I’ve just uploaded to the arXiv my paper “Quantitative bounds for critically bounded solutions to the Navier-Stokes equations“, submitted to the proceedings of the Linde Hall Inaugural Math Symposium. (I unfortunately had to cancel my physical attendance at this symposium for personal reasons, but was still able to contribute to the proceedings.) In recent years I have been interested in working towards establishing the existence of classical solutions for the Navier-Stokes equations
that blow up in finite time, but this time for a change I took a look at the other side of the theory, namely the conditional regularity results for this equation. There are several such results that assert that if a certain norm of the solution stays bounded (or grows at a controlled rate), then the solution stays regular; taken in the contrapositive, they assert that if a solution blows up at a certain finite time 
- (Leray blowup criterion, 1934) If
blows up at a finite time
, then
for an absolute constant
.
- (Prodi–Serrin–Ladyzhenskaya blowup criterion, 1959-1967) If
, where
.
- (Beale-Kato-Majda blowup criterion, 1984) If
, where
is the vorticity.
- (Kato blowup criterion, 1984) If
for some abs}{olute constant
- (Escauriaza-Seregin-Sverak blowup criterion, 2003) If
.
- (Seregin blowup criterion, 2012) If
.
- (Phuc blowup criterion, 2015) If
for any
.
- (Gallagher-Koch-Planchon blowup criterion, 2016) If
for any
.
- (Albritton blowup criterion, 2016) If
for any
My current paper is most closely related to the Escauriaza-Seregin-Sverak blowup criterion, which was the first to show a critical (i.e., scale-invariant, or dimensionless) spatial norm, namely 
On the other hand, it is a general principle that qualitative arguments established using compactness methods ought to have quantitative analogues that replace the use of compactness by more complicated substitutes that give effective bounds; see for instance these previous blog posts for more discussion. I therefore was interested in trying to obtain a quantitative version of this blowup criterion that gave reasonably good effective bounds (in particular, my objective was to avoid truly enormous bounds such as tower-exponential or Ackermann function bounds, which often arise if one “naively” tries to make a compactness argument effective). In particular, I obtained the following triple-exponential quantitative regularity bounds:
Theorem 1 If
with
and
for
and
.
, then
As a corollary, one can now improve the Escauriaza-Seregin-Sverak blowup criterion to
for some absolute constant
The proof uses many of the same quantitative inputs as previous arguments, most notably the Carleman inequalities used to establish unique continuation and backwards uniqueness theorems for backwards heat equations, but also some additional techniques that make the quantitative bounds more efficient. The proof focuses initially on points of concentration of the solution, which we define as points 
for a large absolute constant 
from which the above theorem ends up following from a routine adaptation of the local well-posedness and regularity theory for Navier-Stokes.
The strategy is to show that any concentration such as (2) when 
- Firstly, by using Duhamel’s formula, one can show that a concentration (2) can only occur (with
at some slightly previous point
in spacetime, with
also close to
,
, and
). This can be viewed as a sort of contrapositive of a “local regularity theorem”, such as the ones established by Caffarelli, Kohn, and Nirenberg. A key point here is that the lower bound
in the conclusion (3) is precisely the same as the lower bound in (2), so that this backwards propagation of concentration can be iterated.
- Iterating the previous step, one can find a sequence of concentration points
with the
propagating backwards in time; by using estimates ultimately resulting from the dissipative term in the energy identity, one can extract such a sequence in which the
increase geometrically with time, the
are comparable (up to polynomial factors in
) to the natural frequency scale
, and one has
. Using the “epochs of regularity” theory that ultimately dates back to Leray, and tweaking the
slightly, one can also place the times
(of length comparable to a small multiple of
) in which the solution is quite regular (in particular,
enjoy good
bounds on
).
- The concentration (4) can be used to establish a lower bound for the
norm of the vorticity
near
In the epoch of regularity
of this equation obey good
bounds, allowing the machinery of Carleman estimates to come into play. Using a Carleman estimate that is used to establish unique continuation results for backwards heat equations, one can propagate this lower bound to also give lower
bounds on the vorticity (and its first derivative) in annuli of the form
for various radii
, although the lower bounds decay at a gaussian rate with
.
- Meanwhile, using an energy pigeonholing argument of Bourgain (which, in this Navier-Stokes context, is actually an enstrophy pigeonholing argument), one can locate some annuli
where (a slightly normalised form of) the entrosphy is small at time
; using a version of the localised enstrophy estimates from a previous paper of mine, one can then propagate this sort of control forward in time, obtaining an “annulus of regularity” of the form
in which one has good estimates; in particular, one has
- By intersecting the previous epoch of regularity
, establishing a lower bound for the vorticity on the spatial annulus
. By some basic Littlewood-Paley theory one can parlay this lower bound to a lower bound on the
norm of the velocity
.
- If
The chain of causality is summarised in the following image:
It seems natural to conjecture that similar triply logarithmic improvements can be made to several of the other blowup criteria listed above, but I have not attempted to pursue this question. It seems difficult to improve the triple logarithmic factor using only the techniques here; the Bourgain pigeonholing argument inevitably costs one exponential, the Carleman inequalities cost a second, and the stacking of scales at the end to contradict the
Hermitian matrix, with eigenvalues 
. Let 
be a unit eigenvector corresponding to the eigenvalue 
, and let 
be the 
component of 
is the 
Hermitian matrix formed by deleting the 
, 
-dimensional vector 
, and 
, then we have 
components of 
; for instance from taking traces one sees that 
).
. At the other extreme, if 
and 
, then the 
, and 
, so that the right-hand side of 
and 
be a radius. For each 
, let 
denote a finite family of infinite tubes 
in 
of radius 
. Assume the following axiom: 
is oriented in the direction of a unit vector 
for 
, we have
, where we use the usual Euclidean norm on the wedge product 
.
, one has ![\displaystyle \left\| \prod_{j=1}^d \sum_{T_j \in \mathbb{T}_j} 1_{T_j} \right\|_{L^p({\bf R}^d)} \lesssim_{A,p} \delta^{\frac{d}{p}} \prod_{j \in [d]} \# \mathbb{T}_j. \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleft%5C%7C+%5Cprod_%7Bj%3D1%7D%5Ed+%5Csum_%7BT_j+%5Cin+%5Cmathbb%7BT%7D_j%7D+1_%7BT_j%7D+%5Cright%5C%7C_%7BL%5Ep%28%7B%5Cbf+R%7D%5Ed%29%7D+%5Clesssim_%7BA%2Cp%7D+%5Cdelta%5E%7B%5Cfrac%7Bd%7D%7Bp%7D%7D+%5Cprod_%7Bj+%5Cin+%5Bd%5D%7D+%5C%23+%5Cmathbb%7BT%7D_j.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0 "\displaystyle \left\| \prod_{j=1}^d \sum_{T_j \in \mathbb{T}_j} 1_{T_j} \right\|_{L^p({\bf R}^d)} \lesssim_{A,p} \delta^{\frac{d}{p}} \prod_{j \in [d]} \# \mathbb{T}_j. \ \ \ \ \ (1)")
