arXiv subjects
Publications and source records attributed to M. Ashraf Bhat.
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.
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.
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.