We establish weighted Hardy-Littlewood inequalities for radial derivative and fractional radial derivatives on bounded symmetric domains.

2000 Mathematics Subject Classification: Primary 46E15, 54C55; Secondary 28B20.For weakly compact subsets of Hilbert spaces K, we study the existence of totally disconnected spaces L, such that C(K) is isomorphic to C(L). We prove that the space C(BH ) admits a Pełczyński decomposition and we provide a starshaped weakly compact K, subset of BH with non-empty interior in the norm topology, and such that C(K) ~= C(L) with L totally disconnected.Research partially supported by EPEAEK program “Pythagoras”....

In this paper we consider the regularity problem for the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ where $b$ is a locally integrable function and $(R_j{)}_{1\le j\le n}$ are the Riesz transforms in the $n$-dimensional euclidean space ${ℝ}^n$. More precisely, we prove that these commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded from $L^p$ into the Besov space ${\dot{B}}_p^{s,p}$ for $1\<p\<+\infty$ and $0\<s\<1$ if and only if $b$ is in the $BMO$-Triebel-Lizorkin space ${\dot{F}}_{\infty }^{s,p}$. The reduction of our result to the case $p=2$ gives in particular that the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded form $L^2$ into the Sobolev space ${\dot{H}}^s$ if and only if $b$ is in the $BMO$-Sobolev...

The first section consists of auxiliary results about nondecreasing real functions. In the second section a new characterization of relatively compact sets of regulated functions in the sup-norm topology is brought, and the third section includes, among others, an analogue of Helly's Choice Theorem in the space of regulated functions.

We develop a theory of removable singularities for the weighted Bergman space ${𝒜}_{\mu }^p\left(\Omega \right)=\{f\text{analytic}\text{in}\Omega {\int }_{\Omega }{\left|f\right|}^p\mathrm{d}\mu <\infty \}$, where $\mu$ is a Radon measure on $ℂ$. The set $A$ is weakly removable for ${𝒜}_{\mu }^p(\Omega \setminus A)$ if ${𝒜}_{\mu }^p(\Omega \setminus A)\subset \text{Hol}\left(\Omega \right)$, and strongly removable for ${𝒜}_{\mu }^p(\Omega \setminus A)$ if ${𝒜}_{\mu }^p(\Omega \setminus A)={𝒜}_{\mu }^p\left(\Omega \right)$. The general theory developed is in many ways similar to the theory of removable singularities for Hardy $H^p$ spaces, $\mathrm{B}MO$ and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable....

The classical Riemann Mapping Theorem states that a nontrivial simply connected domain Ω in ℂ is holomorphically homeomorphic to the open unit disc 𝔻. We also know that "similar" one-dimensional Riemann surfaces are "almost" holomorphically equivalent. We discuss the same problem concerning "similar" domains in ℂⁿ in an attempt to find a multidimensional quantitative version of the Riemann Mapping Theorem