arXiv subjects
Publications and source records attributed to G. Sankara Raju Kosuru.
We establish a Wiener-type integral condition for first-order Sobolev functions defined on a complete, doubling metric measure space supporting a Poincaré inequality. It is stronger than the Lebesgue point property, except for a marginal increase in the capacity of the set of non-Lebesgue points.
We establish trace inequalities for Riesz potentials on Herz-type spaces and discuss the optimality of conditions imposed on specific parameters. We also present some applications in the form of Sobolev-type inequalities, including the Gagliardo-Nirenberg-Sobolev inequality and the fractional integration theorem in the Herz space setting. In addition, we obtain a Sobolev embedding theorem for Herz-type Sobolev spaces.
We provide a few characterizations of a strictly convex Banach space. Using this we improve the main theorem of [Digar, Abhik; Kosuru, G. Sankara Raju; Cyclic uniform Lipschitzian mappings and proximal uniform normal structure. Ann. Funct. Anal. 13 (2022)].
In this paper, we introduce a class of function spaces called Köthe-Herz spaces $E(\mathcal{X})$. These spaces are similar to amalgam spaces and are characterized by a local component given by a countable family $\mathcal{X}=\left( X_{α}\right) _{α\in I}$ of quasi-normed function spaces, and a global component $E$, which is a quasi-normed sequence space. We investigate various geometric and topological properties inherited by $E(\mathcal{X})$ from its components, such as their completeness, duality, order continuity, ideal and Fatou properties, in an abstract setting. In addition, we provide a Banach function space characterization for $E(\mathcal{X})$, which allows us to understand its structure and behavior more deeply. Furthermore, by appropriate amalgamation of Lorentz spaces (Orlicz spaces) and Lebesgue sequence spaces, we define Lorentz-Herz spaces (Orlicz-Herz spaces) as a particular case of $E(\mathcal{X})$, which are still generalizations of the classical Herz spaces. In this context (especially Lorentz-Herz spaces), we establish previously studied properties, demonstrate interpolation results, and prove the boundedness of important sublinear integral operators with kernels that satisfy a size condition.
The main objective of this article is to provide an alternative approach to the central result of [Eldred, A. Anthony, Kirk, W. A., Veeramani, P., Proximal normal structure and relatively nonexpansive mappings, Studia Math., vol 171(3), (2005) 283-293] using proximal uniform normal structure. Also we provide characterizations of a strictly convex space. Finally, sufficient conditions for the existence of a non-empty proximal sub-pair for a pair in a Banach space are discussed.
The aim of this paper is to address an open problem given in [Kirk, W. A., Shahzad, Naseer, Normal structure and orbital fixed point conditions, J. Math. Anal. Appl. {\bf{vol 463(2)}}, (2018) 461--476]. We give a characterization of weak proximal normal structure using best proximity pair property. We also introduce a notion of pointwise cyclic contraction wrt orbits and therein prove the existence of a best proximity pair in the setting of reflexive Banach spaces.
For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $ϕ$, we first obtain a Chebyshev type inequality which provides an upper bound for $\displaystyle ϕ(λ_{1}) μ(\{x \in Ω: f(x) \geq λ_{1} \}) + \sum_{k=2}^{n}\left(ϕ(λ_{k})- ϕ(λ_{k-1})\right) μ(\{x \in Ω: f(x) \geq λ_{k}\}) ,$ where $0 < λ_1 < λ_2 \cdots λ_n < \infty$. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaymé-Chebyshev, Cantelli and Hoeffding inequalities are obtained.
This paper has been withdrawn by the author due to a crucial sign error in equation 1