are the usual Lebesgue norms with respect to Lebesgue measure, 
denotes the indicator function of 
denotes the cardinality of 
(and sets 
fashion. If one runs the heat flow monotonicity argument, one now picks up some additional error terms arising from the curvature, but as the spatial scale approaches zero, the tubes become increasingly linear, and as such the error terms end up being an integrable multiple of the main term, at which point one can conclude by Gronwall’s inequality (actually for technical reasons we use a bootstrap argument instead of Gronwall). A key point in this approach is that one obtains optimal bounds (not losing factors of 
or 
), so long as one stays away from the endpoint case 
(which does not seem to be easily treatable by the heat flow methods). Previously, 
be an exponent, and let 
. For ![{j \in [d]}](https://s0.wp.com/latex.php?latex=%7Bj+%5Cin+%5Bd%5D%7D&bg=ffffff&fg=000000&s=0 "{j \in [d]}")
, let 
be an open subset of 
, and let 
be a smooth function obeying the following axioms: 
, one has 
. ![\displaystyle \left| \bigwedge_{j \in [d]} (-\nabla_\xi h_j(\xi_j),1) \right| \geq A^{-1}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleft%7C+%5Cbigwedge_%7Bj+%5Cin+%5Bd%5D%7D+%28-%5Cnabla_%5Cxi+h_j%28%5Cxi_j%29%2C1%29+%5Cright%7C+%5Cgeq+A%5E%7B-1%7D&bg=ffffff&fg=000000&s=0 "\displaystyle \left| \bigwedge_{j \in [d]} (-\nabla_\xi h_j(\xi_j),1) \right| \geq A^{-1}")
for 
be the sets 
![\displaystyle \left\| \prod_{j \in [d]} {\mathcal E}_j f_j \right\|_{L^{2p}({\bf R}^d)} \leq A^{O(1)} \left(d-1-\frac{1}{p}\right)^{-O(1)} \prod_{j \in [d]} \|f_j \|_{L^2(U_{j,1/A})}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleft%5C%7C+%5Cprod_%7Bj+%5Cin+%5Bd%5D%7D+%7B%5Cmathcal+E%7D_j+f_j+%5Cright%5C%7C_%7BL%5E%7B2p%7D%28%7B%5Cbf+R%7D%5Ed%29%7D+%5Cleq+A%5E%7BO%281%29%7D+%5Cleft%28d-1-%5Cfrac%7B1%7D%7Bp%7D%5Cright%29%5E%7B-O%281%29%7D+%5Cprod_%7Bj+%5Cin+%5Bd%5D%7D+%5C%7Cf_j+%5C%7C_%7BL%5E2%28U_%7Bj%2C1%2FA%7D%29%7D&bg=ffffff&fg=000000&s=0 "\displaystyle \left\| \prod_{j \in [d]} {\mathcal E}_j f_j \right\|_{L^{2p}({\bf R}^d)} \leq A^{O(1)} \left(d-1-\frac{1}{p}\right)^{-O(1)} \prod_{j \in [d]} \|f_j \|_{L^2(U_{j,1/A})}")
, 
, and 
denotes the extension operator 
is replaced with 
for some 
, and one accepts a loss of the form 
, were already established by ![\displaystyle \left\| \prod_{j \in [d]} E_{r}[{\mathcal E}_j f_j] \right\|_{L^{p}({\bf R}^d)} \leq A^{O(1)} \left(d-1 - \frac{1}{p}\right)^{O(1)} r^{\frac{d}{p}} \prod_{j \in [d]} \| f_j \|_{L^2(U_{j,1/A})}^2](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleft%5C%7C+%5Cprod_%7Bj+%5Cin+%5Bd%5D%7D+E_%7Br%7D%5B%7B%5Cmathcal+E%7D_j+f_j%5D+%5Cright%5C%7C_%7BL%5E%7Bp%7D%28%7B%5Cbf+R%7D%5Ed%29%7D+%5Cleq+A%5E%7BO%281%29%7D+%5Cleft%28d-1+-+%5Cfrac%7B1%7D%7Bp%7D%5Cright%29%5E%7BO%281%29%7D+r%5E%7B%5Cfrac%7Bd%7D%7Bp%7D%7D+%5Cprod_%7Bj+%5Cin+%5Bd%5D%7D+%5C%7C+f_j+%5C%7C_%7BL%5E2%28U_%7Bj%2C1%2FA%7D%29%7D%5E2+&bg=ffffff&fg=000000&s=0 "\displaystyle \left\| \prod_{j \in [d]} E_{r}[{\mathcal E}_j f_j] \right\|_{L^{p}({\bf R}^d)} \leq A^{O(1)} \left(d-1 - \frac{1}{p}\right)^{O(1)} r^{\frac{d}{p}} \prod_{j \in [d]} \| f_j \|_{L^2(U_{j,1/A})}^2")
denotes the local energies  := \int_{B^{d-1}(x',r)} |f(y',x_d)|^2\ dy'](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++E_%7Br%7D%5Bf%5D%28x%27%2Cx_d%29+%3A%3D+%5Cint_%7BB%5E%7Bd-1%7D%28x%27%2Cr%29%7D+%7Cf%28y%27%2Cx_d%29%7C%5E2%5C+dy%27&bg=ffffff&fg=000000&s=0 "\displaystyle E_{r}[f](x',x_d) := \int_{B^{d-1}(x',r)} |f(y',x_d)|^2\ dy'")
). With logarithmic losses, it is not difficult to establish this estimate by an upward induction on 
. To avoid such losses we use the heat flow monotonicity method. Here we run into the issue that the extension operators 
are complex-valued rather than non-negative, and thus would not be expected to obey many good montonicity properties. However, the local energies ![{E_r[{\mathcal E}_j f_j]}](https://s0.wp.com/latex.php?latex=%7BE_r%5B%7B%5Cmathcal+E%7D_j+f_j%5D%7D&bg=ffffff&fg=000000&s=0 "{E_r[{\mathcal E}_j f_j]}")
can be expressed in terms of the magnitude squared of what is essentially the 
to be the graph with vertex set 
, and with every vertex 
joined to the 
, where 
is the standard basis of 
be a set of at least 
vertices in 
other vertices in 
) is completely sharp, as shown 
be the adjacency matrix of 
), thus 
when 
for some 
, and 
otherwise. The above theorem then asserts that if 
minor 
of 
of the adjacency matrix 
matrix 
which is entrywise dominated by 
. 
-dimensional space 
of functions of 
on which 
of functions on 
in 
of 
of 
, there are at least 
paths of length 
of 
be an eigenfunction of 
, then for any 
is equal to 
; on the other hand, by the triangle inequality, this component is at most 
on all large subsets 
. In this post I would like to give an alternate interpretation in terms of the operation of twisted convolution, which originated in the theory of the 
denotes the ordinary convolution operation 
of the adjacency operator 
is moderately complicated: 
in this expansion comes from combining the two terms 
and 
, which both evaluate to 
.
, one can define the twisted convolution 
. This operation is no longer commutative (unless 
is symmetric). However, it remains associative; indeed, one can easily compute that 
is also a twisted convolution operator 
can be computed as 
for 
and 
for 
, and the above twisted convolution simplifies to 
are 
and 
. The matrix 
. This establishes Proposition 
is actually just a component of ordinary convolution, but not on the original group 
to be the set of pairs 
with group law 
. Now if 
is a function on the original group 
by the formula 
and 
now anticommute rather than commute. This makes the twisted convolution algebra 
isomorphic to a 
(the real or complex algebra generated by formal generators 
subject to the relations 
for 
) rather than the commutative algebra more familiar to abelian Fourier analysis. This connection to Clifford algebra (also observed independently by Tom Mrowka and 
that obeyed the relations 
and 
for 
. However, this additional level of generality does not add much; any such 
(so that 
for all 
, which in characteristic two implied that 
for all 
), and such forms can always be decomposed as 
, where 
. As such, the matrices 
are conjugate, with the conjugation operator being the diagonal matrix with entries 
at each vertex 
that is still of dimensions 
denote the vector space of functions 
from the hypercube to the Clifford algebra; as a real vector space, this is a 
dimensional space, isomorphic to the direct sum of 
matrix. In either case one still has the key algebraic relations 
and 
, ensuring that when viewed as a real 
(i.e., a line) in the Clifford algebra 
(note that this definition is well-defined even if the 
are out of order or contain repetitions). This can be viewed as a discrete line bundle over the hypercube. Since 
for any 
, we see that the 
for all 
, is an invariant subspace of 
, with the left action of 
of 
with the lift 
defined by 
. A somewhat tedious computation using the properties of 
be some domain (such as the real numbers). For any natural number 
, let 
denote the space of symmetric real-valued functions 
on 
, thus 
. For instance, for any natural numbers 
, the 
. With the pointwise product operation, 
, in which case 
consists solely of the real constants.
