Weighted Besov and Triebel--Lizorkin spaces associated to operators

Let $X$ be a space of homogeneous type and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ satisfying Gaussian upper bounds on its heat kernels. In this paper we develop the theory of weighted Besov spaces $\dot{B}^{\alpha,L}_{p,q,w}(X)$ and weighted Triebel--Lizorkin spaces $\dot{F}^{\alpha,L}_{p,q,w}(X)$ associated to the operator $L$ for the full range $0<p,q\le \infty$, $\alpha\in \mathbb R$ and $w$ being in the Muckenhoupt weight class $A_\infty$. Similarly to the classical case in the Euclidean setting, we prove that our new spaces satisfy important features such as continuous charaterizations in terms of square functions, atomic decompositions and the identifications with some well known function spaces such as Hardy type spaces and Sobolev type spaces. Moreover, with extra assumptions on the operator $L$, we prove that the new function spaces associated to $L$ coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of $L$ and the spectral multiplier of $L$ in our new function spaces.Comment: 57 page

On multiplicative subgroups in division rings

Let $D$ be a division ring. In this paper, we investigate properties of subgroups of an arbitrary subnormal subgroup of the multiplicative group $D^*$ of $D$. The new obtained results generalize some previous results on subgroups of $D^*$.Comment: 14 page

The existence of free non-cyclic subgroups in weakly locally finite division rings

In this paper we prove that every non-central subnormal subgroup of the multiplicative group of a weakly locally finite division ring contains free non-cyclic subgroups

On perfect order subsets in finite groups

If $G$ is a finite group and $x\in G$ then the set of all elements of $G$ having the same order as $x$ is called {\em an order subset of $G$ determined by $x$} (see [2]). We say that $G$ is a {\em group with perfect order subsets} or briefly, $G$ is a {\em $POS$-group} if the number of elements in each order subset of $G$ is a divisor of $|G|$. In this paper we prove that for any $n\geq 4$, the symmetric group $S_n$ is not $POS$-group. This gives the positive answer to one of two questions rising from Conjecture 5.2 in [3].Comment: 8 page

Protein escape at the ribosomal exit tunnel: Effects of native interactions, tunnel length and macromolecular crowding

How fast a post-translational nascent protein escapes from the ribosomal exit tunnel is relevant to its folding and protection against aggregation. Here, by using Langevin molecular dynamics, we show that non-local native interactions help decreasing the escape time, and foldable proteins generally escape much faster than same-length self-repulsive homopolymers at low temperatures. The escape process, however, is slowed down by the local interactions that stabilize the {\alpha}-helices. The escape time is found to increase with both the tunnel length and the concentration of macromolecular crowders outside the tunnel. We show that a simple diffusion model described by the Smoluchowski equation with an effective linear potential can be used to map out the escape time distribution for various tunnel lengths and various crowder concentrations. The consistency between the simulation data and the diffusion model however is found only for the tunnel length smaller than a cross-over length of 90 {\AA} to 110 {\AA}, above which the escape time increases much faster with the tunnel length. It is suggested that the length of ribosomal exit tunnel has been selected by evolution to facilitate both the efficient folding and efficient escape of single domain proteins. We show that macromolecular crowders lead to an increase of the escape time, and attractive crowders are unfavorable for the folding of nascent polypeptide

Weighted Hardy spaces associated to operators and boundedness of singular integrals

Let $(X, d, \mu)$ be a space of homogeneous type, i.e. the measure $\mu$ satisfies doubling (volume) property with respect to the balls defined by the metric $d$. Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that the semigroup of $L$ satisfies the Davies-Gaffney estimates. In this paper, we study the weighted Hardy spaces $H^p_{L,w}(X)$, $0 < p \le 1$, associated to the operator $L$ on the space $X$. We establish the atomic and the molecular characterizations of elements in $H^p_{L,w}(X)$. As applications, we obtain the boundedness on \HL for the generalized Riesz transforms associated to $L$ and for the spectral multipliers of $L$.Comment: 26 pages, some minor errors were correcte

