Papers

We show if $\partial \Omega$ is the graph of a parabolic Lipschitz function, then parabolic surface measure $\sigma$ of $\partial \Omega$ is absolutely continuous with respect to its caloric measure if and only if a (square) Dini-$\beta$ condition is satisfied. More specifically, the (square) Dini-$\beta$ condition is that \[\int_0^1 \hat{\beta}(X,t,r)^2 \frac{dr}{r} < \infty, \quad \text{$\sigma$-a.e. } (X,t) \in \partial \Omega.\] Here $\hat{\beta}$ is a parabolic version of the Jones ($L^2$) $\beta$-numbers. We show that these conditions are satisfied if and only if the graph is covered by a countable collection of regular Lipschitz graphs, that is, graphs with additional in-time regularity in the form of a half order time derivative in the parabolic BMO space. This supports the view that covering by regular Lipschitz graphs is the right notion for qualitative parabolic rectifiability in the context of parabolic PDEs. We also show that if \[\int_0^1 \hat{\beta}(X,t,r)^2 \frac{dr}{r} < \infty\] up to a set of caloric measure zero then the caloric measure is absolutely continuous with respect to surface measure.

A formal large-degree phase expansion associated with the zeros of the Jacobi polynomial $P_n^{(\alpha,\beta)}$ contains a sequence of constants $\kappa_1,\kappa_2,\dots$, each of which is a polynomial in $\alpha^2$ and $\beta^2$ with rational coefficients; for instance $\kappa_1=(\alpha^2-\beta^2)/4$. This paper asks a simple arithmetic question: which primes, and to which powers, occur in the denominators of these coefficients? We prove that the odd part of the common denominator of $\kappa_r$ divides $\operatorname{lcm}(1,3,\ldots,2r-1)$, and that the power of $2$ is at most $2^{E_r}$ with $E_r=3r-1+\nu_2((r-1)!)$, where $\nu_2$ is the $2$-adic valuation. The study of the leading coefficient leads to the formula \[ \nu_2\!\left(\sum_{j=0}^{m}\binom mj\frac1{2j+1}\right)=m+\nu_2(m+1)\qquad(m\ge0), \] which we prove by showing that the series $\sum_{k\ge0}k!/(2k+1)!!$, whose real sum is $\pi/2$, converges to $0$ in the field $\mathbb Q_2$ of $2$-adic numbers. Finally, we show that the statement “the exponent $E_r$ is attained for every $r$” is equivalent to an integrality property of the Taylor coefficients of a single explicit power series in one variable. That property is established in a separate paper, and is confirmed here independently, by exact computation, for $r\le48$.

The classical Hamburger moment problem can be viewed as an interpolation problem for Nevanlinna functions. A closely related problem is the Nevanlinna–Pick interpolation problem. Each of these problems is either determinate, with a unique solution, or indeterminate, with infinitely many solutions. Hamburger established a criterion for the indeterminacy of the moment problem in terms of the convergence of two series involving the classical polynomials of the first and second kind. In this paper, we establish an analogous criterion for the matrix Nevanlinna–Pick interpolation problem. The criterion is formulated in terms of the convergence of matrix series involving the rational functions of the first and second kind.

Sparse bounds have proved to be a powerful tool which simplify the understanding of many important operators in harmonic analysis. The existence of a sparse bound for an operator implies the weighted boundedness of the operator for weights within the Muckenhoupt classes. As in the example of Calderón-Zygmund operators, this implication can prove weighted boundedness with the best possible dependency on the characteristic constant of the Muckenhoupt weight. In this paper, we prove a partial converse for sub-additive operators by showing that weighted boundedness, with an arbitrary a priori control of the operator norm in terms of the characteristic constants of the weight, implies the existence of sparse form bounds. This result implies that a near-optimal dependency of the operator norm on characteristic constants of the weights follows from the weighted boundedness itself, rather than relying on the particular structure of the sub-additive operator. Moreover, it allows Rubio de Francia's extrapolation theorem to be extended to the weak-type end-point for sub-linear operators and leads to an interpolation result for sparse form bounds for analytic families of operators. As an application, we obtain new sparse form bounds for rough singular integral operators.