of ![{[F^{(k)}]_{k \rightarrow p} \in L(\Omega^p)_{sym}}](https://s0.wp.com/latex.php?latex=%7B%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D+%5Cin+L%28%5COmega%5Ep%29_%7Bsym%7D%7D&bg=ffffff&fg=000000&s=0 "{[F^{(k)}]_{k \rightarrow p} \in L(\Omega^p)_{sym}}")
of ![\displaystyle [F^{(k)}]_{k \rightarrow p}(x_1,\dots,x_p) = \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq p} F^{(k)}(x_{i_1}, \dots, x_{i_k})](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D%28x_1%2C%5Cdots%2Cx_p%29+%3D+%5Csum_%7B1+%5Cleq+i_1+%3C+i_2+%3C+%5Cdots+%3C+i_k+%5Cleq+p%7D+F%5E%7B%28k%29%7D%28x_%7Bi_1%7D%2C+%5Cdots%2C+x_%7Bi_k%7D%29+&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)}]_{k \rightarrow p}(x_1,\dots,x_p) = \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq p} F^{(k)}(x_{i_1}, \dots, x_{i_k})")
ranges over all injections from 
to 
(the latter formula making it clearer that ![{[F^{(k)}]_{k \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[F^{(k)}]_{k \rightarrow p}}")
is symmetric). Thus for instance ![\displaystyle [F^{(1)}(x_1)]_{1 \rightarrow p} = \sum_{i=1}^p F^{(1)}(x_i)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BF%5E%7B%281%29%7D%28x_1%29%5D_%7B1+%5Crightarrow+p%7D+%3D+%5Csum_%7Bi%3D1%7D%5Ep+F%5E%7B%281%29%7D%28x_i%29&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(1)}(x_1)]_{1 \rightarrow p} = \sum_{i=1}^p F^{(1)}(x_i)")
![\displaystyle [F^{(2)}(x_1,x_2)]_{2 \rightarrow p} = \sum_{1 \leq i < j \leq p} F^{(2)}(x_i,x_j)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BF%5E%7B%282%29%7D%28x_1%2Cx_2%29%5D_%7B2+%5Crightarrow+p%7D+%3D+%5Csum_%7B1+%5Cleq+i+%3C+j+%5Cleq+p%7D+F%5E%7B%282%29%7D%28x_i%2Cx_j%29&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(2)}(x_1,x_2)]_{2 \rightarrow p} = \sum_{1 \leq i < j \leq p} F^{(2)}(x_i,x_j)")
![\displaystyle e_k^{(p)}(x_1,\dots,x_p) = [x_1 \dots x_k]_{k \rightarrow p}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++e_k%5E%7B%28p%29%7D%28x_1%2C%5Cdots%2Cx_p%29+%3D+%5Bx_1+%5Cdots+x_k%5D_%7Bk+%5Crightarrow+p%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle e_k^{(p)}(x_1,\dots,x_p) = [x_1 \dots x_k]_{k \rightarrow p}.")
![\displaystyle [1]_{k \rightarrow p} = \binom{p}{k} = \frac{p(p-1)\dots(p-k+1)}{k!}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5B1%5D_%7Bk+%5Crightarrow+p%7D+%3D+%5Cbinom%7Bp%7D%7Bk%7D+%3D+%5Cfrac%7Bp%28p-1%29%5Cdots%28p-k%2B1%29%7D%7Bk%21%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle [1]_{k \rightarrow p} = \binom{p}{k} = \frac{p(p-1)\dots(p-k+1)}{k!}.")
, and is equal to 
if 
. We also have the transitivity ![\displaystyle [F^{(k)}]_{k \rightarrow p} = \frac{1}{\binom{p-k}{p-l}} [[F^{(k)}]_{k \rightarrow l}]_{l \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D+%3D+%5Cfrac%7B1%7D%7B%5Cbinom%7Bp-k%7D%7Bp-l%7D%7D+%5B%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+l%7D%5D_%7Bl+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)}]_{k \rightarrow p} = \frac{1}{\binom{p-k}{p-l}} [[F^{(k)}]_{k \rightarrow l}]_{l \rightarrow p}")
.![{[]_{k \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7Bk+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{k \rightarrow p}}")
is a linear map from 
to 
, one has ![\displaystyle [x_1]_{1 \rightarrow p} [x_1]_{1 \rightarrow p} = (\sum_{i=1}^p x_i)^2 \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Bx_1%5D_%7B1+%5Crightarrow+p%7D+%5Bx_1%5D_%7B1+%5Crightarrow+p%7D+%3D+%28%5Csum_%7Bi%3D1%7D%5Ep+x_i%29%5E2+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0 "\displaystyle [x_1]_{1 \rightarrow p} [x_1]_{1 \rightarrow p} = (\sum_{i=1}^p x_i)^2 \ \ \ \ \ (1)")
![\displaystyle = [x_1^2]_{1 \rightarrow p} + 2 [x_1 x_2]_{1 \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%5Bx_1%5E2%5D_%7B1+%5Crightarrow+p%7D+%2B+2+%5Bx_1+x_2%5D_%7B1+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle = [x_1^2]_{1 \rightarrow p} + 2 [x_1 x_2]_{1 \rightarrow p}")
![\displaystyle \neq [x_1^2]_{1 \rightarrow p}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cneq+%5Bx_1%5E2%5D_%7B1+%5Crightarrow+p%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle \neq [x_1^2]_{1 \rightarrow p}.")
![\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow p} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow p} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BF%5E%7B%28k%29%7D%28x_1%2C%5Cdots%2Cx_k%29%5D_%7Bk+%5Crightarrow+p%7D+%5BG%5E%7B%28l%29%7D%28x_1%2C%5Cdots%2Cx_l%29%5D_%7Bl+%5Crightarrow+p%7D+%3D+%5Csum_%7Bk%2Cl+%5Cleq+m+%5Cleq+k%2Bl%7D+%5Cfrac%7B1%7D%7Bk%21+l%21%7D+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow p} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow p} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (2)")
![\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%5Csum_%7B%5Cpi%2C+%5Crho%7D+F%5E%7B%28k%29%7D%28x_%7B%5Cpi%281%29%7D%2C%5Cdots%2Cx_%7B%5Cpi%28k%29%7D%29+G%5E%7B%28l%29%7D%28x_%7B%5Crho%281%29%7D%2C%5Cdots%2Cx_%7B%5Crho%28l%29%7D%29%5D_%7Bm+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow p}")
and 
, where 
range over all injections 
, 
with 
. Combinatorially, the identity 
and 
with total image 
of cardinality 
, one has 
, and furthermore there exist precisely 
triples 
of injections 
such that 
and 
.
