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....

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...

The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\left\{b_n\right\}$ of positive elements of $B$, there is a sequence $\left\{{\lambda }_n\right\}$ of positive reals, and $b\in B$, with ${\lambda }_n{b}_n\le b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “$\sigma$” obtains for a Riesz space of continuous real-valued functions $C\left(X\right)$. A basic result is: For discrete $X$, $C\left(X\right)$ has $\sigma$ iff the cardinal $\left|X\right|<𝔟$, Rothberger’s bounding number. Consequences...

Let ${ℒ}_1=-\Delta +V$ be a Schrödinger operator and let ${ℒ}_2={(-\Delta )}^2+V^2$ be a Schrödinger type operator on ${ℝ}^n$ $(n\ge 5)$, where $V\ne 0$ is a nonnegative potential belonging to certain reverse Hölder class $B_s$ for $s\ge n/2$. The Hardy type space $H_{{ℒ}_2}^1$ is defined in terms of the maximal function with respect to the semigroup $\left\{{\mathrm{e}}^{-t{ℒ}_2}\right\}$ and it is identical to the Hardy space $H_{{ℒ}_1}^1$ established by Dziubański and Zienkiewicz. In this article, we prove the $L^p$-boundedness of the commutator ${ℛ}_b=bℛf-ℛ\left(bf\right)$ generated by the Riesz transform $ℛ={\nabla }^2{ℒ}_2^{-1/2}$, where $b\in {\mathrm{BMO}}_{\theta }\left(\rho \right)$, which is larger...

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.

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.

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 $\Omega \in L^s\left({\mathrm{S}}^{n-1}\right)$ for $s\ge 1$ be a homogeneous function of degree zero and $b$ a BMO function. The commutator generated by the Marcinkiewicz integral ${\mu }_{\Omega }$ and $b$ is defined by $${\displaystyle [b,{\mu }_{\Omega }]\left(f\right)\left(x\right)=({\int }_0^{\infty }{\left|{\int }_{|x-y|\le t}\frac{\Omega (x-y)}{{|x-y|}^{n-1}}[b\left(x\right)-b\left(y\right)]f\left(y\right)\mathrm{d}y{|}^2\frac{\mathrm{d}t}{t^3}\right)}^{1/2}.}$$ In this paper, the author proves the $(L^{p(·)}\left({ℝ}^n\right),L^{p(·)}\left({ℝ}^n\right))$-boundedness of the Marcinkiewicz integral operator ${\mu }_{\Omega }$ and its commutator $[b,{\mu }_{\Omega }]$ when $p(·)$ satisfies some conditions. Moreover, the author obtains the corresponding result about ${\mu }_{\Omega }$ and $[b,{\mu }_{\Omega }]$ on Herz spaces with variable exponent.

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1}u=-{\mu }_{1}v$, $L_{2}v=-{\mu }_{2}u$, on a domain $D$ of ${ℝ}^d$, where ${\mu }_1$ and ${\mu }_2$ are suitable measures on $D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${𝒞}^{\infty }$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and...

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 $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...