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().

We study the absolute continuity of the convolution ${\delta }_{e^X}^{♮}*{\delta }_{e^Y}^{♮}$ of two orbital measures on the symmetric space SO₀(p,q)/SO(p)×SO(q), q > p. We prove sharp conditions on X,Y ∈ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for SO₀(p,q)/SO(p)×SO(q) also serves for the spaces SU(p,q)/S(U(p)×U(q)) and Sp(p,q)/Sp(p)×Sp(q), q > p. We moreover apply our results to...

We study convolution operators bounded on the non-normable Lorentz spaces $L^{1,q}$ of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on $L^{1,q}$. In particular, when the positions of the atoms of a discrete measure are linearly independent over...

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 $\alpha -\frac{1}{2}$.

The paper is devoted to infinitely differentiable one-parameter convolution semigroups in the convolution algebra ${}_C^{\text{'}}(ℝⁿ;M_{m\times m})$ of matrix valued rapidly decreasing distributions on ℝⁿ. It is proved that $G{\in }_C^{\text{'}}(ℝⁿ;M_{m\times m})$ is the generating distribution of an i.d.c.s. if and only if the operator ${\partial }_t{\otimes }_{m\times m}-G\ast$ on ${ℝ}^{1+n}$ satisfies the Petrovskiĭ condition for forward evolution. Some consequences are discussed.

Let ${\varphi }_1,\cdots ,{\varphi }_n$ be real homogeneous functions in $C^{\infty }({ℝ}^n-\left\{0\right\})$ of degree $k\ge 2$, let $\varphi \left(x\right)=({\varphi }_1\left(x\right),\cdots ,{\varphi }_n\left(x\right))$ and let $\mu$ be the Borel measure on ${ℝ}^{2n}$ given by $$\mu \left(E\right)={\int }_{{ℝ}^n}{\chi }_E(x,\varphi \left(x\right)){\left|x\right|}^{\gamma -n}\mathrm{d}x$$ where $\mathrm{d}x$ denotes the Lebesgue measure on ${ℝ}^n$ and $\gamma >0$. Let $T_{\mu }$ be the convolution operator $T_{\mu }f\left(x\right)=(\mu *f)\left(x\right)$ and let $$E_{\mu }=\{(1/p,1/q)\parallel T_{\mu }{\parallel }_{p,q}<\infty ,1\le p,q\le \infty \}.$$ Assume that, for $x\ne 0$, the following two conditions hold: $det\left({\mathrm{d}}^2\varphi \left(x\right)h\right)$ vanishes only at $h=0$ and $det\left(\mathrm{d}\varphi \right(x\left)\right)\ne 0$. In this paper we show that if $\gamma >n(k+1)/3$ then $E_{\mu }$ is the empty set and if $\gamma \le n(k+1)/3$ then $E_{\mu }$ is the closed segment with endpoints $D=\left(1-\frac{\gamma }{n(k+1)},1-\frac{2\gamma }{n(k+1)}\right)$ and $D^{\text{'}}=\left(\frac{2\gamma }{n(1+k)},\frac{\gamma }{n(1+k)}\right)$. Also, we give some examples.

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 $\phi \left(w\right)={\left|w\right|}^{2m}$.

We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube $I^M={[0,1)}^M$, with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on $I^M$; (3) the measures have the convolution property that $\mu \ast L^p\subseteq L^{p+\epsilon }$ for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then $\mu \ast L^p\subseteq L^q$ for any measure μ in our...

Let $\tilde{f}$, $\tilde{g}$ be ultradistributions in ${𝒵}^{\text{'}}$ and let ${\tilde{f}}_n=\tilde{f}*{\delta }_n$ and ${\tilde{g}}_n=\tilde{g}*{\sigma }_n$ where $\left\{{\delta }_n\right\}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $\delta$. Then the neutrix product $\tilde{f}\diamond \tilde{g}$ is defined on the space of ultradistributions ${𝒵}^{\text{'}}$ as the neutrix limit of the sequence $\left\{\frac{1}{2}({\tilde{f}}_n\tilde{g}+\tilde{f}{\tilde{g}}_n)\right\}$ provided the limit $\tilde{h}$ exist in the sense that $$\underset{n\to \infty }{\mathrm{N}\text{-}lim}\frac{1}{2}\langle {\tilde{f}}_n\tilde{g}+\tilde{f}{\tilde{g}}_n,\psi \rangle =\langle \tilde{h},\psi \rangle$$ for all $\psi$ in $𝒵$. We also prove that the neutrix convolution product $f♦*g$ exist in ${𝒟}^{\text{'}}$, if and only if the neutrix product $\tilde{f}\diamond \tilde{g}$ exist in ${𝒵}^{\text{'}}$ and the exchange formula $$F\left(f♦*g\right)=\tilde{f}\diamond \tilde{g}$$ is then satisfied.

