We establish several mixed $A_p-A_{\infty }$ bounds for Calderón-Zygmund operators that only involve one supremum. We address both cases when the $A_{\infty }$ part of the constant is measured using the exponential-logarithmic definition and using the Fujii-Wilson definition. In particular, we answer a question of the first author and provide an answer, up to a logarithmic factor, to a conjecture of Hytönen and Lacey. Moreover, we give an example to show that our bounds with the logarithmic factors can be arbitrarily smaller...

Besov spaces of holomorphic functions in tubes over cones have been recently defined by Békollé et al. In this paper we show that Besov p-seminorms are invariant under conformal transformations of the domain when n/r is an integer, at least in the range 2-r/n < p ≤ ∞.

Modeling of repulsive forces is essential to the understanding of certain bio-physical processes, especially for the motion of DNA molecules. These kinds of phenomena seem to be driven by some sort of “energy” which especially prevents the molecules from strongly bending and forming self-intersections. Inspired by a physical toy model, numerous functionals have been defined during the past twenty-five years that aim at modeling self-avoidance. The general idea is to produce “detangled” curves having...

In a series of papers beginning in the late 1990s, Michael Lacey and Christoph Thiele have resolved a longstanding conjecture of Calderón regarding certain very singular integral operators, given a transparent proof of Carleson’s theorem on the almost everywhere convergence of Fourier series, and initiated a slew of further developments. The hallmarks of these problems are multilinearity as opposed to mere linearity, and especially modulation symmetry. By modulation is meant multiplication by characters...

We prove that for symbols in the modulation spaces ${ℳ}^{p,q}$, p ≥ q, the associated multilinear pseudodifferential operators are bounded on products of appropriate modulation spaces. In particular, the symbols we study here are defined without any reference to smoothness, but rather in terms of their time-frequency behavior.

We give a new representation of solutions to a class of time-dependent Schrödinger type equations via the short-time Fourier transform and the method of characteristics. Moreover, we also establish some novel estimates for oscillatory integrals which are associated with the fractional power of negative Laplacian ${(-\Delta )}^{\kappa /2}$ with $1\le \kappa \le 2$. Consequently the classical Hamiltonian corresponding to the previous Schrödinger type equations is studied. As applications, a series of new boundedness results for the corresponding...

Let E be a Banach space. Let $L{¹}_{\left(1\right)}({ℝ}^d,E)$ be the Sobolev space of E-valued functions on ${ℝ}^d$ with the norm ${ʃ}_{{ℝ}^d}\parallel f{\parallel }_{E}dx+{ʃ}_{{ℝ}^d}\parallel \nabla f{\parallel }_{E}dx=\parallel f\parallel ₁+\parallel \nabla f\parallel ₁$. It is proved that if $f\in L{¹}_{\left(1\right)}({ℝ}^d,E)$ then there exists a sequence $\left(g_m\right)\subset L_{\left(1\right)}¹({ℝ}^d,E)$ such that $f={\sum }_m{g}_m$; ${\sum }_m(\parallel g_m\parallel ₁+\parallel \nabla g_m\parallel ₁)<\infty$; and $\parallel g_m{\parallel }_{\infty }^{1/d}\parallel g_m\parallel {₁}^{(d-1)/d}\le b\parallel \nabla g_m\parallel ₁$ for m = 1, 2,..., where b is an absolute constant independent of f and E. The result is applied to prove various refinements of the Sobolev type embedding $L_{\left(1\right)}¹({ℝ}^d,E)\hookrightarrow L²({ℝ}^d,E)$. In particular, the embedding into Besov spaces $L{¹}_{\left(1\right)}({ℝ}^d,E)\hookrightarrow B_{p,1}^{\theta (p,d)}({ℝ}^d,E)$ is proved, where $\theta (p,d)=d(p^{-1}+d^{-1}-1)$ for 1 < p ≤ d/(d-1), d=1,2,... The latter embedding in the scalar case is due to Bourgain and Kolyada....

We study the notion of molecules in coorbit spaces. The main result states that if an operator, originally defined on an appropriate space of test functions, maps atoms to molecules, then it can be extended to a bounded operator on coorbit spaces. For time-frequency molecules we recover some boundedness results on modulation spaces, for time-scale molecules we obtain the boundedness on homogeneous Besov spaces.

Questa è una rassegna di alcuni risultati recenti sui moltiplicatori spettrali dell'operatore di Ornstein-Uhlenbeck, un laplaciano naturale sullo spazio euclideo munito della misura gaussiana. I risultati sono inquadrati nell'ambito della teoria generale dei moltiplicatori spettrali per laplaciani generalizzati.