Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^p$, Sobolev, and some new Hardy spaces naturally associated to $L$. First, we show...

The harmonic Cesàro operator is defined for a function f in $L^p\left(ℝ\right)$ for some 1 ≤ p < ∞ by setting $\left(f\right)\left(x\right):={\int }_x^{\infty }(f\left(u\right)/u)du$ for x > 0 and $\left(f\right)\left(x\right):=-{\int }_{-\infty }^x(f\left(u\right)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L{¹}_{loc}\left(ℝ\right)$ by setting $*\left(f\right)\left(x\right):=(1/x){\int }^{x₀}f\left(u\right)du$ for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If $f\in L^p\left(ℝ\right)$ for some 1 ≤ p ≤ 2, then ${\left(\left(f\right)\right)}^{\wedge }\left(t\right)=*\left(f̂\right)\left(t\right)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f\in L^p\left(ℝ\right)$ for some 1 < p ≤ 2, then...

Let $G$ be a compact group, let $n\in N\setminus \{0,1\}$ be a fixed element and let $\sigma$ be a continuous automorphism on $G$ such that ${\sigma }^n=I$. Using the non-abelian Fourier transform, we determine the non-zero continuous solutions $f:G\to C$ of the functional equation $$f\left(xy\right)+\sum _{k=1}^{n-1}f\left({\sigma }^k\left(y\right)x\right)=nf\left(x\right)f\left(y\right),x,y\in G,$$ in terms of unitary characters of $G$.

Let ${\left(U_t\right)}_{t\ge 0}$ be a Brownian motion valued in the complex projective space $ℂP^{N-1}$. Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of $|U_t^1{|}^2$ and of $\left(\right|U_t^1{|}^2,\left|U_t^2{|}^2\right)$, and express them through Jacobi polynomials in the simplices of $ℝ$ and ${ℝ}^2$ respectively. More generally, the distribution of $\left(\right|U_t^1{|}^2,\cdots ,\left|U_t^k{|}^2\right),2\le k\le N-1$ may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group $𝒰(N-k+1)$ yet computations become tedious. We also revisit the approach initiated in [] and...

Let ${\ell }_j:=-d²/dx²+k²q_j\left(x\right)$, k = const > 0, j = 1,2, $0<essinf{q}_j\left(x\right)\le esssup{q}_j\left(x\right)<\infty$. Suppose that (*) ${\int }_0^{1}p\left(x\right)u₁(x,k)u₂(x,k)dx=0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_j$ solves the problem ${\ell }_j{u}_j=0$, 0 ≤ x ≤ 1, $u_j^{\text{'}}(0,k)=0$, $u_j(0,k)=1$. It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

A famous theorem of Carleson says that, given any function $f\in L^p\left(𝕋\right)$, $p\in (1,+\infty )$, its Fourier series $\left(S_{n}f\left(x\right)\right)$ converges for almost every $x\in 𝕋$. Beside this property, the series may diverge at some point, without exceeding $O\left(n^{1/p}\right)$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_{n}f\left(x\right)=O\left(n^{\beta }\right)$ and we are interested in the size of the exceptional sets $E_{\beta }$, namely the sets of $x\in 𝕋$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p\left(𝕋\right)$ have a multifractal behavior with respect to...

We investigate the behaviour of Fourier coefficients with respect to the system of ultraspherical polynomials. This leads us to the study of the “boundary” Lorentz space ${ℒ}_{\lambda }$ corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients $c{ₙ}^{\left(\lambda \right)}\left(f\right)$ of ${ℒ}_{\lambda }$-functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReH¹. Namely, we prove that for any $f\in {ℒ}_{\lambda }$ the series ${\sum }_{n=1}^{\infty }c{ₙ}^{\left(\lambda \right)}\left(f\right)cosn\theta$ is the Fourier series of some function φ ∈ ReH¹ with ${\left|\right|\phi \left|\right|}_{ReH¹}{\le c\left|\right|f\left|\right|}_{{ℒ}_{\lambda }}$. ...

A result by G. H. Hardy ([11]) says that if f and its Fourier transform f̂ are $O\left(\right|x{|}^m{e}^{-\alpha x²})$ and $O\left(\right|x\left|ⁿe^{-x²/\left(4\alpha \right)}\right)$ respectively for some m,n ≥ 0 and α > 0, then f and f̂ are $P\left(x\right)e^{-\alpha x²}$ and $P^{\text{'}}\left(x\right)e^{-x²/\left(4\alpha \right)}$ respectively for some polynomials P and P’. If in particular f is as above, but f̂ is $o\left(e^{-x²/\left(4\alpha \right)}\right)$, then f = 0. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.

We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{-\beta }$, with $\beta \in [0,\infty )$. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $\beta$. If $\beta \in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $\beta =1$, it is given by a parabolic equation involving an operator...

Let $X\subset {\mathbb{P}}^n$ be an integral and non-degenerate $m$-dimensional variety defined over $ℝ$. For any $P\in {\mathbb{P}}^n\left(ℝ\right)$ the real $X$-rank $r_{X,ℝ}\left(P\right)$ is the minimal cardinality of $S\subset X\left(ℝ\right)$ such that $P\in \langle S\rangle$. Here we extend to the real case an upper bound for the $X$-rank due to Landsberg and Teitler.

The maximal operator S⁎ for the spherical summation operator (or disc multiplier) $S_R$ associated with the Jacobi transform through the defining relation $\widehat{S_{R}f}\left(\lambda \right)=1_{\left|\lambda \right|\le R}f̂\left(t\right)$ for a function f on ℝ is shown to be bounded from $L^p(ℝ₊,d\mu )$ into $L^p(ℝ,d\mu )+L²(ℝ,d\mu )$ for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from $L^{p₀,1}(ℝ₊,d\mu )$ into $L^{p₀,\infty }(ℝ,d\mu )+L²(ℝ,d\mu )$. In particular ${S_{R}f\left(t\right)}_{R>0}$ converges almost everywhere towards f, for $f\in L^p(ℝ₊,d\mu )$, whenever (4α + 4)/(2α + 3) < p ≤ 2.

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...

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...

Let $G$ be a group and $\omega \left(G\right)$ be the set of element orders of $G$. Let $k\in \omega \left(G\right)$ and $m_k\left(G\right)$ be the number of elements of order $k$ in $G$. Let nse$\left(G\right)=\{m_k\left(G\right):k\in \omega \left(G\right)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$\left(G\right)=$ nse$\left(S_r\right)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

Let $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant...

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...

Given 0 < p,q < ∞ and any sequence z = zₙ in the unit disc , we define an operator from functions on to sequences by $T_{z,p}\left(f\right)={(1-|zₙ\left|²\right)}^{1/p}f\left(zₙ\right)$. Necessary and sufficient conditions on zₙ are given such that $T_{z,p}$ maps the Hardy space $H^p$ boundedly into the sequence space ${\ell }^q$. A corresponding result for Bergman spaces is also stated.

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....