Isomorphic groupoid -algebras associated with different Haar systems.
We study, in the context of doubling metric measure spaces, a class of BMO type functions defined by John and Nirenberg. In particular, we present a new version of the Calderón-Zygmund decomposition in metric spaces and use it to prove the corresponding John-Nirenberg inequality.
Some Korovkin-type theorems for spaces containing almost periodic measures are presented. We prove that some sets of almost periodic measures are test sets for some particular nets of positive linear operators on spaces containing almost periodic measures. We consider spaces which contain almost periodic measures defined by densities and measures which can be represented as the convolution between an arbitrary measure with finite support (or an arbitrary bounded measure) and a fixed almost periodic...
A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case.
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 functional calculus for every p ≥ 1.
Let be the singular measure on the Heisenberg group supported on the graph of the quadratic function , where is a real symmetric matrix. If , we prove that the operator of convolution by on the right is bounded from to . We also study the type set of the measures , for , where is a cut-off function around the origin on . Moreover, for we characterize the type set of .
A measure is called -improving if it acts by convolution as a bounded operator from to for some q > p. Positive measures which are -improving are known to have positive Hausdorff dimension. We extend this result to complex -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 -functions.
- boundedness properties are obtained for operators defined by convolution with measures supported on certain curves on the Heisenberg group. We find the curvature condition for which the type set of these operators can be the full optimal trapezoid with vertices A=(0,0), B=(1,1), C=(2/3,1/2), D=(1/2,1/3). We also give notions of right curvature and left curvature which are not mutually equivalent.
- estimates are obtained for convolution operators by finite measures supported on curves in the Heisenberg group whose tangent vector at the origin is parallel to the centre of the group.
Let be a symmetric space of the noncompact type, with Laplace–Beltrami operator , and let be the -spectrum of . For in such that , let be the operator on defined formally as . In this paper, we obtain operator norm estimates for for all , and show that these are optimal when is small and when is bounded below .