Real interpolation with weighted rearrangement invariant Banach function spaces

Abstract

Abstract. Motivated by recent applications of weighted norm inequalities to maximal regularity of first and second order Cauchy problems, we study real interpolation spaces on the basis of general Banach function spaces and, in particular, weighted rearrangement invariant Banach function spaces. We show equivalence of the trace method and the K-method, identify real interpolation spaces between a Banach space and the domain of a sectorial operator, and reprove an extension of Dore's theorem on the boundedness of H ∞ -functional calculus to this general setting.

Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

“…Hence, it only remains to show that D(A) = A s+2m p,r (R n , w; X). But choosing (s 1 , s 2 , s 3 ) = (1, 0, 1) in shows that (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18), we obtain that (1 + λ + A) −1 maps A s p,r (R n , w; X) into A s+2m p,r (R n , w; X). This shows that D(A) = A s+2m p,r (R n , w; X).…”

“…Choosing σ ∈ R such that s − σ < a 1 ( 1+γ p 1 − 1), a combination of (4-4), (4-5), Lemma 4.33, Lemma 4.34 and the lifting property of weighted mixed-norm anisotropic B-and F-spaces (see (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)) gives the desired result. Indeed, we obtain the commutative diagram:…”

“…Proof. This follows directly from the fact that F s,a,d p,q (R n , w; X) and B s,a,d p,q (R n , w; X) are (d , a)-admissible Banach spaces of tempered distributions with (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18).…”

“…References for L q (R)-maximal regularity and L q (R + )-maximal regularity include [4,71] and [27,54]. Works on L q (R + , v)-maximal regularity include [12,13,31].…”

“…Proof. For simplicity of notation we only treat the case A = F. Let N be as in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) for…”