, one has ![\displaystyle [x_1 x_2]_{2 \rightarrow p} [x_1]_{1 \rightarrow p} = [\frac{1}{2! 1!}( 2 x_1^2 x_2 + 2 x_1 x_2^2 )]_{2 \rightarrow p} + [\frac{1}{2! 1!} 6 x_1 x_2 x_3]_{3 \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Bx_1+x_2%5D_%7B2+%5Crightarrow+p%7D+%5Bx_1%5D_%7B1+%5Crightarrow+p%7D+%3D+%5B%5Cfrac%7B1%7D%7B2%21+1%21%7D%28+2+x_1%5E2+x_2+%2B+2+x_1+x_2%5E2+%29%5D_%7B2+%5Crightarrow+p%7D+%2B+%5B%5Cfrac%7B1%7D%7B2%21+1%21%7D+6+x_1+x_2+x_3%5D_%7B3+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [x_1 x_2]_{2 \rightarrow p} [x_1]_{1 \rightarrow p} = [\frac{1}{2! 1!}( 2 x_1^2 x_2 + 2 x_1 x_2^2 )]_{2 \rightarrow p} + [\frac{1}{2! 1!} 6 x_1 x_2 x_3]_{3 \rightarrow p}")
![\displaystyle = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow p} + [3x_1 x_2 x_3]_{3 \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%5Bx_1%5E2+x_2+%2B+x_1+x_2%5E2%5D_%7B2+%5Crightarrow+p%7D+%2B+%5B3x_1+x_2+x_3%5D_%7B3+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow p} + [3x_1 x_2 x_3]_{3 \rightarrow p}")
of formal sums ![\displaystyle F^{(*)} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++F%5E%7B%28%2A%29%7D+%3D+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D&bg=ffffff&fg=000000&s=0 "\displaystyle F^{(*)} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}")
is an element of ![{[]_{k \rightarrow *}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7Bk+%5Crightarrow+%2A%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{k \rightarrow *}}")
being formally linear, thus ![\displaystyle [F^{(k)}]_{k \rightarrow *} + [G^{(k)}]_{k \rightarrow *} := [F^{(k)} + G^{(k)}]_{k \rightarrow *}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D+%2B+%5BG%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D+%3A%3D+%5BF%5E%7B%28k%29%7D+%2B+G%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)}]_{k \rightarrow *} + [G^{(k)}]_{k \rightarrow *} := [F^{(k)} + G^{(k)}]_{k \rightarrow *}")
![\displaystyle c [F^{(k)}]_{k \rightarrow *} := [cF^{(k)}]_{k \rightarrow *}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++c+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D+%3A%3D+%5BcF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D&bg=ffffff&fg=000000&s=0 "\displaystyle c [F^{(k)}]_{k \rightarrow *} := [cF^{(k)}]_{k \rightarrow *}")
and scalars 
, and with multiplication given by the analogue ![\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow *} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow *} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (3)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BF%5E%7B%28k%29%7D%28x_1%2C%5Cdots%2Cx_k%29%5D_%7Bk+%5Crightarrow+%2A%7D+%5BG%5E%7B%28l%29%7D%28x_1%2C%5Cdots%2Cx_l%29%5D_%7Bl+%5Crightarrow+%2A%7D+%3D+%5Csum_%7Bk%2Cl+%5Cleq+m+%5Cleq+k%2Bl%7D+%5Cfrac%7B1%7D%7Bk%21+l%21%7D+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)}(x_1,\dots,x_k)]_{k \rightarrow *} [G^{(l)}(x_1,\dots,x_l)]_{l \rightarrow *} = \sum_{k,l \leq m \leq k+l} \frac{1}{k! l!} \ \ \ \ \ (3)")
![\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow *}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%5Csum_%7B%5Cpi%2C+%5Crho%7D+F%5E%7B%28k%29%7D%28x_%7B%5Cpi%281%29%7D%2C%5Cdots%2Cx_%7B%5Cpi%28k%29%7D%29+G%5E%7B%28l%29%7D%28x_%7B%5Crho%281%29%7D%2C%5Cdots%2Cx_%7B%5Crho%28l%29%7D%29%5D_%7Bm+%5Crightarrow+%2A%7D+&bg=ffffff&fg=000000&s=0 "\displaystyle [\sum_{\pi, \rho} F^{(k)}(x_{\pi(1)},\dots,x_{\pi(k)}) G^{(l)}(x_{\rho(1)},\dots,x_{\rho(l)})]_{m \rightarrow *}")
![\displaystyle [x_1]_{1 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2]_{1 \rightarrow *} + 2 [x_1 x_2]_{2 \rightarrow *}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Bx_1%5D_%7B1+%5Crightarrow+%2A%7D+%5Bx_1%5D_%7B1+%5Crightarrow+%2A%7D+%3D+%5Bx_1%5E2%5D_%7B1+%5Crightarrow+%2A%7D+%2B+2+%5Bx_1+x_2%5D_%7B2+%5Crightarrow+%2A%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [x_1]_{1 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2]_{1 \rightarrow *} + 2 [x_1 x_2]_{2 \rightarrow *}")
![\displaystyle [x_1 x_2]_{2 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow *} + [3 x_1 x_2 x_3]_{3 \rightarrow *}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Bx_1+x_2%5D_%7B2+%5Crightarrow+%2A%7D+%5Bx_1%5D_%7B1+%5Crightarrow+%2A%7D+%3D+%5Bx_1%5E2+x_2+%2B+x_1+x_2%5E2%5D_%7B2+%5Crightarrow+%2A%7D+%2B+%5B3+x_1+x_2+x_3%5D_%7B3+%5Crightarrow+%2A%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle [x_1 x_2]_{2 \rightarrow *} [x_1]_{1 \rightarrow *} = [x_1^2 x_2 + x_1 x_2^2]_{2 \rightarrow *} + [3 x_1 x_2 x_3]_{3 \rightarrow *}.")
![{[1]_{0 \rightarrow *}}](https://s0.wp.com/latex.php?latex=%7B%5B1%5D_%7B0+%5Crightarrow+%2A%7D%7D&bg=ffffff&fg=000000&s=0 "{[1]_{0 \rightarrow *}}")
. (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. )![{[]_{* \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7B%2A+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{* \rightarrow p}}")
from ![\displaystyle [\sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}]_{* \rightarrow p} := \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5B%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+%2A%7D%5D_%7B%2A+%5Crightarrow+p%7D+%3A%3D+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle [\sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow *}]_{* \rightarrow p} := \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}.")
![{[x_1]_{1 \rightarrow *}}](https://s0.wp.com/latex.php?latex=%7B%5Bx_1%5D_%7B1+%5Crightarrow+%2A%7D%7D&bg=ffffff&fg=000000&s=0 "{[x_1]_{1 \rightarrow *}}")
to ![{[x_1]_{1 \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5Bx_1%5D_%7B1+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[x_1]_{1 \rightarrow p}}")
and ![{[x_1 x_2]_{2 \rightarrow *}}](https://s0.wp.com/latex.php?latex=%7B%5Bx_1+x_2%5D_%7B2+%5Crightarrow+%2A%7D%7D&bg=ffffff&fg=000000&s=0 "{[x_1 x_2]_{2 \rightarrow *}}")
to ![{[x_1 x_2]_{2 \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5Bx_1+x_2%5D_%7B2+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[x_1 x_2]_{2 \rightarrow p}}")
. From ![{[]_{* \rightarrow p}: L(\Omega^*)_{sym} \rightarrow L(\Omega^p)_{sym}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7B%2A+%5Crightarrow+p%7D%3A+L%28%5COmega%5E%2A%29_%7Bsym%7D+%5Crightarrow+L%28%5COmega%5Ep%29_%7Bsym%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{* \rightarrow p}: L(\Omega^*)_{sym} \rightarrow L(\Omega^p)_{sym}}")
is an algebra homomorphism, even though the maps ![{[]_{k \rightarrow *}: L(\Omega^k)_{sym} \rightarrow L(\Omega^*)_{sym}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7Bk+%5Crightarrow+%2A%7D%3A+L%28%5COmega%5Ek%29_%7Bsym%7D+%5Crightarrow+L%28%5COmega%5E%2A%29_%7Bsym%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{k \rightarrow *}: L(\Omega^k)_{sym} \rightarrow L(\Omega^*)_{sym}}")
