TY - JOUR

T1 - A noninequality for the fractional gradient

AU - Spector, Daniel

N1 - Funding Information:
The author would like to thank the referees for their careful reading and many comments that have greatly improved the paper. Needless to say the author is responsible for the remaining shortcomings. The author is supported in part by the Taiwan Ministry of Science and Technology under research grant 107-2115-M-009-002-MY2.
Publisher Copyright:
© 2019 European Mathematical Society.

PY - 2019

Y1 - 2019

N2 - In this paper we give a streamlined proof of an inequality recently obtained by the author: For every a a α∈(0,1) there exists a constant C = C(α,d)>0 such that for all u∈ Lq(Rd) for some 1 ≤ q < d/(1-α) such that Dαu :ΔI1-αu∈L1(Rd; Rd). We also give a counterexample which shows that in contrast to the case α=1, the fractional gradient does not admit an L1 trace inequality, i.e. ||Dαu||L1(Rd; Rd) cannot control the integral of u with respect to the Hausdorff content H∞ d-α. The main substance of this counterexample is a result of interest in its own right, that even a weak-type estimate for the Riesz transforms fails on the space L1(H∞ d-α),β∈[1,d).It is an open question whether this failure of a weak-type estimate for the Riesz transforms extends to β∈(0,1).

AB - In this paper we give a streamlined proof of an inequality recently obtained by the author: For every a a α∈(0,1) there exists a constant C = C(α,d)>0 such that for all u∈ Lq(Rd) for some 1 ≤ q < d/(1-α) such that Dαu :ΔI1-αu∈L1(Rd; Rd). We also give a counterexample which shows that in contrast to the case α=1, the fractional gradient does not admit an L1 trace inequality, i.e. ||Dαu||L1(Rd; Rd) cannot control the integral of u with respect to the Hausdorff content H∞ d-α. The main substance of this counterexample is a result of interest in its own right, that even a weak-type estimate for the Riesz transforms fails on the space L1(H∞ d-α),β∈[1,d).It is an open question whether this failure of a weak-type estimate for the Riesz transforms extends to β∈(0,1).

KW - L-Sobolev inequality

KW - Lorentz spaces

KW - Trace inequality

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U2 - 10.4171/PM/2031

DO - 10.4171/PM/2031

M3 - Article

AN - SCOPUS:85081396536

VL - 76

SP - 153

EP - 168

JO - Portugaliae Mathematica

JF - Portugaliae Mathematica

SN - 0032-5155

IS - 2

ER -