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 $L_q$.

Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by $\mu \left(E\right)={\int }_D{\chi }_E(x,\phi \left(x\right))dx$ with D = x ∈ ℝ ²:|x| ≤ 1 and let $T_{\mu }$ be the convolution operator with the measure μ. Let $\phi =\phi {₁}^{e₁}\cdots \phi {ₙ}^{eₙ}$ be the decomposition of φ into irreducible factors. We show that if $e_i\ne m/2$ for each ${\phi }_i$ of degree 1, then the type set $E_{\mu }:=(1/p,1/q)\in [0,1]\times [0,1]:\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty$ can be explicitly described as a closed polygonal region.

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 $p\in [1,2)$.

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.

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 $\left({ℝ}^d\right)$.

Extending previous work by Meise and Vogt, we characterize those convolution operators, defined on the space ${}_{\left(\omega \right)}\left(ℝ\right)$ of (ω)-quasianalytic functions of Beurling type of one variable, which admit a continuous linear right inverse. Also, we characterize those (ω)-ultradifferential operators which admit a continuous linear right inverse on ${}_{\left(\omega \right)}[a,b]$ for each compact interval [a,b] and we show that this property is in fact weaker than the existence of a continuous linear right inverse on ${}_{\left(\omega \right)}\left(ℝ\right)$.

In this paper, we obtain the Fekete-Szego inequalities for the functions of complex order defined by convolution. Also, we find upper bounds for the second Hankel determinant $|a_2{a}_4-a_3^2|$ for functions belonging to the class $S_{\gamma }^b(g\left(z\right);A,B)$.

We study natural measures on sets of $\beta$-expansions and on slices through self similar sets. In the setting of $\beta$-expansions, these allow us to better understand the measure of maximal entropy for the random $\beta$-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing,...

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.

The author investigates the boundedness of the higher order commutator of strongly singular convolution operator, $T_b^m$, on Herz spaces $K{̇}_q^{\alpha ,p}\left(ℝⁿ\right)$ and $K_q^{\alpha ,p}\left(ℝⁿ\right)$, and on a new class of Herz-type Hardy spaces $HK{̇}_{q,b,m}^{\alpha ,p,0}\left(ℝⁿ\right)$ and $H{K}_{q,b,m}^{\alpha ,p,0}\left(ℝⁿ\right)$, where 0 < p ≤ 1 < q < ∞, α = n(1-1/q) and b ∈ BMO(ℝⁿ).

Let p,q be positive integers. The groups $U_p\left(ℂ\right)$ and $U_p\left(ℂ\right)\times U_q\left(ℂ\right)$ act on the Heisenberg group $H_{p,q}:=M_{p,q}\left(ℂ\right)\times ℝ$ canonically as groups of automorphisms, where $M_{p,q}\left(ℂ\right)$ is the vector space of all complex p × q matrices. The associated orbit spaces may be identified with ${\Pi }_q\times ℝ$ and ${\Xi }_q\times ℝ$ respectively, ${\Pi }_q$ being the cone of positive semidefinite matrices and ${\Xi }_q$ the Weyl chamber $x\in {ℝ}^q:x₁\ge \cdots \ge x_q\ge 0$. In this paper we compute the associated convolutions on ${\Pi }_q\times ℝ$ and ${\Xi }_q\times ℝ$ explicitly, depending on p. Moreover, we extend these convolutions by analytic continuation to series...

Let 0 < p ≤ 1, let ω: ℤ → [1,∞) be a weight on ℤ and let f be a nowhere vanishing continuous function on the unit circle Γ whose Fourier series satisfies ${\sum }_{n\in \mathbb{Z}}{\left|f̂\left(n\right)\right|}^p\omega \left(n\right)<\infty$. Then there exists a weight ν on ℤ such that ${\sum }_{n\in \mathbb{Z}}{\left|\widehat{(1/f)}\left(n\right)\right|}^p\nu \left(n\right)<\infty$. Further, ν is non-constant if and only if ω is non-constant; and ν = ω if ω is non-quasianalytic. This includes the classical Wiener theorem (p = 1, ω = 1), Domar theorem (p = 1, ω is non-quasianalytic), Żelazko theorem (ω = 1) and a recent result of Bhatt and Dedania (p = 1). An...

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...

A measure is called $L^p$-improving if it acts by convolution as a bounded operator from $L^p$ to $L^q$ for some q > p. Positive measures which are $L^p$-improving are known to have positive Hausdorff dimension. We extend this result to complex $L^p$-improving measures and show that even their energy dimension is positive. Measures of positive energy dimension are seen to be the Lipschitz measures and are characterized in terms of their improving behaviour on a subset of $L^p$-functions.