A new formula for the L norm

A weak-type expression of the Orlicz modular

“…They complement the Maz'ya-Shaposhnikova formula (3) in the same way the result of Brezis, Van Schaftingen and Yung complements the Bourgain-Brezis-Mironescu formula (2). Namely the result of [7] asserts that, for every…”

“…Following this innovatory approach, Q. Gu and P.-L. Yung established in [7] other, possibly even more unanticipated, formulae. They complement the Maz'ya-Shaposhnikova formula (3) in the same way the result of Brezis, Van Schaftingen and Yung complements the Bourgain-Brezis-Mironescu formula (2).…”

Sobolev spaces revisited

“…Also, in the limiting formulae in Theorem 3 and Theorem 4, we had a constant k(p, n)/|γ|, and here we had a constant 2σ n−1 /|γ|; these should be compared, respectively, to the constant k(p, n)/p in the BBM formula (6), and the constant 2σ n−1 /p in the Maz'ya-Shaposhnikova formula (8). The case γ = n of (21) was proved in [16]. Note that we do not obtain a characterization of L p (R n ), contrarily to Theorem 5: the L p,∞ (ν γ ) norms of Q γ p u are finite (in fact zero) when u is a non-zero constant.…”

“…P.-L.Y. would like to thank Qingsong Gu for sharing his insights in their collaboration on [16]. A.S. and P.-L.Y.…”

Poincaré Inequality Meets Brezis--Van Schaftingen--Yung Formula on Metric Measure Spaces

“…The equivalence (1.2) in particular allows Brezis et al in [7] to derive some surprising alternative estimates of fractional Sobolev and Gagliardo-Nirenberg inequalities in some exceptional cases involving Ẇ1,1 (R n ), where the anticipated fractional Sobolev and Gagliardo-Nirenberg inequalities fail. For more studies on fractional Sobolev spaces, we refer the reader to [18,15,6,8,4,35].…”