We show that a K-quasiregular $$\omega$$ -curve from Euclidean domain to space with respect covector is locally $$(1/K)(\Vert \omega \Vert /|\omega |_{\ell _1})$$ -Hölder continuous. also quasiregular curves enjoy higher integrability.

We consider a nonautonomous systemx?=A(t)x+f(t,x,y),y?=g(t,y) and give conditions under which there is transformation of the form H(t,x,y)=(x+h(t,x,y),y) taking its solutions onto partially linearized systemx?=A(t)x,y?=g(t,y). Shi Xiong [28] proved special case where g(t,y) was linear function y x?=A(t)x had an exponential dichotomy. Our assumptions on A f are general considered by Reinfelds St...

For 0 < α ≤ 2, a super-α-stable motion X in R with branching of index 1 + β ∈ (1, 2) is considered. If d < α/β, a dichotomy for the density of states Xt at fixed times t > 0 holds: the density function is locally Hölder continuous if d = 1 and α > 1 + β, but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal Hölder index.

For 0 < α ≤ 2, a super-α-stable motion X in R with branching of index 1+β ∈ (1, 2) is considered. Fix arbitrary t > 0. If d < α/β, a dichotomy for the density function of the measure Xt holds: the density function is locally Hölder continuous if d = 1 and α > 1 + β, but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local Hölder index.

The Hölder continuity of a harmonic function is characterized by the growth of its gradient. We generalize these results to solutions of certain subelliptic equations in domains in Carnot groups.

We prove the Hölder continuity of the homogeneous gradient of the weak solutions u ∈ W 1,p loc of the p-Laplacian on the Heisenberg group Hn, for 1+ 1 √ 5 < p ≤ 2.