Extension of multilinear fractional integral operators to linear operators on mixed-norm Lebesgue spaces
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Mathematische Annalen
Extension of multilinear fractional integral operators to linear operators on mixed-norm Lebesgue spaces Ting Chen1 · Wenchang Sun1 Received: 4 March 2020 / Revised: 7 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In Kenig and Stein (Math Res Lett 6(1):1–15, 1999, https://doi.org/10.4310/MRL. 1999.v6.n1.a1), the following type of multilinear fractional integral Rmn
f 1 (l1 (x1 , . . . , xm , x)) · · · f m+1 (lm+1 (x1 , . . . , xm , x)) d x1 . . . d xm (|x1 | + · · · + |xm |)λ
was studied, where li are linear maps from R(m+1)n to Rn satisfying certain conditions. They proved the boundedness of such multilinear fractional integral from L p1 × · · · × L pm+1 to L q when the indices satisfy the homogeneity condition. In this paper, we show that the above multilinear fractional integral extends to a linear operator for functions in the mixed-norm Lebesgue space L p which contains L p1 × · · · × L pm+1 as a subset. Under less restrictions on the linear maps li , we give a complete characterization of the indices p, q and λ for which such an operator is bounded from L p to L q . And for m = 1 or n = 1, we give necessary and sufficient conditions on (l1 , . . . , lm+1 ), p = ( p1 , . . . , pm+1 ), q and λ such that the operator is bounded. Keywords Fractional integrals · Riesz potentials · Mixed norms Mathematics Subject Classification 42B20
Communicated by Loukas Grafakos. This work was partially supported by the National Natural Science Foundation of China (11525104, 11531013, 11761131002 and 11801282).
B
Wenchang Sun [email protected] Ting Chen [email protected]
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School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
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T. Chen, W. Sun
1 Introduction and the main results The fractional integral operator is useful in the study of differentiability and smoothness of functions. In [36], Kenig and Stein studied the multilinear fractional integral of the following type, Rmn
f 1 (l1 (x1 , . . . , xm , x)) · · · f m+1 (lm+1 (x1 , . . . , xm , x)) d x1 . . . d xm , (|x1 | + · · · + |xm |)λ
m where li (x1 , . . . , xm , x) = i=1 Ai, j x j + Ai,m+1 x and Ai, j are n × n matrices. They proved that the above fractional integral is bounded from L p1 × · · · × L pm+1 to L q if 1 < pi ≤ ∞, 1 ≤ i ≤ m + 1, 0 < q < ∞, 0 < λ < mn, 1 1 mn − λ 1 + ··· + = + , p1 pm+1 q n
(1.1)
and the coefficient matrices Ai, j satisfy the followings, (i) A = (Ai, j )1≤i, j≤m+1 is an (m + 1)n × (m + 1)n invertible matrix, (ii) each Ai,m+1 is an n × n invertible matrix for 1 ≤ i ≤ m + 1, and (iii) (Ai, j )1≤i≤m+1,i=i0 is an mn × mn invertible matrix for every 1 ≤ i 0 ≤ m + 1. 1≤ j≤m
In this paper, we show that the multilinear operator can be extended to a linear operator defined on the mixed-norm Lebesgue space L p . Recall that for p = ( p1 , . . . , pk ), where 0 < p1 , . . . , pk ≤ ∞ and k ≥ 1, L p consists of all measurable functions f for which f L p := f L xp1 · · · pk < ∞. 1
L xk
For convenience, we also write the
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