We prove the norm estimates for operator-valued functions on free groups supported on the words with fixed length ($f={\sum }_{\left|w\right|=l}{a}_w\otimes \lambda \left(w\right)$). Next, we replace the translations by the free generators with a free family of operators and prove inequalities of the same type.

Starting from a general formulation of the characterization by dyadic crowns of Sobolev spaces, the authors give a result of $L^p$ continuity for pseudodifferential operators whose symbol a(x,ξ) is non smooth with respect to x and whose derivatives with respect to ξ have a decay of order ρ with $0<\rho \le 1$. The algebra property for some classes of weighted Sobolev spaces is proved and an application to multi - quasi - elliptic semilinear equations is given.

We consider a family of non-unimodular rank one NA-groups with roots not all positive, and we show that on these groups there exists a distinguished left invariant sub-Laplacian which admits a differentiable $L^p$ functional calculus for every p ≥ 1.

We consider a double analytic family of fractional integrals $S_z^{\gamma ,\alpha }$ along the curve ${t\mapsto \left|t\right|}^{\alpha }$, introduced for α = 2 by L. Grafakos in 1993 and defined by $\left(S_z^{\gamma ,\alpha }f\right)(x₁,x₂):=1/\Gamma (z+1/2){\int \int |u-1|}^z\psi (u-1){f(x₁-t,x₂-u|t|}^{\alpha }{\left)du\right|t|}^{\gamma }dt/t$, where ψ is a bump function on ℝ supported near the origin, $f{\in }_c^{\infty }\left(ℝ²\right)$, z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that $S_z^{\gamma ,\alpha }$ maps $L^p\left(ℝ²\right)$ to $L^q\left(ℝ²\right)$ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K_{-1+i\theta }^{i\varrho ,\alpha }$ is a product kernel on ℝ², adapted to the curve ${t\mapsto \left|t\right|}^{\alpha }$; as a consequence, we show that the operator...

The commutator of a singular integral operator with homogeneous kernel Ω(x)/|x|ⁿ is studied, where Ω is homogeneous of degree zero and has mean value zero on the unit sphere. It is proved that $\Omega \in L{\left(logL\right)}^{k+1}\left(S^{n-1}\right)$ is a sufficient condition for the kth order commutator to be bounded on $L^p\left(ℝⁿ\right)$ for all 1 < p < ∞. The corresponding maximal operator is also considered.