The space has a natural homomorphism into when is integer-valued (by mapping the virtual function to its actual function interpretation, and extending by homomorphism), but the map is neither surjective nor injective in general. For instance, the image must take values in the space of symmetric functions on (and in fact the image is dense in that space in weak topologies); in particular, the fractional power is more like a quotient space of the Cartesian power by the action of the relevant (product) symmetric group. I had thought about making the notation more complicated to more accurately reflect this feature, but ultimately decided that the notation was already heavy enough as it was. This also explains why is “larger” than , as the functions on the former space have less symmetry demanded of them. (But there is a natural homomorphism from to (mapping to and extending by homomorphism), which can be viewed as a “virtual” projection map from to ).

Injectivity can also fail; for instance, if all the are finite spaces, then is finite dimensional, but is infinite dimensional; there are “accidental” polynomials identities relating various functions in this case that are not picked up by the abstract Fock space. One could potentially quotient out the Fock space by these additional identities to obtain a space that more accurately resembles , but there didn’t seem to be any need for this in any of the applications I was considering (and there was some minor technical convenience in keeping the Fock space “free” of such identities in order to more easily define homomorphisms out of such spaces).

Certainly my exposure to free probability (or more generally noncommutative probability) influenced my thinking when trying to abstract out the key features of integration on product spaces that could extend to the fractional power setting. Indeed the Fock space constructed here could almost be interpreted as in the framework of noncommutative probability theory (though it remains commutative in this case), except for the fact that the positivity property breaks down in the non-integer case (see the discussion before Theorem 3.4); related to this, I did not impose a topology on the Fock space with which to take a suitable weak closure to recover a von Neumann algebra. But one could still view as a sort of “(non)commutative measure space” rather than a “(non)commutative probability space” if one wished, by abandoning the positivity axiom.

]]>Some remarks (please take with a grain of salt in case they’re wrong):

1) The fractional Cartesian product space seems to have the property that

is not .

2) Although is defined for all fractional powers , for integer powers the Fock-type construction is actually a subset of the space the symbol stands for (maybe dense; but as far as I understand it, definitely not equal).

3) The construction of the fractional cartesian product reminded me of free probability, although it seems an element in the fractional cartesian product is identified by what it does with every other element, whereas elements in free probability spaces are identified with their free cumulants if I recall correctly. I have not yet read the section that proves the multi linear Kakeya using this tool but I would like to eventually read it.

]]>Nobody sees a light needle in a dark sea. I am very delighted to see that you solved one of the hardest problems in the world in 2019 . I can not reveal that problem but in disorder letters . May be yourself knows well: N,T,E,M,N,E,I,C,J,T,R,R,C,U,W,E,O,I,P,S. ]]>

There are many good mathematicians in the world , but I only like you.Therefore, one day recent I hope you will stand on the highest stage that receives Clay millenium award. For many reasons including of far distant, I cannot meet you to tell many

secrets, you are very special in 52 Field medalists. Before I die, I want my wish become true, if not I am very sad ]]>

a simple consequence of triangle inequality and proving the estimate for a single intersection of tubes? I.e. proving

where the supremum is taken over all collections of tubes with ? Isn’t this last estimate immediate? I feel that the transversality condition should guarantee the tube intersection is small. What subtlety am I missing?

Given the opportunity, I’d also like to thank you (and others) who make their work available in the form of notes, blog posts, arxiv, etc.

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