A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function

By Hitoshi Tanaka. Bulletin of the Australian Mathematical Society, 2001

100 citations recorded by the source corpus.

Dedicated to Professor Kôzô Yabuta on the occasion of his 60th birthday J. Kinnunen proved that of P > 1, d ≤ 1 and f is a function in the Sobolev space W1,P(Rd), then the first order weak partial derivatives of the Hardy-Littlewood maximal function ℳf belong to LP(Rd). We shall show that, when d = 1, Kinnunen's result can be extended to the case where P = 1.

Subfields: Applied Mathematics

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Citations in the corpus

On approximate differentiability of the maximal function (2010, background): “Let us also mention that the result of Kinnunen [10] has been applied and generalized by many authors ([2], [3], [6], [7], [8], [11], [12], [14], [15], [16], [17], [18], [20], [21], [25]).”
Corrigendum to “On a discrete version of Tanaka’s theorem for maximal functions” (2010, methodology): “ISCRETE VERSION OF TANAKA’S THEOREM FOR MAXIMAL FUNCTIONS JONATHAN BOBER, EMANUEL CARNEIRO, KEVIN HUGHES, AND LILLIAN B. PIERCE Abstract. In this paper we prove a discrete version of Tanaka’s theorem [19] for the Hardy-Littlewood maximal operator in dimension n= 1, both in the non-centered and cente”
On the variation of maximal operators of convolution type (2013, methodology): “1)) Tanaka [17] was the first to observe that, if f ∈W (1,1)(R), then Mrf has a weak derivative and ‖(Mrf) ‖L1(R) ≤ ‖f ‖L1(R), (1.”
On the endpoint regularity of discrete maximal operators (2013, unknown): “In the work [18], Tanaka obtained the bound (1.”
On the variation of the Hardy-Littlewood maximal function (2012, methodology): “[3] J. Bober, E. Carneiro, K. Hughes and L. B. Pierce, On a discrete version of Tanaka’s theorem for maximal functions, Proc.”
Derivative bounds for fractional maximal functions (2015, background): “For the uncentered maximal operator (deﬁned similarly as in (1.1), with the supremum taken over all 4064 E. CARNEIRO AND J. MADRID balls containing the point x in its closure), which we denote here by M , Tanaka [22] showed that if f ∈ W 1 , 1 ( R ), then Mf is weakly diﬀerentiable and This result w”
ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES (2004, methodology): “Is the operator u 7→ |∇Mu| bounded from W (1,1)(R) to L(1)(R)? Tanaka, [18], gave a positive answer to this question in the case of the noncentered maximal function and provided that n = 1.”