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.

We deal with several classes of integral transformations of the form $$f\left(x\right)\to D{\int }_{{ℝ}_{+}^2}\frac{1}{u}({\mathrm{e}}^{-ucosh(x+v)}+{\mathrm{e}}^{-ucosh(x-v)})h\left(u\right)f\left(v\right)\mathrm{d}u\mathrm{d}v,$$ where $D$ is an operator. In case $D$ is the identity operator, we obtain several operator properties on $L_p\left({ℝ}_{+}\right)$ with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on $L_2\left({ℝ}_{+}\right)$ and define the inversion formula. Further, for an other class of differential operators of finite...

We introduce some spaces of generalized functions that are defined as generalized quotients and Boehmians. The spaces provide simple and natural frameworks for extensions of the Fourier transform.

The Stein-Weiss theorem that the distribution function of the Hilbert transform of the characteristic function of E depends only on the measure of E is generalized to the ergodic Hilbert transform.

Mathematics Subject Classification: 33C60, 33C20, 44A15This paper is devoted to an important case of Wright’s hypergeometric function 2Fτ,β1(a, b; c; z) = 2Fτ,β1(z), to studying its basic properties and to application of 2Fτ,β1(z) to the generalization of the associated Legendre functions.

Let $\Gamma$ be a group endowed with a length function $L$, and let $E$ be a linear subspace of $\mathbf{C}\Gamma$. We say that $E$ satisfies the Haagerup inequality if there exists constants $C,s\>0$ such that, for any $f\in E$, the convolutor norm of $f$ on ${\ell }^2\left(\Gamma \right)$ is dominated by $C$ times the ${\ell }^2$ norm of $f(1+L{)}^s$. We show that, for $E=\mathbf{C}\Gamma$, the Haagerup inequality can be expressed in terms of decay of random walks associated with finitely supported symmetric probability measures on $\Gamma$. If $L$ is a word length function on a finitely generated group $\Gamma$, we show that,...

We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove Brascamp-Lieb inequalities and thus prove a new generalised Reverse...