We establish the continuity of the Hardy-Littlewood maximal operator on Sobolev spaces W 1,p (R n) , 1 < p < ∞. As an auxiliary tool we prove an explicit formula for the derivative of the maximal function.

We establish the continuity of the Hardy-Littlewood maximal operator on W1,p(Ω), where Ω ⊂ ℝn is an arbitrary subdomain and 1 < p < ∞. Moreover, boundedness and continuity of the same operator is proved on the Triebel-Lizorkin spaces Fps,q (Ω) for 1 < p,q < ∞ and 0 < s < 1.

Abstract. It is shown for the non-centered Hardy-Littlewood maximal operator M that DM f 1 ≤Cn Df 1 for all radial functions in W 1,1 (R n ) .

In this paper we study the regularity of the noncentered fractional maximal operator M β . As the main result, we prove that there exists C(n, β) such that if q = n/(n − β) andThe corresponding result was previously known only if n = 1 or β = 0. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all f ∈ W 1,1 (R n ).

We give a proof of Lipschitz continuity of p-harmonious functions, that are tug-of-war game analogies of ordinary p-harmonic functions. This result is used to obtain a new proof of Harnack's inequality for p-harmonic functions in the case p > 2 that avoids classical techniques like Moser iteration, but instead relies on suitable choices of strategies for the stochastic tug-of-war game.