446 research outputs found
Regularity of the Hardy-Littlewood maximal operator on block decreasing functions
We study the Hardy-Littlewood maximal operator defined via an unconditional
norm, acting on block decreasing functions. We show that the uncentered maximal
operator maps block decreasing functions of special bounded variation to
functions with integrable distributional derivatives, thus improving their
regularity. In the special case of the maximal operator defined by the
l_infty-norm, that is, by averaging over cubes, the result extends to block
decreasing functions of bounded variation, not necessarily special.Comment: 26 page
Optimality conditions and regularity results for time optimal control problems with differential inclusions
We study the time optimal control problem with a general target
for a class of differential inclusions that satisfy mild smoothness and
controllability assumptions. In particular, we do not require Petrov's
condition at the boundary of . Consequently, the minimum time
function fails to be locally Lipschitz---never mind
semiconcave---near . Instead of such a regularity, we use an
exterior sphere condition for the hypograph of to develop the
analysis. In this way, we obtain dual arc inclusions which we apply to show the
constancy of the Hamiltonian along optimal trajectories and other optimality
conditions in Hamiltonian form. We also prove an upper bound for the Hausdorff
measure of the set of all nonlipschitz points of which implies that
the minimum time function is of special bounded variation.Comment: 23 pages, 1 figur
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