An inequality for the convolutions on unimodular locally compact groups and the optimal constant of Young's inequality

DOI10.1007/S00041-023-09991-5arXiv2204.00742MaRDI QIDQ6395460

Publication date: 1 April 2022

Abstract: Let

m u

be the Haar measure of a unimodular locally compact group

G

and

m (G)

as the infimum of the volumes of all open subgroups of

G

. The main result of this paper is that �egin{align*} int_{G}^{} f circ left( phi_1 * phi_2 ight) left( g ight) dg leq int_{mathbb{R}}^{} f circ left( phi_1^* * phi_2^* ight) left( x ight) dx end{align*} holds for any measurable functions

p h i_{1}, p h i_{2} c o l o n G o m a t h b b R_{g e q 0}

with

m u (m a t h r m s u p p; p h i_{1}) + m u (m a t h r m s u p p; p h i_{2}) l e q m (G)

and any convex function

f c o l o n m a t h b b R_{g e q 0} o m a t h b b R

with

f (0) = 0

. Here

p h i^{*}

is the rearrangement of

p h i

. Let

Y_{O} (P, G)

and

Y_{R} (P, G)

denote the optimal constants of Young's and the reverse Young's inequality, respectively, under the assumption

m u (m a t h r m s u p p; p h i_{1}) + m u (m a t h r m s u p p; p h i_{2}) l e q m (G)

. Then we have

Y_{O} (P, G) l e q Y_{O} (P, m a t h b b R)

and

Y_{R} (P, G) g e q Y_{R} (P, m a t h b b R)

as a corollary. Thus, we obtain that

m (G) = i n f t y

if and only if

H (p, G) l e q H (p, m a t h b b R)

in the case of

p^{'} : = p / (p - 1) i n 2 m a t h b b Z

, where

H (p, G)

is the optimal constant of the Hausdorff--Young inequality.

Mathematics Subject Classification ID

Spaces of measurable functions ((L^p)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) (46E30) Measures on groups and semigroups, etc. (43A05) Functional inequalities, including subadditivity, convexity, etc. (39B62) Convolution, factorization for one variable harmonic analysis (42A85) (L^p)-spaces and other function spaces on groups, semigroups, etc. (43A15) Set functions and measures on topological groups or semigroups, Haar measures, invariant measures (28C10) Group algebras of locally compact groups (22D15)

This page was built for publication: An inequality for the convolutions on unimodular locally compact groups and the optimal constant of Young's inequality

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6395460)