In this paper we bring together the different known ways of establishing the continuity of the integral over a uniformly integrable set of functions endowed with the topology of pointwise convergence. We use these techniques to study Pettis integrability, as well as compactness in C(K) spaces endowed with the topology of pointwise convergence on a dense subset D in K.

It is known that the ring $B\left(ℝ\right)$ of all Baire functions carrying the pointwise convergence yields a sequential completion of the ring $C\left(ℝ\right)$ of all continuous functions. We investigate various sequential convergences related to the pointwise convergence and the process of completion of $C\left(ℝ\right)$. In particular, we prove that the pointwise convergence fails to be strict and prove the existence of the categorical ring completion of $C\left(ℝ\right)$ which differs from $B\left(ℝ\right)$.

Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{-1}{\sum }_{i=0}^{n-1}f\circ {\tau }^i\left(x\right)$ converges almost everywhere to a function f* in $L(p_1,q_1]$, where (pq) and $(p_1,q_1]$ are assumed to be in the set $(r,s):r=s=1,or1<r<\infty and1\le s\le \infty ,orr=s=\infty$. Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified...

Given a distribution $T$ on the sphere we define, in analogy to the work of Łojasiewicz, the value of $T$ at a point $\xi$ of the sphere and we show that if $T$ has the value $\tau$ at $\xi$, then the Fourier-Laplace series of $T$ at $\xi$ is Abel-summable to $\tau$.

We prove that a function belonging to a fractional Sobolev space $L^{\alpha ,p}\left(ℝⁿ\right)$ may be approximated in capacity and norm by smooth functions belonging to $C^{m,\lambda }\left(ℝⁿ\right)$, 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer’s theorem which states that Sobolev functions can be approximated by $C^m$ functions both in norm and capacity.

For a wide class of weights we find the approximative point spectrum and the essential spectrum of the pointwise multiplication operator $M_{\phi }$, $M_{\phi }\left(f\right)=\phi f$, on the weighted Banach spaces of analytic functions on the disc with the sup-norm. Thus we characterize when $M_{\phi }^{\text{'}}$ is Fredholm or is an into isomorphism. We also study cyclic phenomena for the adjoint map $M_{\phi }^{\text{'}}$.

We introduce generalized Campanato spaces ${}_{p,\varphi }$ on a probability space (Ω,ℱ,P), where p ∈ [1,∞) and ϕ: (0,1] → (0,∞). If p = 1 and ϕ ≡ 1, then ${}_{p,\varphi }=BMO$. We give a characterization of the set of all pointwise multipliers on ${}_{p,\varphi }$.

Let E and F be spaces of real- or complex-valued functions defined on a set X. A real- or complex-valued function g defined on X is called a pointwise multiplier from E to F if the pointwise product fg belongs to F for each f ∈ E. We denote by PWM(E,F) the set of all pointwise multipliers from E to F. Let X be a space of homogeneous type in the sense of Coifman-Weiss. For 1 ≤ p < ∞ and for $\varphi :X\times {ℝ}_{+}\to {ℝ}_{+}$, we denote by $bm{o}_{\varphi ,p}\left(X\right)$ the set of all functions $f\in L_{loc}^p\left(X\right)$ such that $su{p}_{a\in X,r>0}1/\varphi (a,r)(1/\mu \left(B(a,r)\right){ʃ}_{B(a,r)}|f\left(x\right)-f_{B(a,r)}{{|}^{p}d\mu )}^{1/p}<\infty$, where B(a,r) is the ball centered at a and of...

In this paper we shall compare three notions of pointwise smoothness: the usual definition, J.M. Bony's two-microlocal spaces Cx0s,s', and the corresponding definition on the wavelet coefficients. The purpose is mainly to show that these two-microlocal spaces provide "good substitutes" for the pointwise Hölder regularity condition; they can be very precisely compared with this condition, they have more functional properties, and can be characterized by conditions on the wavelet coefficients. We...