  We consider fractional online covering problems with $\ell_q$-norm
objectives. The problem of interest is of the form $\min\{ f(x) \,:\, Ax\ge 1,
x\ge 0\}$ where $f(x)=\sum_{e} c_e \|x(S_e)\|_{q_e} $ is the weighted sum of
$\ell_q$-norms and $A$ is a non-negative matrix. The rows of $A$ (i.e. covering
constraints) arrive online over time. We provide an online $O(\log d+\log
\rho)$-competitive algorithm where $\rho = \frac{\max a_{ij}}{\min a_{ij}}$ and
$d$ is the maximum of the row sparsity of $A$ and $\max |S_e|$. This is based
on the online primal-dual framework where we use the dual of the above convex
program. Our result expands the class of convex objectives that admit good
online algorithms: prior results required a monotonicity condition on the
objective $f$ which is not satisfied here. This result is nearly tight even for
the linear special case. As direct applications we obtain (i) improved online
algorithms for non-uniform buy-at-bulk network design and (ii) the first online
algorithm for throughput maximization under $\ell_p$-norm edge capacities.
