Gagliardo representation of norms of ball quasi-Banach function spaces

Let X be a ball quasi-Banach function space on R-n. In this article, under some mild assumptions about both X and the boundedness of the Hardy-Littlewood maximal operator on the associate space of the convexification of X, the authors prove that, for any f is an element of(X/R) boolean AND boolean OR(s is an element of(0,1)). (W) over dot(X)(s,q) (R-n), parallel to f parallel to X/R less than or similar to lim inf(s -> 0+) s(1/q) parallel to[integral(Rn) vertical bar f(center dot) - f(y)vertical bar(q)/vertical bar center dot -y vertical bar(n+sq) dy](1/q)parallel to(X) <= lim sup(s -> 0+) s(1/q) parallel to[integral(Rn) vertical bar f(center dot) - f(y)vertical bar(q)/vertical bar center dot -y vertical bar(n+sq) dy](1/q)parallel to(X) less than or similar to parallel to f parallel to X/R and, for any gamma is an element of R \{0} and f is an element of X/R, sup(lambda is an element of(0, infinity)) lambda parallel to[integral(Rn) 1Ef(lambda, w, gamma) (center dot, y)vertical bar - f(y)vertical bar(q)/vertical bar center dot -y vertical bar(n+sq) dy](1/q)parallel to(X) with the positive equivalence constants independent of f, where f is an element of X/R if and only if there exists a is an element of. Rsuch that f + a is an element of X and where parallel to f parallel to(X/R):= inf(a is an element of R) parallel to f + a parallel to(X) < infinity, the index q is an element of (0, infinity) is related to X, (W) over dot(X)(s,q) (R-n) is the homogeneous fractional Sobolev space associated with the ball quasi-Banach function space X, and E-f(lambda, q, gamma) := {(x, y) is an element of R-n x R-n : vertical bar f(x) - f(y)vertical bar > lambda vertical bar x - y|(gamma/q)}. In the case when X:= L-p(R-n) with 1 <= q = p < infinity and f is an element of X, the first formula is closely related to the celebrated classical formula of V. Maz'ya and T. Shaposhnikova and the second formula is exactly the recent formula of H. Brezis et al. These results are new even when X= L-p(R-n) with 1 <= q < p < infinity and 0 < q <= p < 1. All these results are of quite wide generality and, even when they are applied to various specific function spaces, most of the obtained results are new. To obtain these results, the authors overcome those obstacles caused by the deficiency of both the translation invariance and an explicit expression of the quasi-norm of Xunder consideration via establishing some weighted estimates and new decompositions, which depend on the extrapolation and the exact operator norm of the Hardy-Littlewood maximal operator. (c) 2023 Elsevier Inc. All rights reserved.