We prove for many self-similar, and some more general, sets that if is the Hausdorff dimension of and is Hölder continuous with exponent , then the -dimensional Hausdorff measure of is .
On bounded domains we consider the anisotropic problems in with 1$"> and on and in with and on . Moreover, we generalize these boundary value problems to space-dimensions 2$">. Under geometric conditions on and monotonicity assumption on we prove existence and uniqueness of positive solutions. 相似文献
The vanishing of Van Kampen's obstruction is known to be necessary and sufficient for embeddability of a simplicial -complex into for , and it was recently shown to be incomplete for . We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embedding obstructions for a class of -complexes in .
If we are given real-valued smooth functions on which are in involution, then, under some mild hypotheses, the subset of where these functions are linearly independent is not simply connected.
The Dirichlet-type space ) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space . Let be an analytic self-map of the disc and define for . The operator is bounded (respectively, compact) if and only if a related measure is Carleson (respectively, compact Carleson). If is bounded (or compact) on , then the same behavior holds on ) and on the weighted Dirichlet space . Compactness on implies that is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space . Inner functions which induce bounded composition operators on are discussed briefly.
Let be a convex curve in the plane and let be the arc-length measure of Let us rotate by an angle and let be the corresponding measure. Let . Then This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form where and are conjugate exponents. 相似文献
Let be a commutative ring, let be an indeterminate, and let . There has been much recent work concerned with determining the Dedekind-Mertens number =min , especially on determining when = . In this note we introduce a universal Dedekind-Mertens number , which takes into account the fact that deg() + for any ring containing as a subring, and show that behaves more predictably than .
We consider orthogonal polynomials in two variables whose derivatives with respect to are orthogonal. We show that they satisfy a system of partial differential equations of the form where , , is a vector of polynomials in and for , and is an eigenvalue matrix of order for . Also we obtain several characterizations for these polynomials. Finally, we point out that our results are able to cover more examples than Bertran's.
We give a geometric characterization of those positive finite measures on with the upper density finite at -almost every , such that the principal value of the Cauchy integral of ,
exists for -almost all . This characterization is given in terms of the curvature of the measure . In particular, we get that for , -measurable (where is the Hausdorff -dimensional measure) with , if the principal value of the Cauchy integral of exists -almost everywhere in , then is rectifiable.
It is known that the total (co)-homoloy of a 2-step nilpotent Lie algebra is at least , where is the center of . We improve this result by showing that a better lower bound is , where and is a complement of in . Furthermore, we provide evidence that this is the best possible bound of the form .
Let be the field obtained by adjoining to all -power roots of unity where is a prime number. We prove that the theory of is undecidable.
We prove that the space of equivalence classes of -invariant connections on some -principle bundles over is weakly homotopy equivalent to a component of the second loop space .
We give optimal bounds of the height, curvature, area and volume of -surfaces in bounding a planar curve. The spherical caps are characterized as the unique -surfaces achieving these bounds.
Let be a self-similar probability measure on satisfying where 0$"> and Let be the Fourier transform of A necessary and sufficient condition for to approach zero at infinity is given. In particular, if and for then 0$"> if and only if is a PV-number and is not a factor of . This generalizes the corresponding theorem of Erdös and Salem for the case