Page 150 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 150
CURRENT TOPICS IN COMPLEX ANALYSIS (MS-32)
holds, where ψ(t) is a nonnegative measurable function on (0, +∞) such that
R ∀ R > r0 ,
0 < I(r0, R) = ψ(t)dt < ∞
r0
then
p−n p−1
p−1
κ−p 1 p−n 1
lim L(x0, f, R) I p−n (r0, R) ω n−p c n−p ,
n−1
R→∞
where ωn−1 is an area of the unit sphere Sn−1 = {x ∈ Rn : |x| = 1} in Rn.
Generalizations of Hardy type inequalities via new Green functions
Kristina Krulic´ Himmelreich, kkrulic@ttf.hr
University of Zagreb Faculty of Textile Technology, Croatia
Coauthors: Josip Pecˇaric´, Marjan Praljak
This talk deals with Hardy inequality and its famous generalizations, extensions and refine-
ments, i.e. with Hardy-type inequalities. The classical Hardy inequality reads:
∞ x p p∞
1 f (t) dt dx ≤ p f p(x) dx, p > 1, (1)
x
p−1
00 0
where f is non negative function such that f ∈ Lp(R+) and R+ = (0, ∞). The constant pp
p−1
is sharp. This inequality has been generalized and extended in several directions. In this talk
the Hardy inequality is generalized by using new Green functions.
Almost periodic functions revisited
Juan Matías Sepulcre, JM.Sepulcre@ua.es
University of Alicante, Spain
In the beginnings of the twentieth century, Harald Bohr (1887-1951) gave important steps in
the understanding of Dirichlet series and their regions of convergence, uniform convergence
and absolute convergence. As a result of his investigations concerning those functions which
could be represented by a Dirichlet series, he developed in its main features the theory of almost
periodic functions (both for functions of a real variable and for the case of a complex variable).
Based on a new equivalence relation on these classes of functions, in this talk we will refine
Bochner’s result that characterizes the property of almost periodicity in the Bohr’s sense. Fur-
thermore, we will present a thorough extension of Bohr’s equivalence theorem which states that
two equivalent almost periodic functions take the same set of values on every open half-plane
or open vertical strip included in their common region of almost periodicity.
148
holds, where ψ(t) is a nonnegative measurable function on (0, +∞) such that
R ∀ R > r0 ,
0 < I(r0, R) = ψ(t)dt < ∞
r0
then
p−n p−1
p−1
κ−p 1 p−n 1
lim L(x0, f, R) I p−n (r0, R) ω n−p c n−p ,
n−1
R→∞
where ωn−1 is an area of the unit sphere Sn−1 = {x ∈ Rn : |x| = 1} in Rn.
Generalizations of Hardy type inequalities via new Green functions
Kristina Krulic´ Himmelreich, kkrulic@ttf.hr
University of Zagreb Faculty of Textile Technology, Croatia
Coauthors: Josip Pecˇaric´, Marjan Praljak
This talk deals with Hardy inequality and its famous generalizations, extensions and refine-
ments, i.e. with Hardy-type inequalities. The classical Hardy inequality reads:
∞ x p p∞
1 f (t) dt dx ≤ p f p(x) dx, p > 1, (1)
x
p−1
00 0
where f is non negative function such that f ∈ Lp(R+) and R+ = (0, ∞). The constant pp
p−1
is sharp. This inequality has been generalized and extended in several directions. In this talk
the Hardy inequality is generalized by using new Green functions.
Almost periodic functions revisited
Juan Matías Sepulcre, JM.Sepulcre@ua.es
University of Alicante, Spain
In the beginnings of the twentieth century, Harald Bohr (1887-1951) gave important steps in
the understanding of Dirichlet series and their regions of convergence, uniform convergence
and absolute convergence. As a result of his investigations concerning those functions which
could be represented by a Dirichlet series, he developed in its main features the theory of almost
periodic functions (both for functions of a real variable and for the case of a complex variable).
Based on a new equivalence relation on these classes of functions, in this talk we will refine
Bochner’s result that characterizes the property of almost periodicity in the Bohr’s sense. Fur-
thermore, we will present a thorough extension of Bohr’s equivalence theorem which states that
two equivalent almost periodic functions take the same set of values on every open half-plane
or open vertical strip included in their common region of almost periodicity.
148
