Hardy-Littlewood inequality for quasiregular mappings in certain domains in R^n
Annales Academiae Scientiarum Fennicae Series A I Mathematica
In their paper "Some properties of conjugate functions"  Hardy and Littlewood proved the following Theorem l. If f:vtriu is analytic in a disk DcRz cmtered at zo, thm for O=p=.-, where. C, does not depend on f. Genenlizations of this theorem to solutions of elliptic systems of P.D.E.'s in several variables as well as to more general domains were treated in a recent paper by J. Boman  for l<p<.-; they can be extended to hold for 0<p<-. We will establish similar estimates for the components
... of a quasiregular mapping in domains in R'which satisfy certain geometric conditions (see Theorem 4 and Corollaries 1-5). The main idea we use is based on two geometric results. The first, Theorem 2, states that the exponent can be improved in a weak type reverse Hölder inequality. This should be considered as complementary to Gehring's well known Lemma . The second, Theorem 3, allows us to obtain global estimates from the local inequalities over the cubes in the domain. Both Theorem 2 and Theorem 3 illustrate a self-improving property of some local estimates and theyiseem to be interesting in their own right. The local estimates (Propositions 1, 2, 3\ are obJained from the classical embedding inequalities. We do not appeal to any nonstandard or difrcult result from quasiconformal theory and differential equations. The weighted inequalities as an application give us a version of Theorem 1 over a quasiball (see Corollary 5).