and ![{[]_{k \rightarrow p}: L(\Omega^k)_{sym} \rightarrow L(\Omega^p)_{sym}}](https://s0.wp.com/latex.php?latex=%7B%5B%5D_%7Bk+%5Crightarrow+p%7D%3A+L%28%5COmega%5Ek%29_%7Bsym%7D+%5Crightarrow+L%28%5COmega%5Ep%29_%7Bsym%7D%7D&bg=ffffff&fg=000000&s=0 "{[]_{k \rightarrow p}: L(\Omega^k)_{sym} \rightarrow L(\Omega^p)_{sym}}")
are not homomorphisms. By inspecting the 
component of 
on the space 
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 
![\displaystyle \int_{{\bf R}^p} [x_1]_{1 \rightarrow p}\ d\mu^p = p (\int_{\bf R} x\ d\mu(x)) (\int_{\bf R} \mu)^{p-1} \ \ \ \ \ (4)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%7B%5Cbf+R%7D%5Ep%7D+%5Bx_1%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+%3D+p+%28%5Cint_%7B%5Cbf+R%7D+x%5C+d%5Cmu%28x%29%29+%28%5Cint_%7B%5Cbf+R%7D+%5Cmu%29%5E%7Bp-1%7D+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{{\bf R}^p} [x_1]_{1 \rightarrow p}\ d\mu^p = p (\int_{\bf R} x\ d\mu(x)) (\int_{\bf R} \mu)^{p-1} \ \ \ \ \ (4)")
![\displaystyle \int_{{\bf R}^p} [x_2]_{1 \rightarrow p}\ d\mu^p = \binom{p}{2} (\int_{{\bf R}^2} x_1 x_2\ d\mu(x_1) d\mu(x_2)) (\int_{\bf R} \mu)^{p-2}. \ \ \ \ \ (5)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%7B%5Cbf+R%7D%5Ep%7D+%5Bx_2%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+%3D+%5Cbinom%7Bp%7D%7B2%7D+%28%5Cint_%7B%7B%5Cbf+R%7D%5E2%7D+x_1+x_2%5C+d%5Cmu%28x_1%29+d%5Cmu%28x_2%29%29+%28%5Cint_%7B%5Cbf+R%7D+%5Cmu%29%5E%7Bp-2%7D.+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{{\bf R}^p} [x_2]_{1 \rightarrow p}\ d\mu^p = \binom{p}{2} (\int_{{\bf R}^2} x_1 x_2\ d\mu(x_1) d\mu(x_2)) (\int_{\bf R} \mu)^{p-2}. \ \ \ \ \ (5)")
![\displaystyle \int_{\Omega^p} [F^{(*)}]_{* \rightarrow p}\ d\mu^p = \sum_{k=0}^\infty \binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu^k) (\int_\Omega\ d\mu)^{p-k}. \ \ \ \ \ (6)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%5COmega%5Ep%7D+%5BF%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+%3D+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5Cbinom%7Bp%7D%7Bk%7D+%28%5Cint_%7B%5COmega%5Ek%7D+F%5E%7B%28k%29%7D%5C+d%5Cmu%5Ek%29+%28%5Cint_%5COmega%5C+d%5Cmu%29%5E%7Bp-k%7D.+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [F^{(*)}]_{* \rightarrow p}\ d\mu^p = \sum_{k=0}^\infty \binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu^k) (\int_\Omega\ d\mu)^{p-k}. \ \ \ \ \ (6)")
as a polynomial 
in ![\displaystyle F^{(p)} = [F^{(*)}]_{* \rightarrow p} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++F%5E%7B%28p%29%7D+%3D+%5BF%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%3D+%5Csum_%7Bk%3D0%7D%5E%5Cinfty+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle F^{(p)} = [F^{(*)}]_{* \rightarrow p} = \sum_{k=0}^\infty [F^{(k)}]_{k \rightarrow p}")
and ![\displaystyle [F^{(*)}]_{* \rightarrow p} + [G^{(*)}]_{* \rightarrow p} := [F^{(*)} + G^{(*)}]_{* \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BF%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%2B+%5BG%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%3A%3D+%5BF%5E%7B%28%2A%29%7D+%2B+G%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(*)}]_{* \rightarrow p} + [G^{(*)}]_{* \rightarrow p} := [F^{(*)} + G^{(*)}]_{* \rightarrow p}")
![\displaystyle c [F^{(*)}]_{* \rightarrow p} := [c F^{(*)}]_{* \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++c+%5BF%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%3A%3D+%5Bc+F%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle c [F^{(*)}]_{* \rightarrow p} := [c F^{(*)}]_{* \rightarrow p}")
![\displaystyle [F^{(*)}]_{* \rightarrow p} [G^{(*)}]_{* \rightarrow p} := [F^{(*)} G^{(*)}]_{* \rightarrow p}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BF%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%5BG%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D+%3A%3D+%5BF%5E%7B%28%2A%29%7D+G%5E%7B%28%2A%29%7D%5D_%7B%2A+%5Crightarrow+p%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(*)}]_{* \rightarrow p} [G^{(*)}]_{* \rightarrow p} := [F^{(*)} G^{(*)}]_{* \rightarrow p}.")
of 
no longer makes sense as a set, and the formal measure 
is a random variable; on any power 
, we let 
be the usual independent copies of 
for 
. Then for any real or complex ![\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%5COmega%5Ep%7D+%5BX_1%5D_%7B1+%5Crightarrow+p%7D%5E2%5C+d%5Cmu%5Ep+&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p")
![\displaystyle [X_1]_{1 \rightarrow p}^2 = [X_1^2]_{1 \rightarrow p} + 2[X_1 X_2]_{2 \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BX_1%5D_%7B1+%5Crightarrow+p%7D%5E2+%3D+%5BX_1%5E2%5D_%7B1+%5Crightarrow+p%7D+%2B+2%5BX_1+X_2%5D_%7B2+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [X_1]_{1 \rightarrow p}^2 = [X_1^2]_{1 \rightarrow p} + 2[X_1 X_2]_{2 \rightarrow p}")
to conclude that ![\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = \binom{p}{1} \int_{\Omega} X^2\ d\mu + 2 \binom{p}{2} \int_{\Omega^2} X_1 X_2\ d\mu^2](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%5COmega%5Ep%7D+%5BX_1%5D_%7B1+%5Crightarrow+p%7D%5E2%5C+d%5Cmu%5Ep+%3D+%5Cbinom%7Bp%7D%7B1%7D+%5Cint_%7B%5COmega%7D+X%5E2%5C+d%5Cmu+%2B+2+%5Cbinom%7Bp%7D%7B2%7D+%5Cint_%7B%5COmega%5E2%7D+X_1+X_2%5C+d%5Cmu%5E2&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = \binom{p}{1} \int_{\Omega} X^2\ d\mu + 2 \binom{p}{2} \int_{\Omega^2} X_1 X_2\ d\mu^2")
![\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = p \mathbf{Var}(X) + p^2 \mathbf{E}(X)^2. \ \ \ \ \ (7)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%5COmega%5Ep%7D+%5BX_1%5D_%7B1+%5Crightarrow+p%7D%5E2%5C+d%5Cmu%5Ep+%3D+p+%5Cmathbf%7BVar%7D%28X%29+%2B+p%5E2+%5Cmathbf%7BE%7D%28X%29%5E2.+%5C+%5C+%5C+%5C+%5C+%287%29&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^2\ d\mu^p = p \mathbf{Var}(X) + p^2 \mathbf{E}(X)^2. \ \ \ \ \ (7)")
are jointly independent copies of 
has expectation 
and variance 
. One can thus view ![\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^4\ d\mu^p = p \mathbf{Var}(X^2) + p(3p-2) (\mathbf{E}(X^2))^2](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%5COmega%5Ep%7D+%5BX_1%5D_%7B1+%5Crightarrow+p%7D%5E4%5C+d%5Cmu%5Ep+%3D+p+%5Cmathbf%7BVar%7D%28X%5E2%29+%2B+p%283p-2%29+%28%5Cmathbf%7BE%7D%28X%5E2%29%29%5E2&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [X_1]_{1 \rightarrow p}^4\ d\mu^p = p \mathbf{Var}(X^2) + p(3p-2) (\mathbf{E}(X^2))^2")
. 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 
is non-negative, then so is ![\displaystyle \int_{\Omega^p} [F]_{1 \rightarrow p}\ d\mu^p = p (\int_\Omega F\ d\mu) (\int_\Omega\ d\mu)^{p-1}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%5COmega%5Ep%7D+%5BF%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+%3D+p+%28%5Cint_%5COmega+F%5C+d%5Cmu%29+%28%5Cint_%5COmega%5C+d%5Cmu%29%5E%7Bp-1%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [F]_{1 \rightarrow p}\ d\mu^p = p (\int_\Omega F\ d\mu) (\int_\Omega\ d\mu)^{p-1}.")
