Let ${ℳ}_p={m_k}_{k=0}^{p-1}$ be a finite set of step functions or real valued trigonometric polynomials on = [0,1) satisfying a certain orthonormality condition. We study multiscale generalized Riesz product measures μ defined as weak-* limits of elements ${\mu }_N\in V_N\left(N\in \mathbb{N}\right)$, where $V_N$ are $p^N$-dimensional subspaces of L₂() spanned by an orthonormal set which is produced from dilations and multiplications of elements of ${ℳ}_p$ and $\overline{{\bigcup }_{N\in \mathbb{N}}{V}_N}=L₂\left(\right)$. The results involve mutual absolute continuity or singularity of such Riesz products extending previous...

Let $C_{\alpha }$ be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density $̲(C_{\alpha },E,r)=in{f}_{x\in ℝⁿ}{C}_{\alpha }\left(E\cap B(x,r)\right)/C_{\alpha }\left(B(x,r)\right)$ for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that $li{m}_{r\to \infty }̲(C_{\alpha },E,r)$ is either 0 or 1; the first case occurs if and only if $̲(C_{\alpha },E,r)$ is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also...

For 1 ≤ q ≤ α ≤ p ≤ ∞, ${(L^q,l^p)}^{\alpha }$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^q,l^p)$ and contains the Lebesgue space $L^{\alpha }$. We study the closure ${(L^q,l^p)}_{c,0}^{\alpha }$ in ${(L^q,l^p)}^{\alpha }$ of the space of test functions (infinitely differentiable and with compact support in ${ℝ}^d$) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W¹\left({(L^q,l^p)}^{\alpha }\right)$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...

We establish the following sharp local estimate for the family ${R_j}_{j=1}^d$ of Riesz transforms on ${ℝ}^d$. For any Borel subset A of ${ℝ}^d$ and any function $f:{ℝ}^d\to ℝ$, ${\int }_A|R_{j}f\left(x\right)|dx\le C_p{\left|\right|f\left|\right|}_{L^p\left({ℝ}^d\right)}{\left|A\right|}^{1/q}$, 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_p={[2^{q+2}\Gamma (q+1)/{\pi }^{q+1}{\sum }_{k=0}^{\infty }{(-1)}^k/{(2k+1)}^{q+1}]}^{1/q}$, 1 < p < 2, and $C_p={[4\Gamma (q+1)/{\pi }^q{\sum }_{k=0}^{\infty }1/{(2k+1)}^q]}^{1/q}$, 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

The main aim of this short paper is to study Riesz potentials on one-mode interacting Fock spaces equipped with deformed annihilation, creation, and neutral operators with constants $c_{0,0},c_{1,1}\in ℝ$ and $c_{0,1}>0$, $c_{1,2}\ge 0$ as in equations (1.4)-(1.6). First, to emphasize the importance of these constants, we summarize our previous results on the Hilbert space of analytic L² functions with respect to a probability measure on ℂ. Then we consider the Riesz kernels of order 2α, $\alpha =c_{0,1}/c_{1,2}$, on ℂ if $0<c_{0,1}<c_{1,2}$, which can be derived from...

We study the large time behaviour of the solutions of a nonlocal regularisation of a scalar conservation law. This regularisation is given by a fractional derivative of order $1+\alpha$, with $\alpha \in (0,1)$, which is a Riesz-Feller operator. The nonlinear flux is given by the locally Lipschitz function ${\left|u\right|}^{q-1}u/q$ for $q>1$. We show that in the sub-critical case, $1<q<1+\alpha$, the large time behaviour is governed by the unique entropy solution of the scalar conservation law. Our proof adapts the proofs of the analogous results for...

Let E be a Riesz space. By defining the spaces $L{¹}_E$ and $L_E^{\infty }$ of E, we prove that the center $Z\left(L{¹}_E\right)$ of $L{¹}_E$ is $L_E^{\infty }$ and show that the injectivity of the Arens homomorphism m: Z(E)” → Z(E˜) is equivalent to the equality $L{¹}_E=Z{\left(E\right)}^{\text{'}}$. Finally, we also give some representation of an order continuous Banach lattice E with a weak unit and of the order dual E˜ of E in $L{¹}_E$ which are different from the representations appearing in the literature.

