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Wolff’s inequality in multi-parameter Morrey spaces. (English) Zbl 1255.26006

The nonlinear potential theory for Riesz potentials and more general potentials of functions in the Lebesgue class \(L^p (\mathbb{R}^d)\) is well known [B. O. Turesson, Nonlinear potential theory and weighted Sobolev spaces. Lecture Notes in Mathematics. 1736. Berlin: Springer (2000; Zbl 0949.31006)]. A major step in this theory was made by showing the so-called Wolff’s inequality [L. I. Hedberg and Th. H. Wolff, Ann. Inst. Fourier 33, No. 4, 161–187 (1983; Zbl 0508.31008)]. The corresponding theory for functions in the Morrey spaces is more recent and a part of it was developed in [D. R. Adams and J. Xiao, Indiana Univ. Math. J. 53, No. 6, 1629–1663 (2004; Zbl 1100.31009)].
The author considers multi-parameter Riesz potentials and Wolff potentials in an Euclidean space \(\mathbb{R}^d = \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \times \dots \times \mathbb{R}^{n_k}\). The multi-parameter Riesz potential \(R_{\rho} \mu\) of a nonnegative measure \(\mu\) in \(\mathbb{R}^d\) is defined by \[ R_{\rho} \mu (x) = \int |x_1 -y_1 |^{\rho_1 -n_1} \dots |x_k - y_k |^{\rho_k - n_k } d\mu (y_1, \dots, y_k), \] where \(x= (x_1, \dots, x_k)\). Here \(\rho = (\rho_1, \dots, \rho_k)\) and \(0<\rho_j < n_j\), \(1\leq j \leq k\). The case \(k=1\) gives the classical Riesz potential in \(\mathbb{R}^d\) see [D. R. Adams and L. I. Hedberg, Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften. 314. Berlin: Springer-Verlag (1995; Zbl 0834.46021)]).
The author defines the corresponding dyadic multi-parameter Wolff potential and proves Wolff’s inequality in this setting. The case \(k=1\) was proved by Hedberg and Wolff [loc. cit.], while the case \(k=2\) was proved by the author [Ann. Acad. Sci. Fenn., Math. 22, No. 2, 313–338 (1997; Zbl 0890.31006)]. Adams et al. [loc. cit] studied the potential theory for the Riesz kernel in the Morrey space and proved a Wolff inequality. The author defines the dyadic multi-parameter Morrey-Wolff potential and proves the corresponding Wolff inequality in the multi-parameter Morrey space.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
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References:

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