BV continuity for the uncentered Hardy–Littlewood maximal operator

“…There is ongoing research for the endpoint case p = 1. For example Carneiro et al proved in [11] that f → ∇ M f is continuous W 1,1 (R) → L 1 (R) and in [14] González-Riquelme and Kosz recently improved this to continuity on BV. Carneiro et al proved in [8] that for radial functions f , the operator f → ∇ M f is continuous as a map W 1,1…”

Section: Introductionmentioning

confidence: 99%

Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators

Weigt

2022

Math. Z.

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Let $$0<\alpha <d$$ 0 < α < d and $$1\le p<d/\alpha $$ 1 ≤ p < d / α . We present a proof that for all $$f\in W^{1,p}({\mathbb {R}}^d)$$ f ∈ W 1 , p ( R d ) both the centered and the uncentered Hardy–Littlewood fractional maximal operator $${\mathrm {M}}_\alpha f$$ M α f are weakly differentiable and $$ \Vert \nabla {\mathrm {M}}_\alpha f\Vert _{p^*} \le C_{d,\alpha ,p} \Vert \nabla f\Vert _p , $$ ‖ ∇ M α f ‖ p ∗ ≤ C d , α , p ‖ ∇ f ‖ p , where $$ p^* = (p^{-1}-\alpha /d)^{-1} . $$ p ∗ = ( p - 1 - α / d ) - 1 . In particular it covers the endpoint case $$p=1$$ p = 1 for $$0<\alpha <1$$ 0 < α < 1 where the bound was previously unknown. For $$p=1$$ p = 1 we can replace $$W^{1,1}({\mathbb {R}}^d)$$ W 1 , 1 ( R d ) by $$\mathrm {BV}({\mathbb {R}}^d)$$ BV ( R d ) . The ingredients used are a pointwise estimate for the gradient of the fractional maximal function, the layer cake formula, a Vitali type argument, a reduction from balls to dyadic cubes, the coarea formula, a relative isoperimetric inequality and an earlier established result for $$\alpha =0$$ α = 0 in the dyadic setting. We use that for $$\alpha >0$$ α > 0 the fractional maximal function does not use certain small balls. For $$\alpha =0$$ α = 0 the proof collapses.

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“…In the references [10][11][12][13][14], Question 1 in dimension n = 1 has been completely solved, and in [15,16], partial progress has been made on this issue for the general dimension n ≥ 2. In 2002, Tanaka [14] first observed that if f ∈ W 1,1 (R), then M f is weakly differentiable and…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Beltran and Madrid [15] extended Kurka's result to the fractional version. Other interesting works can be found in [11,13,[20][21][22][23][24][25][26][27], among others. Next, we introduce the basic knowledge of graphs and the regularity properties of maximal operators on the graph settings.…”

Section: Introductionmentioning

confidence: 99%