Hardy spaces, Regularized BMO spaces and the boundedness of Calder\'on-Zygmund operators on non-homogeneous spaces

One defines a non-homogeneous space $(X, \mu)$ as a metric space equipped with a non-doubling measure $\mu$ so that the volume of the ball with center $x$, radius $r$ has an upper bound of the form $r^n$ for some $n> 0$. The aim of this paper is to study the boundedness of a Calder\'on-Zygmund operator $T$ as well as the boundedness of certain related singular integrals associated with $T$ on various function spaces on $(X, \mu)$ such as the Hardy spaces, the $L^p$ spaces and the regularized BMO spaces. This article thus extends the work of X. Tolsa \cite{T1} on the non-homogeneous space $(\mathbb R^n, \mu)$ to the setting of a general non-homogeneous space $(X, \mu)$. While our framework is similar to that of \cite{H}, we are able to obtain quite a few properties similar to those of Calder\'on-Zygmund operators on doubling spaces, including the following for such an operator $T$: weak type $(1,1)$ estimate, boundedness from Hardy space into $L^1$, boundedness from $L^{\infty}$ into the regularized BMO and an interpolation theorem. We also prove that the dual space of the Hardy space is the regularized BMO space, obtain a Calder\'on-Zygmund decomposition on the non-homogeneous space $(X, \mu)$ and use this decomposition to show the boundedness of the maximal operators in the form of Cotlar inequality as well as the boundedness of commutators of Calder\'on-Zygmund operators and BMO functions.Comment: 37 pages; some minor changes; to appear in Journal of Geometric Analysi

Folding and escape of nascent proteins at ribosomal exit tunnel

We investigate the interplay between post-translational folding and escape of two small single-domain proteins at the ribosomal exit tunnel by using Langevin dynamics with coarse-grained models. It is shown that at temperatures lower or near the temperature of the fastest folding, folding proceeds concomitantly with the escape process, resulting in vectorial folding and enhancement of foldability of nascent proteins. The concomitance between the two processes, however, deteriorates as temperature increases. Our folding simulations as well as free energy calculation by using umbrella sampling show that, at low temperatures, folding at the tunnel follows one or two specific pathways without kinetic traps. It is shown that the escape time can be mapped to a one-dimensional diffusion model with two different regimes for temperatures above and below the folding transition temperature. Attractive interactions between amino acids and attractive sites on the tunnel wall lead to a free energy barrier along the escape route of protein. It is suggested that this barrier slows down the escape process and consequently promotes correct folding of the released nascent protein.Comment: to appear in J. Chem. Phy

On subnormal subgroups in division rings containing a non-abelian solvable subgroup

Let $D$ be a division ring with center $F$ and $N$ a subnormal subgroup of the multiplicative group $D^*$ of $D$. Assume that $N$ contains a non-abelian solvable subgroup. In this paper, we study the problem on the existence of non-abelian free subgroups in $N$. In particular, we show that if either $N$ is algebraic over $F$ or $F$ is uncountable, then $N$ contains a non-abelian free subgroup.Comment: 10 page

Regularity estimates for Green operators of Dirichlet and Neumann problems on weighted Hardy spaces

In this paper we first study the generalized weighted Hardy spaces $H^p_{L,w}(X)$ for $0<p\le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of finite and infinite measure. We show that the weighted Hardy spaces defined via maximal functions and atomic decompositions coincide. Then we prove weighted regularity estimates for the Green operators of the inhomogeneous Dirichlet and Neumann problems in suitable bounded or unbounded domains including bounded semiconvex domains, convex regions above a Lipschitz graph and upper half-spaces. Our estimates are in terms of weighted $L^p$ spaces for the range $1<p<\infty$ and in terms of the new weighted Hardy spaces for the range $0<p\le 1$. Our regularity estimates for the Green operators under the weak smoothness assumptions on the boundaries of the domains are new, especially the estimates on Hardy spaces for the full range $0<p\le 1$ and the case of unbounded domains.Comment: 26 page