It is shown that the formula
where and is correct under the restrictions and It is also true if we suppose that and the spaces are functional Banach or quasi-Banach lattices on the same measure space
Let be an integer and let , where denotes the moduli space of compact Riemann surfaces of genus . Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space, we prove that the subloci corresponding to Riemann surfaces with automorphism groups isomorphic to cyclic groups of order 2 and 3 belong to the same connected component. We also prove the connectedness of for and with the exception of the isolated points given by Kulkarni.
The tensor product of two p-harmonic functions is in general not p-harmonic, but we show that it is a quasiminimizer. More generally, we show that the tensor product of two quasiminimizers is a quasiminimizer. Similar results are also obtained for quasisuperminimizers and for tensor sums. This is done in weighted R-n with p-admissible weights. It is also shown that the tensor product of two p-admissible measures is p-admissible. This last result is generalized to metric spaces.
A. Baernstein II (Comparison of p-harmonic measures of subsets of the unit circle, St. Petersburg Math. J. 9 (1998), 543-551, p. 548), posed the following question: If G is a union of m open arcs on the boundary of the unit disc D, then is w _{a,p}(G)=w _{a,p}(G), where w _{a,p} denotes the p-harmonic measure? (Strictly speaking he stated this question for the case m=2.) For p=2 the positive answer to this question is well known. Recall that for p≠2 the p-harmonic measure, being a nonlinear analogue of the harmonic measure, is not a measure in the usual sense.
The purpose of this note is to answer a more general version of Baernstein's question in the affirmative when 1G is the restriction to ∂D of a Sobolev function from W _{1,p}(C).
For p≥2 it is no longer true that X_{G} belongs to the trace class. Nevertheless, we are able to show equality for the case m=1 of one arc for all 1, using a very elementary argument. A similar argument is used to obtain a result for starshaped domains.
Finally we show that in a certain sense the equality holds for almost all relatively open sets.
In this note we show that p-admissible measures in one dimension (i.e. doubling measures admitting a p-Poincaré inequality) are precisely the Muckenhoupt Ap-weights. © 2005 American Mathematical Society.
In this note we construct a family of continuum many hereditarily strongly infinite-dimensional Cantor manifolds such that for every two spaces from this family, no open subset of one is embeddable into the other.
A root ideal arrangement A_I is the set of reflecting hyperplanes corresponding to the roots in an order ideal I of the root poset on the positive roots of a finite crystallographic root system. A characterisation of supersolvable root ideal arrangements is obtained. Namely, A_I is supersolvable if and only if I is chain peelable, meaning that it is possible to reach the empty poset from I by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved under taking subideals. We identify the maximal ideals that correspond to non-supersolvable arrangements. There are essentially two such ideals, one in type D_4 and one in type F_4. By showing that A_I is not line-closed if I contains one of these, we deduce that the Orlik-Solomon algebra OS(A_I) has the Koszul property if and only if A_I is supersolvable.
The Marcinkiewicz integral I_{λ} (x)= ∫ (dist (y, ℝ^{n}\Ω))^{λ}/Ω| x - y | ^{n+λ} dy, where λ > 0, plays a well-known and prominent role in harmonic analysis. In this paper, we estimate the growth of it in the limiting case λ → 0. Throughout, we assume that Ω is convex; it is interesting that this condition cannot be dropped
It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions f in L-p, then we have the sharp estimate
parallel to(I - H)f parallel to(Lp) <= 1/(p - 1)(1/p) parallel to f parallel to(Lp) for p = 2, 3, 4, .... In other words,
parallel to f** - f*parallel to(Lp) <= 1/(p - 1)(1/p) parallel to f parallel to(Lp) for each f is an element of L-p and each integer p >= 2.
It is also shown, via a connection between the operator I - H and Laguerre functions, that
parallel to(1 - alpha)I + Phi(I - H)parallel to(L2 -> L2) = parallel to I - alpha H parallel to(L2 -> L2) = 1 for all a is an element of [ 0, 1].
We generalize the well-known result of É. Cartan on isoparametric cubics by showing that a homogeneous cubic polynomial solution of the eikonal equation must be rotationally equivalent to either a reducible Jordan cubic or to one of four Cartan cubic polynomials in dimensions n=5,8,14,26.
By using a non-associative algebra argument, we prove that any cubic homogeneous polynomial solution to the p-Laplace equation in R^n is identically zero for any n>2 and any p distinct from 1 and 2.