We study deformations of the classical convolution. For every invertible transformation T:μ ↦ Tμ, we are able to define a new associative convolution of measures by $\mu {*}_T\nu =T^{-1}(T\mu *T\nu )$. We deal with the $V_a$-deformation of the classical convolution. We prove the analogue of the classical Lévy-Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the $V_a$-deformed classical convolution and give the orthogonal...

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 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...

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 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...

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

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,...

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

New concepts of Lebesgue measure on ${ℝ}^{\infty }$ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on ${ℝ}^{\infty }$ for α = (1,1,...).

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.

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 $\tilde{f}$, $\tilde{g}$ 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.

We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a $g$-measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique $g$-measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the $g$-measure.

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 $p\in [1,2)$.

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 $L_q$.

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 present a new theorem on the differential inequality $u^{\left(m\right)}\le w\left(u\right)$. Next, we apply this result to obtain existence theorems for the equation $x^{\left(m\right)}=f(t,x)$.