In particular, $L_2+L_{\infty }$ and $L_2\cap L_{\infty }$ are not isomorphic, which is an answer to a question formulated in [2].     

Crossed-products extensions,of Lp-bounds for amenable actions

We will extend earlier transference results due to Neuwirth and Ricard from the context of noncommutative $L_p$-spaces associated with amenable groups to that of noncommutative $L_p$-spaces associated with crossed-products of amenable actions. Namely, if $m:G\to \mathbb{C}$ is a completely bounded Fourier multiplier on $L_p$, then it extends to the crossed-product with similar bounds provided that the action θ is amenable and trace-preserving. Furthermore, our construction also allows to extend G-equivariant completely bounded operators acting on the space part to the crossed-product provided that the generalized Følner sets of the action θ satisfy certain accretivity property. As a corollary we obtain stability results for maximal $L_p$-bounds over crossed products. We derive, using that stability results, an application to the boundedness of smooth multipliers in the $L_p$-spaces of group algebras.     

Cauchy transform and Poisson’s equation

Let $u\in W^{1,p}\cap W_0^{1,p}$, $1?p?\infty$ be a solution of the Poisson equation $\Delta u=h$, $h\in L^p$, in the unit disk. We prove ${6?u6}_{L^p}?a_p{6h6}_{L^p}$ and ${6?u6}_{L^p}?b_p{6h6}_{L^p}$ with sharp constants $a_p$ and $b_p$, for $p=1$, $p=2$, and $p=\infty$. In addition, for $p>2$, with sharp constants $c_p$ and $C_p$, we show ${6?u6}_{L^{\infty }}?c_p{6h6}_{L^p}$ and ${6?u6}_{L^{\infty }}?C_p{6h6}_{L^p}$. We also give an extension to smooth Jordan domains.These problems are equivalent to determining a precise value of the $L^p$ norm of the Cauchy transform of Dirichlet’s problem.     

Global solvability of the Navier–Stokes equations with a free surface in the maximal Lp-Lq regularity class

We consider the motion of incompressible viscous fluids bounded above by a free surface and below by a solid surface in the N-dimensional Euclidean space for $N\ge 2$. The aim of this paper is to show the global solvability of the Navier–Stokes equations with a free surface, describing the above-mentioned motion, in the maximal $L_p\text{-}{L}_q$ regularity class. Our approach is based on the maximal $L_p\text{-}{L}_q$ regularity with exponential stability for the linearized equations, and also it is proved that solutions to the original nonlinear problem are exponentially stable.     

INTERIOR ESTIMATES IN MORREY SPACES FOR SOLUTIONS OF ELLIPTIC EQUATIONS AND WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS

It is proved that, for the nondivergence elliptic equations ∑in, j=1 aijuxixj = f,if f belongs to the generalized Morrey spaces Lp,ψ(ω), then uxixj ∈ Lp,ψ(ω), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp,ψ (ω).     

The reconstruction formula for Banach frames and duality

We study conditions on a Banach frame that ensures the validity of a reconstruction formula. In particular, we show that any Banach frames for (a subspace of) $L_p$ or $L_{p,q}$ ($1\le p<\infty$) with respect to a solid sequence space always satisfies an unconditional reconstruction formula. The existence of reconstruction formulas allows us to prove some James-type results for atomic decompositions: an unconditional atomic decomposition (or unconditional Schauder frame) for $X$ is shrinking (respectively, boundedly complete) if and only if $X$ does not contain an isomorphic copy of ${?}_1$ (respectively, $c_0$).     

On uniqueness of stationary solutions to the Navier–Stokes equations in exterior domains

The aim of this paper is to prove a uniqueness criterion for solutions to the stationary Navier–Stokes equation in 3-dimensional exterior domains within the class $u\in L_{3,\infty }$ with $?u\in L_{3/2,\infty }$, where $L_{3,\infty }$ and $L_{3/2,\infty }$ are the Lorentz spaces. Our criterion asserts that if $u$ and $v$ are the solutions, $u$ is small in $L_{3,\infty }$ and $u,v\in L_p$ for some $p>3$, then $u=v$. The proof is based on analysis of the dual equation with the aid of the bootstrap argument.