Journal of Functional Analysis最新文献

We prove the local Gan–Gross–Prasad conjecture for Fourier–Jacobi models of real unitary groups.

For any $p\in (0,1]$ and $s\in R$, the authors prove two types of characterizations of the pointwise multiplier space $M\left(B_{p,p}^s\right(R^n\left)\right)$ of the Besov space $B_{p,p}^s\left(R^n\right)$. One type is based on wavelet analysis and is an extension of a well-known argument of Yves Meyer. The other type works with Fourier analytic terms. As an application of the above two types of characterizations, the authors further obtain a characterization of bounded functions in the uniform space $B_{p,p,\mathrm{unif}}^{s,\tau }\left(R^n\right)$ via Haar wavelets in the critical index $\tau =\frac{1}{p}-\frac{s}{n}$.

Let A and B be $C^{⁎}$-algebras with A separable, let I be an ideal in B, and let $\psi :A\to B/I$ be a completely positive contractive linear map. We show that there is a continuous family ${\Theta }_t:A\to B$, for $t\in [1,\infty )$, of lifts of ψ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If ψ is of order zero, then ${\Theta }_t$ can be chosen to have this property asymptotically. If A and B carry continuous actions of a second countable locally compact group G such that I is G-invariant and ψ is equivariant, we show that the family ${\Theta }_t$ can be chosen to be asymptotically equivariant. If a linear completely positive lift for ψ exists, we can arrange that ${\Theta }_t$ is linear and completely positive for all $t\in [1,\infty )$. In the equivariant setting, if A, B and ψ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if G is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.

In this paper, we study pointwise estimates for linear and multilinear pseudo-differential operators with exotic symbols in terms of the Fefferman-Stein sharp maximal function and Hardy-Littlewood type maximal function. Especially in the multilinear case, we use a multi-sublinear variant of the classical Hardy-Littlewood maximal function introduced by Lerner, Ombrosi, Pérez, Torres, and Trujillo-González [16], which provides more elaborate and natural weighted estimates in the multilinear setting.

In [1] it is shown that the Bloch space $B$ in the unit disc has the following radicality property: if an analytic function g satisfies that $g^n\in B$, then $g^m\in B$, for all $m\le n$. Since $B$ coincides with the space $T\left(A_{\alpha }^p\right)$ of analytic symbols g such that the Volterra-type operator $T_{g}f\left(z\right)={\int }_0^{z}f\left(\zeta \right)g^{\prime }\left(\zeta \right)d\zeta$ is bounded on the classical weighted Bergman space $A_{\alpha }^p$, the radicality property was used to study the composition of paraproducts $T_g$ and $S_{g}f=T_{f}g$ on $A_{\alpha }^p$. Motivated by this fact, we prove that $T\left(A_{\omega }^p\right)$ also has the radicality property, for any radial weight ω. Unlike the classical case, the lack of a precise description of $T\left(A_{\omega }^p\right)$ for a general radial weight, induces us to prove the radicality property for $A_{\omega }^p$ from precise norm-operator results for compositions of analytic paraproducts.

We consider a completely resonant nonlinear Schrödinger equation on the d-dimensional torus, for any $d\ge 1$, with polynomial nonlinearity of any degree $2p+1$, $p\ge 1$, which is gauge and translation invariant. We study the behaviour of high Sobolev $H^s$-norms of solutions, $s\ge s_1+1>d/2+2$, whose initial datum $u_0\in H^s$ satisfies an appropriate smallness condition on its low $H^{s_1}$ and $L^2$-norms respectively. We prove a polynomial upper bound on the possible growth of the Sobolev norm $H^s$ over finite but long time scale that is exponential in the regularity parameter $s_1$. As a byproduct we get stability of the low $H^{s_1}$-norm over such time interval. A key ingredient in the proof is the introduction of a suitable “modified energy” that provides an a priori upper bound on the growth. This is obtained by combining para-differential techniques and suitable tame estimates.

We investigate $L^p$-boundary representations of hyperbolic groups. We prove that such representations are irreducible if and only if the corresponding Riesz operators are injective.

In the context of the Kuznetsov trace formula, we outline the theory of the Bessel functions on $GL\left(n\right)$ as a series of conjectures designed as a blueprint for the construction of Kuznetsov-type formulas with given ramification at infinity. We are able to prove one of the conjectures at full generality on $GL\left(n\right)$ and most of the conjectures in the particular case of the long Weyl element; as with previous papers, we give some unconditional results on Archimedean Whittaker functions, now on $GL\left(n\right)$ with arbitrary weight. We expect the heuristics here to apply at the level of real reductive groups. A forthcoming paper will address the initial conjectures up to Mellin-Barnes integral representations in the case of $GL\left(4\right)$ Bessel functions.

For the one dimensional Burgers equation with a random and periodic forcing, it is well-known that there exists a family of invariant measures, each corresponding to a different average velocity. In this paper, we consider the coupled invariant measures and study how they change as the velocity parameter varies. We show that the derivative of the invariant measure with respect to the velocity parameter exists, and it can be interpreted as the steady state of a diffusion advected by the Burgers flow.

We consider the validity of Prandtl boundary layer expansion of solutions to the initial boundary value problem for inhomogeneous incompressible magnetohydrodynamics equations in the half-plane when both viscosity and resistivity coefficients tend to zero, where the no-slip boundary condition is imposed on velocity while the perfectly conducting condition is given on magnetic field. Since there exist strong boundary layers, the essential difficulty in establishing the uniform $L^{\infty }$ estimates of the error functions comes from the unboundedness of vorticity of strong boundary layers. Under the assumptions that the viscosity and resistivity coefficients take the same order of a small parameter and the initial tangential component of magnetic field has a positive lower bound near the boundary, we prove the validity of Prandtl boundary layer ansatz in $L^{\infty }$ sense in Sobolev framework. Compared with the homogeneous incompressible case considered in [33], there exists a strong boundary layer of density. Consequently, some suitable functionals should be designed and the elaborated co-normal energy estimates will be involved in analysis due to the variation of density and the interaction between the density and velocity.