Page 493 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 493
HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS (MS-28)
On a control of the maximal truncated Riesz transform by the Riesz
transform; dimension-free estimates
Błaz˙ej Wróbel, blazej.wrobel@uwr.edu.pl
University of Wrocław, Poland
W prove a dimension-free estimate for the L2(Rd) norm of the maximal truncated Riesz trans-
form in terms of the L2(Rd) norm of the Riesz transform. Consequently, the vector of maximal
truncated Riesz transforms has a dimension-free estimate on L2(Rd). We also show that the
maximal function of the vector of truncated Riesz transforms has a dimension-free estimate on
all Lp(Rd) spaces, 1 < p < ∞. The talk is based on joint work with Maciej Kucharski.
Multiplicative inequalities on BMO
Pavel Zatitskii, pavelz@pdmi.ras.ru
St. Petersburg University (SPbU), Russian Federation
We will talk about the so-called multiplicative inequality for BMO functions:
ϕ r ≤ Cp,r ϕ p ϕ r−p O ,
Lr Lp BM
where 1 < p < r < ∞. We will discuss how to find sharp constants in this inequality for the
case of quadratic norm on BMO space based on a segment, circle or a real line. Talking about
cases of segment and circle we assume the average of ϕ to be equal to zero. Also, we prove this
inequality with dimension-free constant for the Garsia-type norm on BMO. The talk is based
on joint work with D. Stolyarov, V. Vasyunin and I. Zlotnikov.
Boundary unique continuation of Dini domains
Zihui Zhao, zhaozh@uchicago.edu
University of Chicago, United States
Coauthor: Carlos Kenig
Let u be a harmonic function in Ω ⊂ Rd. It is known that in the interior, the singular set
S(u) = {u = |∇u| = 0} is (d − 2)-dimensional, and moreover S(u) is (d − 2)-rectifiable and
its Minkowski content is bounded (depending on the frequency of u). We prove the analogue
near the boundary for C1-Dini domains: If the harmonic function u vanishes on an open subset
E of the boundary, then near E the singular set S(u) ∩ Ω is (d − 2)-rectifiable and has bounded
Minkowski content. Dini domain is the optimal domain for which ∇u is continuous towards
the boundary, and in particular every C1,α domain is Dini. The main difficulty is the lack of
monotonicity formula near the boundary of a Dini domain. This is joint work with Carlos
Kenig.
491
On a control of the maximal truncated Riesz transform by the Riesz
transform; dimension-free estimates
Błaz˙ej Wróbel, blazej.wrobel@uwr.edu.pl
University of Wrocław, Poland
W prove a dimension-free estimate for the L2(Rd) norm of the maximal truncated Riesz trans-
form in terms of the L2(Rd) norm of the Riesz transform. Consequently, the vector of maximal
truncated Riesz transforms has a dimension-free estimate on L2(Rd). We also show that the
maximal function of the vector of truncated Riesz transforms has a dimension-free estimate on
all Lp(Rd) spaces, 1 < p < ∞. The talk is based on joint work with Maciej Kucharski.
Multiplicative inequalities on BMO
Pavel Zatitskii, pavelz@pdmi.ras.ru
St. Petersburg University (SPbU), Russian Federation
We will talk about the so-called multiplicative inequality for BMO functions:
ϕ r ≤ Cp,r ϕ p ϕ r−p O ,
Lr Lp BM
where 1 < p < r < ∞. We will discuss how to find sharp constants in this inequality for the
case of quadratic norm on BMO space based on a segment, circle or a real line. Talking about
cases of segment and circle we assume the average of ϕ to be equal to zero. Also, we prove this
inequality with dimension-free constant for the Garsia-type norm on BMO. The talk is based
on joint work with D. Stolyarov, V. Vasyunin and I. Zlotnikov.
Boundary unique continuation of Dini domains
Zihui Zhao, zhaozh@uchicago.edu
University of Chicago, United States
Coauthor: Carlos Kenig
Let u be a harmonic function in Ω ⊂ Rd. It is known that in the interior, the singular set
S(u) = {u = |∇u| = 0} is (d − 2)-dimensional, and moreover S(u) is (d − 2)-rectifiable and
its Minkowski content is bounded (depending on the frequency of u). We prove the analogue
near the boundary for C1-Dini domains: If the harmonic function u vanishes on an open subset
E of the boundary, then near E the singular set S(u) ∩ Ω is (d − 2)-rectifiable and has bounded
Minkowski content. Dini domain is the optimal domain for which ∇u is continuous towards
the boundary, and in particular every C1,α domain is Dini. The main difficulty is the lack of
monotonicity formula near the boundary of a Dini domain. This is joint work with Carlos
Kenig.
491
