Let $\mu \in {\left({ℝ}^d\right)}^{\text{'}}$ be an analytic functional and let $T_{\mu }$ be the corresponding convolution operator on Sato’s space $\left({ℝ}^d\right)$ of hyperfunctions. We show that $T_{\mu }$ is surjective iff $T_{\mu }$ admits an elementary solution in $\left({ℝ}^d\right)$ iff the Fourier transform μ̂ satisfies Kawai’s slowly decreasing condition (S). We also show that there are $0\ne \mu \in {\left({ℝ}^d\right)}^{\text{'}}$ such that $T_{\mu }$ is not surjective on $\left({ℝ}^d\right)$.

For the convolution operators $A_a^{\alpha }$ with symbols ${a\left(\right|\xi \left|\right)\left|\xi \right|}^{-\alpha }expi\left|\xi \right|$, 0 ≤ Re α < n, $a\left(\right|\xi \left|\right)\in L_{\infty }$, we construct integral representations and give the exact description of the set of pairs (1/p,1/q) for which the operators are bounded from $L_p$ to $L_q$.

We consider the Heisenberg group ℍⁿ = ℂⁿ × ℝ. Let ν be the Borel measure on ℍⁿ defined by $\nu \left(E\right)={\int }_{ℂⁿ}{\chi }_E(w,\phi \left(w\right))\eta \left(w\right)dw$, where $\phi \left(w\right)={\sum }_{j=1}^n{a}_j\left|w_j\right|²$, w = (w₁,...,wₙ) ∈ ℂⁿ, $a_j\in ℝ$, and η(w) = η₀(|w|²) with $\eta ₀\in C_c^{\infty }\left(ℝ\right)$. We characterize the set of pairs (p,q) such that the convolution operator with ν is $L^p\left(ℍⁿ\right)-L^q\left(ℍⁿ\right)$ bounded. We also obtain $L^p$-improving properties of measures supported on the graph of the function $\phi \left(w\right)={\left|w\right|}^{2m}$.

A convolution operator, bounded on $L^q({ℝ}^n)$, is bounded on $L^p({ℝ}^n)$, with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace ${ℝ}^n$ with a general non-commutative locally compact group $G$. In this paper we give a simple construction of a convolution operator on a suitable compact group $G$, wich is bounded on $L^q(G)$ for every $q\in [2,\mathrm{\infty })$ and is unbounded on $L^p(G)$ if $p\in [1,2)$.

A convolution operator, bounded on $L^q({ℝ}^n)$, is bounded on $L^p({ℝ}^n)$, with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace ${ℝ}^n$ with a general non-commutative locally compact group $G$. In this paper we give a simple construction of a convolution operator on a suitable compact group $G$, wich is bounded on $L^q(G)$ for every $q\in [2,\mathrm{\infty })$ and is unbounded on $L^p(G)$ if $p\in [1,2)$.

Let WF⁎ be the wave front set with respect to $C^{\infty }$, quasi analyticity or analyticity, and let K be the kernel of a positive operator from $C{₀}^{\infty }$ to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution $u{\ast }_B\phi \left(x\right)=\int u(x-y)\phi \left(y\right)B(x,y)dy$, where $B\in C^{\infty }$ is appropriate, and prove that if $(u{\ast }_B\phi ,\phi )\ge 0$ for every $\phi \in C{₀}^{\infty }$ and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

We introduce the definition of $p$-limited completely continuous operators, $1\le p<\infty$. The question of whether a space of operators has the property that every $p$-limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$-limited completely continuous evaluation operators.

We consider one-parameter (C₀)-semigroups of operators in the space ${}^{\text{'}}(ℝⁿ;{ℂ}^m)$ with infinitesimal generator of the form ${(G*)|}_{{}^{\text{'}}(ℝⁿ;{ℂ}^m)}$ where G is an $M_{m\times m}$-valued rapidly decreasing distribution on ℝⁿ. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces $(ℝⁿ;{ℂ}^m)$, ${}_{L^p}(ℝⁿ;{ℂ}^m)$, p ∈ [1,∞], ${(}_a)(ℝⁿ;{ℂ}^m)$, a ∈ ]0,∞[, or the spaces ${}_{L^q}^{\text{'}}(ℝⁿ;{ℂ}^m)$, q ∈ ]1,∞], of bounded distributions.

If $1\&gt;\alpha \&gt;1/2$, then there exists a probability measure $\mu$ such that the Hausdorff dimension of the support of $\mu$ is $\alpha$ and $\mu *\mu$ is a Lipschitz function of class $\alpha -\frac{1}{2}$.