We completely characterize a two-depth version of Journé's covering lemma in the bi-parameter setting. Let $\mathcal U$ be an incomparable family of bi-parameter dyadic rectangles contained in $\Omega$, and write $e_i(I):=e_i(I;\Omega)$ for the two embeddedness depths. For every coordinatewise nonincreasing $w\colon\mathbb N^2\to[0,\infty)$, the estimate \[ \sum_{I\in\mathcal U}w(e_1(I),e_2(I))|I|\lesssim_w|\Omega| \] holds uniformly in $(\mathcal U,\Omega)$ if and only if the following fixed-total condition holds: \[ \|w\|_{\mathrm{ft}}:=\sup_{N\geq0}\sum_{a=0}^Nw(a,N-a)<\infty. \] For instance, $\|(a+1)^{-s}(b+1)^{-(1-s)}\|_{\mathrm{ft}}\sim[s(1-s)]^{-1}$ for $0<s<1$, so this allows weights that are not summable in either variable, while one-depth theory holds precisely for summable weights. Exactly the same characterization holds for dyadic Zygmund rectangles $I=I^1\times I^2\times I^3$ with $\ell(I^3)=\ell(I^1)\ell(I^2)$. We use probabilistic packing methods to prove the sufficiency, and these techniques also yield sparse refinements of the preceding estimates, which self-improve to various forms of exponential integrability.

We study the radial extremal function $\varphi(|{\,\cdot\,}|)$ arising in the problem of finding the sharp Nikolskii constant $\mathcal C_d$ in $\mathit{PW}_1^1(\mathbb R^d)$ for arbitrary dimension $d\ge1$. We prove a factorization $\varphi=\Phi_1\Phi_2$, where $\Phi_1$ and $\Phi_2$ are entire functions of exponential type $1/2$ satisfying a functional equation and second-order differential equations with polynomial coefficients. As a result, the original extremal problem is reduced to a one-dimensional spectral problem depending on at most $d+1$ parameters. We also obtain a zeta interpretation of the coefficients of the polynomial appearing in the functional equation and a multiplicative equilibrium condition for the zeros of the extremal function. These results can be used to construct several algorithms for computing $\mathcal C_d$.

We obtain a complete description of the class of entrywise preservers of Lorentz-Gram matrices. This resolves, for the case of constant negative curvature, the classification of entrywise preservers obtained by Schoenberg in the zero-curvature (Euclidean) and constant-positive-curvature (spherical) settings. These preservers admit a Lévy–Khintchine-type representation and their asymptotic characteristics are related to Krein's classification of screw lines in Lobachevsky space. Connections with complete Nevanlinna–Pick kernels and Bochner subordination are also obtained.

Let \(\mu\) be a self-similar Cantor measure on \(\mathbb R\) associated with a probability weight vector \(\mathbf p\), let \(K=\operatorname{supp}\mu\), and let \(F\) denote the distribution function of \(\mu\). We characterize the points \(x\in K\) at which the local scaling exponent \[ \lim_{\substack{y\to x, y\in K}} \frac{\log |F(y)-F(x)|}{\log |y-x|} \] exists and assumes a prescribed value. The characterization is formulated in terms of the convergence of the ratio between the accumulated logarithmic mass and geometric scales, together with the sublinear growth of the endpoint runs. Unlike the classical ternary case, our approach applies to arbitrary contraction ratios and probability weights. As an application, we construct a subset \(M\subset K\) of full Hausdorff measure on which the local scaling exponent of \(F\) is ${h(\mathbf q,\mathbf p)}/{\chi(\mathbf q)}$ and establish the exact dimension-distortion formula \[ \dim_{\mathrm H} F(A) = \frac{\chi(\mathbf q)}{h(\mathbf q,\mathbf p)} \dim_{\mathrm H} A \] for every \(A\subset M\), where \(\mathbf q\) is the natural probability vector, \(\chi(\mathbf q)\) is the corresponding Lyapunov exponent, and \(h(\mathbf q,\mathbf p)\) is the cross-entropy of \(\mathbf q\) relative to \(\mathbf p\). A three-branch example illustrates the results.

