## 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:

 [1] Adams D.R., Hedberg L.-I.: Function Spaces and Potential Theory. Springer, Berlin (1995) · Zbl 0834.46021 [2] Adams D.R., Xiao J.: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Math. J. 536, 1629–1663 (2004) · Zbl 1100.31009 [3] Campanato S.: Proprietà di inclusione per spaze di Morrey. Ricerche Mat. 12, 67–86 (1963) · Zbl 0192.22703 [4] Hedberg L.-I., Wolff Th.: Thin sets in nonlinear potential theory. Ann. Inst. Fourier Grenoble 334, 161–187 (1983) · Zbl 0508.31008 · doi:10.5802/aif.944 [5] Morrey C.B. Jr: On the solution of quasi-linear partial differential equations. Trans. Amer. Math. Soc. 43, 126–166 (1938) · JFM 64.0460.02 · doi:10.1090/S0002-9947-1938-1501936-8 [6] Palagachev D.K., Softova L.G.: Singular integral operators, Morrey spaces and fine regularity of solutions of PDE’s. Potential Anal. 20, 237–263 (2004) · Zbl 1036.35045 · doi:10.1023/B:POTA.0000010664.71807.f6 [7] Peetre J.: On the theory of $${$$\backslash$$mathcal L_{p,$$\backslash$$lambda}}$$ spaces. J. Funct. Anal. 4, 71–87 (1969) · Zbl 0175.42602 · doi:10.1016/0022-1236(69)90022-6 [8] Sjödin T.: Thinness in non-linear potential theory for non-isotropic Sobolev spaces. Ann. Sci. Acad. Fenn. 22, 313–338 (1997) · Zbl 0890.31006 [9] Stampaccia G.: The spaces $${$$\backslash$$mathcal L\^{(p, $$\backslash$$lambda)},$$\backslash$$, N\^{(p,$$\backslash$$lambda)}}$$ and interpolation. Ann. Scuola Norm. Sup. Pisa 19, 443–462 (1965) [10] Turesson, B.O.: Nonlinear potential theory and weighted Sobolev spaces. Lecture notes in mathematics, No. 1736, Springer, Berlin (2000) · Zbl 0949.31006
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