Page 490 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 490
HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS (MS-28)
the regularity problems for various non-linear differential equations.
The aim of our talk is to give a complete characterization of a class of measures µ governing
the boundedness of fractional integral operators Iγ defined on a quasi-metric measure space
(X, d, µ) (nonhomogeneous space) from one grand Lebesgue spaces Lpµ),θ1(X) into another one
Lµq),θ2(X). As a corollary, we have a generalization of the Sobolev inequality for potentials with
measure. D. Adams trace inequality (i.e., Lµp),θ1(X) → Lνq),θ2(X) boundedness) is also derived
for these operators in grand Lebesgue spaces. Appropriate problems for grand Morrey spaces
are also studied. In the case of Morrey spaces, we assume that the underlying sets of spaces
might be of infinite measure. Under some additional conditions on a measure, we investigate
the sharpness of the second parameter θ2 in the target space.
Acknowledgement: The work was supported by the Shota Rustaveli National Foundation grant
of Georgia (Project No. DI-18-118).
References
[1] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator, Manuscripta
Math. 92 (1997), no. 2, 249–258.
[2] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypothe-
ses, Arch. Rational Mech. Anal. 119(1992), no. 2, 129–143.
Pointwise ergodic theorems for bilinear polynomial averages
Mariusz Mirek, mariusz.mirek@rutgers.edu
Rutgers University, United States
We shall discuss the proof of pointwise almost everywhere convergence for the non-conventional
(in the sense of Furstenberg and Weiss) bilinear polynomial ergodic averages. This is joint work
with Ben Krause and Terry Tao: arXiv:2008.00857. We will also talk about recent progress in
this area.
Inequalities for noncommutative martingales with applications to
quantum harmonic analysis
Adam Osekowski, ados@mimuw.edu.pl
University of Warsaw, Poland
In the recent twenty years, the theory of noncommutative (or quantum) martingales has gained a
lot interest in the literature, and most of the classical results has been successfully transferred to
this new, operator context. The purpose of the talk is to survey some recent progress in this field
and discuss several applications related to boundedness of Fourier multipliers on some group
von Neumann algebras
488
the regularity problems for various non-linear differential equations.
The aim of our talk is to give a complete characterization of a class of measures µ governing
the boundedness of fractional integral operators Iγ defined on a quasi-metric measure space
(X, d, µ) (nonhomogeneous space) from one grand Lebesgue spaces Lpµ),θ1(X) into another one
Lµq),θ2(X). As a corollary, we have a generalization of the Sobolev inequality for potentials with
measure. D. Adams trace inequality (i.e., Lµp),θ1(X) → Lνq),θ2(X) boundedness) is also derived
for these operators in grand Lebesgue spaces. Appropriate problems for grand Morrey spaces
are also studied. In the case of Morrey spaces, we assume that the underlying sets of spaces
might be of infinite measure. Under some additional conditions on a measure, we investigate
the sharpness of the second parameter θ2 in the target space.
Acknowledgement: The work was supported by the Shota Rustaveli National Foundation grant
of Georgia (Project No. DI-18-118).
References
[1] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator, Manuscripta
Math. 92 (1997), no. 2, 249–258.
[2] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypothe-
ses, Arch. Rational Mech. Anal. 119(1992), no. 2, 129–143.
Pointwise ergodic theorems for bilinear polynomial averages
Mariusz Mirek, mariusz.mirek@rutgers.edu
Rutgers University, United States
We shall discuss the proof of pointwise almost everywhere convergence for the non-conventional
(in the sense of Furstenberg and Weiss) bilinear polynomial ergodic averages. This is joint work
with Ben Krause and Terry Tao: arXiv:2008.00857. We will also talk about recent progress in
this area.
Inequalities for noncommutative martingales with applications to
quantum harmonic analysis
Adam Osekowski, ados@mimuw.edu.pl
University of Warsaw, Poland
In the recent twenty years, the theory of noncommutative (or quantum) martingales has gained a
lot interest in the literature, and most of the classical results has been successfully transferred to
this new, operator context. The purpose of the talk is to survey some recent progress in this field
and discuss several applications related to boundedness of Fourier multipliers on some group
von Neumann algebras
488
