Beurling–Ahlfors Commutators on Weighted Morrey Spaces and Applications to Beltrami Equations

Abstract

Let p ∈ (1, ∞), κ ∈ (0, 1) and w ∈ A p (C). In this article, the authors obtain a boundedness (resp., compactness) characterization of the Buerling-Ahlfors com-where B denotes the Buerling-Ahlfors transform and b ∈ BMO(C) [resp., CMO(C)]. Moreover, an application to the Beltrami equation is also given.2010 Mathematics Subject Classification: Primary 42B20; Secondary 46E35.

A Survey on Function Spaces of John–Nirenberg Type

“…As for the applications of these vanishing subspaces, we know that the commutator [b, T], generated by b ∈ BMO (R n ) and the Calderón-Zygmund operator T, plays an important role in harmonic analysis, complex analysis, partial differential equations, and other fields in mathematics. Here, we only list several typical bilinear results; other linear and multi-linear results can be found, for instance, in [22,100,101] and their references.…”

“…For instance, Fefferman and Stein [2] proved that the dual space of the Hardy space H 1 (R n ) is BMO (R n ); Coifman et al [3] showed an equivalent characterization of the boundedness of Calderón-Zygmund commutators via BMO (R n ); Coifman and Weiss [4,5] introduced the space of homogeneous type and studied the Hardy space and the BMO space in this context; Sarason [6] obtained the equivalent characterization of VMO (R n ), the closure in BMO (R n ) of uniformly continuous functions, and used it to study stationary stochastic processes satisfying the strong mixing condition and the algebra H ∞ + C; Uchiyama [7] established an equivalent characterization of the compactness of Calderón-Zygmund commutators via CMO (R n ) which is defined to be the closure in BMO (R n ) of infinitely differentiable functions on R n with compact support; Nakai and Yabuta [8] studied pointwise multipliers for functions on R n of bounded mean oscillation; and Iwaniec [9] used the compactness theorem in Uchiyama [7] to study linear complex Beltrami equations and the L p (C) theory of quasiregular mappings. All these classical results have wide generalizations as well as applications and have inspired a myriad of further studies in recent years: see, for instance, the References [10][11][12][13] for their applications in singular integral operators as well as their commutators, the References [14][15][16][17][18][19] for their applications in pointwise multipliers, the References [20][21][22] for their applications in partial differential equations, and the References [23][24][25][26][27][28] for more variants and properties of BMO (R n ). In particular, we refer the reader to Chang and Sadosky [29] for an instructive survey on functions of bounded mean oscillation and also Chang et al [25] for BMO spaces on the Lipschitz domain of R n .…”

Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type

“…We mainly combine the ideas in [12] and [35] to prove our main result. We also point out that to obtain the above theorem, we provide an equivalent characterisation of VMO(X), which is stated in Lemma 2.4 below, and is of independent interest.…”

“…Yang and Do. Yang [34,35] for the commutator of the Cauchy integral and Beurling-Ahlfors transformation, respectively. For more results on the boundedness of operators on Morrey spaces in di erent settings, we refer the reader to other studies as in [1, 15, 17, 26-28, 33, 37, 38] for instance.…”

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].…”