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Let be some domain (such as the real numbers). For any natural number
, let
denote the space of symmetric real-valued functions
on
variables
, thus
for any permutation . For instance, for any natural numbers
, the elementary symmetric polynomials
will be an element of . With the pointwise product operation,
becomes a commutative real algebra. We include the case
, in which case
consists solely of the real constants.
Given two natural numbers , one can “lift” a symmetric function
of
variables to a symmetric function
of
variables by the formula
where ranges over all injections from
to
(the latter formula making it clearer that
is symmetric). Thus for instance
and
Also we have
With these conventions, we see that vanishes for
, and is equal to
if
. We also have the transitivity
if .
The lifting map is a linear map from
to
, but it is not a ring homomorphism. For instance, when
, one has
In general, one has the identity
for all natural numbers and
,
, where
range over all injections
,
with
. Combinatorially, the identity (2) follows from the fact that given any injections
and
with total image
of cardinality
, one has
, and furthermore there exist precisely
triples
of injections
,
,
such that
and
.
Example 1 When
, one has
which is just a restatement of the identity
Note that the coefficients appearing in (2) do not depend on the final number of variables . We may therefore abstract the role of
from the law (2) by introducing the real algebra
of formal sums
where for each ,
is an element of
(with only finitely many of the
being non-zero), and with the formal symbol
being formally linear, thus
and
for and scalars
, and with multiplication given by the analogue
of (2). Thus for instance, in this algebra we have
and
Informally, is an abstraction (or “inverse limit”) of the concept of a symmetric function of an unspecified number of variables, which are formed by summing terms that each involve only a bounded number of these variables at a time. One can check (somewhat tediously) that
is indeed a commutative real algebra, with a unit
. (I do not know if this algebra has previously been studied in the literature; it is somewhat analogous to the abstract algebra of finite linear combinations of Schur polynomials, with multiplication given by a Littlewood-Richardson rule. )
For natural numbers , there is an obvious specialisation map
from
to
, defined by the formula
Thus, for instance, maps
to
and
to
. From (2) and (3) we see that this map
is an algebra homomorphism, even though the maps
and
are not homomorphisms. By inspecting the
component of
we see that the homomorphism
is in fact surjective.
Now suppose that we have a measure on the space
, which then induces a product measure
on every product space
. To avoid degeneracies we will assume that the integral
is strictly positive. Assuming suitable measurability and integrability hypotheses, a function
can then be integrated against this product measure to produce a number
In the event that arises as a lift
of another function
, then from Fubini’s theorem we obtain the formula
is an element of the formal algebra , then
Note that by hypothesis, only finitely many terms on the right-hand side are non-zero.
Now for a key observation: whereas the left-hand side of (6) only makes sense when is a natural number, the right-hand side is meaningful when
takes a fractional value (or even when it takes negative or complex values!), interpreting the binomial coefficient
as a polynomial
in
. As such, this suggests a way to introduce a “virtual” concept of a symmetric function on a fractional power space
for such values of
, and even to integrate such functions against product measures
, even if the fractional power
does not exist in the usual set-theoretic sense (and
similarly does not exist in the usual measure-theoretic sense). More precisely, for arbitrary real or complex
, we now define
to be the space of abstract objects
with and
(and
now interpreted as formal symbols, with the structure of a commutative real algebra inherited from
, thus
In particular, the multiplication law (2) continues to hold for such values of , thanks to (3). Given any measure
on
, we formally define a measure
on
with regards to which we can integrate elements
of
by the formula (6) (providing one has sufficient measurability and integrability to make sense of this formula), thus providing a sort of “fractional dimensional integral” for symmetric functions. Thus, for instance, with this formalism the identities (4), (5) now hold for fractional values of
, even though the formal space
no longer makes sense as a set, and the formal measure
no longer makes sense as a measure. (The formalism here is somewhat reminiscent of the technique of dimensional regularisation employed in the physical literature in order to assign values to otherwise divergent integrals. See also this post for an unrelated abstraction of the integration concept involving integration over supercommutative variables (and in particular over fermionic variables).)
Example 2 Suppose
is a probability measure on
, and
is a random variable; on any power
, we let
be the usual independent copies of
on
, thus
for
. Then for any real or complex
, the formal integral
can be evaluated by first using the identity
(cf. (1)) and then using (6) and the probability measure hypothesis
to conclude that
For
a natural number, this identity has the probabilistic interpretation
whenever
are jointly independent copies of
, which reflects the well known fact that the sum
has expectation
and variance
. One can thus view (7) as an abstract generalisation of (8) to the case when
is fractional, negative, or even complex, despite the fact that there is no sensible way in this case to talk about
independent copies
of
in the standard framework of probability theory.
