Strichartz's conjecture on Hardy–Sobolev spaces

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We prove Strichartz's conjecture regarding a characterization of Hardy- Sobolev spaces. Introduction. Hardy-Sobolev spaces arise as an alternative ofL p So- bolev spaces. To describe this notion, letH p denote the real-variable Hardy spaces on R n forp> 0 andIα the Riesz potential operators of orderα> 0 defined via the Fourier transform formula (Iαf)b(ξ) = |ξ| −α b f(ξ) on the class of tempered distributions modulo polynomials. The image spaces of H p underIα, denoted byIα(H p ), are called the homogeneous Hardy-Sobolev spaces. For eachf ∈Iα(H p ) there exists a uniqueg∈H p withf =Iα(g) and we define a quasi-norm kfkI�(Hp) =kgkHp =kΛαfk Hp (0 1, theIα(H p ) are identical to the homogeneousL p Sobolev spacesIα(L p ). For 0< p≤ 1, it is well known that theH p provide an ideal alternative of theL p and thus theIα(H p ) may be thought of as a natural generalization of theIα(L p ). As usual, we may define the inhomogeneous Hardy-Sobolev spaces asH p ∩ Iα(H p ). As for characterizingIα(H p ), let us recall the work of Strichartz which gives us the main motivation. Given a positive integerm and a pointy∈ R n , letm be themth forward difference operator defined inductively as � mf(x) =�y(� m−1 y f)(x), �yf(x) =f(x +y)−f(x) for each locally integrable functionf and consider