diff 91bce0c..f6bf760 annular_triangular_hilbert_v2.txt
diff --git a/annular_triangular_hilbert_v2.txt b/annular_triangular_hilbert_v2.txt
index 0544e83..90cbb51 100644
--- a/annular_triangular_hilbert_v2.txt
+++ b/annular_triangular_hilbert_v2.txt
@@ -27,12 +27,12 @@ BH on a wide range of L^p products is due to Lacey and Thiele; the proof decompo
frequency plane into tilted wave packets and selects coherent
trees.
- ./lacey-thiele-1997.pdf
+ ./pdf/lacey-thiele-1997.pdf
Demeter and Thiele later isolate a two-dimensional analogue in
which the two inputs move along perpendicular axes.
- ./0803.1268.pdf
+ ./pdf/0803.1268.pdf
For complex functions F, G of two real variables and 0 < eps <
R < infinity the annular triangular Hilbert transform is
@@ -66,15 +66,15 @@ of specified structure is constructed by Kovac, Thiele, and
Zorin-Kranich. That work also states the cyclic trace formulation
and the equivalence of the flat and simplex forms.
- ./1506.00861.pdf
+ ./pdf/1506.00861.pdf
Tao proves sublogarithmic cancellation for the one-dimensional
multilinear Hilbert transform, and Zorin-Kranich extends the
technique to the simplex Hilbert transform over finitely many
smooth scales.
- ./1505.06479.pdf
- ./1507.02436.pdf
+ ./pdf/1505.06479.pdf
+ ./pdf/1507.02436.pdf
Durcik, Kovac, Skreb, and Thiele obtain square-function and
norm-variation estimates for the triangular form over finitely many
@@ -85,22 +85,22 @@ sqrt(log(R/eps)) at the symmetric point. Both estimates retain a
dependence on the range of scales, and neither places the supremum
over output-dependent partitions inside the output norm.
- ./1603.00631.pdf
- ./1608.00156.pdf
+ ./pdf/1603.00631.pdf
+ ./pdf/1608.00156.pdf
Durcik's continuous Gaussian telescoping for entangled forms is
the analytic template for the finite-dimensional heat identities
used below.
- ./1412.2384.pdf
- ./1506.08150.pdf
+ ./pdf/1412.2384.pdf
+ ./pdf/1506.08150.pdf
The Bellman function and twisted paraproduct work of Kovac isolate
the obstruction that a triangular cycle presents and motivate the
energy method.
- ./1108.0917.pdf
- ./1011.6140.pdf
+ ./pdf/1108.0917.pdf
+ ./pdf/1011.6140.pdf
Grafakos, Oliveira e Silva, Pramanik, Seeger, and Stovall collect
open problems at the symmetric point. Problem 13 in Section 9 of
@@ -113,17 +113,17 @@ that survey asks precisely for the scalar estimate proved below:
with a universal constant C.
- ./1701.06637.pdf
+ ./pdf/1701.06637.pdf
Durcik-Roos, Christ-Durcik-Roos, Hsu-Lin, and Lin-Slavikova
obtain related bounds by averaging a direction parameter
or by introducing curvature into the kernel. Those mechanisms are
distinct from the flat problem considered below.
- ./1812.11701.pdf
- ./2008.10140.pdf
- ./2410.15791.pdf
- ./2607.20206.pdf
+ ./pdf/1812.11701.pdf
+ ./pdf/2008.10140.pdf
+ ./pdf/2410.15791.pdf
+ ./pdf/2607.20206.pdf
The r-th annular variation of the family { B_{eps,R} }, for real
@@ -529,7 +529,7 @@ The general gradient formula for unitarily invariant matrix
functions is given in Lewis (J. Convex Analysis 2, 1995), of which
the present statement is a special case.
- ./lewis-1995-jca02012-convex-analysis-unitarily-invariant.pdf
+ ./pdf/lewis-1995-jca02012-convex-analysis-unitarily-invariant.pdf
The cyclic mixed-trace inequality is the whole point of collecting
the two incident blocks into one row matrix. For Hermitian
@@ -902,7 +902,7 @@ Its second derivative produces, up to constants, a rescaling of
the truncated triangular Hilbert kernel. The exact comparison is
the identity (hard-smooth) stated below.
-Smooth-switches proposition. Let G_v in C_c^infty(R^2) be
+Smooth-switches proposition. Let G_v in C_c^infinity(R^2) be
complex. At each output pair (a, b), make a step choice of at most
n >= 1 disjoint radius intervals (eps_j, R_j) and complex
coefficients |beta_j| <= 1. For
@@ -1383,7 +1383,7 @@ For
K_{a,b}(t) = sum_k Dil_{2^k} K_{a,b,k}(t)
-and G_v in C_c^infty(R^2),
+and G_v in C_c^infinity(R^2),
| Lambda_K(G_0, G_1, G_2) |
<= C_{A,c} sqrt(n) log(2 + n) ||G_0||_3 ||G_1||_3 ||G_2||_3 .
@@ -1402,11 +1402,11 @@ every derivative up to order three.
Gaussian-weighted primitive. Seek a representation K = (g_{3q} f)'''.
Define
- U(z) = (1/2) int_{-infty}^{z} (z - u)^2 K(u) d u .
+ U(z) = (1/2) int_{-infinity}^{z} (z - u)^2 K(u) d u .
The three moment conditions imply, for j = 0, 1, 2,
- U^{(j)}(z) = -1/(2-j)! int_z^{infty} (z - u)^{2-j} K(u) d u .
+ U^{(j)}(z) = -1/(2-j)! int_z^{infinity} (z - u)^{2-j} K(u) d u .
Use the defining integral for z <= 0 and the tail integral for z >= 0;
the Gaussian tail estimates give |U^{(j)}(z)| <= C exp(-c_1 z^2) for
@@ -1421,7 +1421,7 @@ points bounds the classical derivative of f''' by C exp(-c_2 z^2);
the jumps at the exceptional points are bounded by C, and there are
at most A m of them. Thus
- ||f^{(j)}||_{infty} <= C , Var(f^{(j)}) <= C m , 0 <= j <= 3 .
+ ||f^{(j)}||_{infinity} <= C , Var(f^{(j)}) <= C m , 0 <= j <= 3 .
Fourier bounds. Normalize
f_hat(xi) = (2 pi)^{-1} int f(z) e^{-i xi z} d z. Let
@@ -1442,7 +1442,7 @@ C m / D, using Plancherel f_hat''' = (i xi)^3 f_hat, and using
For D = 1 both bounds follow from the Gaussian decay and m >= 1.
-A finite frequency cutoff. Choose chi in C_c^infty(R) supported on
+A finite frequency cutoff. Choose chi in C_c^infinity(R) supported on
[-2, 2] and equal to one on [-1, 1], let P_* = (2 + n)^8, and put
f_hat_*(xi) = f_hat(xi) chi(xi / P_*) ,
@@ -1737,7 +1737,7 @@ operator gives, with the maximal functions of the tail,
By density and continuity of B_* in the pair (F, G), the same tail
convergence extends to all complex F, G in L^3(R^2). Uniqueness of
the bounded bilinear extension follows from the density of
-C_c^infty and from the maximal bound.
+C_c^infinity and from the maximal bound.
Optimal scalar constant. Let C_S be the common optimal constant of
the uniform family of individual bilinear bounds
@@ -1774,4 +1774,4 @@ supplies the gradient formula across matrix rank changes. Kwong
(Proc. Amer. Math. Soc. 51, 1975) supplies the Loewner order facts
for the square root of a positive matrix.
- ./0910.1555.pdf
+ ./pdf/0910.1555.pdf