We prove that if the exponent function $p(\cdot)$ satisfies log-Hölder continuity conditions locally and at infinity, then the fractional maximal operator $M_\alpha$, $0 < \alpha < n$, maps $L^{p(\cdot)}$ to $L^{q(\cdot)}$, where $\frac{1}{p(x)}-\frac{1}{q(x)}=\frac{\alpha}{n}$. We also prove a weak-type inequality corresponding to the weak $(1,n/(n-\alpha))$ inequality for $M_\alpha$. We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [The maximal function on variable $L^p$ spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003), 223-238]. As a consequence of these results for $M_\alpha$, we show that the fractional integral operator $I_\alpha$ satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable $L^p$ spaces.

We show that the Lusin area function and essentially all of its real-variable generalizations are pointwise dominated by an \lq\lq intrinsic\rq\rq\ square function, and that this latter function is, for all practical purposes, no larger than a \lq\lq generic\rq\rq square function.

Let $\gamma, r, s$, $ \geq 1$ be non-negative integers. If $p$ is a prime sufficiently large relative to the values $\gamma$, $r$ and $s$, then a group $H$ of conformal automorphisms of a closed Riemann surface $S$ of order $p^{s}$ so that $S/H$ has signature $(\gamma,r)$ is the unique such subgroup in $\mathrm{Aut}(S)$. Explicit sharp lower bounds for $p$ in the case $(\gamma,r,s) \in \{(1,2,1),(0,4,1)\}$ are provided. Some consequences are also derived.

We prove that a locally compact metric space that supports a doubling measure and a weak $p$-Poincaré inequality for some $1\le p < \infty$ is a $\mathrm{MEC}_p$-space. The methods developed for this purpose include measurability considerations and lead to interesting consequences. For example, we verify that each extended real valued function having a $p$-integrable upper gradient is locally $p$-integrable.

It is proved that if $u$ is a real valued function harmonic in the open unit ball $\mathbb B_N\subset \mathbb R^N$ and continuous on the closed ball, then the following conditions are equivalent, for $0 < \alpha < 1$: \begin{itemize} \item $|u(x)-u(w)|\le C|x-w|^\alpha, \quad x, w\in \mathbb B_N$; \item $| |u(y)|-|u(\zeta) | |\le C|y-\zeta|^\alpha, \quad y, \zeta\in \partial\mathbb B_N$; \item $| |u(y)|-|u(ry)| |\le C(1-r)^\alpha, \quad y\in \partial\mathbb B_N,\ 0 < r < 1$. \end{itemize} The Lipschitz condition on $|u|^p$ is also considered.

Let $\Omega$ and $\Pi$ be two simply connected proper subdomains of the complex plane $\mathbb{C}$. We are concerned with the set $A(\Omega,\Pi)$ of functions $f: \Omega\longrightarrow\Pi$ holomorphic on $\Omega$ and we prove estimates for $|f^{(n)}(z)|, f \in A(\Omega,\Pi), z \in \Omega$, of the following type. Let $\lambda_{\Omega}(z)$ and $\lambda_{\Pi}(w)$ denote the density of the Poincaré metric with curvature $K=-4$ of $\Omega$ at $z$ and of $\Pi$ at $w$, respectively. Then for any pair $(\Omega,\Pi)$ of convex domains, $f \in A(\Omega,\Pi), z \in \Omega$, and $n\geq 2$ the inequality $$ \frac{|f^{(n)}(z)|}{n!}\leq 2^{n-1}\frac{(\lambda_{\Omega}(z))^n}{\lambda_{\Pi}(f(z))} $$ is valid. The constant $2^{n-1}$ is best possible for any pair $(\Omega,\Pi)$ of convex domains. For any pair $(\Omega,\Pi)$, where $\Omega$ is convex and $\Pi$ linearly accessible, $f,z,n$ as above, we prove $$ \frac{|f^{(n)}(z)|}{(n+1)!}\leq 2^{n-2}\frac{(\lambda_{\Omega}(z))^n}{\lambda_{\Pi}(f(z))}. $$ The constant $2^{n-2}$ is best possible for certain admissible pairs $(\Omega,\Pi)$. These considerations lead to a new, nonanalytic, characterization of bijective convex functions $h:\Delta\to\Omega$ not using the second derivative of $h$.

A metric space $U$ has the universal Lipschitz extension property if for an arbitrary metric space $M$ and every subspace $S$ of $M$ isometric to a subspace of $U$ there exists a continuous linear extension of Banach-valued Lipschitz functions on $S$ to those on all of $M$. We show that the finite direct sum of Gromov hyperbolic spaces of bounded geometry is universal in the sense of this definition.

We shall discuss various points on solutions of the 3D Navier-Stokes equations from the point of view of Morrey-Campanato spaces (global solutions, strong-weak uniqueness, the role of real interpolation, regularity).

