In this paper, we consider the modified quasi-geostrophic equation ∂tθ + (u · ∇) θ + κΛ θ = 0 u = ΛRθ. with κ > 0, α ∈ (0, 1] and θ0 ∈ L(R). We remark that the extra Λ is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasi-geostrophic equations. In this paper, we use Besov space techniques to prove global existence and regula...

In this paper we study differentiability properties of the map T 7→ φ(T ), where φ is a given function in the disk-algebra and T ranges over the set of contractions on Hilbert space. We obtain sharp conditions (in terms of Besov spaces) for differentiability and existence of higher derivatives. We also find explicit formulae for directional derivatives (and higher derivatives) in terms of doubl...

This paper is concerned with the regularity of the solutions to Dirichlet and Neumann problems in smooth and polyhedral cones contained in R. Especially, we consider the speci c scale Bs τ (Lτ ), 1/τ = s/3 + 1/2, of Besov spaces. The regularity of the solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We show th...

We introduce a decoupling method on the Wiener space to define wide class of anisotropic Besov spaces. The is based general distributional approach and not restricted space. spaces we contains traditional isotropic obtained by real interpolation method, but also new that are designed investigate backwards stochastic differential equations (BSDEs). As examples discuss regularity (in sense our sp...

The classical Haar wavelet system of L2(R) is commonly considered to be very local in space. We introduce and study in this paper piecewise-constant framelets (PCF) that include the Haar system as a special case. We show that any bi-framelet pair consisting of PCFs provides the same Besov space characterizations as the Haar system. In particular, it has Jackson-type performance sJ = 1 and Berns...

Let H(S) be the Hardy space on the unit sphere S in C, n ≥ 2. Consider the Hankel operator Hf = (1 − P )Mf |H(S), where the symbol function f is allowed to be arbitrary in L(S, dσ). We show that for p > 2n, Hf is in the Schatten class Cp if and only if f − Pf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if ...

We study the regularity of discontinuous entropy solutions to scalar hyperbolic conservation laws with uniformly convex fluxes posed as initial value problems on R. For positive α we show that if the initial data has bounded variation and the flux is smooth enough then the solution u( · , t) is in the Besov space Bα σ (L σ) where σ = 1/(α + 1) whenever the initial data is in this space. As a co...

In this article we study bounded operators T on a Banach space X which satisfy the discrete Gomilko–Shi-Feng condition Z 2? 0 |hR(re it, T) 2 x, x i|dt ? C (r ? 1) kxk kx k , r > 1, ? X, x? We show that it is equivalent to certain derivative functional calculus and also relative Besov space. Also Hilbert spaces powerboundedness. Finally discuss last equivalence general involving concept of ?-bo...

In nonparametric statistics an optimality criterion for estimation procedures is provided by the minimax rate of convergence. However this classical point of view is subject to controversy as it requires to look for the worst behaviour reached by an estimation procedure in a given space. The purpose of this paper is to show that this is not justified as the minimax risk often coincides with a g...