Ask Mr. Smarty Plants

I'm not sure what palms you have.  The only palm native to California is Washingtonia filifera (California fan palm), but there are other palms commonly found there [e.g., Phoenix canariensis (Canary Island Palm) from the Canary Islands, Phoenix dactylifera (Date palm) from Asia and Washingtonia robusta (Mexican palm fan)].

I searched for references to palms and allelopathy and below are the only references I found. 

M. I. Khan in an article, [Allelopathic potential of dry fruits of Washingtonia filifera: Inhibition of seed germination.  Physiologia Plantarum, 54(3):323-328, 1982] says that the leachates of the fruits of this palm inhibited the germination of the seeds of lettuce and cucumbers.

Another article by Khan [Allelopathic potential of dry fruits of Washingtonia filifera (L. LINDEN) H. WENDL.  II.  Inhibition of seedling growth.  Biologia Plantarum, 24(4), 1982] reports the allelopathic effect of the dry palm fruits on growth of the seedlings of lettuce, red cabbage, tomato and cucumber.

Finally, A. K. Mukherjee [Allelopathy in Phoenix paludosa pure stand formation.  Journal of Economic and Taxonomic Botany, 12(2), 1988] reported that the roots of Pheonix paludosa (Mangrove date palm), an Asian species related to date palms found in California, delayed germination and retarded the growth of gram (a kind of bean) seed.   Mukherjee speculates that a growth retardant from the roots of the palm eliminates competing plants and results in pure stands of the palm in India.

These references are to experiments performed 20 years or more ago and no one has repeated them, as far as I know, nor has anyone reported allelopathy associated with any other palm trees.  That doesn't mean that your palm seeds or fruits don't have any allelopathic properties.  Since palms are prevalent in lots or landscapes, however, it seems that it would be reported by homeowners and landscapers.  

Just in case the seeds or any other parts of the above ground tree is inhibiting the growth of plants underneath your palm, you can alleviate the effect by keeping the area under and around the tree free of fruits, seeds and fronds.  Also, since the palm is going to provide a good deal of shade, this could be part of your problem.  Try choosing shade-loving plants for the area under the palms.