In this paper, we study the smoothness of restrictions of Besov functions. It is known that for any $f\in B\_{p,q}^s(\mathbb{R}^N)$ with $q\leq p$ we have $f(\cdot,y)\in B\_{p,q}^s(\mathbb{R}^d)$ for a.e. $y\in \mathbb{R}^{N-d}$. We prove that this is no longer true when $p\<q$. Namely, we construct a function $f\in B\_{p,q}^s(\mathbb{R}^N)$ such that $f(\cdot,y)\notin B\_{p,q}^s(\mathbb{R}^d)$ for a.e. $y\in \mathbb{R}^{N-d}$. We show that, in fact, $f(\cdot,y)$ belong to $B\_{p,q}^{(s,Ψ)}(\mathbb{R}^d)$ for a.e. $y\in\mathbb{R}^{N-d}$, a Besov space of generalized smoothness, and, when $q=\infty$, we find the optimal condition on the function $Ψ$ for this to hold. The natural generalization of these results to Besov spaces of generalized smoothness is also investigated.
This paper is concerned with proving some embeddings of the form \begin{equation*} F_{p_{1},q}^{s_{1}}\cdot B_{p_{2},\infty }^{s_{2}}\cdot ...\cdot B_{p_{m},\infty }^{s_{m}}\hookrightarrow F_{p,q}^{s_{1}},\quad m\geq 2. \end{equation*} The different embeddings obtained here are under certain restrictions on the parameters. In particular, we improve some results of pointwise multiplication on Triebel-Lizorkin spaces. Franke-Jawerth embeddings, the $\pm $ - method of Gustaffson-Peetre and the relation between Hardy spaces and Triebel-Lizorkin spaces are the main tools.
In this papae we introduce and investigate new 2-microlocal spaces associated with Besov type and Triebel-LIzorkin type spaces. We establish characterizations of these function spaces via the phi transform, the atom and molecular decomposition and wavelet decomposition. As applications we consider the boundedness of the Calderon-Zygmund operator and the pseudo-differential operator on these function spaces
We give a sufficient (and, in the case of a compact domain, a necessary) condition for the embedding of Sobolev space of functions with integrable gradient into Besov-Orlicz spaces to be bounded. The condition has a form of a simple integral inequality involving Young and weight functions. We provide an example with Matuszewska-Orlicz indices of involved Orlicz norm equal to one. The main tool is the molecular decomposition of functions from a BV space.
We construct generalised shift-invariant systems of functions of several real variables for anisotropic Besov spaces that can be generated by the decomposition method using any given expansive matrix and establish the conditions on those systems under which they will constitute Banach frames or sets of atoms for the anisotropic homo- or inhomogeneous Besov spaces.
In this paper we introduce Besov-type spaces with variable smoothness and integrability. We show that these spaces are characterized by the $φ$-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces. Moreover the Sobolev embeddings for these function spaces are obtained.
In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted $L_{q}$-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt $A_{\infty}$-class. In Besov space case we have the restriction that the microscopic parameter equals to $q$. Going beyond the $A_{p}$-range, where $p$ is the integrability paramet…
On sharp embeddings of Besov and Triebel-Lizorkin spaces in the subcritical case
讨论了Fourier解析定义的Besov和Triebel-Lizorkin空间$B^s_{p,q}(\R^n)$和$F^s_{p,q}(\R^n)$对于$s=σ_p=n\max(\frac 1p-1,0)$的增长包络.这些结果也可以被重新表述为Lorentz空间$L_{p,q}(\R^n)$尺度的最优嵌入。我们关闭几个开放的问题已经概述了H。Triebel在[H. Triebel，The structure of functions，Birkhäuser，Basel，2001.]并由D. D. Haroske [D. D. Haroske，Envelopes and sharp embeddings of函数空间，Chapman & Hall / CRC，Boca Raton，2007.].
For an interpolation pair $(E_0,E_1)$ of Banach spaces with $E_1 \hookrightarrow E_0$ we use vectors $b_1,b_2,\ldots \in E_1$ that satisfy an extremal property with respect to the $J$- and $K$-functional to construct sub-spaces that are isometric to $\ell_q^{(θ)}$. The construction is based on a randomisation using independent Rademacher variables. We verify that systems obtained by re-scaling a function with a certain periodicity property share this extreme property. This implies the sharpness of natural embeddings of Hölder spaces obtained by the real interpolation into the corresponding Gau…
Using partial derivatives $\partial_zf$ and $\partial_{\ol z}f$, we introduce Besov spaces of polyanalytic functions on the unit disk and on the upper half-plane. We then prove that the dilatations of each function in polyanalytic Besov spaces converge to the same function in norm. This opens the way for the norm approximation of functions in polyanalytic Besov spaces by polyanalytic polynomials.
