Can aggregate demand fall if rates fall?

Some commentators ask if it is possible that aggregate demand could actually fall if real rates are pushed down. As Keynes showed, anything is possible in economics provided you make the right assumptions. The question really is whether the assumptions are plausible. Take a situation where the beneficiaries of cheap savings are unwilling (firms and banks) or unable (young households) to spend for exogenous reasons. So spenders do not contribute to aggregate demand but save any windfall they get from low rates. Typically, one would imagine that savers, seeing the price paid for savings is too low, would go out and spend, thus reducing savings and boosting demand. But it is equally possible that in these straightened times, fearing a bleak retirement with meager savings, savers respond to low rates by saving even more in order to have enough for the future. In economic jargon, the income effect dominates the substitution effect. Indeed, this is one reason economists have been advocating higher deposit rates in China, arguing that higher incomes to depositors will lead to higher spending rather than higher savings.
More broadly, monetary policy is cloaked in a lot of mystery, which is why I tried to give  a simple analogy of a tax and subsidy. To the extent that the Fed can push equilibrium real rates below market levels (if the claim is that negative real for the short rate is the market equilibrium level today, then we do not need to have a debate about monetary policy -- the Fed is not setting anything), it is taxing someone and subsidizing someone else. Taxes and subsidies can be expansionary or contractionary dependning on incidence. I don't think that there is anything here that violates Economics 101 (yes I did attend that class!).  
Do I know that income effects outweigh substitution effects today? I don't and I am willing to accept that they may not. My goal was not to assert a particular view (there is too much of that based on ideology, not economics) but to argue that there were other theoretical possibilities than the commandments Keynes handed down to us. In the face of uncertainty about theory, and when the normal channels of transmission are not working,  I would be wary of pushing any policy too hard, especially if there is little evidence that it is working -- the unintended consequences could outweigh the meager benefits, if any. Paul Krugman warns us about the recession within the depression in 1937, caused by the Fed raising rates too soon (of course, there is a fair amount of disgreement among economists about what caused that recession). But one could equally point to recent history in 2002-2004 when the Fed kept rates too low for too long and helped fuel the housing bubble (admittedly, economists disagree here also). More generally, risk management in policy making might suggest not going to extremes when policy makers are faced with high levels of uncertainty about policy effectiveness (this is common sense, I don't have a model here).  
Finally, do I advocate raising rates immediately? No, of course not. Given the expectations the Fed has built up, this would roil the markets. But I do think that going back to zero real, sooner rather than later, and preparing the way for that as soon as indicators start strengthening again, would be a reasonable (and my preferred) course of action.