We prove new irrationality measures with restricted denominators of the form $\mathrm{d}_{\lfloor\nu m\rfloor}^s B^m$ (where $B, m \in\mathbb{N}, \nu > 0$, $s\in\{0,1\}$ and $\mathrm{d}_m=\textup{lcm}\{1,2, \ldots, m\}$) for values of the logarithm at certain rational numbers $r>0$. In particular, we show that such an irrationality measure of $\log(r)$ is arbitrarily close to 1 provided $r$ is sufficiently close to 1. This implies certain results on the number of non-zero digits in the $b$--ary expansion of $\log(r)$ and on the structure of the denominators of convergents of $\log(r)$. No simple method for calculating the latter is known. For example, we show that, given integers $a,c\ge 1$, for all large enough $b, n$, the denominator $q_n$ of the $n$--th convergent of $\log(1\pm a/b)$ cannot be written under the form $\dd_{\lfloor\nu m\rfloor}^s (bc)^m$: this is true for $a=c=1$, $b \ge 12$ when $s=0$, resp. $b \ge 2$ when $s=1$ and $\nu=1$. Our method rests on a detailed diophantine analysis of the upper Padé table $([p/q])_{p\ge q\ge 0}$ of the function $\log(1-x)$. Finally, we remark that worse results (of this form) are currently provable for the exponential function, despite the fact that the complete Padé table $([p/q])_{p, q\ge 0}$ of $\exp(x)$ and the convergents of $\exp(1/b)$, for $\vert b\vert \ge 1$, are well-known, for example.

In this paper we relate the minimal annulus of a planar convex body $K$ with its inradius, obtaining all the upper and lower bounds, in terms of these quantities, for the classic geometric measures associated with the set: area, perimeter, diameter, minimal width and circumradius. We prove the optimal inequalities for each one of those problems, determining also its corresponding extremal sets.

The Littlewood-Paley theory for a symmetric diffusion semigroup $T^t$, as developed by Stein, is here generalized to deal with the tensor extensions of these operators on the Bochner spaces $L^p(\mu,X)$, where $X$ is a Banach space. The $g$-functions in this situation are formulated as expectations of vector-valued stochastic integrals with respect to a Brownian motion. A two-sided $g$-function estimate is then shown to be equivalent to the UMD property of $X$. As in the classical context, such estimates are used to prove the boundedness of various operators derived from the semigroup $T^t$, such as the imaginary powers of the generator.

Given a probability space $(\Omega, \mathsf{A}, \mu)$, let $\mathsf{A}_1, \mathsf{A}_2, \ldots$ be a filtration of $\sigma$-subalgebras of $\mathsf{A}$ and let $\mathsf{E}_1, \mathsf{E}_2, \ldots$ denote the corresponding family of conditional expectations. Given a martingale $f = (f_1, f_2, \ldots)$ adapted to this filtration and bounded in $L_p(\Omega)$ for some $2 \le p < \infty$, Burkholder's inequality claims that $$ \|f\|_p \sim_{\mathrm{c}_p} \Big\| \Big( \sum_{k=1}^\infty \mathsf{E}_{k-1}(|df_k|^2) \Big)^\frac12 \Big\|_p + \Big( \sum_{k=1}^\infty \|df_k\|_p^p \Big)^\frac1p. $$ Motivated by quantum probability, Junge and Xu recently extended this result to the range $1 < p < 2$. In this paper we study Burkholder's inequality for $p=1$, for which the techniques must be different. Quite surprisingly, we obtain two non-equivalent estimates which play the role of the weak type $(1,1)$ analog of Burkholder's inequality. As application we obtain new properties of Davis decomposition for martingales.

We prove the existence of positive solutions concentrating simultaneously on some higher dimensional manifolds near and on the boundary of the domain for a nonlinear singularly perturbed elliptic Neumann problem.

This article studies strong $A_{\infty}$-weights and Besov capacities as well as their relationship to Hausdorff measures. It is shown that in the Euclidean space ${\mathbb{R}}^n$ with $n\ge 2$, whenever $n-1 < s \le n$, a function $u$ yields a strong $A_\infty$-weight of the form $w=e^{nu}$ if the distributional gradient $\nabla u$ has sufficiently small $||\cdot||_{{\mathcal L}^{s,n-s}}({\mathbb{R}}^n; {\mathbb{R}}^n)$-norm. Similarly, it is proved that if $2\le n < p < \infty$, then $w=e^{nu}$ is a strong $A_\infty$-weight whenever the Besov $B_p$-seminorm $[u]_{B_p({\mathbb{R}}^n)}$ of $u$ is sufficiently small. Lower estimates of the Besov $B_p$-capacities are obtained in terms of the Hausdorff content associated with gauge functions $h$ satisfying the condition $\int_0^1 h(t)^{p'-1} \frac{dt}{t} < \infty$.

We prove Serre's conjecture for the case of Galois representations of Serre's weight $2$ and level $1$. We do this by combining the potential modularity results of Taylor and lowering the level for Hilbert modular forms with a Galois descent argument, properties of universal deformation rings, and the non-existence of $p$-adic Barsotti-Tate conductor $1$ Galois representations proved in [Dieulefait, L.: Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture. J. Reine Angew. Math. 577 (2004), 147-151].

We show rough path estimates for smooth $L^p$ functions whose derivatives are in $L^q$. We also give applications to Fourier analysis.