![{[1]_{0 \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7B%5B1%5D_%7B0+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{[1]_{0 \rightarrow p}}")
against 
on 
and varies by the formula 
. Let ![\displaystyle \frac{d}{dt} \int_{\Omega^p} F^{(p)}\ d\mu(t)^p = \int_{\Omega^p} F^{(p)} [a(t)]_{1 \rightarrow p}\ d\mu(t)^p \ \ \ \ \ (10)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7Bd%7D%7Bdt%7D+%5Cint_%7B%5COmega%5Ep%7D+F%5E%7B%28p%29%7D%5C+d%5Cmu%28t%29%5Ep+%3D+%5Cint_%7B%5COmega%5Ep%7D+F%5E%7B%28p%29%7D+%5Ba%28t%29%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%29%5Ep+%5C+%5C+%5C+%5C+%5C+%2810%29&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{d}{dt} \int_{\Omega^p} F^{(p)}\ d\mu(t)^p = \int_{\Omega^p} F^{(p)} [a(t)]_{1 \rightarrow p}\ d\mu(t)^p \ \ \ \ \ (10)")
that are independent of 
to now depend on 
![\displaystyle = \int_{\Omega^p} \frac{d}{dt} F^{(p)}(t) + F^{(p)}(t) [a(t)]_{1 \rightarrow p}\ d\mu(t)^p,](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%5Cint_%7B%5COmega%5Ep%7D+%5Cfrac%7Bd%7D%7Bdt%7D+F%5E%7B%28p%29%7D%28t%29+%2B+F%5E%7B%28p%29%7D%28t%29+%5Ba%28t%29%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%29%5Ep%2C+&bg=ffffff&fg=000000&s=0 "\displaystyle = \int_{\Omega^p} \frac{d}{dt} F^{(p)}(t) + F^{(p)}(t) [a(t)]_{1 \rightarrow p}\ d\mu(t)^p,")
![{F^{(p)} = [F^{(k)}]_{k \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7BF%5E%7B%28p%29%7D+%3D+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{F^{(p)} = [F^{(k)}]_{k \rightarrow p}}")
for some symmetric function ![\displaystyle \frac{d}{dt} [\binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu(t)^k) (\int_\Omega\ d\mu(t))^{p-k}]. \ \ \ \ \ (12)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7Bd%7D%7Bdt%7D+%5B%5Cbinom%7Bp%7D%7Bk%7D+%28%5Cint_%7B%5COmega%5Ek%7D+F%5E%7B%28k%29%7D%5C+d%5Cmu%28t%29%5Ek%29+%28%5Cint_%5COmega%5C+d%5Cmu%28t%29%29%5E%7Bp-k%7D%5D.+%5C+%5C+%5C+%5C+%5C+%2812%29&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{d}{dt} [\binom{p}{k} (\int_{\Omega^k} F^{(k)}\ d\mu(t)^k) (\int_\Omega\ d\mu(t))^{p-k}]. \ \ \ \ \ (12)")
are the standard 
on 
, we can write this expression using ![\displaystyle \int_{\Omega^p} [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p}\ d\mu^p](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%5COmega%5Ep%7D+%5BF%5E%7B%28k%29%7D+%28a_1+%2B+%5Cdots+%2B+a_k%29%5D_%7Bk+%5Crightarrow+p%7D+%2B+%5B+F%5E%7B%28k%29%7D+%5Cast+a+%5D_%7Bk%2B1+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{\Omega^p} [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p}\ d\mu^p")
is the symmetric function 
![\displaystyle [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p} = [F^{(k)}]_{k \rightarrow p} [a]_{1 \rightarrow p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BF%5E%7B%28k%29%7D+%28a_1+%2B+%5Cdots+%2B+a_k%29%5D_%7Bk+%5Crightarrow+p%7D+%2B+%5B+F%5E%7B%28k%29%7D+%5Cast+a+%5D_%7Bk%2B1+%5Crightarrow+p%7D+%3D+%5BF%5E%7B%28k%29%7D%5D_%7Bk+%5Crightarrow+p%7D+%5Ba%5D_%7B1+%5Crightarrow+p%7D&bg=ffffff&fg=000000&s=0 "\displaystyle [F^{(k)} (a_1 + \dots + a_k)]_{k \rightarrow p} + [ F^{(k)} \ast a ]_{k+1 \rightarrow p} = [F^{(k)}]_{k \rightarrow p} [a]_{1 \rightarrow p}")
can be interpreted as a classical integral. In this case, the identity ![\displaystyle \frac{d}{dt} d\mu(t)^p = [a(t)]_{1 \rightarrow p} d\mu(t)^p.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7Bd%7D%7Bdt%7D+d%5Cmu%28t%29%5Ep+%3D+%5Ba%28t%29%5D_%7B1+%5Crightarrow+p%7D+d%5Cmu%28t%29%5Ep.&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{d}{dt} d\mu(t)^p = [a(t)]_{1 \rightarrow p} d\mu(t)^p.")
be a solution to the heat equation 
with initial data 
a rapidly decreasing finite non-negative Radon measure, or more explicitly 
. Then for any 
, the quantity 
for 
, constant for 
, and monotone non-increasing for 
. 
is absolutely continuous, with Radon-Nikodym derivative a test function; this is more than enough regularity to justify the arguments below.
, let 
denote the Radon measure 
can be written as a fractional dimensional integral 
![\displaystyle \frac{d}{dt} Q_p(t) = -\frac{d}{2t} Q_p(t) + t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [\frac{|x-y|^2}{4t^2}]_{1 \rightarrow p} d\mu(t,x)^p\ dx \ \ \ \ \ (13)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7Bd%7D%7Bdt%7D+Q_p%28t%29+%3D+-%5Cfrac%7Bd%7D%7B2t%7D+Q_p%28t%29+%2B+t%5E%7B-d%2F2%7D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5B%5Cfrac%7B%7Cx-y%7C%5E2%7D%7B4t%5E2%7D%5D_%7B1+%5Crightarrow+p%7D+d%5Cmu%28t%2Cx%29%5Ep%5C+dx+%5C+%5C+%5C+%5C+%5C+%2813%29&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{d}{dt} Q_p(t) = -\frac{d}{2t} Q_p(t) + t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [\frac{|x-y|^2}{4t^2}]_{1 \rightarrow p} d\mu(t,x)^p\ dx \ \ \ \ \ (13)")
for the variable of integration in the factor space 
.
, we have 
![\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}\ d\mu(t,x)^p\ dx = - \int_{({\bf R}^d)^p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x_j%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx+%3D+-+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5B%5Cfrac%7Bx_j-y_j%7D%7B2t%7D%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx.&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}\ d\mu(t,x)^p\ dx = - \int_{({\bf R}^d)^p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.")
![\displaystyle d Q_p(t) = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d+Q_p%28t%29+%3D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+x_j+%5B%5Cfrac%7Bx_j-y_j%7D%7B2t%7D%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx&bg=ffffff&fg=000000&s=0 "\displaystyle d Q_p(t) = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx")
![\displaystyle = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+x_j+%5B%5Cfrac%7Bx_j-y_j%7D%7B2t%7D%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx&bg=ffffff&fg=000000&s=0 "\displaystyle = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} x_j [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx")
. Similarly, if 
is any reasonable function depending only on ![\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}[F_j(y)]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x_j%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D%5BF_j%28y%29%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx+&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{\partial}{\partial x_j} \int_{({\bf R}^d)^p}[F_j(y)]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx")
![\displaystyle = - \int_{({\bf R}^d)^p} [F_j(y)]_{1 \rightarrow p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%3D+-+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5BF_j%28y%29%5D_%7B1+%5Crightarrow+p%7D+%5B%5Cfrac%7Bx_j-y_j%7D%7B2t%7D%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx&bg=ffffff&fg=000000&s=0 "\displaystyle = - \int_{({\bf R}^d)^p} [F_j(y)]_{1 \rightarrow p} [\frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx")
![\displaystyle 0 = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [F_j(y) \frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++0+%3D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5BF_j%28y%29+%5Cfrac%7Bx_j-y_j%7D%7B2t%7D%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx.&bg=ffffff&fg=000000&s=0 "\displaystyle 0 = \int_{{\bf R}^d} \int_{({\bf R}^d)^p} [F_j(y) \frac{x_j-y_j}{2t}]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.")
![\displaystyle \frac{d}{2t} Q_p(t) = t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} (x_j - [F_j(y)]_{1 \rightarrow p}) [\frac{(x_j-y_j)}{4t}]_{1 \rightarrow p} d\mu(t,x)^p\ dx](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cfrac%7Bd%7D%7B2t%7D+Q_p%28t%29+%3D+t%5E%7B-d%2F2%7D+%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D+%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%28x_j+-+%5BF_j%28y%29%5D_%7B1+%5Crightarrow+p%7D%29+%5B%5Cfrac%7B%28x_j-y_j%29%7D%7B4t%7D%5D_%7B1+%5Crightarrow+p%7D+d%5Cmu%28t%2Cx%29%5Ep%5C+dx+&bg=ffffff&fg=000000&s=0 "\displaystyle \frac{d}{2t} Q_p(t) = t^{-d/2} \int_{{\bf R}^d} \int_{({\bf R}^d)^p} (x_j - [F_j(y)]_{1 \rightarrow p}) [\frac{(x_j-y_j)}{4t}]_{1 \rightarrow p} d\mu(t,x)^p\ dx")
![\displaystyle [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p} - (x_j - [F_j(y)]_{1 \rightarrow p}) [x_j - y_j]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%28x_j-y_j%29%28x_j-y_j%29%5D_%7B1+%5Crightarrow+p%7D+-+%28x_j+-+%5BF_j%28y%29%5D_%7B1+%5Crightarrow+p%7D%29+%5Bx_j+-+y_j%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%28t%2Cx%29%5Ep%5C+dx.&bg=ffffff&fg=000000&s=0 "\displaystyle [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p} - (x_j - [F_j(y)]_{1 \rightarrow p}) [x_j - y_j]_{1 \rightarrow p}\ d\mu(t,x)^p\ dx.")
