We show that in Euclidean space and other regular metric spaces, the notions of dimensions defined by Assouad and Aikawa coincide. In addition, in more general metric spaces, we study the relationship between these two dimensions and a related codimension and give an application of the Aikawa (co)dimension for the Hardy inequalities.

Let $R$ be a Gorenstein local ring which is locally a hypersurface on the punctured spectrum. In this paper, we classify thick subcategories of the bounded derived category of finitely generated $R$-modules. Moreover, using this classification, we also classify thick subcategories of finitely generated $R$-modules, and find out the relationships with thick subcategories of Cohen-Macaulay $R$-modules.

In this paper, we study some dynamical properties of fixed-point free homeomorphisms of separable metric spaces. For each natural number $p$, we define eventual colorings within $p$ of homeomorphisms which are generalized notions of colorings of fixed-point free homeomorphisms, and we investigate the eventual coloring number $C(f,p)$ of a fixed-point free homeomorphism $f: X \to X$ with zero-dimensional set of periodic points. In particular, we show that if $\dim X$ < $\infty$, then there is a natural number $p$, which depends on $\dim X$, and $X$ can be divided into two closed regions $C_{1}$ and $C_{2}$ such that for each point $x\in X$, the orbit $\{f^{k}(x)\}_{k=0}^{\infty}$ of $x$ goes back and forth between $C_1-C_2$ and $C_2-C_1$ within the time $p$.

Let $(X, d, \mu)$ be a metric measure space endowed with a metric $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $X$. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables $x$ and $y$. Also assume that $L$ satisfies a Plancherel type estimate. Under these conditions, we show the $L^p$ bounds for Stein's square functions arising from Bochner-Riesz means associated to the operator $L$. We then use the $L^p$ estimates on Stein's square functions to obtain a Hörmander-type criterion for spectral multipliers of $L$. These results are applicable for large classes of operators including sub-Laplacians acting on Lie groups of polynomial growth and Schrödinger operators with rough potentials.

We denote by $\delta_g$ (resp. $\delta_g^+$), the minimal dilatation for pseudo-Anosovs (resp. pseudo-Anosovs with orientable invariant foliations) on a closed surface of genus $g$. This paper concerns the pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the Whitehead sister link exterior $W$ by Dehn filling two cusps, where the fillings are on the boundary slopes of fibers of $W$. We give upper bounds of $\delta_g$ for $g \equiv 0,1,5,6,7,9 \pmod{10}$, $\delta_g^+$ for $g \equiv 1,5,7,9 \pmod{10}$. Our bounds improve the previous one given by Hironaka. We note that the monodromies of fibrations on $W$ were also studied by Aaber and Dunfield independently.

In this paper a counterpart of the classical Poisson integral formula is found for a class of standard weighted Laplace differential operators in the unit disc. In the process the corresponding Dirichlet boundary value problem is solved for arbitrary distributional boundary data. Boundary limits and representations of the associated solutions are studied within a framework of homogeneous Banach spaces. Special emphasis is put on the so-called relative completion of a homogeneous Banach space.

Let $H$ be the upper half-space of the $(n+1)$-dimensional Euclidean space. Let 0 < $\alpha \le 1$ and $m(\alpha)=\min \{1, 1/(2\alpha) \}$. For $\sigma$ > $-m(\alpha)$, the $\alpha$-parabolic Bloch type space ${\cal B}_{\alpha}(\sigma)$ on $H$ is the set of all solutions $u$ of the equation $( \partial/\partial t+(-\Delta_{x})^{\alpha} )u=0$ with finite Bloch norm $\| u \|_{{\cal B}_{\alpha}(\sigma)}$ of a weight $t^{\sigma}$. It is known that ${\cal B}_{1/2}(0)$ coincides with the classical harmonic Bloch space on $H$. We extend the notion of harmonic conjugate functions to functions in the $\alpha$-parabolic Bloch type space ${\cal B}_{\alpha}(\sigma)$. We study properties of $\alpha$-parabolic conjugate functions and give an application to the estimates of tangential derivative norms on ${\cal B}_{\alpha}(\sigma)$. Inversion theorems for $\alpha$-parabolic conjugate functions are also given.

The asymptotic behavior at infinity of oscillatory integrals is in detail investigated by using the Newton polyhedra of the phase and the amplitude. We are especially interested in the case that the amplitude has a zero at a critical point of the phase. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation.

We consider Schrödinger equations $i\partial_t u = (-\Delta + V)u$ in ${\mathbb R}^3$ with a real potential $V$ such that, for an integer $k\geq 0$, $\langle x \rangle^k V(x)$ belongs to an amalgam space $\ell^p(L^q)$ for some $1\leq p$ < 3/2 < $q \leq \infty$, where $\langle x \rangle=(1+|x|^2)^{1/2}$. Let $H = -\Delta + V$ and let $P_{ac}$ be the projector onto the absolutely continuous subspace of $L^2({\mathbb R}^3)$ for $H$. Assuming that zero is not an eigenvalue nor a resonance of $H$, we show that solutions $u(t) = \exp(-itH)P_{ac}\varphi$ admit asymptotic expansions as $t \to \infty$ of the form

$$\bigg\| \langle x \rangle^{-k-\varepsilon} \bigg( u(t)- \sum_{j=0}^{[k/2]}t^{-\frac32-j}P_j \varphi \bigg) \bigg\|_{\infty} \leq C |t|^{-\frac{k+3+\varepsilon}2} \big\| \langle x \rangle^{k+\varepsilon}\varphi \big\|_1$$

for 0 < $\varepsilon$ < $3(1/p-2/3)$, where $P_0, \dots, P_{[k/2]}$ are operators of finite rank and $[k/2]$ is the integral part of $k/2$. The proof is based upon estimates of boundary values on the reals of the resolvent $(-\Delta -\lambda^2)^{-1}$ as an operator-valued function between certain weighted amalgam spaces.

We analyze in weighted $L^q$-spaces the linearized system of partial differential equations arising from the motion of a rotating obstacle in a fluid. We prove some existence, uniqueness and regularity results of decaying or growing weak solutions. Two auxiliary equations are also considered and treated.

Our aim in this paper is to prove the Gagliardo-Nirenberg inequality for Riesz potentials of functions in variable exponent Lebesgue spaces, which are called Musielak-Orlicz spaces with respect to $\Phi(x,t)=t^{p(x)}(\log(c_0+t))^{q(x)}$ for $t$ > 0 and $x \in {\mathbb R}^n$, via the Littlewood-Paley decomposition.

For a genus-1 1-bridge knot in $S^3$, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained by Goda, Hayashi, and Ishihara. In a previous paper, we generalized their construction and calculated the slope invariants for the resulting examples. We give an iterated version of the construction that produces many more examples, and calculate their slope invariants. If one starts with the trivial knot, the iterated constructions produce all the 2-bridge knots, giving a new calculation of the slope invariants of their tunnels. In the final section we compile a list of the known possibilities for the set of tunnels of a given tunnel number 1 knot.