We prove F. Riesz’ inequality assuming the boundedness of the norm of the first arithmetic mean of the functions ${\left|\phi ₙ\right|}^p$ with p ≥ 2 instead of boundedness of the functions φₙ of an orthonormal system.

For $1\le n<d$ integers and $\rho >2$, we prove that an $n$-dimensional Ahlfors-David regular measure $\mu$ in ${ℝ}^d$ is uniformly $n$-rectifiable if and only if the $\rho$-variation for the Riesz transform with respect to $\mu$ is a bounded operator in $L^2\left(\mu \right)$. This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the $L^2\left(\mu \right)$ boundedness of the Riesz transform to the uniform rectifiability of $\mu$.

We consider the multiplier $m_{\mu }$ defined for ξ ∈ ℝ by $m_{\mu }\left(\xi \right)\doteq {((1-\xi ₁²-\xi ₂²)/(1-\xi ₁))}^{\mu }{1}_D\left(\xi \right)$, D denoting the open unit disk in ℝ. Given p ∈ ]1,∞[, we show that the optimal range of μ’s for which $m_{\mu }$ is a Fourier multiplier on $L^p$ is the same as for Bochner-Riesz means. The key ingredient is a lemma about some modifications of Bochner-Riesz means inside convex regions with smooth boundary and non-vanishing curvature, providing a more flexible version of a result by Iosevich et al. [Publ. Mat. 46 (2002)]. As an application, we show...

Let $E$ be a Riesz space, $F$ a Hausdorff topological vector space (t.v.s.). We prove, under a certain separation condition, that any orthosymmetric bilinear map $T:E\times E\to F$ is automatically symmetric. This generalizes in certain way an earlier result by F. Ben Amor [On orthosymmetric bilinear maps, Positivity 14 (2010), 123–134]. As an application, we show that under a certain separation condition, any orthogonally additive homogeneous polynomial $P:E\to F$ is linearly represented. This fits in the type of...

Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form $C_{\Lambda }\left(G\right)$ or $L{¹}_{\Lambda }\left(G\right)$ and which quotients of the form $C\left(G\right)/C_{\Lambda }\left(G\right)$ or $L¹\left(G\right)/L{¹}_{\Lambda }\left(G\right)$ have the Daugavet property. We show that $C_{\Lambda }\left(G\right)$ is a rich subspace of C(G) if and only if $\Gamma \setminus {\Lambda }^{-1}$ is a semi-Riesz set. If $L{¹}_{\Lambda }\left(G\right)$ is a rich subspace of L¹(G), then $C_{\Lambda }\left(G\right)$ is a rich subspace of C(G) as well. Concerning quotients, we prove that $C\left(G\right)/C_{\Lambda }\left(G\right)$ has the Daugavet property if Λ is a Rosenthal set, and that $L{¹}_{\Lambda }\left(G\right)$ is a poor subspace of L¹(G)...

Let F be a Siegel cusp form of integral weight k on the Siegel modular group Sp₂(ℤ) of genus 2. The coefficients of the spinor zeta function $Z_F\left(s\right)$ are denoted by cₙ. Let $D_{\rho }(x;Z_F)$ be the Riesz mean of cₙ. Kohnen and Wang obtained the truncated Voronoï-type formula for $D_{\rho }(x;Z_F)$ under the Ramanujan-Petersson conjecture. In this paper, we study the higher power moments of $D_{\rho }(x;Z_F)$, and then derive an asymptotic formula for the hth (h=3,4,5) power moments of $D₁(x;Z_F)$ by using Ivić’s large value arguments and other techniques. ...

Let ϕ(n) denote the Euler totient function. We study the error term of the general kth Riesz mean of the arithmetical function n/ϕ(n) for any positive integer k ≥ 1, namely the error term $E_k\left(x\right)$ where $1/k!{\sum }_{n\le x}n/\varphi \left(n\right){(1-n/x)}^k=M_k\left(x\right)+E_k\left(x\right)$. For instance, the upper bound for |Ek(x)| established here improves the earlier known upper bounds for all integers k satisfying $k\gg {\left(logx\right)}^{1+ϵ}$.