In this paper we establish a criterion on weak type $(1,\,1)$ bound of the following rough singular integral $$T_{\Omega,K,a}f(x)={\rm p.v.}\int_{\mathbb{R}^n}\Omega(x-y)K(x,y)m_{x,y}a f(y)dy,$$ where $m_{x,y}a=\int_0^1a(sx+(1-s)y)ds$ with $a\in L^1(\mathbb{R}^n)$ and $\hat{a}\in L^1(\mathbb{R}^n)$, $\Omega$ is homogeneous of degree zero, integrable in $\mathbb{S}^{n-1}$ and satisfies the cancellation condition $\int_{\mathbb{S}^{n-1}}\Omega(\theta)d\sigma(\theta)=0$ and $K$ is a measurable function defined on $\mathbb{R}^n\times\mathbb{R}^n\setminus \{(x,x):x\in\mathbb{R}^n\}$ and satisfies a Hölder condition. By assuming that $\Omega\in L\log L(\mathbb{S}^{n-1})$ and the operator $T_{\Omega, K}f(x)={\rm p.v.}\int_{\mathbb{R}^n}\Omega(x-y)K(x,y)f(y)dy$ is bounded on $L^2(\mathbb{R}^n)$, we prove the weak type (1,1) bound of $T_{\Omega,K,a}$. As several applications, we obtain a large class of singular integral operators which possess weak type $(1,\,1)$ bound. The main results of this paper essentially extend and generalize some known ones.

We prove the existence of infinitely many lemniscate-like, butterfly-like, and circular-wave critical points for the length-penalised ideal energy. For the lemniscate and butterfly family, we use a staged direct minimisation process to establish existence. A boundary-layer analysis reveals critical points near two Fresnel phases: $3\pi/4$ modulo $2\pi$, corresponding to lemniscates, and $7\pi/4$ modulo $2\pi$, which are the butterflies. The family of circular waves bifurcates, in a sense, from multiply-covered circles. We use an adapted shooting method to establish their existence.

We develop a componentwise geometric and monodromy theory for the one-support generalized Lamé equation on an elliptic curve, with singularities at \(0\) and \(\pm p\). Its log-free curve decomposes canonically into irreducible even and non-even components, the former being governed by elliptic Painlevé~VI. For the non-even component, we construct the hyperelliptic spectral curve, Baker–Akhiezer functions, and addition map, and prove that \[ \deg \sigma_{n,p}^{(1)}=n(n+1). \] After quotienting by the involution \(T\mapsto -T\), we identify the non-even spectral curve with the classical Lamé spectral curve of weight \(n\), compatibly with the addition map and the rational function \(\kappa\). This identification is realized by \[ \widetilde B=T^2-n(n+1)\wp(p), \] and associates every non-even generalized Lamé equation with a unique classical Lamé equation on the same elliptic curve having equivalent period monodromy. Fixing \(\widetilde B\) yields an isomonodromic deformation with \(\tau\) fixed. Together with the Painlevé-VI deformation on the even component, it gives a componentwise interpretation of the collision \(p\to0\), and yields a finite descent on the admissible completely reducible locus. The classical spectral, finite-gap, finite-monodromy, and curvature theories consequently transfer to the non-even component. Finally, within the symmetric family $\left(n_0,n_1,n_2,n_3,\frac12,\frac12\right),$ the one-support case forms an affine genus-zero hierarchy, whereas for $\left(1,1,0,0,\frac12,\frac12\right)$, the non-even normalization is generically elliptic and becomes rational on the discriminant locus, while the full compactified log-free curve retains arithmetic genus two. This first genus jump marks the boundary of the affine theory and motivates a genus-dependent componentwise geometry.

A non-technical exposition of the Kakeya conjecture, why it matters, and the road to the solution of this conjecture in three dimensions by Hong Wang and Joshua Zahl.

We study the problem of finding the norm of the point evaluation operator in the Paley–Wiener space $PW^{1}(\r^d)$, consisting of $d$-variable functions of spherical exponential type that are integrable on $\r^d$. The extremal functions can be taken radial, which naturally leads us to consider a related extremal problem in a weighted Paley–Wiener space of single-variable functions. We establish that the radial extremal function must satisfy a third-order linear ODE with polynomial coefficients for every $d \geq 1$, extending Gorbachev's recent odd-dimensional result. Along the way, we prove interpolation and reciprocal formulas involving the zeros of the extremizer.

