We present a counter-example to a certain conjecture that is related to Whitney's extension problems.

In this work, cyclic Blaschke products for composition operators induced by disc automorphisms are studied. In particular, we obtain interpolating Blaschke products that are cyclic for nonelliptic automorphisms and we obtain a new characterization of Blaschke products that are not finite products of interpolating Blaschke products.

We investigate the $L^2$ boundedness of the triple Hilbert transform along the surface given by the graph of a real polynomial $P$ of three variables. We are interested in understanding the relationship between the geometric properties of the Newton polyhedron of $P$ and the analytic property of $L^2$ boundedness.

We prove a quantitative version of the following statement. Given a Lipschitz function $f$ from the k-dimensional unit cube into a general metric space, one can be decomposed $f$ into a finite number of BiLipschitz functions $f|_{F_i}$ so that the k-Hausdorff content of $f([0,1]^k\smallsetminus \cup F_i)$ is small. We thus generalize a theorem of P. Jones [Lipschitz and bi-Lipschitz functions. Rev. Mat. Iberoamericana 4 (1988), no. 1, 115-121] from the setting of $\mathbb{R}^d$ to the setting of a general metric space. This positively answers problem 11.13 in "Fractured Fractals and Broken Dreams" by G. David and S. Semmes, or equivalently, question 9 from "Thirty-three yes or no questions about mappings, measures, and metrics" by J. Heinonen and S. Semmes. Our statements extend to the case of {\it coarse} Lipschitz functions.

In this paper, we prove the Adams inequality in complete metric spaces supporting a Poincaré inequality with a doubling measure. We also prove the trace inequalities for the Riesz potentials.

Coifman and Fefferman established that the class of Muckenhoupt weights is equivalent to the class of weights satisfying the "reverse Hölder inequality". In a recent paper V. Vasyunin [The exact constant in the inverse Hölder inequality for Muckenhoupt weights. St. Petersburg Math. J. 15 (2004), no. 1, 49-79] presented a proof of the reverse Hölder inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the inverse, that is, we use the Bellman function technique to find the sharp $A_p$ constants for weights in a reverse-Hölder class on an interval; we also find the sharp constants for the higher-integrability result of Gehring [The $L_p$-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265-277]. Additionally, we find sharp bounds for the $A_p$ constants of reverse-Hölder-class weights defined on rectangles in $\mathbb{R}^n$, as well as bounds on the $A_p$ constants for reverse-Hölder weights defined on cubes in $\mathbb{R}^n$, without claiming the sharpness.

Let $\mu$ be a nonnegative Radon measure on ${\mathbb R}^d$ which satisfies the growth condition that there exist constants $C_0 > 0$ and $n\in(0,d]$ such that for all $x\in{\mathbb R}^d$ and $r > 0$, $\mu(B(x,\,r)) \le C_0 r^n$, where $B(x,r)$ is the open ball centered at $x$ and having radius $r$. In this paper, we introduce a local atomic Hardy space ${h_{\rm atb}^{1,\infty}(\mu)}$, a local BMO-type space ${\mathop\mathrm{rbmo}(\mu)}$ and a local BLO-type space ${\mathop\mathrm{rblo}(\mu)}$ in the spirit of Goldberg and establish some useful characterizations for these spaces. Especially, we prove that the space ${\mathop\mathrm{rbmo}(\mu)}$ satisfies a John-Nirenberg inequality and its predual is ${h_{\rm atb}^{1,\infty}(\mu)}$. We also establish some useful properties of ${\mathop\mathrm{RBLO}\,(\mu)}$ and improve the known characterization theorems of ${\mathop\mathrm{RBLO}(\mu)}$ in terms of the natural maximal function by removing the assumption on the regularity condition. Moreover, the relations of these local spaces with known corresponding function spaces are also presented. As applications, we prove that the inhomogeneous Littlewood-Paley $g$-function $g(f)$ of Tolsa is bounded from ${h_{\rm atb}^{1,\infty}(\mu)}$ to ${L^1(\mu)}$, and that $[g(f)]^2$ is bounded from ${\mathop\mathrm{rbmo}(\mu)}$ to ${\mathop\mathrm{rblo}(\mu)}$.

The famous Nyquist-Shannon sampling theorem has been recently improved by A. Olevskii and A. Ulanovskii. The present contribution is aimed at bridging the gap between their advance on irregular sampling and what is named "compressed sensing" in signal processing.

This paper investigates the problem of global existence of small solutions to semi-linear wave equations with radially symmetric data of critical regularity. Under radial symmetry we focus on the case where the power of nonlinear term is somewhat smaller than the conformal power. Our result covers the case where the power is strictly larger than the John-Glassey exponent in two or three space dimensions. In higher dimension it applies to the equation whose power is strictly larger than the $L^2$-critical exponent. The main theorem is therefore an improvement over a previous result due to Lindblad and Sogge. The new ingredient in our proof is an effective use of some weighted estimates of radially symmetric solutions to inhomogeneous wave equations.

We use explicit solutions to a drifted Laplace equation in warped product model spaces as comparison constructions to show $p$-hyperbolicity of a large class of submanifolds for $p\ge 2$. The condition for $p$-hyperbolicity is expressed in terms of upper support functions for the radial sectional curvatures of the ambient space and for the radial convexity of the submanifold. In the process of showing $p$-hyperbolicity we also obtain explicit lower bounds on the $p$-capacity of finite annular domains of the submanifolds in terms of the drifted $2$-capacity of the corresponding annuli in the respective comparison spaces.

How different is the universal cover of a given finite $2$-complex from a $3$-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group $G$ is said to be properly $3$-realizable if there exists a compact $2$-polyhedron $K$ with $\pi_1(K) \cong G$ whose universal cover $\tilde{K}$ has the proper homotopy type of a PL $3$-manifold (with boundary). In this paper, we study the asymptotic behavior of finitely generated one-relator groups and show that those having finitely many ends are properly $3$-realizable, by describing what the fundamental pro-group looks like, showing a property of one-relator groups which is stronger than the QSF property of Brick (from the proper homotopy viewpoint) and giving an alternative proof of the fact that one-relator groups are semistable at infinity.

We link the overconvergence properties of certain Taylor series in the unit disk to the maximality of their cluster sets, so connecting outer wild behavior to inner wild behavior. Specifically, it is proved the existence of a dense linear manifold of holomorphic functions in the disk that are, except for zero, universal Taylor series in the sense of Nestoridis and, simultaneously, have maximal cluster sets along many curves tending to the boundary. Moreover, it is constructed a dense linear manifold of universal Taylor series having, for each boundary point, limit zero along some path which is tangent to the corresponding radius. Finally, it is proved the existence of a closed infinite dimensional manifold of holomorphic functions enjoying the two-fold wild behavior specified at the beginning.

In this paper we prove Hilbert Nullstellensatz for real coherent analytic surfaces and we give a precise description of the obstruction to get it in general. Refering the first, we prove that the ideals of global functions vanishing on analytic subsets are exactly the real saturated ones. For $\mathbb{R}^3$ we prove that the real Nullstellensatz holds for real saturated ideals if and only if no principal ideal generated by a function whose zero set is a curve (indeed, a special function) is real. This led us to compare the Nullstellensatz problem with the Hilbert 17th one, also in its weaker form involving infinite sums of squares, proving that they share in fact the same obstruction.