that then achieves the most cancellation turns out to be 
(this cancels the terms that are linear or quadratic in the ![{x_j - [F_j(y)]_{1 \rightarrow p} = \frac{1}{p} [x_j - y_j]_{1 \rightarrow p}}](https://s0.wp.com/latex.php?latex=%7Bx_j+-+%5BF_j%28y%29%5D_%7B1+%5Crightarrow+p%7D+%3D+%5Cfrac%7B1%7D%7Bp%7D+%5Bx_j+-+y_j%5D_%7B1+%5Crightarrow+p%7D%7D&bg=ffffff&fg=000000&s=0 "{x_j - [F_j(y)]_{1 \rightarrow p} = \frac{1}{p} [x_j - y_j]_{1 \rightarrow p}}")
. Repeating the calculations establishing ![\displaystyle \int_{({\bf R}^d)^p} [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p}\ d\mu^p = p \mathop{\bf E} |x-Y|^2 (\int_{{\bf R}^d}\ d\mu)^{p}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5B%28x_j-y_j%29%28x_j-y_j%29%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+%3D+p+%5Cmathop%7B%5Cbf+E%7D+%7Cx-Y%7C%5E2+%28%5Cint_%7B%7B%5Cbf+R%7D%5Ed%7D%5C+d%5Cmu%29%5E%7Bp%7D&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{({\bf R}^d)^p} [(x_j-y_j)(x_j-y_j)]_{1 \rightarrow p}\ d\mu^p = p \mathop{\bf E} |x-Y|^2 (\int_{{\bf R}^d}\ d\mu)^{p}")
![\displaystyle \int_{({\bf R}^d)^p} [x_j-y_j]_{1 \rightarrow p} [x_j-y_j]_{1 \rightarrow p}\ d\mu^p](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_%7B%28%7B%5Cbf+R%7D%5Ed%29%5Ep%7D+%5Bx_j-y_j%5D_%7B1+%5Crightarrow+p%7D+%5Bx_j-y_j%5D_%7B1+%5Crightarrow+p%7D%5C+d%5Cmu%5Ep+&bg=ffffff&fg=000000&s=0 "\displaystyle \int_{({\bf R}^d)^p} [x_j-y_j]_{1 \rightarrow p} [x_j-y_j]_{1 \rightarrow p}\ d\mu^p")
is the random variable drawn from 
. Since 
, one thus has 
, equal to zero for 
here as 
if desired, though it is not strictly necessary to do so for the proof.) 
(sometimes written as 
in the literature for some small scale parameter 
for some large radius 
(e.g. all sufficiently large real numbers 
depending on 
, and one wishes to obtain a subpolynomial upper bound for 
and all 
can depend on 
but is independent of 
be a fixed dimension. For each 
, we pick a maximal 
-separated set of directions 
. We let 
is a 
-tube oriented in the direction 
. 
on the unit sphere 
equipped with surface measure 
, where 
is the ball of radius 
be compact subsets of the sphere 
for the wedge product of directions 
for 
(equivalently, there is no hyperplane through the origin that intersects all of the 
). For each 
is a collection of infinite tubes in 
in 
is our notation for the geometric mean 
on 
. ![{[0,1]^{d-1}}](https://s0.wp.com/latex.php?latex=%7B%5B0%2C1%5D%5E%7Bd-1%7D%7D&bg=ffffff&fg=000000&s=0 "{[0,1]^{d-1}}")
into 
cubes 
of sidelength 
. For any ![{g \in L^1([0,1]^{d-1})}](https://s0.wp.com/latex.php?latex=%7Bg+%5Cin+L%5E1%28%5B0%2C1%5D%5E%7Bd-1%7D%29%7D&bg=ffffff&fg=000000&s=0 "{g \in L^1([0,1]^{d-1})}")
, define the extension operators ![\displaystyle E_{[0,1]^{d-1}} g( x', x_d ) := \int_{[0,1]^{d-1}} e^{2\pi i (x' \cdot \xi + x_d |\xi|^2)} g(\xi)\ d\xi](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++E_%7B%5B0%2C1%5D%5E%7Bd-1%7D%7D+g%28+x%27%2C+x_d+%29+%3A%3D+%5Cint_%7B%5B0%2C1%5D%5E%7Bd-1%7D%7D+e%5E%7B2%5Cpi+i+%28x%27+%5Ccdot+%5Cxi+%2B+x_d+%7C%5Cxi%7C%5E2%29%7D+g%28%5Cxi%29%5C+d%5Cxi&bg=ffffff&fg=000000&s=0 "\displaystyle E_{[0,1]^{d-1}} g( x', x_d ) := \int_{[0,1]^{d-1}} e^{2\pi i (x' \cdot \xi + x_d |\xi|^2)} g(\xi)\ d\xi")
and 
. We also introduce the weight function 
be the smallest constant for which one has the decoupling inequality ![\displaystyle \| E_{[0,1]^{d-1}} g \|_{L^p(w_{B(0,N)})} \leq D_p(N) (\sum_Q \| E_Q g \|_{L^p(w_{B(0,N)})}^2)^{1/2}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+E_%7B%5B0%2C1%5D%5E%7Bd-1%7D%7D+g+%5C%7C_%7BL%5Ep%28w_%7BB%280%2CN%29%7D%29%7D+%5Cleq+D_p%28N%29+%28%5Csum_Q+%5C%7C+E_Q+g+%5C%7C_%7BL%5Ep%28w_%7BB%280%2CN%29%7D%29%7D%5E2%29%5E%7B1%2F2%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle \| E_{[0,1]^{d-1}} g \|_{L^p(w_{B(0,N)})} \leq D_p(N) (\sum_Q \| E_Q g \|_{L^p(w_{B(0,N)})}^2)^{1/2}.")
. ![{[0,1]}](https://s0.wp.com/latex.php?latex=%7B%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0 "{[0,1]}")
into 
of length ![{g \in L^1([0,1])}](https://s0.wp.com/latex.php?latex=%7Bg+%5Cin+L%5E1%28%5B0%2C1%5D%29%7D&bg=ffffff&fg=000000&s=0 "{g \in L^1([0,1])}")
, define the extension operators ![\displaystyle E_{[0,1]} g(x_1,\dots,x_d) = \int_{[0,1]} e^{2\pi i ( x_1 \xi + x_2 \xi^2 + \dots + x_d \xi^d} g(\xi)\ d\xi](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++E_%7B%5B0%2C1%5D%7D+g%28x_1%2C%5Cdots%2Cx_d%29+%3D+%5Cint_%7B%5B0%2C1%5D%7D+e%5E%7B2%5Cpi+i+%28+x_1+%5Cxi+%2B+x_2+%5Cxi%5E2+%2B+%5Cdots+%2B+x_d+%5Cxi%5Ed%7D+g%28%5Cxi%29%5C+d%5Cxi&bg=ffffff&fg=000000&s=0 "\displaystyle E_{[0,1]} g(x_1,\dots,x_d) = \int_{[0,1]} e^{2\pi i ( x_1 \xi + x_2 \xi^2 + \dots + x_d \xi^d} g(\xi)\ d\xi")
![\displaystyle E_J g(x_1,\dots,x_d) = \int_{[0,1]} e^{2\pi i ( x_1 \xi + x_2 \xi^2 + \dots + x_d \xi^d} g(\xi)\ d\xi](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++E_J+g%28x_1%2C%5Cdots%2Cx_d%29+%3D+%5Cint_%7B%5B0%2C1%5D%7D+e%5E%7B2%5Cpi+i+%28+x_1+%5Cxi+%2B+x_2+%5Cxi%5E2+%2B+%5Cdots+%2B+x_d+%5Cxi%5Ed%7D+g%28%5Cxi%29%5C+d%5Cxi&bg=ffffff&fg=000000&s=0 "\displaystyle E_J g(x_1,\dots,x_d) = \int_{[0,1]} e^{2\pi i ( x_1 \xi + x_2 \xi^2 + \dots + x_d \xi^d} g(\xi)\ d\xi")
. For any ![\displaystyle \| E_{[0,1]} g \|_{L^p(w_{B(0,N^d)})} \leq D_p(N) (\sum_J \| E_J g \|_{L^p(w_{B(0,N^d)})}^2)^{1/2}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+E_%7B%5B0%2C1%5D%7D+g+%5C%7C_%7BL%5Ep%28w_%7BB%280%2CN%5Ed%29%7D%29%7D+%5Cleq+D_p%28N%29+%28%5Csum_J+%5C%7C+E_J+g+%5C%7C_%7BL%5Ep%28w_%7BB%280%2CN%5Ed%29%7D%29%7D%5E2%29%5E%7B1%2F2%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle \| E_{[0,1]} g \|_{L^p(w_{B(0,N^d)})} \leq D_p(N) (\sum_J \| E_J g \|_{L^p(w_{B(0,N^d)})}^2)^{1/2}.")