In this particular case, the quantity (7) is non-negative for every nonnegative
, which looks plausible given the form of the left-hand side. Unfortunately, this sort of non-negativity does not always hold; for instance, if
has mean zero, one can check that
and the right-hand side can become negative for
. This is a shame, because otherwise one could hope to start endowing
with some sort of commutative von Neumann algebra type structure (or the abstract probability structure discussed in this previous post) and then interpret it as a genuine measure space rather than as a virtual one. (This failure of positivity is related to the fact that the characteristic function of a random variable, when raised to the
power, need not be a characteristic function of any random variable once
is no longer a natural number: “fractional convolution” does not preserve positivity!) However, one vestige of positivity remains: if
is non-negative, then so is
One can wonder what the point is to all of this abstract formalism and how it relates to the rest of mathematics. For me, this formalism originated implicitly in an old paper I wrote with Jon Bennett and Tony Carbery on the multilinear restriction and Kakeya conjectures, though we did not have a good language for working with it at the time, instead working first with the case of natural number exponents and appealing to a general extrapolation theorem to then obtain various identities in the fractional
case. The connection between these fractional dimensional integrals and more traditional integrals ultimately arises from the simple identity
(where the right-hand side should be viewed as the fractional dimensional integral of the unit against
). As such, one can manipulate
powers of ordinary integrals using the machinery of fractional dimensional integrals. A key lemma in this regard is
Lemma 3 (Differentiation formula) Suppose that a positive measure
on
depends on some parameter
and varies by the formula
for some function
. Let
be any real or complex number. Then, assuming sufficient smoothness and integrability of all quantities involved, we have
for all
that are independent of
. If we allow
to now depend on
also, then we have the more general total derivative formula
again assuming sufficient amounts of smoothness and regularity.
Proof: We just prove (10), as (11) then follows by same argument used to prove the usual product rule. By linearity it suffices to verify this identity in the case for some symmetric function
for a natural number
. By (6), the left-hand side of (10) is then
Differentiating under the integral sign using (9) we have
and similarly
where are the standard
copies of
on
:
By the product rule, we can thus expand (12) as
where we have suppressed the dependence on for brevity. Since
, we can write this expression using (6) as
where is the symmetric function
But from (2) one has
and the claim follows.
Remark 4 It is also instructive to prove this lemma in the special case when
is a natural number, in which case the fractional dimensional integral
can be interpreted as a classical integral. In this case, the identity (10) is immediate from applying the product rule to (9) to conclude that
One could in fact derive (10) for arbitrary real or complex
from the case when
is a natural number by an extrapolation argument; see the appendix of my paper with Bennett and Carbery for details.
Let us give a simple PDE application of this lemma as illustration:
Proposition 5 (Heat flow monotonicity) Let
be a solution to the heat equation
with initial data
a rapidly decreasing finite non-negative Radon measure, or more explicitly
for al
. Then for any
, the quantity
is monotone non-decreasing in
for
, constant for
, and monotone non-increasing for
.
Proof: By a limiting argument we may assume that is absolutely continuous, with Radon-Nikodym derivative a test function; this is more than enough regularity to justify the arguments below.
For any , let
denote the Radon measure
Then the quantity can be written as a fractional dimensional integral
Observe that
and thus by Lemma 3 and the product rule
where we use for the variable of integration in the factor space
of
.
To simplify this expression we will take advantage of integration by parts in the variable. Specifically, in any direction
, we have
and hence by Lemma 3
Multiplying by and integrating by parts, we see that
where we use the Einstein summation convention in . Similarly, if
is any reasonable function depending only on
, we have
and hence on integration by parts
We conclude that
and thus by (13)
The choice of that then achieves the most cancellation turns out to be
(this cancels the terms that are linear or quadratic in the
), so that
. Repeating the calculations establishing (7), one has
and
where is the random variable drawn from
with the normalised probability measure
. Since
, one thus has
This expression is clearly non-negative for , equal to zero for
, and positive for
, giving the claim. (One could simplify
here as
if desired, though it is not strictly necessary to do so for the proof.)
Remark 6 As with Remark 4, one can also establish the identity (14) first for natural numbers
by direct computation avoiding the theory of fractional dimensional integrals, and then extrapolate to the case of more general values of
. This particular identity is also simple enough that it can be directly established by integration by parts without much difficulty, even for fractional values of
.
A more complicated version of this argument establishes the non-endpoint multilinear Kakeya inequality (without any logarithmic loss in a scale parameter ); this was established in my previous paper with Jon Bennett and Tony Carbery, but using the “natural number
first” approach rather than using the current formalism of fractional dimensional integration. However, the arguments can be translated into this formalism without much difficulty; we do so below the fold. (To simplify the exposition slightly we will not address issues of establishing enough regularity and integrability to justify all the manipulations, though in practice this can be done by standard limiting arguments.)
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