Maximizing subgraph density in graphs of bounded degree and clique number

We asymptotically determine the maximum density of subgraphs isomorphic to $H$, where $H$ is any graph containing a dominating vertex, in graphs $G$ on $n$ vertices with bounded maximum degree and bounded clique number. That is, we asymptotically determine the constant $c=c(H,\Delta,\omega)$ such that ex$(n,H,\{K_{1,\Delta+1},K_{\omega+1}\})=(1-o_n(1))cn$. More generally, if $H$ has at least $u$ dominating vertices, then we find the maximum density of subgraphs isomorphic to $H$ in graphs $G$ that have $p$ cliques of size $u$, have bounded clique number, and are $K_u\vee I_{\Delta+1}$-free. For example, we asymptotically determine the constant $d=d(H,\Delta,\omega)$ such that mex$(m,H,\{K_{1,1,\Delta+1},K_{\omega+1}\})=(1-o_m(1))dm$. Then we localize these results, proving a tight inequality involving the sizes of the locally largest cliques and complete split graphs.