We study when a compactly supported isotropic positive definite function on a Euclidean space remains positive definite after Euclidean distance is replaced by spherical geodesic distance. The corresponding assertion is known in odd dimensions. We prove that it fails in every even dimension, no matter how small a positive upper bound is imposed on the support. An explicit counterexample is also given in dimension two.

In this paper, we study $L^p$-estimates for maximal averaging operators $\mathcal M$ along hypersurfaces $S$ in $\mathbb{R}^{3}$ which are the graph of a mixed homogeneous function $\Phi$ which is analytic away from the origin. The closure of such a surface will pass through the origin, so that the usual transversality condition that had been imposed in many previous works on maximal averages along hypersurfaces will not hold even when $\Phi$ is analytic at the origin. As our main result, under mild assumptions which are satisfied for instance for every mixed homogeneous polynomial $\Phi,$ we determine the critical Lebesgue exponent $p_c$ for which $\mathcal M$ is $L^p$-bounded for every $p>p_c,$ but unbounded for $p<p_c,$ in terms of multiplicities of the real roots of the Hessian determinant of $\Phi.$ It turns out that the study of the contributions by neighborhoods of a certain type of roots is closely related to recent work by Dendrinos, Ikromov and the first and third author on sharp estimates for a maximal averaging operator along a transversal hypersurface of an “exceptional” class, whose $L^p$-boundedness had been an open problem for a long time and which has recently been established by means of their new theory of FIO-cone multipliers.

In this article, we introduce and study a set of generalized Gaussian beam Radon transforms (GbRt) acting on tensor fields in $\mathbb{R}^2$. The operators considered include longitudinal, transverse, mixed GbRts, along with their integral moments. These operators extend the corresponding notions of the classical generalized Radon transforms for tensor fields. We establish reconstruction results for vector and symmetric 2-tensor fields using appropriate combinations of the defined transforms. This work extends a recent study on the recovery of scalar functions from their GbRt to the recovery of vector and tensor fields from analogously defined generalized GbRts.

We establish the orthonormal Strichartz estimates in the abstract Keel–Tao framework, under precisely the same hypotheses as in their original theorem. As an important consequence, we resolve the endpoint conjecture for the Schrödinger equation in dimensions $d\ge 2$, which was first raised by Frank, Lewin, Lieb, and Seiringer. This consequence also yields an affirmative answer to the conjecture for the free transport equation, which was stated in dual form by Bennett, Bez, Gutiérrez, and Lee. As a further application, we also establish a refinement of the Strichartz estimate for a single function.

Let $1\leq m<n$, and let $\gamma:[0,1]\to\mathbb R^n$ be a smooth non-degenerate curve. For $\theta\in[0,1]$, set \[ V_\theta := \operatorname{span} \{\gamma'(\theta),\ldots,\gamma^{(m)}(\theta)\}, \] and let $\pi_\theta$ denote the orthogonal projection onto $V_\theta$. We prove that if $E\subset\mathbb R^n$ is a Borel set with $\dim_{\mathrm H}E>m$, then \[ \mathcal H^m(\pi_\theta(E))>0 \] for almost every $\theta\in[0,1]$.

We classify the differential operators \(T=z^j\partial_z^j+z^m\partial_z^\ell\), where \(0\le m<\ell\) and \(1\le j<\ell\), whose monic eigenpolynomials satisfy a finite-term recurrence relation. Writing \(k=\ell-m\), such a recurrence exists if and only if \(j=1\) and \(k\mid\ell\). In that case we determine all recurrence coefficients in closed form and prove that the associated difference operator has order \(\ell\). We also give an explicit factorization showing that every admissible operator is of Type~(2) in Conjecture~1.10 of Horozov–Shapiro–Tater; the equality of the differential and difference orders is the conclusion predicted by their Conjecture~1.9.

