Page 481 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 481
GEOMETRIC-FUNCTIONAL INEQUALITIES AND RELATED TOPICS (MS-23)
Minimal Conditions to define BMO
Carlos Perez Moreno, cperez@bcamath.org
University of Basque Country and BCAM, Spain
It is well known the importance of the BMO space of functions with bounded mean oscillation
especially due to the famous John-Nirenberg theorem of the early 60’s of the last century. This
result is the archetypical self-improving result in Analysis. In this talk we will show that there
is another self-improving phenomenon attached to this class of functions which roughly gives
a way of defining BMO using much weaker conditions than the usual L1 oscillation. These
results improved a recent work by Logunov-Slavin-Stolyarov-Vasyunin-Zatitskiy. Our method
is more flexible yielding sharp results under rougher geometries.
If there is enough time I will show some self-improving phenomenon considered for first
time by B. Muckenhoupt and R. Wheeden with weights which turned out to be very useful in
different situations like in the extrapolation theory.
This joint work with J. Canto and E. Rela.
Blaschke–Santaló inequalities for Minkowski endomorphisms
Franz Schuster, franz.schuster@tuwien.ac.at
Vienna University of Technology, Austria
In this talk we explain how each monotone Minkowski endomorphism of convex bodies gives
rise to an isoperimetric inequality which directly implies the classical Urysohn inequality.
Among this large family of new inequalities, the only affine invariant one – the Blaschke–
Santaló inequality – turns out to be the strongest one. A further extension of these inequalities
to merely weakly monotone Minkowski endomorphisms is proven to be impossible. Moreover,
for functional analogues of monotone Minkowski endomorphisms, a family of analytic inequal-
ities for log-concave functions is established which generalizes the functional Blaschke–Santaló
inequality.
Classical multiplier theorems and their sharp variants
Lenka Slavíková, slavikova@karlin.mff.cuni.cz
Charles University, Czech Republic
The question of finding good sufficient conditions on a bounded function m guaranteeing the
Lp-boundedness of the associated Fourier multiplier operator is a long-standing open problem
in harmonic analysis. In this talk we recall the classical multiplier theorems of Hörmander
and Marcinkiewicz and present their sharp variants in which the multiplier belongs to a certain
fractional Lorentz-Sobolev space. The talk is based on a joint work with L. Grafakos and M.
Mastyło.
479
Minimal Conditions to define BMO
Carlos Perez Moreno, cperez@bcamath.org
University of Basque Country and BCAM, Spain
It is well known the importance of the BMO space of functions with bounded mean oscillation
especially due to the famous John-Nirenberg theorem of the early 60’s of the last century. This
result is the archetypical self-improving result in Analysis. In this talk we will show that there
is another self-improving phenomenon attached to this class of functions which roughly gives
a way of defining BMO using much weaker conditions than the usual L1 oscillation. These
results improved a recent work by Logunov-Slavin-Stolyarov-Vasyunin-Zatitskiy. Our method
is more flexible yielding sharp results under rougher geometries.
If there is enough time I will show some self-improving phenomenon considered for first
time by B. Muckenhoupt and R. Wheeden with weights which turned out to be very useful in
different situations like in the extrapolation theory.
This joint work with J. Canto and E. Rela.
Blaschke–Santaló inequalities for Minkowski endomorphisms
Franz Schuster, franz.schuster@tuwien.ac.at
Vienna University of Technology, Austria
In this talk we explain how each monotone Minkowski endomorphism of convex bodies gives
rise to an isoperimetric inequality which directly implies the classical Urysohn inequality.
Among this large family of new inequalities, the only affine invariant one – the Blaschke–
Santaló inequality – turns out to be the strongest one. A further extension of these inequalities
to merely weakly monotone Minkowski endomorphisms is proven to be impossible. Moreover,
for functional analogues of monotone Minkowski endomorphisms, a family of analytic inequal-
ities for log-concave functions is established which generalizes the functional Blaschke–Santaló
inequality.
Classical multiplier theorems and their sharp variants
Lenka Slavíková, slavikova@karlin.mff.cuni.cz
Charles University, Czech Republic
The question of finding good sufficient conditions on a bounded function m guaranteeing the
Lp-boundedness of the associated Fourier multiplier operator is a long-standing open problem
in harmonic analysis. In this talk we recall the classical multiplier theorems of Hörmander
and Marcinkiewicz and present their sharp variants in which the multiplier belongs to a certain
fractional Lorentz-Sobolev space. The talk is based on a joint work with L. Grafakos and M.
Mastyło.
479
