Gagliardo-Nirenberg Inequalities for Differential Forms in Heisenberg Groups

The L 1-Sobolev inequality states that the L n/(n--1)-norm of a compactly supported function on Euclidean n-space is controlled by the L 1-norm of its gradient. The generalization to differential forms (due to Lanzani & Stein and Bourgain & Brezis) is recent, and states that a the L n/(n--1)-norm of a compactly supported differential h-form is controlled by the L 1-norm of its exterior differential du and its exterior codifferential $\delta$u (in special cases the L 1-norm must be replaced by the H 1-Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms

L1L^1-Poincar\'e and Sobolev inequalities for differential forms in Euclidean spaces

In this paper, we prove Poincar\'e and Sobolev inequalities for differential forms in $L^1(\mathbb R^n)$. The singular integral estimates that it is possible to use for $L^p$, $p>1$, are replaced here with inequalities which go back to Bourgain-Brezis.Comment: Accepted for publication in Science China Mathematics. arXiv admin note: text overlap with arXiv:1902.0481

A robust multivariate long run analysis of European electricity prices

This paper analyses the interdependencies existing in wholesale electricity prices in six major European countries. The results of our robust multivariate long run dynamic analysis reveal the presence of four highly integrated central European markets (France, Germany, the Netherlands and Austria). The trend shared by these four electricity markets appears to be common also to gas prices, but not to oil prices. The existence of long term dynamics among electricity prices and between electricity prices and gas prices may prove to be important for long term hedging operations to be conducted even in countries where well established and liquid electricity derivatives markets are not present. Since standard unit root and cointegration tests are not robust to the peculiar characteristics of electricity prices time series, we adapt and further develop a battery of robust inference procedures that should assure the reliability of our results.European electricity prices, Cointegration, Interdependencies, Equilibrium Correction model, Oil prices, Robustness

Deregulated Wholesale Electricity Prices in Europe

This paper analyses the interdependencies existing in the European electricity prices. The results of a multivariate dynamic analysis of weekly median prices reveal the presence of strong integration (but not perfect integration) among the markets considered in the sample and the existence of a common trend among electricity prices and oil prices. This implies that there are no long-run arbitrage opportunities. The latter result appears to be relevant also in the context of the discussion of efficient hedging instruments to be used by medium-long term investors.European electricity prices, Cointegration, Interdependencies, Equilibrium Correction model, Oil prices

A Robust Multivariate Long Run Analysis of European Electricity Prices

This paper analyses the interdependencies existing in wholesale European electricity prices. The results of a multivariate long run dynamic analysis of weekly median prices reveal the presence of a strong although not perfect integration among some neighboring markets considered in the sample and the existence of common long-term dynamics of electricity prices and gas prices but not oil prices. The existence of long-term dynamics among gas prices and electricity prices may prove to be important for long-term hedging operations to be conducted even in markets where there are no electricity derivatives.European Electricity Prices, Cointegration, Interdependencies, Equilibrium Correction Model, Oil Prices

L1L^1-Poincar\'e inequalities for differential forms on Euclidean spaces and Heisenberg groups

In this paper, we prove interior Poincar{\'e} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree. The singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis in Euclidean spaces, and to Chanillo-van Schaftingen in Heisenberg groups

Orlicz spaces and endpoint Sobolev-Poincaré inequalities for differential forms in Heisenberg groups

In this paper we prove Poincar´e and Sobolev inequalities for differential forms in the Rumin’s contact complex on Heisenberg groups. In particular, we deal with endpoint values of the exponents, obtaining finally estimates akin to exponential Trudinger inequalities for scalar function. These results complete previous results obtained by the authors away from the exponential case. From the geometric point of view, Poincaré and Sobolev inequalities for differential forms provide a quantitative formulation of the vanishing of the cohomology. They have also applications to regularity issues for partial differential equations

L1-Poincar\ue9 and Sobolev inequalities for differential forms in Euclidean spaces

In this paper, we prove Poincar\ue9 and Sobolev inequalities for differential forms in L1(\u211dn). The singular integral estimates that it is possible to use for Lp, p > 1, are replaced here with inequalities which go back to Bourgain and Brezis (2007)