Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userClassic and exotic Besov spaces induced by good grids
87
0
0.0
(
0
)
Authors
Daniel Smania
Ask ChatGPT about the research
No Arabic abstract
In a previous work we introduced Besov spaces $mathcal{B}^s_{p,q}$ defined on a measure spaces with a good grid, with $pin [1,infty)$, $qin [1,infty]$ and $0< s< 1/p$. Here we show that classical Besov spaces on compact homogeneous spaces are examples of such Besov spaces. On the other hand we show that even Besov spaces defined by a good grid made of partitions by intervals may differ from a classical Besov space, giving birth to exotic Besov spaces.
rate research
Read More
We use the method of atomic decomposition to build new families of function spaces, similar to Besov spaces, in measure spaces with grids, a very mild assumption. Besov spaces with low regularity are considered in measure spaces with good grids, and results on multipliers and left compositions are obtained.
Some Besov-type spaces $B^{s,tau}_{p,q}(mathbb{R}^n)$ can be characterized in terms of the behavior of the Fourier--Haar coefficients. In this article, the authors discuss some necessary restrictions for the parameters $s$, $tau$, $p$, $q$ and $n$ of this characterization. Therefore, the authors measure the regularity of the characteristic function $mathcal X$ of the unit cube in $mathbb{R}^n$ via the Besov-type spaces $B^{s,tau}_{p,q}(mathbb{R}^n)$. Furthermore, the authors study necessary and sufficient conditions such that the operation $langle f, mathcal{X} rangle$ generates a continuous linear functional on $B^{s,tau}_{p,q}(mathbb{R}^n)$.
Let $1le p<infty$, $0<q<infty$ and $ u$ be a two-sided doubling weight satisfying $$sup_{0le r<1}frac{(1-r)^q}{int_r^1 u(t),dt}int_0^rfrac{ u(s)}{(1-s)^q},ds<infty.$$ The weighted Besov space $mathcal{B}_{ u}^{p,q}$ consists of those $fin H^p$ such that $$int_0^1 left(int_{0}^{2pi} |f(re^{itheta})|^p,dthetaright)^{q/p} u(r),dr<infty.$$ Our main result gives a characterization for $fin mathcal{B}_{ u}^{p,q}$ depending only on $|f|$, $p$, $q$ and $ u$. As a consequence of the main result and inner-outer factorization, we obtain several interesting by-products. In particular, we show the following modification of a classical factorization by F. and R. Nevanlinna: If $fin mathcal{B}_{ u}^{p,q}$, then there exist $f_1,f_2in mathcal{B}_{ u}^{p,q} cap H^infty$ such that $f=f_1/f_2$. In addition, we give a sufficient and necessary condition guaranteeing that the product of $fin H^p$ and an inner function belongs to $mathcal{B}_{ u}^{p,q}$. Applying this result, we make some observations on zero sets of $mathcal{B}_{ u}^{p,p}$.
In this article, the authors introduce Besov-type spaces with variable smoothness and integrability. The authors then establish their characterizations, respectively, in terms of $varphi$-transforms in the sense of Frazier and Jawerth, smooth atoms or Peetre maximal functions, as well as a Sobolev-type embedding. As an application of their atomic characterization, the authors obtain a trace theorem of these variable Besov-type spaces.
In this paper, we consider the trace theorem for modulation spaces, alpha modulation spaces and Besov spaces. For the modulation space, we obtain the sharp results.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
