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.

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 ) .

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.

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.

We establish regularity for functions satisfying a dynamic programming equation, which may arise for example from stochastic games or discretization schemes. Our results can also be utilized in obtaining regularity and existence results for the corresponding partial differential equations.2010 Mathematics Subject Classification. 91A15, 35J92, 35B65, 35J60, 49N60.

Abstract. In this note we establish the boundedness properties of local maximal operators M G on the fractional Sobolev spaces W s,p (G) whenever G is an open set in R n , 0 < s < 1 and 1 < p < ∞. As an application, we characterize the fractional (s, p)-Hardy inequality on a bounded open set G by a Maz'ya-type testing condition localized to Whitney cubes.

No abstract