We consider a multidimensional version of an inequality due to Leray as a substitute for Hardy’s inequality in the case $p = n ≥ 2$. In this paper we provide an optimal Sobolev-type improvement of this substitute, analogous to the corresponding improvements obtained for $p = 2 < n$ in S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (1) (2002) 186–233, and for $p > n ≥ 1$ in G. Psaradakis, An optimal Hardy-Morrey inequality, Calc. Var. Partial Differential Equations 45 (3-4) (2012) 421–441.

Moduli of path families are widely used to mark curves which may be neglected for some applications. We introduce ordinary and approximation modulus with respect to Banach function spaces. While these moduli lead to the same result in reflexive spaces, we show that there are important path families (like paths tangent to a given set) which can be labeled as negligible by the approximation modulus with respect to the Lorentz Lp,1-space for an appropriate p, in particular, to the ordinary L1-space if p=1, but not by the ordinary modulus with respect to the same space.

Let $\mathcal{M} =\{m_{j}\}_{j=1}^{\infty}$ be a family of Marcinkiewicz multipliers of sufficient uniform smoothness in $\mathbb{R}^{n}$ . We show that the$L$^{$p$}norm, 1<$p$<∞, of the related maximal operator $$M_Nf(x)= \sup_{1\leq j \leq N} |\mathcal{F}^{-1} ( m_j \mathcal{F} f)|(x) $$ is at most$C$(log($N$+2))^{$n$/2}. We show that this bound is sharp.

We consider weighted ray-transforms PW (weighted Radon transforms along oriented straight lines) in Rd,d≥2, with strictly positive weights W. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on Rd. In addition, the constructed weight W is rotation-invariant continuous and is infinitely smooth almost everywhere on Rd×Sd−1. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of W is slightly relaxed. We also give examples of continous strictly positive W such that dimkerPW≥n in the space of infinitely smooth compactly supported functions on Rd for arbitrary n∈N∪{∞}, where W are infinitely smooth for d=2 and infinitely smooth almost everywhere for d≥3.

In this paper, we give a variational analysis to the planar dual Minkowski problem in the Sobolev space. With the new variational characterization, we can deal with existence results for prescribed not necessarily positive data. Meanwhile, functional inequalities and multiple solutions are also obtained.

It is shown that for any$t$, 0<$t$<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that $\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\}$ and with the property that the analytic polynomials are dense in the Bergman space $\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma)$ . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in $H^{t}(\mathbb{D}\setminus\Gamma)$ ; improving upon a result in an earlier paper.

The geometric descriptions of the (essential) spectra of Toeplitz operators with piecewise continuous symbols are among the most beautiful results about Toeplitz operators on Hardy spaces Hp with 1<p<∞. In the Hardy space H1, the essential spectra of Toeplitz operators are known for continuous symbols and symbols in the Douglas algebra C+H∞. It is natural to ask whether the theory for piecewise continuous symbols can also be extended to H1. We answer this question in the negative and show in particular that the Toeplitz operator is never bounded on H1 if its symbol has a jump discontinuity.

We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ^{$n$}and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.

Let$f$be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, $${f \in \mathcal{A}(\bar{\mathbb{C}} \setminus A)}$$ , $${\# A< \infty}$$ . J. Nuttall has put forward the important relation between the$maximal domain$of$f$where the function has a single-valued branch and the$domain of convergence$of the diagonal Padé approximants for$f$. The Padé approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of$f$in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function$f$single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has$minimal logarithmic capacity$among all other systems converting the function$f$to a single-valued branch. Thus the domain of convergence corresponds to the$maximal$(in the sense of$minimal$boundary) domain of single-valued holomorphy for the analytic function $${f\in\mathcal{A}(\bar{\mathbb{C}} \setminus A)}$$ . The complete proof of Nuttall’s conjecture (even in a more general setting where the set$A$has logarithmic capacity 0) was obtained by H. Stahl. In this work, we derive$strong asymptotics$for the denominators of the diagonal Padé approximants for this problem in a rather general setting. We assume that$A$is a finite set of branch points of$f$which have the$algebro-logarithmic character$and which are placed in a$generic position$. The last restriction means that we exclude from our consideration some degenerated “constellations” of the branch points.

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This paper deals with the moment problem on a (not necessarily finitely generated) commutative unital real algebra A. We define moment functionals on A as linear functionals which can be written as integrals over characters of A with respect to cylinder measures. Our main results provide such integral representations for A+–positive linear functionals (generalized Haviland theorem) and for positive functionals fulfilling Carleman conditions. As an application, we solve the moment problem for the symmetric algebra S(V) of a real vector space V. As a byproduct, we obtain new approaches to the moment problem on S(V) for a nuclear space V and to the integral decomposition of continuous positive functionals on a barrelled nuclear topological algebra A.

