\(L^{s}(\mu)\)-averaging domains. (English) Zbl 1027.30053

Ding, Shusen; Nolder, Craig A.

J. Math. Anal. Appl. 283, No. 1, 85-99 (2003).

After introducing \(L^s(\mu)\)-averaging domains the authors obtain a similar characterization of these domains as in the special case \(\mu=m =n\)-dimensional Lebesgue measure which are given by S. G. Staples [Ann. Acad. Sci. Fenn., Ser. A I 14, 103–127 (1989; Zbl 0706.26010)]. As an application of their results they prove norm inequalities for conjugate \(A\)-harmonic tensors which are generalizations of the following theorem being due to G. H. Hardy and J. E. Littlewood [J. Reine Angew. Math. 167, 405–423 (1932; Zbl 0003.20203)]. Theorem (G. H. Hardy and J. E. Littlewood). For each \(p>0\) there exists a constant \(C=C(p)\) such that \[ \iint_D\bigl|u(x,y)- u(0,0)\bigr|^p dx dy\leq C\iint_D\bigr|v(x,y)-v(0,0)\bigr |^p dx dy \] holds for all real functions \(u=u(x,y)\), \(v=v(x,y)\), where \(f=f(z)= f(x+iy)= u(x,y)+iv(x,y)\) is analytic in the unit disk \(D\). Furthermore, it should be noted that the consideration of \(L^s(\mu)\)-averaging domains allows the choice \(d\mu=J_fdm\), where \(J_f\) denotes the Jacobian determinant of a quasiconformal mapping \(f\) in \(\mathbb{R}^n\). This provides the results for quasiconformal mappings and quasiregular functions at the end of the paper.

Reviewer: Erich Hoy (Friedberg)

MSC:

30C65	Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions
26D20	Other analytical inequalities
35J60	Nonlinear elliptic equations
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

Poincaré inequality; \(L^p\)-averaging domains; \(A\)-harmonic tensor; quasiregular function

Citations:

Zbl 0706.26010; Zbl 0003.20203

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