Olsen-type inequalities for the generalized commutator of multilinear fractional integrals

Abstract

In this paper, we study certain multilinear operators of fractional integral type defined by I ⃗ A α ⃗ f (x) = ∫ (R n) m f1(y1) • • • fm(ym) |(x − y1, • • • , x − ym)| mn−α+ m ∑ i=1 (N i −1) m ∏ i=1 RN i (Ai; x, yi)d⃗ y, where 0 < α < mn and RN i (Ai; x, yi) = Ai(x) − ∑ |γ|<N i 1 γ! D γ Ai(yi)(x − yi) γ. For the operator I ⃗ A α , we obtain an Olsen-type inequality on the Morrey space. Our proof is based on a complicated multiple dyadic decomposition.

Some estimates for the bilinear fractional integrals on the Morrey space

“…Here we would like to mention that Theorems 5.2 is not an easy consequence of Corollary 5.3 and the Hölder inequality for functions on the Morrey spaces (see (2.1) in [26, p.1377]). Readers may see [37,38,42] for details. In fact, from Corollary 5.3 and the Hölder inequality for functions on the Morrey spaces, there is…”

Boundedness for the Modified Fractional Integral Operator from Mixed Morrey Spaces to the Bounded Mean Oscillation Space and Lipschitz Spaces

“…e mapping properties of 􏽢 I α on Morrey spaces were first studied by Peetre [2] and further generalized by Adams [4]. We refer readers to [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein for more studies about boundedness of the fractional integral operator on Morrey-type and anisotropic spaces. Recently, the mapping properties of 􏽢 I α from Morrey spaces to BMO(R n ) and Lipschitz spaces were also obtained in [20][21][22].…”