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Search: All articles in the CMB digital archive with keyword function space

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1. CMB Online first

Bu, Shangquan; Cai, Gang
Hölder continuous solutions of degenerate differential equations with finite delay
Using known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely characterize the $C^\alpha$-well-posedness of the first order degenerate differential equations with finite delay $(Mu)'(t) = Au(t) + Fu_t + f(t)$ for $t\in\mathbb R$ by the boundedness of the $(M, F)$-resolvent of $A$ under suitable assumption on the delay operator $F$, where $A, M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\cap D(M) \not=\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r, 0]; X)$ to $X$ and $r \gt 0$ is fixed.

Keywords:well-posedness, degenerate differential equation, $\dot{C}^\alpha$-multiplier, Hölder continuous function space
Categories:34N05, 34G10, 47D06, 47A10, 34K30

2. CMB Online first

Bao, Guanlong; Göğüş, Nıhat Gökhan; Pouliasis, Stamatis
$\mathcal{Q}_p$ spaces and Dirichlet type spaces
In this paper, we show that the Möbius invariant function space $\mathcal {Q}_p$ can be generated by variant Dirichlet type spaces $\mathcal{D}_{\mu, p}$ induced by finite positive Borel measures $\mu$ on the open unit disk. A criterion for the equality between the space $\mathcal{D}_{\mu, p}$ and the usual Dirichlet type space $\mathcal {D}_p$ is given. We obtain a sufficient condition to construct different $\mathcal{D}_{\mu, p}$ spaces and we provide examples. We establish decomposition theorems for $\mathcal{D}_{\mu, p}$ spaces, and prove that the non-Hilbert space $\mathcal {Q}_p$ is equal to the intersection of Hilbert spaces $\mathcal{D}_{\mu, p}$. As an application of the relation between $\mathcal {Q}_p$ and $\mathcal{D}_{\mu, p}$ spaces, we also obtain that there exist different $\mathcal{D}_{\mu, p}$ spaces; this is a trick to prove the existence without constructing examples.

Keywords:$\mathcal {Q}_p$ space, Dirichlet type space, Möbius invariant function space
Categories:30H25, 31C25, 46E15

3. CMB 2015 (vol 58 pp. 757)

Han, Yanchang
Embedding Theorem for Inhomogeneous Besov and Triebel-Lizorkin Spaces on RD-spaces
In this article we prove the embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces. The crucial idea is to use the geometric density condition on the measure.

Keywords:spaces of homogeneous type, test function space, distributions, Calderón reproducing formula, Besov and Triebel-Lizorkin spaces, embedding
Categories:42B25, 46F05, 46E35

4. CMB 2008 (vol 51 pp. 570)

Lutzer, D. J.; Mill, J. van; Tkachuk, V. V.
Amsterdam Properties of $C_p(X)$ Imply Discreteness of $X$
We prove, among other things, that if $C_p(X)$ is subcompact in the sense of de Groot, then the space $X$ is discrete. This generalizes a series of previous results on completeness properties of function spaces.

Keywords:regular filterbase, subcompact space, function space, discrete space
Categories:54B10, 54C05, 54D30

5. CMB 1999 (vol 42 pp. 321)

Kikuchi, Masato
Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces
We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section~2 the equivalence between Shimogaki's theorem and some martingale inequalities will be established, and in Section~3 the equivalence between Boyd's theorem and martingale inequalities with change of probability measure will be established.

Keywords:martingale inequalities, rearrangement invariant function spaces
Categories:60G44, 60G46, 46E30

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