Research article:

Nate Segerlind, Samuel R. Buss, and Russell Impagliazzo.
"A Switching Lemma for Small Restrictions and Lower Bounds for k-DNF Resolution."
SIAM Journal on Computing 33, no 5 (2004) 1171-1200 .

Abstract: We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the $\Resk$ propositional proof system, an extension of resolution which works with k-DNFs instead of clauses. We also obtain an exponential separation between depth d circuits of bottom fan-in k and depth d circuits of bottom fan-in k+1.
Our results for $\Resk$ are:

1. The  2n  to  n  weak pigeonhole principle requires exponential size to refute in $\Resk$, for $k \leq \sqrt{\log n / \log \log n }$
2. For each constant  k, there exists a constant $w>k$ so that random $w$-CNFs require exponential size to refute in ${\mbox{Res}}(k)$.
3. For each constant  k, there are sets of clauses which have polynomial size $\Res{k+1}$ refutations, but which require exponential size $\Resk$ refutations.

Earlier conference proceedings paper:

Nate Segerlind, Samuel R. Buss, and Russell Impagliazzo.
"A Switching Lemma for Small Restrictions and Lower Bounds for k-DNF Resolution (Extended Abstract)."
In: Proceedings of the 43rd Annual Symposium on Foundations of Computer Science (FOCS'02),
IEEE Computer Society, 2002, pp 604-613.