, which among other things implies the Vinogradov main conjecture (as discussed in 
, 
, or 
to denote the inequality 
for some constant 
for 
. We write 
to denote a bound of the form 
where 
as 
along the given range of 
for 
, and 
for 
. Then the statement that 
for some exponents 
. For instance, suppose one is somehow able to establish the inequality 
is also known to be of polynomial size. Then this implies that 
for any 
notation is independent of 
similarly to 
, except that we now also require that the diameter of each set 
. One can then observe the following estimates: 
, we have 
, one has 
) and 
exponent does not depend on 
decoupling of the paraboloid by 
(the argument is also more or less summarised in this previous post). (In the cited papers, the argument was phrased only for the non-endpoint case 
, but it has been observed independently by many experts that the argument extends with only minor modifications to the endpoint 
and 
, one can define an auxiliary quantity 
. The precise definitions of 
and 
respectively, setting 
and 
) but will not be of importance to us for this discussion. Suffice to say that the following estimates are known: 
) 
. 
are sufficiently small (e.g. both less than 
), then 
are independent of 
notation can depend on both 
, 
, or 
might not be an integer (and might not divide evenly into 
for which we have the bound 
to be the minimal exponent for which this bound is attained: thus 
. We assume for contradiction that 
. Let 
that is independent of 
will dominate the 
for some 
strictly less than 
, so long as we also improve 
of the decoupling theorem for the paraboloid.) 
for sake of discussion, an even more sophisticated induction on scales argument was deployed 
, some auxiliary quantities 
are introduced for various exponents 
and 
and 
and 
one has 
, where 
. 
and 
, then 
, then 
.
, the quantity 
is also of polynomial size and hence has some upper exponent 
; meanwhile 
. Also, comparing the upper exponents of both sides of the axioms 
, which when combined with the second inequality gives 
, which on combination with the first estimate gives 
. Sending 
, the quantity 
), and hence has some upper exponent 
; similarly 
so we have to temporarily admit the possibility that 
, though this will soon be eliminated anyway.) Taking upper exponents of all the axioms we then conclude that 
.
, and so the statement 
. We take advantage of this by taking a further limit superior (or “upper derivative”) in the limit 
to eliminate the role of 
is the best constant for which 
as 
, each 
(which for now we can permit to be 
). Taking upper exponents of all the various axioms we can now eliminate 
, introducing 
for 
. By the axioms 
. We can then eliminate the role of 
by taking another limit superior 
, and 
, thus 
and 
. From the convexity 
; the same bound holds for 
if we drop the term with the 
factor, thanks to 
(arranged in whatever order one pleases), then the above system of inequalities 
in 
with non-negative coefficients, where the inequality denotes pointwise domination, and 
is an explicit vector with non-positive coefficients that reflects the effect of 
would now grow exponentially; this is typically the situation for “non-endpoint” applications such as proving decoupling inequalities away from the endpoint. In the endpoint situation discussed above, the Perron-Frobenius eigenvalue is 
now grows at least linearly, which still gives the required contradiction for any 
is the spectral radius of 
is a left Perron-Frobenius eigenvector, that is to say a non-negative vector, not identically zero, such that 
, then by taking inner products of 
we obtain 
this leads to a contradiction since 
is negative and 
is non-positive. When 
one still gets a contradiction as long as 
, 
as coming from the left Perron-Frobenius eigenvector of the matrix 
is bounded by 
term is absent); this simplifies after some calculation to the bound 
. (One will need to establish the claim 
for 
.) 
by dropping some of the steps.
is a bijection. The answer to this question is almost surely “no”, but it is remarkable how hard this problem resists any attempt at rigorous proof. (MathOverflow users with enough privileges to see deleted answers will find that there are no fewer than seventeen deleted attempts at a proof in response to this question!)
of one variable, the surface 
lifts to exactly one rational point in 
. I believe that for “typical” 
that their projection 
to the third coordinate is “large”. In particular, I believe
is birational to an elliptic curve, then the number of elements of 
of 
. 
for some rational 
, then then the number of elements of 
(in fact I think it can only grow like 
).
of 
nonlinear). Assuming these assertions, this would mean that there is a curve of the form 
that captures a “positive fraction” of the rational points of 
to lie below a large threshold 
for arbitrary polynomials 
of one variable) but I believe that the only way that such identities hold for a “positive fraction” of 
with rational coefficients (in which case we would have 
and 
). But such an identity would contradict the hypothesis that 
outside of the curve 
, and set 
, in which case we have 
violating the injective nature of 
, since I would expect the equation 
to describe a surface of general type for “generic” choices of 
as a fibre product of the simpler surface 
and the curve 
over the line 
. In any event, I believe the way to handle (a) is to show that the failure of general type of 
in the plane, the distances 
between them have to be non-negative and obey the triangle inequalities 
in the plane, then there is an additional constraint connecting the six distances 
between them, coming from the 
, then we have the usual cosine rule 
, and similarly for all the other distances. The 
matrix appearing in 
, where 
is the (augmented) 
is also. The Gram matrix 
, where 
is the 
matrix with rows 
, and thus has rank at most 
, and hence has determinant zero as claimed. 
and 
, then in order for 
has to obey the equation 
, so the non-negative quantity is constrained to equal either 
. The former happens when 
; the latter happens when 
form the vertices of a unit square traversed in that order. Any other value for 
of radius 
now measured as lengths of arcs on the sphere. There is a spherical analogue of the Cayley-Menger determinant:
, then the spherical Cayley-Menger determinant 
from the origin is 
, so by the cosine rule we have 
, where 
where 
is the 
matrix with rows 
and thus the determinant vanishes as required. 
are all equal to 
, then the above proposition gives 
; as 
lies between 
, the only choices for this distance are then 
, 
are points on the equator with longitudes differing by 90 degrees, and 
is also equal to the north pole; the latter occurs when 
is instead placed on the south pole.
. In the limit 
(so that the curvature of the sphere 
tends to 
, and by Taylor expansion 
tends to 
; similarly for the other distances. Now we see that the planar Cayley-Menger determinant emerges as the limit of 
) if one is given the six distances 
between four points 
kilometers, I chose the change of variables 
(so that 
corresponds to the flat Earth), and obtained the following plot for 
km. There is another radius that would also be compatible with this data at 
(corresponding to an Earth of radius about 
km), but presumably one could rule out this as a spurious coincidence by experimenting with other quadruples of cities than the ones I selected. On the other hand, these distances are highly incompatible with the flat Earth model 
, 
, etc.).
kilometers per hour, the true radius of the Earth should be about 
of flight time. If one then uses the normalisation 
, one obtains the following plot:
. (The website for the flight calculator does say it calculates short and long haul flight times slightly differently, which may be the cause of the slight discrepancies between this figure and the previous one.)
as before):
greater than the calculator predicts, while the flight time from LA to Dubai is about 
less; these sorts of errors seem to pile up in this calculation.) Nevertheless, it does seem that flight time data is (barely) enough to establish the roundness of the Earth and obtain a somewhat ballpark estimate for its radius. (I assume that the fit would be better if one could include some Southern Hemisphere cities such as Sydney or Santiago, but I was not able to find a good quadruple of widely spaced cities on both hemispheres for which there were direct flights between all six pairs.)
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