Mathematics

The paper alluded to in the title contains the following striking result: Let I be the unit interval and ? the Cantor set. If X is a quasi Banach space containing no copy of c 0 which is isomorphic to a closed subspace of a space with a basis and C(I,X) is linearly homeomorphic to C(?,X) , then X is locally convex, i.e., a Banach space. It is shown that Kalton result is sharp by exhibiting non locally convex quasi Banach spaces X with a basis for which C(I,X) and C(?,X) are isomorphic. Our examples are rather specific and actually in all cases X is isomorphic to C(?,X) if K is a metric compactum of finite covering dimension.

We propose a new viewpoint on Hilbert scales extending them by means of all Hilbert spaces that are interpolation ones between spaces on the scale. We prove that this extension admits an explicit description with the help of OR -varying functions of the operator generating the scale. We also show that this extended Hilbert scale is obtained by the quadratic interpolation (with function parameter) between the above spaces and is closed with respect to the quadratic interpolation between Hilbert spaces. We give applications of the extended Hilbert scale to interpolational inequalities, generalized Sobolev spaces, and spectral expansions induced by abstract and elliptic operators.

We provide an extension operator for weighted Sobolev spaces on bounded polyhedral cones K involving a mixture of weights, which measure the distance to the vertex and the edges of the cone, respectively. Our results are based on Stein's extension operator for Sobolev spaces.

In this paper we study an extension problem for the Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type and use the solution to prove Hardy-type inequalities for fractional powers of the Laplace-Beltrami operator. Next, we study the mapping properties of the extension operator. In the last part we prove Poincaré-Sobolev inequalities on these spaces.

It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality H β ∞ ({x∈Ω:| I α f(x)|>t})≤C e −c t q ′ for all ∥f ∥ L N/α,q (Ω) ≤1 and any β∈(0,N] , where Ω⊂ R N , H β ∞ is the Hausdorff content, L N/α,q (Ω) is a Lorentz space with q∈(1,∞] , q ′ =q/(q−1) is the Hölder conjugate to q , and I α f denotes the Riesz potential of f of order α∈(0,N) .

In this paper, we present an integral Suzuki-type fixed point theorem for multivalued mappings defined on a complete metric space in terms of the Ćirić integral contractions. As an application, we will prove an existence and uniqueness theorem for a functional equation arising in dynamic programming of continuous multistage decision processes.

We give an accessible introduction and elaboration on the methods used in obtaining a geodesic, which is the curve of shortest length connecting two points lying on the surface of a function. This is found through computing what's known as the variation of a functional, a "function of functions" of sorts. Geodesics are of great importance with wide applications, e.g. dictating the path followed by aircraft (great-circles), how light travels through space, assist in the process of mapping a 2D image to a 3D surface, and robot motion planning.

A polygon is derived that contains the numerical range of a bounded linear operator on a complex Hilbert space, using only norms. In its most general form, the polygon is an octagon, symmetric with respect to the origin, and tangent to the closure of the numerical range in at least four points when the spectral norm is used.

The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical `` T(1)∈BMO '' assumptions that are not easily checkable. Recently, Hong, Liu and Mei (J. Funct. Anal. 2020) observed that the situation improves remarkably for singular integrals with a symmetry assumption, so that a classical T(1) criterion still guarantees their L 2 -boundedness on Hilbert space -valued functions. Here, these results are extended to general UMD (unconditional martingale differences) spaces with the same natural BMO condition for symmetrised paraproducts, and requiring in addition only the usual replacement of uniform bounds by R -bounds in the case of general singular integrals. In particular, under these assumptions, we obtain boundedness results on non-commutative L p spaces for all 1<p<∞ , without the need to replace the domain or the target by a related non-commutative Hardy space as in the results of Hong et al. for p≠2 .

For n>k≥0 , λ>0 , and p,r>1 , we establish the following optimal Hardy-Littlewood-Sobolev inequality ∣ ∣ ∬ R n × R n−k f(x)g(y) |x−y | λ | y ′′ | β dxdy ∣ ∣ ≲∥f ∥ L p ( R n−k ) ∥g ∥ L r ( R n ) with y=( y ′ , y ′′ )∈ R n−k × R k under the two conditions β<{ k−k/r n−λ−k/r if 0<λ≤n−k, if n−k<λ, and n−k n 1 p + 1 r + β+λ n =2− k n . Remarkably, there is no upper bound for λ , which is quite different from the case with the weight |y | −β , commonly known as Stein-Weiss inequalities. We also show that the above condition for β is sharp. Apparently, the above inequality includes the classical Hardy-Littlewood-Sobolev inequality when k=0 and the HLS inequality on the upper half space R n + when k=1 . In the unweighted case, namely β=0 , our finding immediately leads to the sharp HLS inequality on R n−k × R n with the \textit{optimal} range 0<λ<n−k/r, which has not been observed before, even for the case k=1 . Improvement to the Stein-Weiss inequality in the context of R n−k × R n is also considered. The existence of an optimal pair for this new inequality is also studied.