In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $𝕂={𝕂}^{+}\oplus {𝕂}^{-}$ of the Krein space $𝕂$ with ${𝕂}^{+}$ and ${𝕂}^{-}$ invariant under $T$.

Let $f_s$ and $f_t$ be tent maps on the unit interval. In this paper we give a new proof of the fact that if the critical points of $f_s$ and $f_t$ are periodic and the inverse limit spaces $(I,f_s)$ and $(I,f_t)$ are homeomorphic, then s = t. This theorem was first proved by Kailhofer. The new proof in this paper simplifies the proof of Kailhofer. Using the techniques of the paper we are also able to identify certain isotopies between homeomorphisms on the inverse limit space.

For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from $L^p\left(\right)$ to $L^q\left(\right)$. We also give a condition on p which is necessary if this operator maps $L^p\left(\right)$ into L²().

Let $R$ be a unital $*$-ring. For any $a,s,t,v,w\in R$ we define the weighted $w$-core inverse and the weighted dual $s$-core inverse, extending the $w$-core inverse and the dual $s$-core inverse, respectively. An element $a\in R$ has a weighted $w$-core inverse with the weight $v$ if there exists some $x\in R$ such that $awxvx=x$, $xvawa=a$ and ${\left(awx\right)}^{*}=awx$. Dually, an element $a\in R$ has a weighted dual $s$-core inverse with the weight $t$ if there exists some $y\in R$ such that $ytysa=y$, $asaty=a$ and ${\left(ysa\right)}^{*}=ysa$. Several characterizations of weighted $w$-core invertible and weighted dual $s$-core...

Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w*}(X*,Y)$, as well as the complementation of the space $K_{w*}(X*,Y)$ of w*-w compact operators in the space $L_{w*}(X*,Y)$ of w*-w operators from X* to Y.

We consider $L_G\left(s\right)$ to be the $L$-function attached to a particular automorphic form $G$ on $GL\left(6\right)$. We establish an upper bound for the mean square estimate on the critical line of Rankin-Selberg $L$-function $L_{G\times G}\left(s\right)$. As an application of this result, we give an asymptotic formula for the discrete sum of coefficients of $L_{G\times G}\left(s\right)$.

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

Let ${\alpha }_i,{\beta }_i>0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t•x=(t^{\alpha ₁}x₁,...,t^{\alpha ₙ}xₙ)$, $t\circ x=(t^{\beta ₁}x₁,...,t^{\beta ₙ}xₙ)$ and $\left|\right|x\left|\right|={\sum }_{i=1}^n{\left|x_i\right|}^{1/{\alpha }_i}$. Let φ₁,...,φₙ be real functions in $C^{\infty }(ℝⁿ-0)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on ${ℝ}^{2n}$ given by $\mu \left(E\right)={\int }_{ℝⁿ}{\chi }_E{(x,\phi \left(x\right))\left|\right|x\left|\right|}^{\gamma -\alpha }dx$, where $\alpha ={\sum }_{i=1}^n{\alpha }_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_{\mu }f=\mu \ast f$ and let $\left|\right|T_{\mu }{\left|\right|}_{p,q}$ be the operator norm of $T_{\mu }$ from $L^p\left({ℝ}^{2n}\right)$ into $L^q\left({ℝ}^{2n}\right)$, where the $L^p$ spaces are taken with respect to the Lebesgue measure. The type set $E_{\mu }$ is defined by $E_{\mu }=(1/p,1/q):\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty ,1\le p,q\le \infty$. In the case ${\alpha }_i\ne {\beta }_k$ for 1 ≤ i,k ≤ n we characterize the...

We provide a deep investigation of the notions of - and ${}^{\infty }$-hypoellipticity for partial differential operators with constant Colombeau coefficients. This involves generalized polynomials and fundamental solutions in the dual of a Colombeau algebra. Sufficient conditions and necessary conditions for - and ${}^{\infty }$-hypoellipticity are given

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 1 ≤ p < ∞, $={\left(Xₙ\right)}_{n\in \mathbb{N}}$ be a sequence of Banach spaces and $l_p\left(\right)$ the coresponding vector valued sequence space. Let $={\left(Xₙ\right)}_{n\in \mathbb{N}}$, $={\left(Yₙ\right)}_{n\in \mathbb{N}}$ be two sequences of Banach spaces, $={\left(Vₙ\right)}_{n\in \mathbb{N}}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_{}:l_p\left(\right)\to l_q\left(\right)$ by $M_{}\left({\left(xₙ\right)}_{n\in \mathbb{N}}\right):={\left(Vₙ\left(xₙ\right)\right)}_{n\in \mathbb{N}}$. We give necessary and sufficient conditions for $M_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞. ...

Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...