REGULARITY OF SOME NONLINEAR QUANTITIES ON SUPERHARMONIC FUNCTIONS IN LOCAL HERZ-TYPE HARDY SPACES

In this paper, the authors introduce a kind of local Hardy spaces in Rn associated with the local Herz spaces. Then the authors investigate the regularity in these local Hardy spaces of some nonlinear quantities on superharmonic functions on R2. The main results of the authors extend the corresponding results of Evans and Müller in a recent paper.


Introduction
In recent years, Hardy space methods have lead to remarkable progress on nonlinear partial differential equations with critical growth; see [3], [15], [1] and [16].The main reason is as follows: a typical difficulty when studying such equations is that the nonlinear term is a priori only known to be in L 1 (R n ) in which there is no good elliptic theory; however, there is a well-established regularity theory in a particular subspace of L 1 (R n ), namely the standard Hardy space H 1 (R n ).In other words, Calderón-Zygmund singular integral operators are not bounded on L 1 (R n ); but, they are indeed bounded on H 1 (R n ) if they satisfy the vanishing moment condition; see [5].
Moreover, Coifman, Meyer, Lions and Semmes in [1] proved that certain nonlinear quantities mentioned above are in fact in H 1 (R n ).In particular, L. C. Evans and S. Müller in [3] established the local boundedness in standard Hardy space H 1 (R n ) of some nonlinear quantities on the weakly superharmonic functions.Applying their results to the two-dimensional Euler equations with nonnegative vorticity, Evans and Müller gave a new proof of Delort's results in [2].
On the other hand, a theory of the Hardy spaces associated with Herz spaces has been developed considerably in recent years; see [6], [11] and [12].These Herz-type Hardy spaces are good substitutes of the usual Hardy spaces when studying boundedness of non-translation invariant operators (see [13]) and can be regarded as the local version at the origin of the usual Hardy spaces.There is also a good regularity theory in these spaces; see [12] and [14].However, it is still not clear how to apply the theory on these spaces to partial differential equations.The main purpose of this paper is to establish a local version at the origin of Evans and Müller result in [3].That is, we will study the local boundedness on the Hardy spaces associated with Herz spaces of some nonlinear quantities on weakly superharmonic functions.Our main results also extend the corresponding results of Evans and Müller, Theorem 1.1 and Theorem 5.4 in [3].We hope this will be helpful to finding more applications of the Herz-type Hardy spaces in the study on partial differential equations; see also [14].
Let us first introduce some definitions.
We introduce the local Herz spaces as follows.
Obviously, by the above definition, f ∈ Kα,p q,loc (R n ) is equivalent to the fact that for any k ∈ Z, χ B k f ∈ Kα,p q (R n ).It is also easy to observe that f ∈ Kα,p q,loc (R n ) if and only if for any k ∈ N, χ B k f ∈ Kα,p q (R n ).We easily verify that Kn(1/p−1/q),p ).Now choose η : R n −→ R to be any smooth function satisfying supp η ⊆ B(0, 1) and x − y r dy .
In [11] (and independently in [6]), S. Lu and D. Yang introduce the following Herz-type Hardy spaces, H Kn(1−1/q),1 q (R n ), which can be regarded as the local version at the origin of the standard Hardy space, H 1 (R n ), studied by C. Fefferman and E. M. Stein in [5].
According to Lu-Yang [12], this definition does not depend on the particular choice of η.Obviously, when q = 1, H Kn(1−1/q),1 the standard Hardy space studied by Fefferman and Stein in [5].
In [4], Fan and Yang introduce the local version, the space h Kn(1−1/q),1 ) be a weak solution of the partial differential equation where w ∈ K2(1−1/q),1 q,loc (R 2 ) and In addition, for each φ ∈ C ∞ c (R 2 ), we have the estimate for some constant c and some radius R depending only on φ.
If we take q = 1, then Theorem 1 is just Theorem 1.1 in [3].Thus, Theorem 1 is a generalization of Theorem 1.1 in [3].In fact, Theorem 1 can be regarded as some kind local version of that theorem at the origin.See also Semmes [16] for several different versions of Evans and Müller's theorem.
It has been pointed by Evans and Müller in [3] that without the sign condition (1.1), Theorem 1 is false.However, in the radical case, the nonnegativity of w is not required.To be precise, we have

be a weak solution of the partial differential equation
, we have the estimate ), where k ∈ N depends only on φ.

Finally, we point that we also obtain a local version on Herz-type Hardy spaces h
Kn(1−1/q),1 q (R n ) of Jones-Journé's result on convergence a.e. and convergence in the distribution sense for a sequence bounded in the Hardy space H 1 (R n ); see [9], [1] and see also [3] for the local version on h 1 (R n ). (1.4) in the sense of distributions.

Proofs of Theorems
We begin with the proof of Proposition 1.
Proof of Proposition 1: When q = 1, this proposition is just Lemma 5.1 in [3].Now we restrict that 1 <q <∞.Suppose f ∈H loc Kn(1−1/q),1 Thus, we have By the hypothesis, we know that χ B l 0 f * * Kn(1−1/q),1 q (R n ) < ∞.We still need to show that χ B l 0 g Kn(1−1/q),1 q (R n ) < ∞.In fact, we have For I 1 , by the trivial estimate g l χ k L q (R n ) ≤ c f l L q (R n ) and the hypothesis, we obtain To estimate I 2 , we first note that if l ≥ 0 and k ≥ l + 2, then g l (x) ≡ 0 when x ∈ A k .Thus, By substituting this into I 2 , we obtain This finishes the proof of Proposition 1.
Proof of Theorem 1: Fix R > 8. Let B(0, R) denote the closed ball with center 0 and radius R, and set