Suppose that $X$ and $Y$ are Banach spaces and that the Banach space $X{\widehat{\otimes }}_{\tau }Y$ is their complete tensor product with respect to some tensor product topology $\tau$. A uniformly bounded $X$-valued function need not be integrable in $X{\widehat{\otimes }}_{\tau }Y$ with respect to a $Y$-valued measure, unless, say, $X$ and $Y$ are Hilbert spaces and $\tau$ is the Hilbert space tensor product topology, in which case Grothendieck’s theorem may be applied. In this paper, we take an index $1\le p<\infty$ and suppose that $X$ and $Y$ are $L^p$-spaces with ${\tau }_p$ the associated...

Let $\left\{T\right(t):t>0\}$ be a strongly continuous semigroup of linear contractions in $L_p$, $1<p<\infty$, of a $\sigma$-finite measure space. In this paper we prove that if there corresponds to each $t>0$ a positive linear contraction $P\left(t\right)$ in $L_p$ such that $\left|T\right(t\left)f\right|\le P\left(t\right)\left|f\right|$ for all $f\in L_p$, then there exists a strongly continuous semigroup $\left\{S\right(t):t>0\}$ of positive linear contractions in $L_p$ such that $\left|T\right(t\left)f\right|\le S\left(t\right)\left|f\right|$ for all $t>0$ and $f\in L_p$. Using this and Akcoglu’s dominated ergodic theorem for positive linear contractions in $L_p$, we also prove multiparameter pointwise ergodic and local ergodic...

This paper gives the local existence of mild solutions to the Cauchy problem for the complex Ginzburg-Landau type equation $${\displaystyle \frac{\partial u}{\partial t}}-(\lambda +\mathrm{i}\alpha )\Delta u+(\kappa +\mathrm{i}\beta ){\left|u\right|}^{q-1}u-\gamma u=0$$ in ${ℝ}^N\times (0,\infty )$ with $L^p$-initial data $u_0$ in the subcritical case ($1\le q<1+2p/N$), where $u$ is a complex-valued unknown function, $\alpha$, $\beta$, $\gamma$, $\kappa \in ℝ$, $\lambda >0$, $p>1$, $\mathrm{i}=\sqrt{-1}$ and $N\in \mathbb{N}$. The proof is based on the $L^p$-$L^q$ estimates of the linear semigroup $\{exp(t(\lambda +\mathrm{i}\alpha )\Delta \left)\right\}$ and usual fixed-point argument.

If $(\Omega ,\Sigma ,\mu )$ is a finite measure space and $X$ a Banach space, in this note we show that $L_{w^{*}}^1(\mu ,X^{*})$, the Banach space of all classes of weak* equivalent $X^{*}$-valued weak* measurable functions $f$ defined on $\Omega$ such that $\parallel f\left(\omega \right)\parallel \le g\left(\omega \right)$ a.e. for some $g\in L_1\left(\mu \right)$ equipped with its usual norm, contains a copy of $c_0$ if and only if $X^{*}$ contains a copy of $c_0$.

The main purpose of this paper is to prove that the boundedness of the commutator [...] Mκ,b∗ ${ℳ}_{\kappa ,b}^{*}$ generated by the Littlewood-Paley operator [...] Mκ∗ ${ℳ}_{\kappa }^{*}$ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of [...] Mκ∗ ${ℳ}_{\kappa }^{*}$ satisfies a certain Hörmander-type condition, the authors prove that [...] Mκ,b∗ ${ℳ}_{\kappa ,b}^{*}$ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from...

Commuting multi-contractions of class $C_{0·}$ and having a regular isometric dilation are studied. We prove that a polydisc contraction of class $C_{0·}$ is the restriction of a backwards multi-shift to an invariant subspace, extending a particular case of a result by R. E. Curto and F.-H. Vasilescu. A new condition on a commuting multi-operator, which is equivalent to the existence of a regular isometric dilation and improves a recent result of A. Olofsson, is obtained as a consequence. ...