Given a real-valued function $f$, let $\mathcal{N}_f(\delta, Q)$ be the number of rational points with denominators at most $Q\ge 1$ in the $(\delta/Q)$-tubular neighbourhood of the graph of the function $f$. A heuristic predicts that the number of such points grows like the area of the neighbourhood provided that $\delta$ is big enough (in a suitable sense). Considerable efforts have been committed to prove this heuristic for regular curves. This culminated in the works by Vaughan & Velani~(2006) and by Huang~(2015) establishing an asymptotic expansion for $\mathcal{N}_f(\delta, Q)$ provided that $\delta\gg Q^{-1+\epsilon}$ for some $\epsilon>0$ when the map $f$ is, among other assumptions, twice continuously differentiable. The present work deals with the thus-far unexplored regime where minimal regularity conditions are imposed on the curve. More precisely, it is concerned with the case where the map $f$ is an a.s. realisation of the graph of Brownian motion. The main result establishes the existence of an almost sure asymptotic expansion for the counting function for all values of $\delta$, with the exception of a critical regime, thereby going well beyond the theory currently available for regular curves. A key ingredient in the proof is the derivation of the area heuristic, which relies on establishing the a.s. asymptotics of the area of the tubular neighbourhood of the graph of Brownian motion. This result has two main consequences: firstly, it completes the counting aspect of the theory of Diophantine approximation on the graph of Brownian motion initiated by Sprindžuk (1979). Secondly, it hints at the existence of a theory unifying the analysis of rational points near a curve on the one hand and, on the other, its local Hölder regularity and fine-scale oscillations. It thus builds a seemingly new bridge between Number Theory and Multifractal Analysis.

Following a hint of Semmes, we employ Stong's bijections between integer lattices to construct space-filling surfaces, which are higher-dimensional analogues of space-filling curves. For each $m\geq 2$ we build $\alpha$-Hölder continuous parameterizations $f:[0,1]^m\rightarrow[0,1]^{m+1}$ with sharp exponent $\alpha=m/(m+1)$. In particular, there exist $(2/3)$-Hölder continuous surjections from squares to cubes. This solves Arnold's problem 1988–5.

We investigate the variational properties of Riemann's ”nondifferentiable” function $R$. We show that $R$ has finite $p$-variation for $p>4/3$ and infinite $p$-variation for $p<4/3$. As an application of our results, we provide a new look on an upper estimate of the Hausdorff dimension of the image of $R$, due to Eceizabarrena.

We provide an adjustment of the Hardy–Littlewood–Sobolev inequality for p=1 to the anisotropic setting. Several examples of anisotropic Bourgain–Brezis inequalities are obtained as corollaries of the main theorem.

Let $\alpha>-1/2$, $\alpha\ne0$, and let $$ \Delta_\alpha=-\frac{d^2}{dx^2}-\frac{2\alpha}{x}\frac{d}{dx} $$ be the Bessel operator on $\mathbb R_+=(0,\infty)$. We characterize boundedness and compactness of commutators of $R_\alpha=\frac{d}{dx}\Delta_\alpha^{-1/2}$ on the Andersen–Kerman two-weight setting. For $1<p<\infty$ and $\mu,\lambda\in A_{p,\alpha}$, put $ \nu=\left(\frac{\mu}{\lambda}\right)^{1/p}, \ d\rho_\alpha(x)=x^{2\alpha+1}\,dx. $ For real-valued symbols, $$ \|[b,R_\alpha]\|_{L^p(\mu\,dx)\to L^p(\lambda\,dx)} \simeq \|b\|_{\rm{BMO}_{\nu,\alpha}}, $$ where the Bloom oscillation is computed with respect to $d\rho_\alpha$. Moreover $$ [b,R_\alpha]:L^p(\mu\,dx)\to L^p(\lambda\,dx) \text{ is compact} \quad\Longleftrightarrow\quad b\in\rm{VMO}_{\nu,\alpha}. $$ The proof uses the exact conjugation $ U_w(x)=x^{p-2\alpha-1}w(x), \ [U_w]_{A_p(\rho_\alpha)}=[w]_{A_{p,\alpha}}, $ which transfers the problem to the quotient Bessel kernel on $(\mathbb R_+,|x-y|,\rho_\alpha)$. We isolate the required Calderón–Zygmund estimates and one-sided non-degeneracy in this quotient normalization. The measure $d\rho_\alpha$ is distinct from the usual Bessel measure $dm_\alpha=x^{2\alpha}\,dx$. It is the Bloom base measure selected by the Andersen–Kerman conjugation and is used throughout the two-weight theory below.

