Let L(ν)⊗̂γqY = Lq(ν, Y ) and X⊗̂∆pL(μ) = Lp(μ,X). It is shown that any Lp(μ)-valued measure has finite L2(ν)-semivariation with respect to the tensor norm L(ν)⊗̂∆pL(μ) for 1 ≤ p < ∞ and finite Lq(ν)semivariation with respect to the tensor norm L(ν)⊗̂γqL(μ) whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor no...

In this paper, we study controllability for a parabolic system of chemotaxis. With one control only, the local exact controllability to positive trajectory of the system is obtained by applying Kakutani’s fixed point theorem and the null controllability of associated linearized parabolic system. The positivity of the state is shown to be remained in the state space. The control function is show...

We consider the linear thermoelastic plate equations with free boundary conditions in the Lp in time and Lq in space setting. We obtain unique solvability with optimal regularity for the inhomogeneous problem in a uniform C-domain, which includes the cases of a bounded domain and of an exterior domain with C-boundary. Moreover, we prove uniform a priori-estimates for the solution. The proof is ...

This paper characterizes the possible blow-up of solutions for 3D magneto-hydrodynamics (MHD short) equations. We first establish some ϵ-regularity criteria in Lq,∞ spaces suitable weak solutions, and then together with an embedding theorem from Lp,∞ space into a Morrey type we characterize local behaviors near potential singular point. More precisely, show that if z0=(t0,x0) is point, any r>0 ...

We prove an analogue of the Lp version of Hardy’s theorem on semisimple Lie groups. The theorem says that on a semisimple Lie group, a function and its Fourier transform cannot decay very rapidly on an average.

The idea of replacing functions by linear operators, the process of quantization, goes back to the foundations of quantum mechanics and has a great impact in mathematics. This applies for instance to representation theory, operator algebra, noncommutative geometry, quantum and free probability or operator space theory. The quantization of measure theory leads to the theory of Lp spaces defined ...

We give necessary and sufficient conditions for the Hardy operator to be bounded on a rearrangement invariant quasi-Banach space in terms of its Boyd indices. MAIN RESULTS A rearrangement invariant space X on R is a set of measurable functions (modulo functions equal almost everywhere) with a complete quasi-norm ‖ · ‖X such that the following holds: i) if g ≤ f ∗ and f ∈ X, then g ∈ X with ‖g‖X...