Narrow Results

We extend and unify several Hardy type inequalities to functions whose values are positive semi-definite operators. In particular, our methods lead to the operator versions of Hardy–Hilbert's and Godunova's inequalities. While classical Hardy type inequalities hold for parameter values p > 1, it is typical that the operator versions hold only for 1 < p ≤ 2, even for functions with values in 2 × 2 matrices.

In this paper, a two-dimensional Hardy type inequality for fuzzy integrals is proved. Also, some illustrative examples are presented.

Some properties of distributions f satisfying x · ∇ f ∈ L^p (ℝⁿ), 1 ≤ p < ∞, are studied. The operator x · ∇ is the generator of a semi-group of dilations. We first give Sobolev type inequalities with respect to the operator x · ∇. Using the inequalities, we also show that if , x · ∇ f ∈ L^p (ℝⁿ) and |x|^n/p|f(x)| vanishes at infinity, then f belongs to L^p (ℝⁿ). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L² (ℝⁿ).

In continuation to the recent study “Coupled Fractional Wigner Distribution with Applications to LFM Signals” in [Fractals, World Scientific; 2022], we formulate some important uncertainty inequalities including, the Hausdorff–Young, Lieb’s and Pitts inequalities for the coupled fractional Wigner distribution. In the sequel, we also establish Heisenberg’s, logarithmic, Hardy’s and Beurling’s uncertainty principles. Towards the end, we derive some Sobolev-type inequalities for the coupled fractional Wigner distribution.

We consider the strategy of realizing the solution of a Cauchy problem (CP) with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the three-dimensional (3d), linear wave equation ${\square }_{1+3}U=0$ with radial Cauchy data $U(0,x)=\mathrm{\Phi }(x)=\phi (|x|)$, $U_t(0,x)=\mathrm{\Psi }(x)=\psi (|x|)$. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available for the radial symmetric CP. In treating the 3d wave equation, we therefore insist on robust arguments based on energy methods and strong convergence. (In particular, this work does not address what can be established via solution formulae.) Our findings for the 3d wave equation show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. The standard existence result for the linear wave equation guarantees a unique solution in $C([0,T);H^s({ℝ}^3))$ whenever $(\mathrm{\Phi },\mathrm{\Psi })\in H^s\times H^{s-1}({ℝ}^3)$. However, within the constrained framework outlined above, we obtain strictly lower regularity for solutions obtained as limits of exterior solutions. We also show that external Neumann solutions yield better regularity than external Dirichlet solutions. Specifically, for Cauchy data in $H^2\times H^1({ℝ}^3)$, we obtain $H^1$-solutions via exterior Neumann solutions, and only $L^2$-solutions via exterior Dirichlet solutions.