We investigate the asymptotic behavior of Jacobi-Piñeiro polynomials of degree $2n$ orthogonal on $[0,1]$ with respect to weights $w_j(x) = x^{\alpha_j}(1-x)^{\beta}$, $j=1,2$ where $\alpha_1,\alpha_2, \beta>-1$, and $\alpha_1-\alpha_2 \notin \mathbb{Z}$. These polynomials are characterized by a Riemann-Hilbert problem for a $3 \times 3$ matrix-valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics in the complex plane. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

Let $\mathbb{G}$ be a stratified Lie group and $\mathcal L$ be its sub-Laplacian. We prove that the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homogeneous dimension, the step, and the underlying group structure of $\mathbb{G}$. Our result provides a noncommutative generalization of the dimension-free Euclidean theorem of Ouyang, Spector, and Stockdale arXiv:2608.18068, with the same universal constant. Our proof relies upon a fractional obstacle problem adapted to stratified Lie groups by using the functional calculus of $\mathcal L$ instead of the Fourier transform. As a consequence, by interpolation we obtain uniform $L^p$ bounds for the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ for $p \in (1,2]$. By contrast, for every $p > 2$, we construct a sequence of stratified Lie groups with fixed horizontal dimension $5$ and steps tending to infinity for which the $L^p$ norms of the horizontal Riesz transforms diverge.

Let $ C \subseteq \mathbb{R}^n $ be a convex set. The mapping $ f : C \longrightarrow \mathbb{R}^n $ is strictly increasing, if $ \langle f(x)-f(y), x-y \rangle > 0 $ for all distinct elements $ x,y \in C $. Applying classical theorems of finite dimensional convex geometry and convex analysis, we show that the inverse functions has a unique extension $ f^{(-1)} : \mathrm{conv} (f(C)) \longrightarrow C $ such that $ f^{(-1)} $ is monotone, continuous and it acts as a left-inverse of $ f $. As an application we introduce the concept of vector-valued weighted quasi-arithmetic means and discuss their equality problem.

We consider the natural family of Fourier inequalities for the Paley–Wiener space $\mathrm{PW}^q(\Omega)$, consisting of $L^q$-functions with Fourier support in a convex set $\Omega \subset \mathbb{R}^n$, $n \geq 2$, free of affine lines. Namely, \[ \int_{\Omega}\dfrac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}dx\leq C\|f\|_{L^q}^p,\quad f\in \mathrm{PW}^q(\Omega). \] Here $\hat{f}$ is the Fourier transform of $f$, $1 \leq p, q < \infty$, $d \in \mathbb{R}$, and $\omega_\Omega$ is the frequency multiplier weight associated with the Paley–Wiener space of $\Omega$, \[ \omega_{\Omega}(x)=m(\Omega\cap (2x-\Omega)), \qquad x \in \Omega. \] For an arbitrary polyhedron $P$, we completely characterize the triples $(p,q,d)$ which yield valid Fourier inequalities. For a ball $B$, we characterize the valid triples when $p \geq 2$. When $p < 2$, the situation is different for the ball, and natural critical inequalities fail. However, we show that the spherical restriction conjecture implies a family of subcritical Fourier inequalities for the ball, which in turn imply the Kakeya conjecture (in its Minkowski-form). Finally, we link our family of Fourier inequalities to the theory of truncated Hankel operators acting on the Paley–Wiener space of $\Omega$.

We prove the cyclic-product conjecture of Holmes, Holroyd and Ramírez: among all circular arrangements of a decreasing positive sequence, the circular-symmetric “greedy” arrangement maximises every consecutive cyclic product sum $P_r$.

In this work, we establish a strong factorization for Hardy-Orlicz spaces on the upper half-plane. We show that the product of two functions belonging respectively to Hardy-Orlicz spaces $H^{\Phi_{1}}$ and $H^{\Phi_{2}}$ lies in a third space $H^{\Phi_{3}}$, and every holomorphic function in $H^{\Phi_{3}}$ admits such a decomposition. We then provide an atomic decomposition for certain Hardy-Orlicz spaces, which allows us to describe the topological dual of these spaces when the associated function is concave. Finally, these results are applied to the study of the continuity of the Hankel operators in this setting.