Let A1, A2 be (not necessarily unital or closed) standard operator algebras on locally convex spaces X1, X2, respectively. For k 2, define different kinds of products T1 Tk on elements in Ai, which covers the usual product T1 Tk= T1 Tk, and the Jordan triple product T1 T2= T2T1T2. Let : A1 A2 be a (not necessarily linear) map satisfying that ( (A1) (Ak))= (A1 Ak) whenever any one of Ais is of rank zero or one. It is shown that if the range of contains all rank one and rank two operators then it must be a Jordan isomorphism multiplied by a root of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.

The analyticity of the Stokes semigroup with the Dirichlet boundary condition is established in spaces of bounded functions when the domain occupied with fluid is bounded or more generally admissible which admits a special estimate for the Helmholtz decomposition. The proof is based on a blow-up argument. This is the first proof of the analyticity in spaces of bounded functions which was left open more than thirty years.

Let Ω be a Greenian domain in ℝ^{$d$},$d$≥2, or—more generally—let Ω be a connected $\mathcal{P}$ -Brelot space satisfying axiom D, and let$u$be a numerical function on Ω, $u\not\equiv\infty$ , which is locally bounded from below. A short proof yields the following result: The function$u$is the infimum of its superharmonic majorants if and only if each set {$x$:$u$($x$)>$t$},$t$∈ℝ, differs from an analytic set only by a polar set and $\int u\,d\mu_{x}^{V}\le u(x)$ , whenever$V$is a relatively compact open set in Ω and$x$∈$V$.

In our previous paper “The primes contain arbitrarily long polynomial progressions” [$Acta Math$., 201 (2008), 213–305] there were some minor errors in our definition of the polynomial forms and polynomial correlation conditions, which is unreasonably strong as written due to the overly loose bound on the degree of the polynomials involved. In this erratum we repair this issue by capping the degree of the polynomials more strongly, and adjusting some other relevant components of the argument accordingly.

On montre que la fonction maximale centrée de Hardy–Littlewood,$M$, sur les espaces hyperboliques réels $\mathbb{H}^{n} = \mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}}{\raise .17ex\hbox {$\scriptstyle +$}}{\raise .1ex\hbox {$\scriptscriptstyle +$}}{\scriptscriptstyle +}}} \times \mathbb{R}^{n - 1}$ , satisfait l’inégalité de type faible $\| M f \|_{L^{1, \infty}} \leq A (n \log {n}) \| f\|_{1}$ pour toute$f$∈$L$^{1}(ℍ^{$n$}), où$A$>0 est une constante indépendante de la dimension$n$.

Let$D$($μ$) be the Dirichlet space weighted by the Poisson integral of the positive measure$μ$. We give a characterization of the measures$μ$equal to a countable sum of atoms for which the Blaschke condition is a necessary and sufficient condition for a sequence to be a zero set for$D$($μ$).

In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the dual Orlicz-Minkowski problem regarding the dual Orlicz curvature measure is provided.

We prove a mean ergodic theorem for amenable discrete quantum groups. As an application, we prove a Wiener type theorem for continuous measures on compact metrizable groups.

If$X$is a closed subset of the real line, denote by$G$_{$X$}the supremum of the size of the gap in the Fourier spectrum of a measure, taken over all non-trivial finite complex measures supported on$X$. In this paper we attempt to find$G$_{$X$}in terms of$X$.

For a wide class of 2-dimensional periodic elliptic operators, we show that the global extrema of all spectral band functions are isolated.

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the $(p,q)$-dual curvature measures when $p>0$ and $q<0$. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the $(p,q)$-mixed volume.

We investigate compact quantum group actions on unital $C^*$-algebras by analyzing invariant subsets and invariant states. In particular, we come up with the concept of compact quantum group orbits and use it to show that countable compact metrizable spaces with infinitely many points are not quantum homogeneous spaces.

Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,\Gamma)$, the space of (ergodic) $\Gamma$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\rtimes\Gamma$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $\Gamma$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type of irreducible representations of $C(X)\rtimes \Gamma$.

We consider real interpolation methods defined by means of slowly varying functions$b$and symmetric spaces$E$, for which we present extreme reiteration theorems. As an application we identify, for all possible values of$θ$∈[0,1], the interpolation spaces ($L$_{1},$L$log$L$)_{$θ$,$b$,$E$}and ($L$_{exp},$L$_{∞})_{$θ$,$b$,$E$}.