Mapping properties of operator-valued pseudo-differential operators

This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

Pseudo-differential operators, Wigner transform and Weyl transform on the affine Poincaré group

The pseudo-differential Mellin-edge-approach is a tool for studying differential and pseudo-differential operators on manifolds with corners. The Mellin transform, acting on the corner axis ℝ+, is a substitute for the Fourier transform along edge variables in the calculus of wedge pseudo-differential operators. The basic elements of that theory (cf. Schulze [6,8]) are extended to edges like ℝ+ ∋ t with a control of symbols and smoothing operators near the vertext=0. The authors study the weighted Mellin wedge Sobolev spaces, the operator-valued Mellin convention translating Fourier symbols into Mellin ones under preserved smoothness up tot=0, and develop an operator calculus with its characterization on the level of symbols. Throughout the theory, there are involved one-parameter groups of isomorphisms acting on the Banach spaces that are the abstract analogues of the weighted cone Sobolev spaces.

The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg–Whitham equation in Besov spaces

This paper is the second part of a two-paper series whose aim is to give a detailed description of Connes’ pseudodifferential calculus on noncommutative [Formula: see text]-tori, [Formula: see text]. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension [Formula: see text]. This includes the main results mentioned in [2, 5, 11]. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negative order, including a trace-formula for pseudodifferential operators of order [Formula: see text].

Non-uniform continuous dependence on initial data for a two-component Novikov system in Besov space

We investigate global microlocal properties of localization operators and Shubin pseudodifferential operators. The microlocal regularity is measured in terms of a scale of Shubin-type Sobolev spaces. In particular, we prove microlocality and microellipticity of these operators.

Spectral properties of zeroth-order pseudodifferential operators

Hypercyclic composition operators on the little Bloch space and the Besov spaces

Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space

Characterizations of logarithmic Besov spaces in terms of differences, Fourier-analytical decompositions, wavelets and semi-groups

Characterization of inclusion relations between Wiener amalgam and some classical spaces

In this paper we obtain an estimate of the norm of the Bergman projection from Lp(D, dλ) onto the Besov space Bp, 1 < p < + ∞. The result is asymptotically sharp when p → + ∞. Further for the case P : L1(D, dλ) → B1, we consider some weak type inequalities with the corresponding spaces.

Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation

We study the surface plasmon modes of an arbitrarily shaped nanoparticle in the electrostatic limit. We first deduce an eigenvalue equation for these modes, expressed in terms of the Dirichlet–Neumann operators. We then use the properties of these pseudo-differential operators for deriving the limit of the high-order modes.