Weak-type and end-point norm estimates for Hardy operators
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Weak‑type and end‑point norm estimates for Hardy operators Santiago Boza1 · Javier Soria2 Received: 9 October 2019 / Accepted: 29 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We explicitly calculate the best constants for weak-type and other end-point estimates for the Hardy operator and its adjoint. In particular, we find the right value for decreasing power weights, fixing some previously unclear results. Keywords Hardy operator · Dual operator · Best constants · Decreasing functions Mathematics Subject Classification 26D10 · 46E30
1 Introduction Let f be a locally integrable function on [0, ∞) . The classical Hardy operator S acting on f is defined as x
Sf (x) =
1 f (t) dt, x ∫0
and its adjoint operator is given by
S∗ f (x) =
∫x
∞
f (t) dt. t
Both authors have been partially supported by the Spanish Government Grant MTM2016-75196-P (MINECO/FEDER, UE) and the Catalan Autonomous Government Grant 2017SGR358. * Santiago Boza [email protected] Javier Soria [email protected] 1
Department of Mathematics, EETAC, Polytechnical University of Catalonia, 08860 Castelldefels, Spain
2
Department of Analysis and Applied Mathematics, Instituto de Matemática Interdisciplinar, Complutense University of Madrid, 28040 Madrid, Spain
13
Vol.:(0123456789)
S. Boza, J. Soria
For w(t) a nonnegative locally integrable function on (0, ∞) , usually called a weight, let us consider the weighted Lp-norm for 1 ≤ p < ∞
and, if p = ∞,
‖f ‖Lp (w) =
�
∫0
∞
�f (t)� w(t) dt p
�1∕p
,
‖f ‖L∞ (w) = sup �f (t)�w(t). t>0
The corresponding weighted weak-type L -norm of f is defined as p
� �1∕p . ‖f ‖Lp,∞ (w) = sup 𝜆 w(t) dt ∫{t>0∶ �f (t)�>𝜆} 𝜆>0
(Even though, for p = 1 , L1,∞ (w) might not be normable, for simplicity we will use the term norm for all values of p ≥ 1.) Boundedness of S and S∗ on Lp (t𝛼 ) (i.e., when w(t) = t𝛼 is a power weight) is the wellknown Hardy’s inequality. In recent years, extensions of these estimates to more general weights, different classes of functions (positive or decreasing) and related operators, have been thoroughly considered: In [19] the author finds necessary and sufficient conditions for the boundedness S ∶ Lp (w) → Lp (w) . The characterization for the restriction of S to decreasing functions in Lp (w) was first obtained in [2] (in terms of the so-called Bp condition) and further extended (to the non-diagonal case, the full range of the exponent p > 0 or weak-type estimates) in [11, 20, 21] (see also [18] and the references therein). From the pioneering result of [9] regarding the isometric properties in L2 of the Hardy operator minus the identity S − Id (which measures the oscillation of a function and has many applications in harmonic analysis, like Sobolev embeddings or rearrangement estimates of BMO functions), there has also been a great interest in determining the optimal bounds for this difference in other settings. In particular, in [17] the authors found the norm of S − Id o
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