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.

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...

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...

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 $1\le n<d$ 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 $\mu$.

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.

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)...

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 $(X,d,\mu )$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $L^2\left(X\right)$. Assume that the semigroup ${\mathrm{e}}^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimate of order $m$ and $L$ satisfies the Plancherel type estimate. Let $H_L^p\left(X\right)$ be the Hardy space associated with $L.$ We show the boundedness of Stein’s square function ${𝒢}_{\delta }\left(L\right)$ arising from Bochner-Riesz means associated to $L$ from Hardy spaces $H_L^p\left(X\right)$ to $L^p\left(X\right)$, and also study...

We prove an interpolatory estimate linking the directional Haar projection $P^{\left(\epsilon \right)}$ to the Riesz transform in the context of Bochner-Lebesgue spaces $L^p(ℝⁿ;X)$, 1 < p < ∞, provided X is a UMD-space. If ${\epsilon }_{i₀}=1$, the result is the inequality $\left|\right|P^{\left(\epsilon \right)}{u\left|\right|}_{L^p(ℝⁿ;X)}{\le C\left|\right|u\left|\right|}_{L^p(ℝⁿ;X)}^{1/}\left|\right|R_{i₀}u{\left|\right|}_{L^p(ℝⁿ;X)}^{1-1/}$, (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of $L^p(ℝⁿ;X)$. In order to obtain the interpolatory result (1) we analyze stripe operators $S_{\lambda }$, λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

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...

Denote by $C$ the commutator $AB-BA$ of two bounded operators $A$ and $B$ acting on a locally convex topological vector space. If $AC-CA=0$, we show that $C$ is a quasinilpotent operator and we prove that if $AC-CA$ is a compact operator, then $C$ is a Riesz operator.

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...