Daulti Verma - Academia.edu (original) (raw)

Papers by Daulti Verma

Real Analysis Exchange, 2008

arXiv (Cornell University), Apr 12, 2024

In this paper, we discuss the Hardy inequality with bilinear operators on general metric measure ... more In this paper, we discuss the Hardy inequality with bilinear operators on general metric measure spaces. We give the characterization of weights for the bilinear Hardy inequality to hold on general metric measure spaces having polar decompositions. We also provide several examples of the results, finding conditions on the weights for integral Hardy inequalities on homogeneous Lie groups, as well as on hyperbolic spaces and more generally on Cartan-Hadamard manifolds.

Journal of Mathematical Analysis and Applications, 2007

Boundedness of the Hardy operator (Hf)(x) = x 0 f (t) dt and its adjoint (H * f)(x) = ∞ x f (t) d... more Boundedness of the Hardy operator (Hf)(x) = x 0 f (t) dt and its adjoint (H * f)(x) = ∞ x f (t) dt is characterized between Banach function spaces X q and L p. By applying a limiting procedure, corresponding boundedness of the geometric mean operator (Gf)(x) = exp(1 x x 0 ln f (t) dt) is also derived.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019

In this note, we give several characterizations of weights for two-weight Hardy inequalities to h... more In this note, we give several characterizations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. We give examples obtaining new weighted Hardy inequalities on R n , on homogeneous groups, on hyperbolic spaces and on Cartan–Hadamard manifolds. We note that doubling conditions are not required for our analysis.

In this note we give several characterisations of weights for two-weight Hardy inequalities to ho... more In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. We give examples obtaining new weighted Hardy inequalities on R^n, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

In this paper, we continue our investigations giving the characterization of weights for two-weig... more In this paper, we continue our investigations giving the characterization of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy’s original inequality. This is a continuation of our paper (Ruzhansky & Verma 2018. Proc. R. Soc. A 475 , 20180310 ( doi:10.1098/rspa.2018.0310 )) where we treated the case p ≤ q . Here the remaining range p > q is considered, namely, 0

Weight characterization is obtained for the L p-X q boundedness of the two-dimensional Hardy oper... more Weight characterization is obtained for the L p-X q boundedness of the two-dimensional Hardy operator (H2f)(x1, x2) = R x 1 0 R x 2 0 f (t1, t2) dt1 dt2. By using a limiting procedure as well as by a direct method, the corresponding boundedness of the two-dimensional geometric mean operator (G2f)(x1, x2) = exp " 1 x1x2 R x 1 0 R x 2 0 ln f (t1, t2) dt1 dt2 « is obtained. 1 Introduction. Let Ω ⊂ R n. A real normed linear space X = {f : f X < ∞} of measurable functions on Ω is called a Banach function space (BFS), if in addition to the usual norm axioms, f X satisfies the following. (1) f X = |f | X for all f ∈ X. (2) 0 ≤ f ≤ g a.e. ⇒ f X ≤ g X. (3) 0 < f n ↑ f a.e. ⇒ f n X ↑ f X. (4) mes E < ∞ ⇒ χ E X < ∞.