for some l > 0 and for all x E 0 and C; E R" . We also assume a;j = aj;. For such operators, the Dirichlet problem is solvable in n if and only if it is solvable for the Laplace operator, according to a theorem of Littman, Stampacchia and Weinberger [I]. This means that if n ~ R" is a sufficiently nice bounded region (the unit ball, B, is an example) and J is a given continuous function on the ...

In this paper we give simpler proofs of the classical continuity and lower semicontinuity theorems of Reshetnyak. 1. Main result In 1968, Reshetnyak [20] proved two important results concerning the continuity and lower semicontinuity of functionals with respect to weak-star convergence of measures. These theorems are used in a variety of areas in the calculus of variations, ranging from problem...

We consider maximal operators associated to singular averages along finite subsets Σ of the Grassmannian Gr(d,n) d-dimensional subspaces Rn. The well studied d=1 case corresponds directional function with respect arbitrary Gr(1,n)=Sn−1. provide a systematic study all cases 1≤d<n and prove essentially sharp L2(Rn) bounds for subspace averaging operator in terms cardinality Σ, no assumption on st...

In the signal-processing literature, a frame is mechanism for performing analysis and reconstruction in Hilbert space. By contrast, quantum theory, positive operator-valued measure (POVM) decomposes Hilbert-space vector purpose of computing measurement probabilities. Frames their most common generalizations can be seen to give rise POVMs, but does every reasonable POVM arise from type frame? We...

Abstract It is proved that each Gaussian cocycle over a mildly mixing transformation either coboundary or sharply weak mixing. The class of non-singular infinite direct products T transformations $T_n$ , $n\in \mathbb N$ finite type studied. shown if mixing, the sequence Radon–Nikodym derivatives asymptotically translation quasi-invariant and conservative then Maharam extension This technique p...

Araci et al. introduced a $$p$$ -adic $$(\rho,q)$$ -analogue of the Haar distribution. By means distribution, they constructed -Volkenborn integral. In this paper, by virtue Mahler expansion continuous functions, author gives Radon-Nikodym theorem with respect to -distribution on $$\Bbb Z_p$$ .