We introduce function spaces $B^{p,\lambda }$ with Morrey-Campanato norms, which unify $B^{p,\lambda }$, $CM{O}^{p,\lambda }$ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator $I_{\alpha }$ on these spaces.

We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.

We study the notion of fractional $L^p$-differentiability of order $s\in (0,1)$ along vector fields satisfying the Hörmander condition on ${ℝ}^n$. We prove a modified version of the celebrated structure theorem for the Carnot-Carathéodory balls originally due to Nagel, Stein and Wainger. This result enables us to demonstrate that different $W^{s,p}$-norms are equivalent. We also prove a local embedding $W^{1,p}\subset W^{s,q}$, where q is a suitable exponent greater than p.

Let K be a compact Hausdorff space, $C_p\left(K\right)$ the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and $t_p\left(D\right)$ the topology in C(K) of pointwise convergence on D. It is proved that when $C_p\left(K\right)$ is Lindelöf the $t_p\left(D\right)$-compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and $C_p\left(K\right)$ is Lindelöf, then K is metrizable if, and only if, there is a countable and dense...

We construct wavelet-type frames associated with the expansive matrix dilation on the Anisotropic Triebel-Lizorkin spaces. We also show the a.e. convergence of the frame expansion which includes multi-wavelet expansion as a special case.

We characterize all Fréchet quotients of the space (Ω) of (complex-valued) real-analytic functions on an arbitrary open set $\Omega \subseteq {ℝ}^d$. We also characterize those Fréchet spaces E such that every short exact sequence of the form 0 → E → X → (Ω) → 0 splits.

Fréchet spaces of strongly, weakly and weak*-continuous Fréchet space valued functions are considered. Complete solutions are given to the problems of their injectivity or embeddability as complemented subspaces in dual Fréchet spaces.

Let X be a quasi-Banach rearrangement invariant space and let T be an (ε,δ)-atomic operator for which a restricted type estimate of the form $\parallel T{\chi }_E{\parallel }_X\le D\left(\right|E\left|\right)$ for some positive function D and every measurable set E is known. We show that this estimate can be extended to the set of all positive functions f ∈ L¹ such that ${\left|\right|f\left|\right|}_{\infty }\le 1$, in the sense that $\parallel Tf{\parallel }_X\le D\left(\right|\left|f\right|\left|₁\right)$. This inequality allows us to obtain strong type estimates for T on several classes of spaces as soon as some information about the galb of the space X is known. In this paper...

Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space ${ℒ}_E^1$ to be norm convergent (resp. relatively norm compact), thus extending the known results for ${ℒ}_{ℝ}^1$. Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in ${ℒ}_E^1$. It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence....

In this paper we study the role that unimodular functions play in deciding the uniform boundedness of sets of continuous linear functionals on various function spaces. For instance, inner functions are a UBD-set in H∞ with the weak-star topology.

We prove that in the homogeneous Besov-type space the set of bounded functions constitutes a unital quasi-Banach algebra for the pointwise product. The same result holds for the homogeneous Triebel-Lizorkin-type space.